Session1. Introduction - Daniel Andrei · 2013-02-06 · Course Overview Course Essentials Pedagogical Material Classnotes Textbooks I Bodie,KaneandMarcus,Investments I Hull,Options,Futures,

Investments

Session 1. Introduction

EPFL - Master in Financial EngineeringDaniel Andrei

Spring 2010

Introduction (Session 1) Investments Spring 2010 1 / 29

OutlineI. Course Overview

Course EssentialsProgram

II. Market OverviewThe Money MarketThe Bond MarketEquity SecuritiesStock and Bond Market IndexesDerivatives Markets & CurrenciesFinancial Innovation and Securitization

III. Return and RiskRandom EventsProbability DistributionsLognormality, Return and Risk

IV. Summary & Further Reading


Course Overview







Course Overview Course Essentials

Schedule

Dates: February 22 - June 01, 2010Location: EXTRANEF 126Clasroom sessions: Mondays, 9h15 - 12h00Exercise sessions: Tuesdays, 13h15 - 15h00



Grading

There is a midterm (MID) and a final exam (FE ). Documentation isallowed. You can use a calculator.You will have to solve problem sets but DO NOT NEED to hand themin. Each Tuesday, some of you will help me to solve problems. Thiswill grant you a participation bonus (PB), going from 0% to 20%.Everyone starts at PB = 0%.The final grade will be

GRADE = min[6,

MID +FE2

(1+PB)

]Midterm date: April 13, 2010 (Tuesday)



Pedagogical Material

Class notesTextbooks

I Bodie, Kane and Marcus, InvestmentsI Hull, Options, Futures, and Other DerivativesI Kritzmann, The Portable Financial Analyst: What Practitioners Need

to KnowI Bernstein, Peter, Capital Ideas: The Improbable Origins of Modern

Wall Street.

Journal ArticlesWebsite: www.danielandrei.net → Teaching → Investments [usernameand password to be given in class]


Course Overview Program

Two Parts

1 Part One (lectures 1-7)I Asset Allocation, CAPM, APT, Security Analysis, Applications.

2 Part Two (lectures 8-14)I Interest Rates,Term Structure, Bond Portfolios, Credit Risk, Alternative

Investments.


Market Overview







Market Overview The Money Market

Short-Term Debt Securities

The Money Market consists of very short-term debt securities thatusually are highly marketable. Some examples below (definitions areform the Oxford Dictionary of Finance and Banking)

I Treasury Bills

F Definition: a bill of exchange issued by the US Treasury or the Bank ofEngland on the authority of the US and UK governments respectivelythat is repayable in three months. They bear no interest, the yield beingthe difference between the purchase price and the redemption value.

F Simplest form of borrowing: the government raises money by sellingbills to the public.


Market Overview The Money Market

Short-Term Debt Securities (cont.)

I Certificates of Deposit

F Definition: a negotiable certificate issued by a bank in return for a termdeposit of up to five years.

F Time deposits with a bank. These time-deposits may not be withdrawnon demand.

I Commercial Paper

F Definition: a relatively low-risk short-term (maturing 60 days or less inthe US) unsecured form of borrowing. Commercial paper is oftenregarded as a reasonable substitute for Treasury bills, certificates ofdeposits.

F Short-term unsecured debt issued by large, well-known companies.


Market Overview The Bond Market

Longer-Term Borrowing Instruments

The bond market is composed of longer-term borrowing instrumentsthan those that trade in the money market:

I Treasury Notes and Bonds: issued by the US Treasury.I Corporate Bonds: issued by private corporations.I Municipal Bonds: issued by local governments authorities, especially in

the US.I Mortgage Securities: securities in which cash flows derive form an

underlying pool of mortgages.



Longer-Term Borrowing Instruments (cont.)



Longer-Term Borrowing Instruments (cont.)


Market Overview Equity Securities

Common Stock as Ownership Shares

Common stocks, also known as equity securities or equities, representownership shares in a corporation. Each share of common stockentitles its owner to one vote on any matters of corporate governancethat are put to a vote at the corporation’s annual meeting and to ashare in the financial benefits of ownership.

The common stock of most large corporations can be bought or soldfreely on one or more stock exchanges.

Two important characteristics of common stock: residual claim andlimited liability.


Market Overview Equity Securities

Common Stock as Ownership Shares (cont.)

Calculating returns: buy at time 0 and pay P0, sell at time T andreceive PT and dividend DT .

I The percentage return is calculated as

rT =PT +DT −P0

P0(1)

I The log-return is calculated as

rT = ln(

PT +DT

P0

)(2)


Market Overview Stock and Bond Market Indexes

Averages

Indexes represent measures of the performance of the stock market.They consists of several stocks from different sectors of the economy(the average can be computed equally-weighted or value-weighted).Examples: Dow Jones Industrial Average, Standard & Poor’sComposite 500, FTSE, DAX, NIKKEI 300, etc.

Bond market indexes measure the performance of various categories ofbonds.The three most well-known groups of indexes are those ofMerrill Lynch, Lehman Brothers, and Salomon Smith Barney.


Market Overview Stock and Bond Market Indexes

Averages (cont.)


Market Overview Derivatives Markets & Currencies

Futures, Options and other Derivatives

These instruments provide payoffs that depend on the values of otherassets such as commodity prices, bond and stock prices, or marketindex values. For this reason they are called derivative assets,orcontingent claims.Overview of derivatives by underlying:

I Equity Derivatives: stock options, index futures, futures options, etc.;I Fixed-Income Derivatives: caps/floors, swaps, swaptions, etc.;I Credit Derivatives: credit swap, collateralized loan obligations, etc.;I Other Derivatives: FX, weather, “exotics”, etc.


Market Overview Derivatives Markets & Currencies

Currencies

Currencies are the most liquid financial instrument.Currency instruments have generally spoken the same type ofparameters. However, characteristics may differ, e.g. currency returnvolatility, which is not shaped in the same form as the equity returnvolatiltiy.Currency positions usually have a much shorter maturity. Proprietarytraders on Wall Street take positions up to half an hour.


Market Overview Financial Innovation and Securitization

Does Financial Innovation Add Value?

Empirical evidence supports the statement that financial innovationdoes provide social wealth:

I It caters to the investment diversity desired by the investors;I It improves the opportunities for investors to receive efficient risk-return

trade-offs;I It provides risk management tools for all market participants;I It promotes broad distribution and liquidity to economic resources.

Securitization is a structured finance process, which involves poolingand repackaging of cash flow producing financial assets into securitiesthat are then sold to investors. Example: securitization of themortgage market.


Return and Risk







Return and Risk Random Events

An Example

Imagine a series of bets on fair and independent coin tosses at times1/2, 3/4, 7/8, and so on.Suppose one’s goal is to earn a riskless profit of α by time 1, where α

is some arbitrarily large number.One can bet α on heads by time 1/2. If the first toss comes up heads,one stops. Otherwise one owes α to one’s opponent.A bet of 2α on heads for the second toss at time 3/4 produces thedesired profit if heads comes up at that time.Otherwise, one is down 3α and bets 4α on the third toss, and so on.Because there is an infinite number of potential tosses, one willeventually stop with a riskless profit of α (almost surely), because theprobability of losing every one of an infinite number of tosses is(1/2) · (1/2) · ... = 0.This is a classic “doubling strategy”.


Return and Risk Probability Distributions

The Normal Distribution

The normal distribution is a continuous probability distribution: itassumes there are an infinite number of observations covering allpossible values along a continuous scale.Characteristics:

1 A normal distribution can be fully characterized by only twoparameters: mean and variance,

2 It is symmetric around its mean,3 The area enclosed within one standard deviation on either side of the

mean encompasses 68 percent of the total area under the curve,4 95 percent for two standard deviations,5 99.7 percent for three standard deviations,

It fails to capture large movements in stock prices (mathematically,the tail distribution is too thin).


Return and Risk Lognormality, Return and Risk

Simple and log returns

In probability and statistics, the log-normal distribution is thesingle-tailed probability distribution of any random variable whoselogarithm is normally distributed. If X is a random variable with anormal distribution, then Y = eX has a log-normal distribution.In many cases returns are assumed to be lognormally distributed.Difference between simple and log return [to be shown in class].Return and Risk [to be shown in class].



Some History

The question whether investors can successfully forecast stock priceswas adressed for the first time by Luis Bachelier in his 1900dissertation on the “Theory of Speculation”.He ended up with a mathematical formula that describes the Brownianmotion.In Bachelier’s words: “The mathematical expectation of the speculatoris zero”. He describes this condition as a “fair game”.Bachelier was far ahead of his time. It took sixty years before FisherBlack, Myron Scholes and Bob Merton worked out the Black Scholesoption pricing formula, by using the geometric Brownian motion.Additionally (!) the derived formula anticipated Enstein’s research intothe behavior of particles subject to random shocks in space.



Random Walks in Stock Market PricesPaul Samuelson (economist and Nobel laureate): “It is not easy to getrich in Las Vegas, at Churchill Downs, or at the local Merrill Lynchoffice”.Many empirical tests of the random walk theory have been performed.Here is a simple visual test which might take you only a few minutesto perform [Matlab code to be shown in class]

−10 −5 0 5 10−10

−8

−6

−4

−2

0

2

4

6

8

10

Return yesterday

Ret

urn

toda

y

MSCI France Index


Summary & Further Reading








Summary

Broad overview of the the financial system (markets, intermediaries,instruments, clients, etc.)Empirical evidence supports the statement that financial innovationdoes provide social wealth. However, sometimes abusive usage offinancial innovation may cause disruptions.We use probability distributions to characterize and evaluate randomevents.There is a long tradition of using the normal distribution tocharacterize the fluctuations in stock prices. However, the normaldistribution is not adequate to capture large surprises.The theory of random walks in stock market prices presents importantchallenges to both the chartist and the fundamentalist.



For Further Reading

Fama, Eugene, “Random Walks in Stock Prices,” FAJ, 1965I why common techniques for predicting stock market prices fail

(because of the theory of random walks). Both the chartist and thefundamentalist will have a hard time to add value.

Sharpe, William, “Risk, Market Sensitivity, and Diversification,” FAJ,1972

I about market and non-market risk. On how diversification reduces risk.On why some securities are more sensitive to market changes thanothers.

Bernstein, Peter, “Capital Ideas: The Improbable Origins of ModernWall Street,” Wiley & Sons, 1992

I chapter 1, about Bachelier, the Dow Jones Average, the birth of thejournal Econometrica, etc.


Investments

Session 2. Asset Allocation (part I)


Spring 2010

Asset Allocation I (Session 2) Investments Spring 2010 1 / 34

Outline

I. Portfolio ChoicesExpected Utility MaximizationThe Mean-Variance Criterion

II. Mean-Variance Portfolio AnalysisMinimum-Variance and Efficient PortfoliosThe Mean-Variance Portfolio ProblemProperties of Minimum-Variance PortfoliosDiversification

III. Summary & Further Reading

IV. Formula Sheet


Portfolio Choices

Outline




IV. Formula Sheet


Portfolio Choices Expected Utility Maximization

The Portfolio Choice Problem

Let us begin with a simple case: an individual has an initial wealth ofW0 and he must choose his current consumption C0 and invest hissavings W0−C0 among N assets.Two constituent elements are necessary to formulate the problem:

1 The time scale: We are dealing with a static situation characterized bytwo instants t = 0 and t = 1.

2 The hypothesis regarding the attitude of individuals towards risk: weassume that the investor builds his portfolio so as to maximize hisexpected utility E

[U(C0, C1

)].

The investor’s (random) consumption in period 1, C1, will depend onhow much he has saved in period 0, and on which assets he hasinvested in.




Let ωn denote the fraction of wealth invested in the nth asset and Rnthe random return on the nth asset in period 1, i.e. the payoff of oneunit of account invested in asset n in period 0.When formulating the investor’s maximization problem, we must taketwo things into account:

1 The size of the investor’s savings in period 0, W0−C0, constrains howmuch he can invest, and

2 His portfolio holdings must sum to 1.

In matrix notation, the formal statement of the portfolio problem istherefore

maxC0,wE[U(C0,(W0−C0)w ′R

)]s.t. 1′w = 1 (1)




To solve this problem, form the Lagrangian

L = E[U(C0,(W0−C0)w ′R

)]+ λ

(1−1′w

)(2)

The first order conditions are

∂L∂C0

= E[U1−U2w ′R

]= 0

∂L∂w

= E[U2 (W0−C0) R

]−λ1 = 0

∂L∂λ

= 1−1′w = 0 (3)



Interpretation

Replace third FOC in the second to see that λ is the expectedmarginal utility of wealth:

λ = w ′E[U2 (W0−C0) R

](4)

Here is the economic interpretation of the portfolio solution:1 The first condition states that the investor consumes from wealth until

the expected marginal utilities of consumption and savings are equal,E [U1] = E

[U2w ′R

].

2 The second condition says that the savings are allocated among theassets until each gives an equal contribution to expected marginalutility, E

[U2 (W0−C0) Rn

]= λ .

In static finance, one ignores the consumption part of the decision andonly considers the investment decision aspect of the problem.Moreover, one considers utility over returns.



InterpretationLet W denote wealth available for consumption in period 1,W = C1 = (W0−C0)w ′R . Then, the overall return on the portfolio,Rp, is given by Rp = W

W0−C0= w ′R .

Writing u (Rp) = U (C0,W ), we have

u′ (Rp) =∂u

∂Rp=

∂U∂W

dWdRp

= U2 (W0−C0) (5)

Thus, we can rewrite the condition E[U2 (W0−C0) R

]−λ1 = 0 as

E[u′(w ′R

)R]

= E[u′(Rp

)R]

= λ1 (6)

Without some additional structure, the portfolio problem is hard toanalyze. To make it tractable, restrictions are imposed on

I the investor’s utility function u, and/orI the distribution of asset returns, R.

This brings us to the Mean-Variance Criterion.Asset Allocation I (Session 2) Investments Spring 2010 8 / 34

Portfolio Choices The Mean-Variance Criterion

Expected Utility Maximization and Mean-Variance

The mean-variance criterion is widely used in practice. Under thiscriterion, a portfolio w can be optimal if it has minimum variance σ2

p ,given expected return µp.Formally, letting µ denote the vector of expected returns and Σ thevariance-covariance matrix of returns, the portfolio’s expected return isµp = µ ′w and its variance σ2

p = w ′Σw . Then, the minimum-varianceportfolio with expected return µp is the solution to

minw12w ′Σw s.t. µ

′w = µp (7)

The main question is: under which conditions on (i) the distributionof asset returns and (ii) the investor’s preferences is the mean-variancecriterion consistent with expected utility maximization?



Expected Utility Maximization and Mean-Variance

Answer: there are two particular cases1 The case of quadratic utility, which can be written as

u (Rp) = Rp−bR2

p

2(8)

(computations for this case in class).2 The case of multivariate normally distributed asset returns and

increasing and concave utility. The reason is that a normal distributioncan be fully characterized by only two parameters: mean and variance.Thus, if the return on all assets is normally distributed, then the returnon any portfolio of these assets will also be normally distributed, andmean and variance will be sufficient to describe it.



Proof of Equivalence in the Second Case

Let f(Rp; µ,σ2) denote the normal density of the portfolio return.

Note that we drop p subscripts on µ and σ here. Then, expectedutility is given by

V(µ,σ2)= E

[u(Rp

)]=∫

∞

−∞

u (Rp) f(Rp; µ,σ2)dRp

=∫

∞

−∞

u (µ + σε)n (ε)dε (9)

where ε =Rp−µ

σis standard normal with density n (ε) = 1√

2πe−

ε22 .

Using Leibniz’ rule, we see that nonsatiated investors will prefer higherexpected returns:

∂V∂ µ

=∫

∞

−∞

u′ (µ + σε)n (ε)dε > 0 (10)



Proof of Equivalence in the Second Case

To derive the dependence of expected utility on portfolio variance,note that

∂V∂σ2 =

12σ

∂V∂σ

=12σ

∫∞

−∞

u′ (µ + σε)εn (ε)dε

=12σ

∫∞

−∞

u”(µ−σε)σn (ε)dε < 0 (11)

The last term is negative because u is concave. Thus, for normallydistributed returns, the mean-variance criterion is consistent withexpected utility maximization for all non-satiated risk-averse investors.



The Case of Exponential Utility

Example(to be solved in class) Consider a risk-averse investor with exponentialutility:

u (Rp) =−e−aRp , a > 0 (12)

If the distribution of returns is normal wiht mean µ and variance σ2, then,maximizing expected utility is equivalent to maximizing µ− a

2σ2. Notethat a is the investor’s degree of risk aversion.



InterpretationMaximizing µ− a

2σ2 means that the investor is maximizing theportfolio’s expected return minus a risk premium which is proportionalto the variance of portfolio returns.Therefore, the investor’s indifference curves are the solution toV = µ− a

2σ2. Graphically, they are half-parabolas in mean-standarddeviation space:

0.05 0.10 0.15 0.20Standard deviation

0.12

0.14

0.16

0.18

Expected Return

Figure 1: Indifference curves for a = 2 and a = 5.


Mean-Variance Portfolio Analysis

Outline




IV. Formula Sheet


Mean-Variance Portfolio Analysis Minimum-Variance and Efficient Portfolios

Efficient Portfolios

The investor’s problem is to choose a portfolio such that1 He minimizes the risk for a given expected return, and simultaneously,2 He maximizes the expected return for a given risk.

We will call the set of the optimal portfolios mean-variance efficient.


Mean-Variance Portfolio Analysis Minimum-Variance and Efficient Portfolios

Minimum-Variance vs Efficient PortfoliosWe will describe the properties of mean-variance efficient portfolios.To start with, we will consider the broader class of minimum-varianceportfolios, i.e. the set that includes the single portfolio with thesmallest variance at every level of expected return (it is easy to workwith analytically).Graphically, the distinction between efficient and minimum-varianceportfolios translates to the following:

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Standard Deviation

Exp

ecte

d R

etur

n

Minimum−variance setEfficient frontier


Mean-Variance Portfolio Analysis The Mean-Variance Portfolio Problem

The Mean-Variance Portfolio Problem

We first consider the case without a riskless asset.Suppose there are N risky assets available for investment, and let µ

denote the vector of asset expected returns and Σ thevariance-covariance matrix of returns. Then, for any portfolio w ,expected return is given by

µp = w ′µ (13)

and portfolio variance equals

σ2p = w ′Σw (14)

The minimum-variance portfolio with expected return µp is thesolution w (µp) to

minw12w ′Σw s.t. 1′w = 1, µ

′w = µp (15)



The Mean-Variance Portfolio Problem (cont.)To solve this problem, set up the Lagrangian

L =w ′Σw

2+ λ

(1−1′w

)+ γ(µp−µ

′w)

(16)

The first-order conditions are

∂L∂w

= Σw −λ1− γµ = 0

∂L∂λ

= 1−1′w = 0

∂L∂γ

= µp−µ′w = 0 (17)

Hence, all minimum-variance portfolios are of the form

w = λ Σ−11+ γΣ−1µ (18)



The Mean-Variance Portfolio Problem (cont.)

In order to determine the constants λ and γ , just use the twoconstraints and require that they be satisfied.

For the first constraint, we have

1′w = 1′(λ Σ−11+ γΣ−1

µ)

= 1 (19)

or1′Σ−11λ +1′Σ−1

µγ = Aλ +Bγ = 1 (20)

For the second constraint, we have

µ′w = µ

′ (λ Σ−11+ γΣ−1

µ)

= µp (21)

orµ′Σ−11λ + µ

′Σ−1µγ = Bλ +Cγ = µp (22)



The Mean-Variance Portfolio Problem (cont.)Therefore, the composition of the minimum-variance portfolio withexpected return µp is given by

w = λ Σ−11+ γΣ−1µ

λ =C −µpB

∆, γ =

µpA−B∆

A = 1′Σ−11 > 0, B = 1′Σ−1µ

C = µ′Σ−1

µ > 0, ∆ = AC −B2 > 0 (23)

Using the previous results, the relationship between expected returnand variance on the minimum-variance set is given by

σ2p (µp) = w ′Σw = w ′Σ

(λ Σ−11+ γΣ−1

µ)

= λw ′1+ γw ′µ =Aµ2

p −2Bµp +C∆

(24)



Graphically

In the mean-standard deviation space this is the equation of ahyperbola.It is important to note that the shape of the minimum-variance setdepends not only on assets’ expected returns and variances, but alsoon the correlation between their returns:

0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Efficient Frontier When Correlation Coefficient Changes

Standard Deviation

Exp

ecte

d R

etur

nRho=0Rho=0.25Rho=0.5

Figure 2: Minimum-variance set as a function of ρ


Mean-Variance Portfolio Analysis Properties of Minimum-Variance Portfolios

Properties and Graphical Description

Let us analyze the properties of the minimum-variance set hyperbola

The expected return on the global minimum variance portfolio wg canbe found by minimizing the expression for σ2

p (µp):

dσ2p

dµp=

2Aµp−2B∆

= 0 (25)

yielding µg = B/A.

Inserting this value into the expression for σ2p yields

σ2g =

1A

(26)



Properties and Graphical Description (cont.)Using

λ =C −µgB

∆=

C −B2/A∆

=1A

(27)

andγ =

µgA−B∆

= 0 (28)

the composition of the global minimum-variance portfolio wg is givenby

wg = λ Σ−11 =Σ−11

A=

Σ−111Σ−11

(29)

Let us now determine the slope of the asymptotes of the hyperbola.First, compute the slope of the hyperbola at any point,

dµp

dσp=

dµp

dσ2p

dσ2p

dσp=

∆

2Aµp−2B2σp =

∆

Aµp−B

√Aµ2

p −2Bµp +C∆

(30)Asset Allocation I (Session 2) Investments Spring 2010 24 / 34


Properties and Graphical Description (cont.)

Taking the limit of this expression as µp→±∞ then yields

limµp→±∞

dµp

dσp=±

√∆

A(31)

This completes our description of the hyperbola.

Note that the covariance of any asset or portfolio with the globalminimum variance portfolio wg is 1/A:

Cov(Rg , Rp

)= w ′g Σwp =

1′Σ−1

AΣwp =

1′wp

A=

1A

(32)

All minimum-variance portfolios can be seen as portfolio combinationsof only two distinct portfolios. This result is called two-fund separation[to be shown in class]


Mean-Variance Portfolio Analysis Diversification

Systematic vs Idiosyncratic Risk

So far, we have been concerned with the properties ofminimum-variance portfolos. given µ and Σ, we saw that we can fullydescribe the mean-variance efficient set.We mentioned earlier that correlation between asset returns is a keydriver of the shape of the efficient set. Low or negative correlationseemed to lead to a more favorable efficient set in terms of theavailable risk-return menu.

Example(to be solved in class ). Let us analyze the issue of diversification in moredetail. To do so, consider an equally-weighted portfolio. This portfolio willnot be efficient in most cases, but let us analyze its properties to get someintuition for what is actually happening. The weight of each of the Nassets in an equally-weighted portfolio is ωn = 1/N. The figure belowshows that most of the benefits of diversification arise with 20 to 30 assets.



Systematic vs Idiosyncratic Risk (cont.)

0 10 20 30 40 50Number of assets0.00

0.05

0.10

0.15

0.20Portfolio Variance

Figure 3: Diversification

Therefore, one can distinguish two kinds of risk:



Systematic vs Idiosyncratic Risk (cont.)

1 idiosyncratic risk is specific to a given asset and can be diversified,2 systematic risk arises from the correlation/covariance in asset returns

and cannot be diversified

A very important result of asset pricing theory is that the market paysno risk premium for bearing idiosyncratic risk because it can be avoidedby diversification. We will return to this point later in the course.



Outline




IV. Formula Sheet



Summary

If returns are log-normal and utility is exponential ⇒ mean-varianceframework.The investment set:

I is a hyperbola with focal point the global minimum-variance portfolio.I depends on µand Σ and implicitely on ρ.

Two-fund separation: all minimum-variance portfolios can be obtainedas a combination of the global minimum-variance portfolio wg and theportfolio wd . Soon we will see why this result is so powerful.Market pays no risk premium for bearing idiosyncratic risk.



For Further Reading

1 Peter Bernstein, “Capital Ideas: The Improbable Origins of ModernWall Street”, 1992, Chapter 2.

I About Harry Markowitz, the Nobel Prize, etc.I I was struck with the notion that you should be interested in risk as

well as return.

2 Elton, Edwin, and Gruber, Martin, “The Rationality of AssetAllocation Recommendations”. The Journal of Financial andQuantitative Analysis, 2000.

I Helps to better understand the efficient set mathematics.I The recommendations on asset allocation presented by the investment

advisors are consistent with modern portfolio theory.



For Further Reading (cont.)3 Solnik, Bruno, “Why Not Diversify Internationally Rather Than

Domestically?”, Financial Analysts Journal, 1974.I Substantial advantages in risk reduction can be attained through

portfolio diversification in foreign securities.


Formula Sheet

Outline




IV. Formula Sheet


Formula Sheet

The composition of the minimum-variance portfolio

w = λ Σ−11+ γΣ−1µ

λ =C −µpB

∆, γ =

µpA−B∆

A = 1′Σ−11 > 0, B = 1′Σ−1µ

C = µ′Σ−1

µ > 0, ∆ = AC −B2 > 0

Relation between expected return and variance for the minimum varianceset

σ2p (µp) =

Aµ2p −2Bµp +C

∆

The expected return, variance and composition of the globalminimum-variance portfolio

µg = B/A, σ2g = 1/A, wg =

Σ−111Σ−11


Investments

Session 3. Asset Allocation (part II)


Spring 2010

Asset Allocation II (Session 3) Investments Spring 2010 1 / 33

Outline

I. Optimal Portfolios with no Riskless AssetGraphicallyOptimal Portfolios

II. Optimal Portfolios with a Riskless AssetA New Minimum Variance SetThe Tangency PortfolioThe Optimal PortfolioThe Separation Result

III. Examples


V. Formula Sheet


Optimal Portfolios with no Riskless Asset

Outline



III. Examples


V. Formula Sheet


Optimal Portfolios with no Riskless Asset Graphically

Graphically, optimal portfolios are such that the investor’s indifferencecurves are tangent to the mean-variance efficient set:

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

Portfolio Standard Deviation

Por

tfolio

Exp

ecte

d R

etur

n

Figure 1: Optimal Portfolio with no Riskless Asset


Optimal Portfolios with no Riskless Asset Optimal Portfolios

The Portfolio Choice Problem in the Mean-Variance Case

Consider a mean-variance investor with risk aversion a seeking todetermine his optimal portfolio. His problem is

maxw

(w ′µ− a

2w ′Σw

)s.t. w ′1 = 1 (1)

Set up the Lagrangian

L = w ′µ− a2w ′Σw + λ

(1−1′w

)(2)

The first order condition is

∂L∂w

= µ−aΣw −λ1 = 0 (3)



The Portfolio Choice Problem in the Mean-Variance Case(cont.)

This condition says that the marginal utility of investing in each assetis the same and equals λ .

The investor’s optimal portfolio as a function of λ is therefore

w =1a

Σ−1 (µ−λ1) (4)

To obtain λ , use the constraint w ′1 = 1,

1a1′Σ−1 (µ−λ1) = 1 (5)

solving,

λ =1′Σ−1µ−a

1′Σ−11=

B−aA

(6)



The Portfolio Choice Problem in the Mean-Variance Case(cont.)

Inserting the solution for λ in the equation for w yields thecomposition of the investor’s optimal portfolio as a function of theproperties of the different assets and the investor’s risk aversion,

wo =1a

Σ−1 (µ−λ1) =1−B/a

AΣ−11+

1a

Σ−1µ (7)

The expected return and the variance of the optimal portfolio are

µo = µ′wo =

1−B/aA

B +Ca

(8)

σ2o = w ′oΣwo =

1−B/aA

+µo

a(9)



ExampleConsider an economy with 15 risky assets, and an investor withmean-variance utility and a risk aversion coefficient of 8. Graphically, theoptimal portfolio is: [Matlab code to be shown in class]

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Standard Deviation

Exp

ecte

d R

etur

n


Optimal Portfolios with a Riskless Asset

Outline



III. Examples


V. Formula Sheet


Optimal Portfolios with a Riskless Asset A New Minimum Variance Set

A New Efficient Frontier

Suppose now that in addition to the N risky assets, a riskless assetwith a rate of return of R is also available for investment.

Then, the constraint 1′w = 1 disappears, as any excess or shortage of1′w compared to 1 can be offset by a riskless asset position. That is,the investment in the riskless asset is just ω0 = 1−1′w , where positivevalues denote lending and negative values borrowing (leverage).

Taking the riskless asset position into account, the overall expectedreturn on the portfolio is given by

µp = µ′w +

(1−1′w

)R (10)

yielding the constraint

(µ−R1)′w = µp−R (11)



A New Efficient Frontier (cont.)

Therefore, the variance minimization problem with a riskless assetreads

minw

12w ′Σw s.t. (µ−R1)′w = µp−R (12)

To solve this problem, set up the Lagrangian

L =w ′Σw

2+ γ

(µp−R− (µ−R1)′w

)(13)

The first order conditions are

∂L∂w

= Σw − γ (µ−R1) = 0

∂L∂γ

= µp−R− (µ−R1)′w = 0 (14)




yielding the set of minimum variance portfolios

w = γΣ−1 (µ−R1) , ω0 = 1−1′w (15)

In order to determine γ , use the constraint (µ−R1)′w = µp−R :

(µ−R1)′w = γ (µ−R1)′Σ−1 (µ−R1) = µp−R (16)

yielding

γ =µp−R

(µ−R1)′Σ−1 (µ−R1)=

µp−RC −2RB +R2A

(17)

with A, B, and C defined as before.




The relationship between the expected return and variance of theminimum-variance portfolios when there is a riskless asset is given by:

σ2p (µp) = w ′Σw = γ (µp−R) =

(µp−R)2

C −2RB +R2A(18)

In standard deviation space, the minimum variance set is a pair of rayswith intercepts R and slopes ±

√C −2RB +R2A:




0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25


Por

tfolio

Exp

ecte

d R

etur

n

Figure 2: Minimum variance set with a riskless asset


Optimal Portfolios with a Riskless Asset The Tangency Portfolio

Two Distinct Portfolios

All minimum-variance portfolios are combinations of only two distinctportfolios (“mutual funds”).

Any two minimum-variance portfolios will span the set of allminimum-variance portfolios.

Thus, a natural choice of portfolios is:1 The riskless asset, wR = 0, ω0R = 1, and2 The tangency portfolio

wt =Σ−1 (µ−R1)

B−AR, ω0t = 0 (19)

Let us consider the properties of the tangency portfolio. Note firstthat since ω0t = 0, the tangency portfolio is a member of the originalrisky-asset-only minimum-variance set.


Optimal Portfolios with a Riskless Asset The Tangency Portfolio

Two Distinct Portfolios (cont.)

Its expected return is given by

µt = µ′wt =

C −BRB−AR

(20)

Its variance is given by

σ2t = w ′tΣwt =

C −2RB +R2A(B−AR)2 (21)

Whenever R < µg = B/A, tangency is on the upper limb of thehyperbola, and µt > µg [proof in class].


Optimal Portfolios with a Riskless Asset The Optimal Portfolio

A New Optimal Portfolio

Consider a mean-variance investor with risk aversion a seeking todetermine his optimal portfolio.

Recall that with a riskless asset, the portfolio’s expected return isgiven by

µp = R +w ′ (µ−R1) (22)

The expression for the variance is unchanged and reads

σ2p = w ′Σw (23)

Hence, the investor’s problem is

maxw

(µp−

a2

σ2p

)= max

w

(R +w ′ (µ−R1)− a

2w ′Σw

)(24)



A New Optimal Portfolio (cont.)

The first-order condition is

µ−R1 = aΣw (25)

yielding the optimal portfolio

w =1a

Σ−1 (µ−R1) (26)

Observe that this portfolio is proportional to the tangency portfolio

wt =Σ−1 (µ−R1)

B−AR, ω0t = 0 (27)



Graphically

Graphically, the optimal portfolio is shown below:

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25


Por

tfolio

Exp

ecte

d R

etur

n

Figure 3: Optimal portfolio with a riskless asset


Optimal Portfolios with a Riskless Asset The Separation Result

The Capital Market Line

Hence, under homogeneous beliefs and mean-variance preferences, thefollowing “separation” result holds

I The optimal risky portfolio is identical for all investors, it is thetangency portfolio.

I All investors hold a combination of the risk-free asset and the tangencyportfolio, independently of preferences.

I Individual preferences only determines the share of those two portfoliosin the investor’s overall investment. Investors with low risk-aversionmake levered purchases of the tangency portfolio, investors with highrisk-aversion invest part of their money in the risk-free asset, part inthe tangency portfolio.

The line that goes through the riskless asset and the tangencyportfolio is the mean-variance efficient set and is called the CapitalMarket Line (CML).



The Capital Market Line (cont.)It gives the risk-return tradeoff available on the market. There is nobetter risk-return tradeoff than the one offered by the Capital MarketLine.

As a result, the portfolio selection process for all investors can be seenas consisting of two stages:

1 Selection of the optimal risky portfolio (the tangency portfolio), and2 Based on individual risk tolerance, choice of the optimal combination

between the riskless asset and the investment in the tangency portfolio.

An interpretation of the tangency portfolio in terms of the riskaversion is instructive:

I Suppose we have i = 1, ...I investors and that investor i ’s risk aversionis ai . Investor i ’s optimal portfolio is therefore

wi =1ai

Σ−1 (µ−R1) (28)



The Capital Market Line (cont.)

I The tangency portfolio is given by

wt =Σ−1 (µ−R1)

B−AR, ω0t = 0 (29)

I Hence, we can view B−AR as the “market’s” risk aversion, aM . Then,using Σ−1 (µ−R1) = (B−AR)wt = aMwt implies

wi = wtB−AR

ai= wt

aM

ai(30)


Examples

Outline



III. Examples


V. Formula Sheet


Examples

Examples[Matlab codes for these examples will be shown in class]

Take the previous example with 15 risky assets and add the risk-freeasset with R = 0.05. The graph below shows the investor’s utility gain:

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Standard Deviation

Exp

ecte

d R

etur

n


Examples

Examples (cont.)Let’s go back to the case without the risk-free asset, and assume thatshort sales are not allowed. We are forced to solve numerically for thenew optimal portfolio. The solution is:

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Standard Deviation

Exp

ecte

d R

etur

n


Examples

Examples (cont.)

Two more examples

on the importance of correctly estimating expected return,

on time diversification.



Outline



III. Examples


V. Formula Sheet



Summary

The Markowitz Portfolio Selection Model1 The first step is to determine the risk-return opportunities available to

the investor, summarized by the minimum-variance frontier. The partof the frontier that lies above the global minimum-variance portfolio isthe efficient frontier.

2 The point of tangency between the efficient frontier and the CapitalMarket Line is the optimal risky portfolio, identical for all investors.

3 All investors hold a combination of the risk-free asset and thetangency portfolio, independently of preferences.

4 The mix of risky investments and risk-free investment vary with thedegree of risk aversion of the investor



For Further Reading

1 Black, Fischer, “Estimating Expected Return”. Financial AnalystsJournal, 1993.

I Estimating expected return is hard. Past average return is normally ahighly innacurate estimate. Moreover, in finance we are interested in"ex ante" i.e., the future not "ex post" the past. The article outlinesthe problems of using theory or data to estimate expected return.

2 Brinson et al., “Determinants of Portfolio Performance”, FinancialAnalysts Journal, 1986.

I Much time is spent evaluationg individual portfolio managers or stocks- Is it worth it?

I Brinson et al. measured the importance of asset allocation. Conclusion:in typical cases, variability in portfolio performance is driven by theasset allocation.

I Similar conclusions in Ibbotson and Kaplan (2000).



For Further Reading (cont.)

3 Peter Bernstein, “Capital Ideas: The Improbable Origins of ModernWall Street”, 1992, Chapter 3.

I About James Tobin and the logic of the Separation Theorem.I The convenient fact that has just been proved is that the proportionate

composition of the non-cash [i.e., risky] assets is independent of theiraggregate share of the investment balance.


Formula Sheet

Outline



III. Examples


V. Formula Sheet


Formula Sheet

Optimal portfolios with no riskless assets

wo =1−B/a

AΣ−11+

1a

Σ−1µ (31)

µo =1−B/a

AB +

Ca

(32)

σ2o =

1−B/aA

+µo

a(33)

The new efficient set with a riskless asset

w = γΣ−1 (µ−R1) , ω0 = 1−1′w (34)

γ =µp−R

C −2RB +R2A(35)

σ2p (µp) =

(µp−R)2

C −2RB +R2A(36)


Formula Sheet

The tangency portfolio

wt =Σ−1 (µ−R1)

B−AR, ω0t = 0 (37)

µt =C −BRB−AR

(38)

σ2t =

C −2RB +R2A(B−AR)2 (39)

Optimal portfolios with a riskless asset

wo =1a

Σ−1 (µ−R1) (40)


Investments

Session 4. The Capital Asset Pricing Model


Spring 2010

CAPM (Session 4) Investments Spring 2010 1 / 37

Outline

I. Introduction & DerivationIntroductionIntuitive DerivationFormal Derivation

II. InterpretationFrom the CML to the SML

III. Nonstandard FormsRelaxing Assumptions

IV. Limitations

V. Summary & Further Reading

VI. Formula Sheet


Introduction & Derivation

Outline




IV. Limitations


VI. Formula Sheet


Introduction & Derivation Introduction

Previous Results

We have seen that the Capital Market Line (the line that goes throughthe riskless asset and the tangency portfolio) gives the best risk-returntradeoff available to investors in a mean-variance world.As a result, all investors’ optimal portfolio consisted in a combinationof the riskless asset and the tangency portfolio. We know now thatportfolio choice could be made in two steps:

1 Selection of the optimal risky portfolio (the tangency portfolio).2 Based on individual risk tolerance, choice of the optimal combination

between the riskless asset and the investment in the tangency portfolio.

The question we whish to answer now is: what does this “separation”imply for capital asset prices in equilibrium?


Introduction & Derivation Introduction

A Nobel Prize idea: William SharpeSharpe was the first to answer this question and developed the capitalasset pricing model. He shared the 1990 Nobel Prize in economicswith Harry Markowitz.“I said what if everyone was optimizing? They’ve all got their copies ofMarkowitz and they’re doing what he says. Then some people decidethey want to hold more IBM, but there aren’t enough shares to satisfydemand. So they put price pressure on IBM and up it goes, at whichpoint they have to change their estimates of risk and return, becausenow they’re paying more for the stock. That process of upward anddownward pressure on prices continues until prices reach anequilibrium and everyone collectively wants to hold what’s available.At that point, what can you say about the relationship between riskand return? The answer is that expected return is proportionate tobeta relative to the market.”Source: J. Burton (1998), “Revisiting the CAPM”, Dow Jones AssetManager, 1998.CAPM (Session 4) Investments Spring 2010 5 / 37

Introduction & Derivation Intuitive Derivation

In a mean-variance world, all investors optimally hold a combination ofthe riskless asset and of the tangency portfolio.

Suppose an investor chooses to hold another combination (i.e., to bearidiosyncratic risk). Will he earn a higher return?

Just looking at the graph, we can see that the answer is no.Themarket pays no risk premium for being inside the frontier.

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25


Por

tfolio

Exp

ecte

d R

etur

n



Using the CML, the expected return on any efficient portfolio we (theones on the CML) is given by

µe = R +µt −R

σtσe (1)

The first term (R) is compensation for the time value of money, thesecond term µt−R

σtσe is compensation for risk.

This second term is the product of the market price of risk,

λ =µt −R

σt(2)

and of the amount of risk borne by the investor, σe . The market priceof risk λ tells us how much the market compensates investors for eachunit of risk. The risk premium µt−R

σtσe is therefore linear in the

amount of systematic risk borne by the investor.



Now, recall that as claimed earlier, only systematic risk is iscompensated. Bearing idiosyncratic risk does not entitle to a riskpremium.Consider an asset or portfolio with total risk σj . Letting ρjt denote thecorrelation of its returns with the tangency portfolio, the risk σj canbe composed into

1 systematic risk ρjtσj and2 idiosyncratic risk

√1−ρ2

jtσj .

Therefore, using the market price for risk λ , the risk premium on theasset must be

µj −R = λρjtσj =µt −R

σtρjtσj = (µt −R)ρjt

σj

σt

= (µt −R)Cov

(Rj ,Rt

)Var

(Rt

) (3)


Introduction & Derivation Formal Derivation

The CAPM relationship

In this section we will derive the CAPM more rigorously. To beginwith, make the following assumptions:

1 Individual investors cannot affect prices. That is, there are many buyersand sellers (perfect competition).

2 All investors plan for one identical holding period and they aremean-variance optimizers. That is, they choose the portfolio wi thatmaximizes µ− ai

2 σ2, where ai denotes investor i ’s degree of riskaversion (any other mean-variance preferences would work as well).

3 There is a riskless asset with return R and N risky assets with expectedreturn µ and variance-covariance matrix Σ.

4 Investors pay neither taxes on returns nor transactions costs.5 Each asset is perfectly divisible.6 The riskless asset can be bought or sold in unlimited amounts.7 All investors have homogenous expectations about µ and Σ.



The CAPM relationship (cont.)

Under these assumption, each investor’s portfolio is the solution to

maxwi

(R +w ′i (µ−R1)− ai

2w ′i Σwi

)(4)

which yields the first-order condition (this becomes standard by now)

µ−R1 = aiΣwi (5)

and the optimal portfolio

wi =1ai

Σ−1 (µ−R1) (6)




Now, sum over all investors, weighting each investor by his relativewealth Wi

∑Ii=1 Wi

= WiW to get

I

∑i=1

(Wi

Wwi

)=

I

∑i=1

(Wi

W1ai

)Σ−1 (µ−R1) (7)

Since the riskless asset is in zero net supply, the left hand side of thisexpression must be the tangency portfolio, or the market portfolio.Therefore, we can write

wt = wM =I

∑i=1

(Wi

W1ai

)Σ−1 (µ−R1) (8)




or, solving for µ−R1,

µ−R1 =1

∑Ii=1

(WiW

1ai

)ΣwM (9)

The problem we face here is that we do not know ∑Ii=1

(WiW

1ai

). To

find it, pre-multiply the above expression by w ′M to get

w ′M (µ−R1) =1

∑Ii=1

(WiW

1ai

)w ′MΣwM (10)




Using w ′M1 = 1, rewrite this expression as

µM −R =1

∑Ii=1

(WiW

1ai

)σ2M (11)

to get1

∑Ii=1

(WiW

1ai

) =µM −R

σ2M

(12)




Then, substitute into µ−R1 = 1∑

Ii=1

(WiW

1ai

)ΣwM to obtain

µ−R1 =µM −R

σ2M

ΣwM (13)

Finally, note that ΣwM = Cov(R, RM

)and write the CAPM

relationship as

µ−R1 =Cov

(R, RM

)Var

(RM

) (µM −R) (14)


Interpretation

Outline




IV. Limitations


VI. Formula Sheet


Interpretation From the CML to the SML

Interpretation

The CAPM relationship says that the equilibrium risk premium on anyasset j , µj −R , equals

µj −R = βj (µM −R) , βj =Cov

(Rj , RM

)Var

(RM

) = ρjMσj

σM(15)

Thus, the market only compensates investors for bearing systematic(i.e., market) risk, not idiosyncratic risk. There is a single source ofsystematic risk: covariance with the market portfolio. The expectedreturn on an asset with returns uncorrelated with those of the marketportfolio is R , even if the asset is risky.



Interpretation (cont.)

The relationship between the expected return on an asset and its β islinear.

The risk premium on an asset is proportional to the excess return onthe market portfolio, µM −R .

What the CAPM achieves is to move from the capital market line(CML) to the security market line (SML):

µj = R + βj (µM −R) (16)

In equilibrium, all assets plot on the security market line.



Interpretation (cont.)

The CML versus the SML.I The CML graphs the risk premiums of efficient portfolios (complete

portfolios of risky securities and the risk-free asset) as a function ofportfolio standard deviation.

I The SML graphs the risk premium of individual assets, thus measuringthe risk contribution that an asset would make to the standarddeviation of a portfolio. This contribution is measured by the asset’sbeta.

I “Fairly priced” assets lie on the SML; alpha measures the distancebetween the fair and the actually expected return on the asset. Assetsplotting below the SML are overpriced; those plotting above the SMLare underpriced.



Beta and expected returns

Why should a positive dependence between β and the expected returnon an asset arise?Consider an asset with a high β . It is procyclical: whenever themarket does well, it tends to do very well. But if the market does well,then overall wealth is high, and the marginal utility of consumption islow. In a sense, this asset pays off a lot when money is least valuable,and very little when money is very valuable. Therefore its price will below and its expected return high.On the other hand, an asset with negative β is anticyclical: it pays offa lot when aggregate wealth is low (and money very valuable), andlittle when overall wealth is high (and money less valuable). As aresult, its price will be high, and expected return low (an example ofsuch an asset is gold, which has a correlation of -0.4 with stocks).


Nonstandard Forms

Outline




IV. Limitations


VI. Formula Sheet


Nonstandard Forms Relaxing Assumptions

Extensions of the CAPM

In the above analysis, we made a number of restrictive assumptions,such as

1 No short-selling constraints,2 Existence of a riskless asset allowing unlimited lending and borrowing

at the same rate,3 No taxes,4 All assets can be marketed,5 Heterogeneous expectations,6 Non-price-taking behavior,7 Single investment period.

We will now consider how robust the CAPM equation is to relaxationof these assumptions. Only short-selling constraints and the no risklessasset case will be discussed here.



Short-Selling Constraints

What is the effect of short-selling constraints on equilibrum assetprices?Recall that in the CAPM, all investors hold the market portfolio inequilibrium. Therefore, in equilibrium, no investor sells any securityshort.As a result, the short-selling constraints is non-binding and equilibriumprices are unaffected by it. The standard version of the CAPM holdshere as well.



The Zero-Beta CAPM

When investors can no longer borrow or lend at a common risk-freerate, they may choose risky portfolios from the entire set of efficientfrontier portfolios, according to how much risk they choose to bear.

An equilibrium expected return-beta relationship in the case ofrestricted risk-free investments has been developed by Fischer Black.The model is based on the three following properties of mean-varianceefficient portfolios:

1 any portfolio constructed by combining efficient portfolios is on theefficient frontier.

2 every portfolio on the efficient frontier has a “companion” portfolio onthe inefficient portion of the minimum-variance frontier with which it isuncorrelated.

3 the expected return of any asset can be expressed as an exact, linearfunction of the expected return on any two frontier portfolios.



The Zero-Beta CAPM (cont.)We assume that the risky assets can be sold short. The result wederive below also obtains if we allow lending and borrowing, but atdifferent rates.If there is no riskless asset, each investor solves

maxwi

(w ′i µ− ai

2w ′i Σwi

)s.t. w ′i 1 = 1 (17)

yielding the first-order condition µ−aiΣwi −λ1 = 0 and the optimalportfolio

wi =1ai

Σ−1 (µ−λ1) = biwd + (1−bi )wg (18)

with wd = Σ−1µ

1′Σ−1µand wg = Σ−11

1′Σ−11 from the two-fund separation result.

Since all investors’ portfolios are a combination of portfolios wg andwd , the market portfolio will be such a combination as well, i.e. wehave wM = bMwd + (1−bM)wg .



The Zero-Beta CAPM (cont.)

Given this, note that

Cov(R, RM

)= ΣwM = Σ[bMwd + (1−bM)wg ]

= bMΣΣ−1µ

1′Σ−1µ+ (1−bM)

ΣΣ−111′Σ−11

=bM

Bµ +

1−bM

A1 (19)

Solving for µ implies the expected returns relation

µ =BbM

(ΣwM −

1−bM

A1)

=BbM

(Cov

(R, RM

)− 1−bM

A1)

(20)

As in the CAPM case, our problem is to determine unknown quantitiesfrom market data (here bM/B and (1−bM)/A).



The Zero-Beta CAPM (cont.)Using the above relationship, for the market portfolio, we have

σ2M = w ′MΣwM =

bM

Bw ′M µ +

1−bM

Aw ′M1 =

bM

BµM +

1−bM

A(21)

Now, take another minimum-variance portfolio wz that is uncorrelatedwith the market portfolio. This yields

σzM =bM

Bµz +

1−bM

A= 0 (22)

Solving these two equations, we have

bM

B=

σ2M

µM −µz(23)

and1−bM

A=−bM

Bµz =−

µzσ2M

µM −µz(24)



The Zero-Beta CAPM (cont.)Inserting this into the expected returns relationµ = B

bM

(ΣwM − 1−bM

A 1)yields

µ =µM −µz

σ2M

(ΣwM +

µzσ2M

µM −µz1)

= µz1+ (µM −µz)ΣwM

σ2M

= µz1+Cov

(R, RM

)σ2

M= µz1+ β (µM −µz) (25)

This result is known as the zero-beta CAPM (or the Black CAPM ortwo-factor model). When there is no riskless asset, R is replaced withµz in the pricing equations, where µz is the expected return on aportfolio that is uncorrelated with the market portfolio.CAPM (Session 4) Investments Spring 2010 27 / 37


The Zero-Beta CAPM (cont.)Note the difference between the usual CAPM and the zero-betaCAPM:

I In the standard CAPM, all investors hold the market portfolio andtherefore short sale restrictions are not binding.

I In the zero-beta CAPM, investors may want to hold any portfolio onthe mean-variance efficient set (the upper limb of the hyperbola). Forthese combinations to be attainable, short-selling of risky assets mustbe allowed.

One can show that the expected return of the zero-beta portfolio islower than the expected return of the global minimum-varianceportfolio wg , as follows.Recall that any minimum-variance portfolio can be formed as thecombination of the market portfolio and the zero-beta portfolio. Sincethe two are uncorrelated,

σ2g = ω

2z σ

2z + (1−ωz)2

σ2M (26)



The Zero-Beta CAPM (cont.)

Minimizing this expression with respect to ωz yields the first-ordercondition

dσ2g

dωz= 2ωzσ

2z −2(1−ωz)σ

2M = 0 (27)

or, since both variances are positive,

0 < ωz =σ2

Mσ2

M + σ2z

< 1 (28)

Therefore , the expected return on the global minimum varianceportfolio,

µg = ωz µz + (1−ωz) µM (29)

will lie between that on the market portfolio and that on the zero-betaportfolio, implying that the latter is on the lower limb of the hyperbola.


Limitations

Outline




IV. Limitations


VI. Formula Sheet


Limitations

Shortcomings of CAPM

The model assumes just two dates. In reality, people make multiperioddecisions, so they should have the opportunity to consume andrebalance portfolios repeatedly over time. For such extensions, see theintertemporal CAPM (ICAPM) of Robert Merton, and theconsumption CAPM (CCAPM) of Douglas Breeden and MarkRubinstein.

Individuals have imperfect information and heterogeneousexpectations.

The specification of the market portfolio is difficult. In theory, themarket portfolio should include all types of assets that are held byanyone - including works of art, real estate, human capital, etc. (Roll’scritique). This brings a severe limitation with respect to empiricaltesting.


Limitations

Shortcomings of CAPM (cont.)

The model assumes that there are no taxes or transaction costs.

Investors could have biased expectations. They can be, for example,overconfident. Also, some investors, like casino gamblers, will acceptlower returns for higher risk.

The model assumes that asset returns are jointly normally distributed.

The model assumes that the variance of returns is an adequatemeasurement of risk.



Outline




IV. Limitations


VI. Formula Sheet



Summary

In equilibrium, the tangency portfolio becomes the market portfolio.The expected return of the market portfolio depends on the averagerisk aversion in the market.The intuition of the CAPM: expected return of any risky assetdepends linearly on its exposure to the market risk, measured by β .Diversification is an important concept in finance. It builds on apowerful mathematical machine called Strong Law of Large Numbers.Beyond the CAPM:

I Is β a good measure of risk exposure? What about the risk associatedwith negative skewness?

I Could there be other risk factors?I Time varying risk aversion, and time varying β?



For Further Reading

1 Jagannathan and McGrattan, The CAPM Debate, 1995I a nice review of studies that support or challenge the CAPM.

2 Bernstein, Peter, Capital Ideas: The Improbable Origins of ModernWall Street, 1992, Chapter 4.

I about William Sharpe and his enormous single influence (he didhowever his work under the supervision of Harry Markowitz).

3 Burton, Jonathan, Revisiting the CAPM, 1998.I an eye-opening interview with William Sharpe.


Formula Sheet

Outline




IV. Limitations


VI. Formula Sheet


Formula Sheet

The Capital Market Line

µe = R +µt −R

σtσe (30)

The Security Market Line (CAPM relationship):

µj −R = βj (µM −R) , βj =Cov

(Rj , RM

)Var

(RM

) = ρjMσj

σM(31)

The Zero-Beta CAPM

µ = µz1+ β (µM −µz) (32)


Investments

Session 5. Arbitrage Pricing Theory


Spring 2010

APT (Session 5) Investments Spring 2010 1 / 36

Outline

I. Intro & AssumptionsIntroductionAssumptions

II. Derivation & InterpretationDerivationInterpretation

III. CAPM versus APTCAPM versus APT

IV. Using APT in PracticeSeveral ApproachesAn ExampleOther Uses



Intro & Assumptions

Outline







Intro & Assumptions Introduction

Novel Feature

Arbitrage pricing theory is a new and different approach todetermining asset prices. It is based on the law of one price: two itemsthat are the same can’t sell at different prices.The CAPM predicts than only one type of nondiversifiable riskinfluences expected security returns: market risk i.e. covariance withthe market portfolio.On the other hand, the APT accepts a variety of different risk sources,such as the business cycle, interest rates and inflation.Like the CAPM, the APT distinguishes between idiosyncratic andsystematic sources of risk. However, in the APT there are severalsources of systematic risk. The expected return on an asset is drivenby its exposure to the different sources of systematic risk.


Intro & Assumptions Assumptions

The APT requires that te returns on any stock be linearly related to aset of indices (or factors) as shown below:

Rj = aj +bj1I1 +bj2I2 + ...+bjK IK + εj (1)

whereI aj = the expected level of return for stock j if all indices have a value

of zeroI Ik = the value of the kth index that impacts the return on stock jI bjk = the sensitivity of stock j ’s return to the kth indexI εj = a random error term with mean zero and variance equal to σ2

εj .

Some assumptions are required to fully describe the process-generatingsecurity returns:

E [εiεj ] = 0, for all i and jwhere i 6= jE[ej(Ik − Ik

)]= 0, for all stocks and indices

with Ik = E [Ik ].APT (Session 5) Investments Spring 2010 5 / 36


Taking the expected value of equation (1) and substracting it fromequation (1), we have

Rj −µj = bj1f1 + ...+bjK fK + εj (2)

where fk = Ik − Ik . The difference between the realized return and theexpected return for any asset is

1 the sum, over all risk factors k , of the asset’s risk exposure (bjk) tothat factor, multiplied by the realization for that risk factor, fk ,

2 plus an asset-specific idiosyncratic error term, εj .

Note that the risk factors themselves may be correlated, as may theasset-specific shocks for different assets.

To derive the APT, one postulates that pure arbitrage profits areimpossible.



The Notion of Arbitrage

The principal strength of the APT approach is that it is based on theno arbitrage condition.Intuitively, arbitrage means “there is no such thing as a free lunch”.Two assets with identical attributes should sell for the same price, andso should an identical asset trading in two different markets (Law ofone Price).Arbitrage is a common feature of competitive markets. Even touristsignorants of the theory of finance can turn into arbitrageurs (exchangerate example).Arbitrage has been elevated to the level of a driving force byModigliani and Miller in 1958. They used the arbitrage argument toprove that the value of a firm as a whole is independent of its capitalstructure (MM theorems).


Derivation & Interpretation

Outline







Derivation & Interpretation Derivation

To derive the APT equilibrium pricing relationship, one constructs aportfolio w with the following properties:

w ′1 =N

∑n=1

ωn = 0 (3)

w ′bk =N

∑n=1

ωnbnk = 0 ∀k (4)

w ′ε =N

∑n=1

ωnεn ≈ 0 (5)

These conditions can be met if there is a sufficient number ofsecurities available on the market.



Since the portfolio has zero initial cost and a risk of zero, it must havean expected return of zero, i.e. we must have

w ′µ =N

∑n=1

ωnµn = 0 (6)

In other words: if w is orthogonal to a vector of ones (the firstcondition) and orthogonal to K vectors of bk ’s, then it is alsoorthogonal to the vector of expected returns, µ .

There is a theorem in linear algebra that states the following: if thefact that a vector is orthogonal to M−1 vectors implies that it is alsoorthogonal to the Mth vector, then the Mth vector can be expressedas a linear combination of the M−1 vectors.



Here is some intuition for why this result holds. Write µ as some linearcombination of 1 and the bk ’s, plus an error term d (which will benonzero if µ cannot be written as a linear combination of the vectors),

µ = P01+K

∑k=1

Pkbk +d (7)

Then,

w ′µ = P0w ′1+K

∑k=1

Pkw ′bk +w ′d = w ′d (8)

Note that none of the restrictions on w imposed by the constraintsthat w ′1 = 0 and w ′bk = 0 allow us to say anything about theremaining term w ′d . Therefore, the only way to ensure that w ′d = 0so that w ′µ = 0 is to set d = 0, which says that µ can indeed bewritten as a linear combination of 1 and the bk ’s.



We have just shown that for there to be no arbitrage, one must beable to write µ as a linear combination of 1 and the K vectors bk ,

µ = P01+K

∑k=1

Pkbk (9)

This is the main APT theorem: under the above assumptions, thereexist K +1 numbers P0,P1, ...,PK , not all zero, such that theexpected return on asset j is approximately equal to

µj ≈ P0 +bj1P1 + ...+bjKPK (10)

Under the additional assumptions that (i) there exists a portfolio withno nonsystematic risk and that (ii) some investor considers it hisoptimal portfolio, one can show that the above equation holds withequality (Chen and Ingersoll, JF 1983),

µj = P0 +bj1P1 + ...+bjKPK (11)



Pk is the price of risk (the risk premim) for the kth risk factor, anddetermines the risk-return tradeoff.

Consider a portfolio p that is perfectly diversified (εp = 0) and with nofactor exposures

(bpk = 0, ∀k

). Such portfolio has zero risk, and its

expected return is P0. Therefore, P0 must equal the risk-free rate ofreturn R .

Similarly, the risk premium for the kth risk factor, Pk , is the return, inexcess of the risk-free rate, earned on an asset that has an exposure ofbjk = 1 to the kth factor and zero risk exposure to all other factors(bjh = 0, ∀h 6= k

):

Pk = µF ,k −R (12)

Thus Pk are returns for bearing the risks associated with the indicesIk , or factor risk premiums.



Substituting the expected return relationship (11) into the multi-factormodel specification (2) yields

Rj −µj = Rj − (P0 +bj1P1 + ...+bjKPK )

= bj1f1 + ...+bjK fK + εj (13)

which, using P0 = R , can be rewritten to yield the full APT equation

Rj −R = bj1 (P1 + f1) + ...+bjK (PK + fK ) + εj (14)

This says that the realized return on an asset in excess of the risk-freerate is the sum of 3 components

1 expected macroeconomic factor return (the P’s), i.e. the reward for therisks taken,

2 unexpected macroeconomic factor return (the f ’s), and3 an idiosyncratic component (ε).


Derivation & Interpretation Interpretation

Interpretation: Several Risk Factors

Taking expectations on the full APT equation yields

µj −R = bj1P1 + ...+bjKPK (15)

which says that the expected excess return on an asset is the sum overall factors k of the product of the factor’s risk premium Pk and of theasset’s risk exposure to that factor bjk .As the exposure of a portfolio to a particular factor k is increased, theexpected return on the portfolio is increased if Pk > 0.Note that the big difference between the CAPM and the APT is thatthe CAPM postulates that the risk premium on an asset depends on asingle factor: covariance with the market portfolio. In the APT, on theother hand, several factors may drive expected returns (but the APTdoes not say what they are).


CAPM versus APT

Outline







CAPM versus APT CAPM versus APT

Equivalence

We should discuss the fact that the APT model and, in fact, theexistence of a multifactor model, is not necessarily inconsistent withthe CAPM.

The APT equation is

µj = R +K

∑k=1

bjkPk (16)

Recall that if the CAPM is the equilibrium model, it holds for allsecurities, as well as all portfolios of securities. We have seen that Pkis the excess return on a portfolio with a bjk of one on one index andbjk of zero on all other indices, thus the indices can be represented byportfolios of securities.

If the CAPM holds, the equilibrium return on each Pk is given by

Pk = µF ,k −R = βF ,k (µM −R) , ∀k (17)



Equivalence (cont.)

Substituting into equation (16) yields

µj −R = (µM −R)

(K

∑k=1

bjkβF ,k

)(18)

Defining βj as ∑Kk=1 bjkβF ,k results in the expected return of security j

being priced by the CAPM

µj = R + βj (µM −R) (19)

The APT solution with multiple factors appropriately priced is fullyconsistent with the CAPM. Conversely, if the APT is true and the Krestrictions on the Pk ’s hold, then the CAPM is also true.



Equivalence (cont.)

This result is important for empirical testing: employing statisticaltechniques to estimate the Pk ’s and finding that more than onecoefficient is significantly different from zero is not sufficient proof toreject any CAPM. If the Pk ’s are not significantly different formβF ,k (µM −R), the empirical results could be fully consistent with theCAPM.

Thus, it is perfectly possible that more than one index explains thecovariance between security returns but that the CAPM holds.

However, in empirical tests, the restrictions on APT coefficientsimposed by the CAPM are rejected.


Using APT in Practice

Outline







Using APT in Practice Several Approaches

The proof of any economic theory is how well it describes reality. Letus review the structure of APT that will enter any test procedure.

We can write the multifactor return-generating process as

Rj = aj +K

∑k=1

bjk Ik + εj (20)

The APT model that arises from this return-generating process can bewritten as

µj = R +K

∑k=1

bjkPk (21)

Notice from equation (20) that each security j has a unique sensitivityto each Ik but that any Ik has a value that is the same for allsecurities. Any Ik affects more than one security (if it did not, it wouldhave been compounded in the residual term εj).



These Ik ’s have generally been given the name factors in the APTliterature. The factors affect the returns of more than one security andare the sources of covariance between securities.

The bjk ’s are unique to each security and represent attributes or acharacteristics of the security.

Finally, from equation (21) we see that Pk is the extra expected returnrequired because of a security’s sensitivity to the kth attribute.

Recall that the APT does not say what the K factors are. Therefore,in order to test the APT, one must test equation (21), which meansthat one must have estimates of the bjk ’s. However, to estimate thebjk ’s we must have definitions of the relevant Ik ’s.



Three approaches can be used to estimate and test the APT:1 The most general approach is to use statistical techniques and estimate

simultaneously factors (Ik ’s) and firm attributes (bjk ’s). The resultsthus obtained have the drawback that the estimated factors are difficultto interpret because they are non-unique linear combinations of morefundamental underlying economic forces.

2 Specify a set of characteristics (bjk ’s) a priori. Then the values of thePk ’s would be estimated via regression analysis.

3 The drawback of the first two methods is that they use stock returnsto explain stock returns. A third approach would be to use economictheory and knowledge of financial markets to specify K risk factors thatcan be measured from available macroeconomic and financial data.This is the preferred approach.


Using APT in Practice An Example

In Practice

Let us take a look at the third approach. This part is followingBurmeister, Roll and Ross, (2003) “Using Macroeconomic Factors toControl Portfolio Risk”. The factors should

I be easy to interpret,I be robust over time, andI explain as much as possible of the variation in stock returns.

Empirical research has established that one set of five factors meetingthese criteria is the following:



In Practice (cont.)Name Measure Risk Premium

Confidence Risk: Investors’willingness to take risks

Rate of return on riskycorporate bonds minus rate ofreturn on government bonds(positive values meansincreased investor confidence)

P1 = 2.59%,

bj1 > 0

Time Horizon Risk: change ininvestors’ desired time topayouts

Return on 20-year governmentbonds minus return on 30-dayTreasury bills (f2 > 0 when theprice of long-term bonds risesrelative to the t-bill price)

P2 =−0.66%,

bj2 > 0

Inflation Risk: unexpected inflation Inflation surprise: actualinflation minus expectedinflation

P3 =−4.32%,

usually bj3 < 0

Business Cycle Risk:unanticipated changes in realbusiness activity

Change in the index of businessactivity (i.e. value in monthT +1 minus value in month T

P4 = 1.49%,

usually bj4 > 0

Market-Timing Risk Part of the S&P 500 totalreturn that is not explained bythe first four factors plus aconstant

P5 = 3.61%,

bJ5 > 0



In Practice (cont.)

For any asset or portfolio, we therefore have

µj −R = 2.59bj1−0.66bj2−4.32bj3 +1.49bj4 +3.61bj5 (22)

This says that the risk premium on any asset or portfolio is the sum ofthe product, over all K risk factors, of the asset’s exposure and of thecorresponding price of risk.

As an example, for the S&P 500, the exposures are b1 = 0.27,b2 = 0.56, b3 =−0.37, b4 = 1.71, b5 = 1.

Therefore, using the factor risk premia, the expected excess return onthe S&P is

µM −R = 8.09% (23)

Computing the expected return on some assets or portfolios is onlyone of the many uses of the APT. We now briefly discuss the others.


Using APT in Practice Other Uses

Tilting & Other Strategies

Determining Risk Exposure: Using the APT, one can determine theexposure of one’s portfolio to the different factors. Let w denote thevector of portfolio weights and

B =

b11 ... bN1...b1K ... bNK

(24)

the (stacked) matrix of factor exposures of the N assets. Then, the(column) vector of the portfolio’s factor exposures is given by

bp = B ·w (25)

The exposure of the portfolio to the kth factor is simply a weightedaverage of the individual assets’ exposure to that factor.



Tilting & Other Strategies (cont.)Tilting (Making a Factor Bet): If you consider that you have superiorknowledge about the future evolution of some of the factors, you canincrease the exposure of your portfolio to the factors that are expectedto lead to improvements in returns and reduce the exposure to thosefactors that are expected to lead to a deterioration in returns. To doso, construct factor portfolios with an exposure of 1 to the kth factorand 0 to all other factors. Let 1kdenote this target exposure pattern.Then, the portfolio weights must solve

B ·wk = 1k (26)

If the number of assets is equal to the number of factors, then one haswk = B−11k . If there are more assets than factors, then there will bean inifinite number of factor portfolios solving B ·wk = 1k .

In order to find factor portfolios in this more general case, we can usethe following results from linear algebra:



Tilting & Other Strategies (cont.)

I An n×m matrix X is the pseudoinverse of an m×n matrix A if thefollowing four conditions hold: AXA = A, XAX = X , (AX )′ = AX , and(XA)′ = XA. We will denote the pseudoinverse of a matrix A by A+.The matlab command for the pseudoinverse is “pinv”.

I A necessary and sufficient condition for the vector equation Ax = b tohave a solution is that AA+b = b, in which case the general solution is

x = A+b+(I −A+A

)q (27)

where q is an arbitrary vector (see, for example, Magnus andNeudecker, Matrix Differential Calculs with Applications to Statisticsand Econometrics, Chapter 2, Theorem 12).

Hence, the set of factor portfolios for the kth factor is given by

wk = B+1k +(I −B+B

)q (28)




Similarly, the set of portfolios with a factor exposure of bp is

w = B+bp +(I −B+B

)q (29)

If one is looking for the portfolio that has minimum risk subject tomeeting the target factor exposure, one needs to solve

minq

w ′Σw (30)

with w given by (29).

Long-Short Strategies: If one has stock selection skills but nomacroeconomic prediction skills, one can still use the APT, as follows:

1 buy stocks with high expected idiosyncratic return ε,2 short stocks with negative expected idiosyncratic return.




If the long and the short portfolio are constructed so as to haveopposite exposures to each of the risk factors, the systematic risk willbe zero and expected return will lie above the risk-ree rate. The APThelps ensure that the portfolios are appropriately constructed.

Return Attribution: After observing the returns on one’s portfolio, onecan determine their source:

1 expected macroeconomic factor return (the P’s), i.e. the reward for therisks taken,

2 unexpected macroeconomic factor return (the f ’s), and3 anything that remains (ε), which one can attribue to luck or to stock

selection.



Outline








Summary

Like the CAPM, the basic concept of the APT is that differences inexpected return must be driven by differences in non-diversifiable risk.

The APT is based purely on no-arbitrage condition. It is not anequilibrium concept, and does not depend on having a marketportfolio.

Through the use of arbitrage, APT provides investors with strategiesfor betting on their forecasts of the factors that shape stock returns.

The construction of APT enables it to avoid the rigid and oftenunrealistic assumptions required by CAPM.

CAPM specifies where asset prices will settle, given investorpreferences, but it is silent about what produces the returns thatinvestors expect. It also identifies only one factor as the dominantinfluence on stock returns.



Summary (cont.)

APT fills those gaps by providing a method to measure how stockprices will respond to changes in the multitude of economic factorsthat influence them, such as economic growth, inflation , interest ratepatterns, etc.

The CAPM assumes an unobservable “market” portfolio. The APT isbased on the assumption of no arbitrage profits in well-diversifiedportfolios.

The APT provides no guidance for identification of the various marketfactors and appropriate risk premiums for these factors.



For Further Reading

1 Roll and Ross, The Arbitrage Pricing Theory Approach to StrategicPortfolio Planning, Financial Analysts Journal 1984.

I intuitive description of APT and a discussion ot its merits for portfoliomanagement.

2 Burmeister, Roll and Ross, Using Macroeconomic Factors to ControlPortfolio Risk, 2003.

I understanding the macroeconomic forces impacting stock returns.



Formula Sheet

The multifactor return-generating process

Rj = aj +K

∑k=1

bjk Ik + εj (31)

The APT model that arises from this return-generating process

µj = R +K

∑k=1

bjkPk (32)


Investments

Session 6. Practical Issues in Portfolio Management


Spring 2010

Practical Issues (Session 6) Investments Spring 2010 1 / 33

OutlineI. Portfolio Strategies

Buy & HoldConstant MixConstant-Proportion Portfolio InsuranceOption-Based Portfolio InsuranceQuantitative Market-Timing Models

II. Other Practical IssuesConstraints on PortfoliosInternational Diversification

III. Investments Styles and Types of Funds AvailablePassive Fund Management StylesActive Fund Management StylesAlternative Investments

IV. For Further Reading


Portfolio Strategies







Portfolio Strategies Buy & Hold

Buy-and-Hold Strategy

Characterized by an initial mix (e.g., 60/40 stocks/bills) that isbought and then held.No matter what happens to relative values, no rebalancing is required.The portfolio’s value is linearly related to that of the stock market.Portfolio value will never fall below the value of the initial investmentin bills.Upside potential is unlimited.[Diagram in class]


Portfolio Strategies Constant Mix

Constant-Mix Strategies

Maintain an exposure to stocks that is a constant proportion of wealth.Dynamic strategies: whenever the relative values of assets change,purchases and sales are required to return to the desired mix.Rebalancing to a constant mix requires the purchase of stocks as theyfall in value, and sale of stocks as they rise in value.Concave payoff curve [Diagram in class].Concave strategies will perform well when there are reversals in stockreturns.[Example in class]


Portfolio Strategies Constant-Proportion Portfolio Insurance

Constant-Proportion Strategies

Constant-proportion strategies take the following form

Dollars in stocks=m(Assets-Floor) (1)

where m is a fixed multiplier.Three special cases:

1 If m > 1, the strategy is called the constant-proportion portfolioinsurance strategy (CPPI).

2 If m = 1, floor = value of bills, this strategy is the buy-and-holdstrategy.

3 If 0 < m < 1, floor = 0, the strategy is the constant-mix strategy.

Sell stocks as they fall and buy stocks as they rise.In a bear market, the portfolio, will do as well as the floor. In a bullmarket, the CPPI will do very well.Convex payoff curve [Diagram in class].


Portfolio Strategies Constant-Proportion Portfolio Insurance

Constant-Proportion Strategies (cont.)

In a flat market, a CPPI strategy will do relatively poorly, whileconstant-mix strategies will perform well.

[Simulation example in class].


Portfolio Strategies Option-Based Portfolio Insurance

OBPI

Start by secifying an investment horizon and a desired floor value atthat horizon.The OBPI strategy consists of a set of rules designed to give the samepayoff at the horizon as would a portfolio composed of bills and calloptions.One instant prior to the horizon, OBPI involves investing entirely inbills if the asset equals the floor, and entirely in stocks if the assetexceed the floor.With more than just one an instant to go before “expiration”, we useoption pricing formulas to find amounts invested in stocks and bills.OBPI strategies are “sell stock as they fall...”. They must thus provideconvex payoff diagrams [Diagram and example in class].


Portfolio Strategies Quantitative Market-Timing Models

Trend Following

0

0.5

1

1.5

2

2.5

3

log

retu

rncumulative log returns

S&P500Timing

65 70 75 80 85 90 95 00 05year

posi

tion


Other Practical Issues







Other Practical Issues Constraints on Portfolios

Constraints

So far, we assumed that any optimal portfolio could be held.

In practice, this will not be the case. Examples:I In many countries, short-selling is forbidden.I For pension funds, there is an upper limit on the proportion that can be

invested in a given stock.

How can we incorporate these kind of constraints in our portfolioanalysis?

To make things as intuititve as possible, let us take our investorseeking to maximize expected utility V

(µp,σ

2p)

= µp− a2σ2

p . Formally,his problem is

maxw

w ′µ− a2w ′Σw s.t. w ′1 = 1,w ≥ wL,w ≤ wU (2)



Constraints (cont.)To solve this problem, we rewrite the constrained problem as anunconstrained one using the Lagrangian

L = w ′µ− a2w ′Σw + λ

(1−w ′1

)+ φ

′L (w −wL) + φ

′U (wU −w) (3)

This is the same as what we usually did, except that we haveincorporated the inequality constraints as well.The Kuhn-Tucker optimality conditions are

∂L∂w

= µ−aΣw −λ1+ φL−φU = 0

∂L∂λ

= 1−w ′1 = 0

∂L∂φL

= w −wL ≥ 0, φL ≥ 0, φ′L (w −wL) = 0

∂L∂φU

= wU −w ≥ 0, φU ≥ 0, φ′U (wU −w) = 0 (4)



Constraints (cont.)Thus, for assets for which the lower bound ωL is binding, φL > 0,meaning that the marginal utility of this asset is less than λ , youwould want do reduce your holdings. Conversely, for assets for whichthe upper bound ωU is binding, φU > 0 and marginal utility exceeds λ ,you would like to increase your holdings.

There is a neat iterative procedure to solve this problem:1 Start with a feasible allocation (one that satisfies all the constraints

and such that the sum of holdings is 1).2 For this allocaiton w , compute marginal utility from increasing asset

holdings, µ−aΣw (we can ignore λ here).3 Find the asset i with ωi > ωLi that has the lowest marginal utility and

the asset j with ωj < ωUi that has the highest marginal utility.4 If the difference in marginal utility between the best and the worst

asset exceeds a certain level, increase the holding of asset j , decreasethat of asset i , compute the new portfolio w and goto step 2.Otherwise, optimization is complete.



Constraints (cont.)

This recipe is almost all we need to write an optimizer, except for twothings: how to determine the initial allocation and how much of theholdings of assets i and j to swap.

For the initial allocation, we could enter it “by hand”. But we can alsocompute one automatically. There are several ways to do this. Onewhich is simple and works well is the following:

1 Make sure that wL ≤ wU , w ′L1 < 1, and w ′U1 > 1. Otherwise, eitherthere is no solution satisfying the constraints or the solution is alreadyknown.

2 Set

w = wL +1−w ′L1

w ′U1−w ′L1(wU −wL) (5)

(note that w ′1 = 1 and that all the constraints are satisfied.)



Constraints (cont.)In order to determine the amount by which we should reduce theholding of the worst asset and increase that of the best asset,remember that we are trying to increase expected utility by increasingour holding of j and reducing our holding of i .Therefore, we will change holdings from w to w + cs. Here, s is aN-vector telling us which holding we will increase and which holdingwe will reduce, i.e. containing +1 at position j , −1 at position i andzero elsewhere; c is the size of the increase/decrease.

Our problem is to determine c . To do this, let us compare the utilityin the new allocation, w + cs, with that in the old one, w :

∆V = (w + cs)′ µ− a2

(w + cs)′Σ(w + cs)−(w ′µ− a

2w ′Σw

)(6)

Simplifying,∆V = cs ′µ− a

2(c2s ′Σs +2cs ′Σw

)(7)



Constraints (cont.)

To find the maximum increase in utility, differentiate ∆V with respectto c to get

∂ ∆V∂c

= s ′µ−as ′Σw −acs ′Σs = 0 (8)

Therefore, our tentative value for c is given by

c =s ′µ−as ′Σw

as ′Σs(9)

Incorporating the constraints on holdings, we have

c = min(

s ′µ−as ′Σwas ′Σs

,ωUj −ωj ,ωi −ωLi

)(10)



Constraints (cont.)

Graphically, there will be a difference between the constrained and theunconstrained efficient set (in the example µ =

[0.1 0.2 0.15

]′,Σ =

0.2 0.1 0.050.1 0.2 0.10.05 0.1 0.3

, wL =[0.3 0.3 0.3

]′,wU =

[1 1 1

]′ ):



Constraints (cont.)

0.36 0.365 0.37 0.375 0.38 0.385 0.39 0.395 0.4

0.145

0.15

0.155

0.16

0.165

0.17

0.175

0.18

Standard Deviation

Exp

ecte

d R

etur

n


Other Practical Issues International Diversification

International Diversification

Correlation in returns between countries is much lower than correlationbetween stock returns within a country. Consider the following datafrom Solink for the period 1971-1998 (International Investments, 4thed.):

Country F D I NL SP CH GB USFrance 1Germany 0.61 1Italy 0.44 0.4 1

Netherlands 0.61 0.69 0.38 1Spain 0.43 0.42 0.42 0.43 1

Switzerland 0.60 0.67 0.35 0.69 0.39 1United Kingdom 0.54 0.44 0.34 0.64 0.35 0.54 1

U.S.A. 0.45 0.38 0.26 0.59 0.33 0.48 0.52 1



International Diversification (cont.)As a result, for a given level of expected return, the risk of a stockportfolio can be further reduced by considering international stocks.Solnik (JF 1974) has shown that the residual risk by diversifyingacross U.S. stocks only is 27% of intial risk, whereas internationaldiversification can reduce it to 11.7%.

0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20

0.25



International Diversification (cont.)

When considering international stocks, it is important to takeexchange rate risk into account.

Suppose that exchange rate risk is not hedged. How should portfoliooptimization be performed?

Things can actually be done in the usual way, but all returns, variancesand covariances must be translated in the home currency.

Let Rm denote the return on the foreign market (or stock) and Rs thereturn on the foreign currency. Then, the return on the foreign markettranslated in home currency, Rf , is given byRf = Rm +Rs +RmRs ≈ Rm +Rs . For expected returns,

µf ≈ µm + µs (11)




Similarly, let σm denote the standard deviation of returns on theforeign market, σs the exchange rate volatility and ρm,s the correlationcoefficient between the two. Then, the risk of the foreign market whenreturns are translated in home currency, σf , is given by

σf =√

σ2m +2ρm,sσsσm + σ2

s (12)

Since the two risks are not perfectly correlated, σf < σm + σs .

Over short periods of time, exchange rate risk can dominate capitalgains or losses on the foreign market.

Over long periods of time, however, exchange rate risk is smallcompared to market risk on the foreign markets. This is because thecorrelation between currencies is low and currencies tend to bemean-reverting.




The consequence is that the country of domicile affects the expectedreturns and risk from international diversification (so, if you are anasset manager, you should think about which currency your customeris consuming in before deciding how to invest).

Suppose you want to decide wether to invest in a number of foreignmarkets or not.This can be done by letting each of the countries(including the domestic country) be one asset in your decision problemand solving it as usual.

Your vector of expected returns would be

µ =[

µd µf 1 ... µfK]′ (13)

where µd denotes expected return on domestic stocks,µfk = µmk + µsk expected return in foreign country k .



International Diversification (cont.)The variance-covariance matrix of returns would be

Σ =

σ2

d ρd ,f 1σdσf 1 ... ρd ,fK σdσfKρd ,f 1σdσf 1 σ2

f 1 ... ρf 1,fK σf 1σfK...

ρd ,fK σdσfK ρf 1,fK σf 1σfK ... σ2fK

(14)

Then, you could solve for the optimal risky portfolio

w =1a

Σ−1 (µ−R1) (15)

A positive weight indicates you should invest in a given market, anegative weight that you should short it (of course, short-saleconstraints can be accounted for here as well).

As was noted when we discussed multi-factor models, industry facorsare an important determinant of the covariation in stock returns.




A controversial issue that arises in the context of internationalinvestments is the extent to which industry sectors matter for assetallocation or whether one would do “well enough” by diversifyingacross countries.

Most studies come to the conclusion that little of the variation incountry index returns can be explained by their industrial composition.

Thus, most of the benefits of diversification can be achieved bydiversifying across countries; also diversifying across industries onlybrings a minor additional reduction in portfolio risk.


Investments Styles and Types of Funds Available







Investments Styles and Types of Funds Available Passive Fund Management Styles

Tracker Funds and ETFsIndex tracking is a version of the buy and hold strategy that eliminatesdiversifiable risk.It is virtually impossible to be exactly indexed at all times ⇒ trackingerror (due to addition and deletion of securities that constitute theindex, collection of dividends, infusion of new contributions).Transaction costs will also eat into the performance figures.Exchange traded funds (ETFs) are basically index funds that are listedon exchanges, and trade like stocks.ETFs can track every type of equity or bond index. Both institutionaland retail investors can invest in ETFs. They can be used toimplement sector rotation and sector allocation strategies and toadjust sector or country exposure.Tracker funds and ETFs have grown in popularity worldwide, especiallyafter periods of time when actively managed funds underperformedtheir indices. See next graph from The Economist, January 2010 .


Investments Styles and Types of Funds Available Passive Fund Management Styles

A Fast-Growing Asset Class


Investments Styles and Types of Funds Available Active Fund Management Styles

Active Investing

Within the active fund management segment of portfoliomanagement, fund managers pursue a variety of investmenttechniques known as styles of investing.Some popular styles:

I Value investing: stock that trade at low multiples of price to measuresof fundamental value. In recent years value stocks have concentrated inestablished, stable industries, such as manufacturing utilities and food.

I Growth investing : companies with strong growth expectations. Theycommonly trade at prices that are high relative to current earnings,dividends, or book values.

I Large-cap: the biggest companies, also known as blue chips. Large-capinvestors prefer safety.

I Small-cap: companies with small market capitalizations.Small-capinvestors hope for relatively good performance.


Investments Styles and Types of Funds Available Alternative Investments

Hedge Funds and Private Equity

The object of alternative invseting (or absolute return investing) is totarget an absolute return range and not returns relative to apredermined index.While traditional funds are organized around styles, hedge funds areorganized around strategies:

I Non-directional strategies, commonly reffered to as “market neutral”strategies. Do not depend on the direction of any specific marketmovement. Examples include fixed income arbitrage, event-driven,merger arbitrage.

I Directional strategies, designed to take advantage of broad marketmovements. Examples include macro, emerging markets, short selling.

Private equity funds invest in securities which are not publicly traded.Types of private equity investing include leveraged buyouts, venturecapital investments, distressed debt investments.


Investments Styles and Types of Funds Available Alternative Investments

Hedge Funds

The Economist, January 2010: Hedge fundsmade their biggest gains in a decade last year,according to the Hedge Fund Research Index, anindustry benchmark. Funds returned an average20% in 2009, having had their worst year ever in2008. Investors withdrew $131 billion fromhedge funds last year, but the healthy gainsmade on the money left in meant that assetsunder management increased to $1.6 trillion.That was $193 billion more than at the end of2008, though still below the 2007 peak. Around2,000 hedge funds have closed since thefinancial crisis started but an estimated 9,000remain worldwide. Investors were charged lessthan the “two-and-twenty” benchmark:management fees averaged 1.6% of assets.Incentive fees were 19.2%.


For Further Reading







For Further Reading

1 Brentani, Christine, Portfolio management in Practice, 2004, ElsevierI Provides an overview of the day-to-day aspects with which a portfolio

manager must be concerned.

2 Perold and Sharpe, Dynamic Strategies for Asset Allocation, FAJ 1988I Analysis of different investment strategies.


Investments

Session 7. Security Analysis


Spring 2010

Security Analysis (Session 7) Investments Spring 2010 1 / 26

Outline

I. Macroeconomic and Industry AnalysisThe Global and the Domestic EconomyDemand and Supply ShocksFederal Government PolicyBusiness CyclesIndustry Analysis

II. Equity Valuation ModelsThe Present Value ModelA Simple Two-Period ModelThe Gordon ModelStock Prices and Investment OpportunitiesThe Price-Earnings Ratio

III. Financial Statement AnalysisThe Ins and Outs of Cash Flow


Macroeconomic and Industry Analysis

Outline





Macroeconomic and Industry Analysis The Global and the Domestic Economy

The Global Economy

The international economy might affect a firm’s export prospects, theprice competition it faces from competitors, or the profits it makes oninvestments abroad.National economic environment can be a crucial determinant ofindustry performance.The global environment presents political risks of far greatermagnitude than are typically encountered in U.S.-based investments.Other political issues important to economic growth:

I protectionism and trade policyI the free flow of capitalI the status of a nation’s work force.



The Domestic Macroeconomy

The first step in forecasting the performance of the broad market is toassess the status of the economy as a whole.

The ability to forecast the macroeconomy better than yourcompetitors can translate into spectacular investment performance.

Some of the key economic statistics used to describe the state of themacroeconomy

I Gross Domestic Product (GDP): measure of the economy’s totalproduction of goods and services. Related measure: IndustrialProduction (focused more on the manufacturing side of the economy).

I Unemployment Rate: the percentage of the total labor force yet to findwork. Related measure: Capacity Utilization Rate (ratio of actualoutput from factories to potential output).



The Domestic Macroeconomy (cont.)

I Inflation: the rate at which the general level of prices is rising. Highrates of inflation often are associed with “overheated” economies. Thereis a trade-off between Inflation and unemployment. Related measure:The Output Gap (the difference between the economy’s actual outputand the level of production it can achieve with existing labour, capital,and technology without putting sustained upward pressure on inflation).

I Interest Rates: real interest rates are key determinants of businessinvestment opportunities.

I Budget Deficit: (the difference between government spending andrevenues.

I Sentiment: consumers’ and producers’ optimism or pessimismconcerning the economy.


Macroeconomic and Industry Analysis Demand and Supply Shocks

Demand and Supply Shocks

A demand shock is an event that affects the demand for goods andservices in the economy. Demand shocks are usually characterized byaggregate output moving in the same direction as interest rates andinflation.A supply shock is an event that influences production capacity andcosts. Supply shocks are usually characterized by aggregate outputmoving in the opposite direction of inflation and interest rates.Typically, you want to identify the industries that will be most helpedor hurt in any macroeconomic scenario you envision. Examples:

I if you forecast a tightening of the money supply, you might want toavoid industries such as automobile producers that might be hurt bythe likely increase in interest rates, and favor industries withbelow-average sensitivity to macroeconomic conditions.

I if you have an optimistic view of the business cycle, you might favorinvestments with a greater sensitivity to the business cycle.


Macroeconomic and Industry Analysis Federal Government Policy

Fiscal Policy

The government has two broad classes of macroeconomic tools: thosethat affect the demand for goods and services and those that affectthe supply.Fiscal policy is a demand-oriented tool. It refers to the government’sspending and tax actions.Decreases in government spending directly deflate the demand forgoods and services. Similarly, increases in tax rates immediately siphonincome from consumers and result in fairly rapid decreases inconsumption.Fiscal policy has the most immediate impact on the economy.However, it requires enormous amounts of compromise between theexecutive and legislative branches. Thus, the formulation andimplementation of fiscal policy is usually painfully slow and involved.



Monetary Policy

Monetary policy refers to the manipulation of the money supply toaffect the macroeconomy and is the other main leg of demand-sidepolicy.It works largely through its impact on interest rates. Increases in themoney supply lower short-term interest rates, ultimately encouraginginvestment and consumption demand.Monetary policy is easily formulated and implemented but has a lessdirect impact than the fiscal policy.The most widely used tool is the open market operation, in which theFed buys or sells bonds for its own account. When the Fed buyssecurities, it simply “writes a check”, thereby increasing the moneysupply.Oppen market opperations occur daily, allowing the Fed to fine-tuneits monetary policy.



Supply-Side Policies

Supply-side policies treat the issue of the productive capacity of theeconomy.The goal is to create an environment in which workers and owners ofcapital have the maximum incentive and ability to produce anddevelop goods.Whereas demand-siders look at the effect of taxes on consumptiondemand, supply-siders focus on incentives and marginal tax rates.In some situations, reductions in tax rates can lead to increases in taxrevenues because the lower tax rates will cause the economy and therevenue tax base to grow by more than the tax rate is reduced.


Macroeconomic and Industry Analysis Business Cycles

Business Cycles

Economies repeatedly seem to pass through good and bad times. Thisrecurring pattern of recession and recovery is called the business cycle.

The transition points across cycles are called peaks and troughs. Apeak is the transition from the end of an expansion to the start of acontraction. A trough occurs at the bottom of a recession just as theeconomy enters a recovery.

One determinant of the broad asset allocation decision of manyanalysts is a forecast of whether the macroeconomy is improving ordeteriorating.

As the economy passes through different stages of the business cycle,the relative performance of different industry groups might beexpected to vary:



Business Cycles (cont.)

I cyclical industries have above-average sensitivity to the state of theeconomy. Cyclical industries outperform other industires duringrecoveries. Examples: producers of durable goods, producers of capitalgoods.

I defensive industries have little sensitivity to the business cycle. Theseindustries will outperform others when the economy enters a recession.Examples: food producers and processors, pharmaceutical firms, andpublic utilities.

The cyclical/defensive classification corresponds well to the notion ofsystematic or market risk introduced in our discussion of portfoliotheory.

Unfortunately, it is not so easy to determine when the economy ispassing through a peak or a trough. For this, we use economicindicators, released to the public on a regular “economic calendar”.



Business Cycles (cont.)

I leading indicatorsF manufacturer’s new ordersF stock pricesF money supplyF index of consumer expectationsF new orders for nondefense capital goods

I coincident indicatorsF industrial productionF employees on nonagricultural payrolls

I lagging indicatorsF average duration of unemploymentF ratio of trade inventories to salesF change in index of labor cost per unit of output


Macroeconomic and Industry Analysis Industry Analysis

Sensitivity to the Business Cycle

Once the analyst forecasts the state of the macroeconomy, it isnecessary to determine the implication of that forecast for specificindustries.Three factors will determine the sensitivity of a firm’s earnings to thebusiness cycle:

1 Sensitivity of sales. Example: necessities will show little sensitivity tobusiness conditions (food, drugs, medical services). Other examples:tobacco industry, movies. On the opposite side, machine tools, steel,autos, transportation are highly sensitive to the state of the economy.

2 Operating leverage (division between fixed and variable costs). Firmswith greater amounts of variable as opposed to fixed costs will be lesssensitive to business conditions. Profits from firms with high fixed costswill swing more widely with sales.

3 Financial leverage. Interest payments on debt must be paid regardlessof sales. They are fixed costs that also increase the sensitivity of profitsto business conditions.


Equity Valuation Models

Outline





Equity Valuation Models The Present Value Model

The Present Value Model

To determine the price of a treasury bond, we calculate the presentvalue of future cash flows.

In principle, one can apply the same approach to stock valuation,thinking of the future dividend as a stream of coupon payments.Additional issues need to be addressed:

I Unlike the coupon payments, the dividend payouts are uncertain. Whatare the appropriate discount rates?

I Dividends are known to be sticky, and some firms do not even paydividends. Where do we get information about the growth componentof a firm?

I Unlike fixed-income securities, stocks do not have maturity dates. Howdo we take care of dividend payments that are postponed into theinfinite future?


Equity Valuation Models A Simple Two-Period Model

A Simple Two-Period Model

By definition

R1 =P1 + D1−P0

P0(1)

Letting I0 be the collection of public information available at time 0, itmust be

E[R1/I0

]=

E[P1/I0

]+E

[D1/I0

]−P0

P0(2)

where we take our expectation conditioning on the informationavailable in I0.

Define the intrinsic value of the firm

V0 =E[P1/I0

]+E

[D1/I0

]1+E

[R1/I0

] (3)


Equity Valuation Models A Simple Two-Period Model

A Simple Two-Period Model (cont.)If the market is efficient, accurately reflecting E

[P1/I0

], E[D1/I0

],

and E[R1/I0

], then the market price must agree with the intrinsic

value of the firm.

We can move into the future by applying the two-period recursively:

V0 =E[D1

]1+E

[R1

] +E[D2

](1+E

[R1

])(1+E

[R2

]) + ...

...+E[Dn

]∏

ni=1

(1+E

[Ri

]) + ... (4)

where all expectations are taken wiht respect to the informationavailable at time 0, I0.


Equity Valuation Models The Gordon Model

The Gordon Model

Using all of the information available at time 0, the marketparticipants agree that:

1 the dividend growth is constant

E[Dn

]= D0 (1+g)n (5)

2 the expected return is constant

E[Rn

]= k (6)

where g ≥ 0 and k ≥ 0.

This implies that

V0 = D0

∞

∑n=1

(1+g1+k

)n

(7)


Equity Valuation Models The Gordon Model

The Gordon Model (cont.)Suppose that the expected rate of return is always higher than theexpected growth rate (if dividends were expected to grow forever at arate faster than k , the value of the stock would be infinite -unsustainable):

k > g (8)

Letting x = 1+g1+k < 1, we have

V0 = D0(x + x2 + x3 + ...

)= D0x

(1+ x + x2 + ...

)= D0x

11− x

(9)

Now plugging x = 1+g1+k back in:

V0 = D01+g1+k

11− 1+g

1+k

= D01+gk−g

(10)


Equity Valuation Models Stock Prices and Investment Opportunities

Example

Consider two companies:1 X: 100% dividend payout ratio. It provides a stream of dividends

E [Dn] = $5, maintaining a zero dividend growth g = 0 (a “Cash Cow”firm).

2 Y: 40% dividend payout ratio. In any given year, it plows back(1−40%) of its earnings to a project generating an expected return ofK per year. That is

E [D1] = $5 ·40%E [D2] = ($5+$5 · (1−40%) ·K ) ·40% (11)

maintaining a dividend growth rate of g = (1−40%)K (a “GrowthProspects” firm):

The share of Y could rise if K is larger than the market capitalizationrate [More in class]


Equity Valuation Models The Price-Earnings Ratio

Price-Earnings Ratio

The P/E ratio (price-to-earnings ratio) of a stock (also called its"P/E", "PER", "earnings multiple," or simply "multiple") is ameasure of the price paid for a share relative to the annual net incomeor profit earned by the firm per share.In our previous example, X had a P/E multiple of 8, whereas Y had aP/E larger than 8.This observation suggests the P/E ratio might serve as a usefulindicator of expectations of growth opportunities.[Diagrams and discussion in class]


Financial Statement Analysis

Outline





Financial Statement Analysis The Ins and Outs of Cash Flow

The Major Financial Statements

The income statement is a summary of the profitability of the firmover a period of time.

It presents revenues generated during the operating period, theexpenses incurred during that same period, and the company’s netearnings or profits.

The balance sheet provides a “snapshot” of the financial condition ofthe firm at a particular moment.

It is a list of the firm’s assets and liabilities at that moment.

The statement of cash flows details the cash flow generateed by thethe firm’s operations, investments, and financial activities. It is thusorganized into three parts:

1 Cash from operating activities, can signal when a company is havingtrouble selling inventory or collecting cash it is owed.



The Major Financial Statements (cont.)2 Cash from investing activities, gives information on how much the

company earned in the stock market, or whether it’s cutting back oncapital expenditures.

3 Cash from financing activities, to see if a company receives cashinfusions from outsiders, such as banks or shareholders.

Ideally, a company’s operations should generate excess cash, while itsinvesting and financing sections show negative cash balances. Why? Aself-sustaining business can pay down debt and finance newinvestments internally.

Negative cash flow from operations isn’t always bad. Because of thehigh costs of building a business, it’s perfectly normal - even desirable- for fast-growing companies to consume more cash than theygenerate. Typically, such companies tide themselves over with bankloans or equity sales. In other words, they run a surplus in “financing”cash flows.



The Major Financial Statements (cont.)

Some examples:I The cash-flow statement can shine a light on earnings quality. The

cash-flow statement reflects how much cash is actually collected. So ifearnings soar, but cash collections stall, be wary: Future earnings couldbe at risk of being dragged bown by bad debt.

I When inventories grow faster than sales, it might mean demand issoftening. Since buying inventory requires cash, an increase here causescash to fall.

I Conversely, when liabilities such as accounts payable increase, so do acompany’s cash balances. By delaying payments to creditors,management frees up cash.

General rule: “if operating cash doesn’t pick up, bail out”. Example: Adebt-ridden firm is raising cash the only way it can - by selling assets.When investing cash flow is positive because of asset sales, that’susually a sick company.


Investments

Session 8. Interest Rates and Related Contracts

EPFL - Master in Financial Engineering

Philip Valta

Spring 2010

Interest Rates (Session 8) Investments Spring 2010 1 / 44

Outline of the lecture

1 Fixed Income Securities2 Interest Rates3 Bond Pricing4 Credit Risk


Fixed Income Securities De�nition

Fixed Income Security

A debt security is a claim on a speci�ed periodic stream of income.

Debt securities are often called �xed income securities.

De�nitionA �xed-income security is a �nancial claim which provides a return in theform of �xed periodic payments and an eventual return of principal atmaturity

Unlike a variable-income security, such as a stock or an option, thepayments of a �xed-income security are known in advance.


Fixed Income Securities Classi�cation of Securities

Classi�cation of Securities

1 TreasuriesI U.S. bills, notes, and bondsI German Bunds, French OATs, UK Gilts,...

2 CorporateI Commercial paper and midterm notesI Corporate bonds

3 Munies4 Mortgage and asset backed5 Catastrophe bonds6 Indexed bonds7 Agencies

I Federal �nancing banksI Fannie Mae, Freddie Mac


Fixed Income Securities Market Capitalization

Market Capitalization

US: 2007 Value (in trillions $) %Treasuries 4.5 0.18Corporate 5.8 0.23Mortgage 7.2 0.28Agencies 2.9 0.11Munies 2.6 0.10Asset-backed 2.5 0.10Total Value 25.2 100Source: FINRA

World: 2007 Value (in trillions $)Credit Default Swaps 57.89Source: BIS


Fixed Income Securities The Players

The Players

1 Issuers of securitiesI Government, states, and municipalitiesI Corporations, commercial banks...I Special purpose vehicles

2 IntermediariesI Primary and secondary dealersI Rating agencies (S&P, Moody�s, Fitch)I Investment banks

3 InvestorsI Pension fundsI Insurance companiesI Commercial banks and mutual fundsI Individual investors


Interest Rates Evolution

Evolution

Example: Suppose that you invest 1 dollar at a rate of 5% per annumcapitalized every 6 months.

The evolution of your semester account balance over the �rst twoyears of investment is given by:

(1+ 12R) = (1+

120.05) = 1.0250

(1+ 12R)(1+

12R) = (1+

120.05)

2 = 1.0506

(1+ 12R)(1+

12R)(1+

12R) = (1+

120.05)

3 = 1.0769

(1+ 12R)(1+

12R)(1+

12R)(1+

12R) = (1+

120.05)

4 = 1.1038


Interest Rates Some De�nitions

Some De�nitions

Annual Percentage Rate (APR): amount of simple interest,ignoring compounding.

E¤ective Annual Rate (EAR): indicates the total amount of interest.The EAR corresponding to an APR with k compounding periodssatis�es:

1+ EAR =�1+ APR

k

�kAs APR ignores compounding we have: APR < EAR

Example (con�t): APR = 5% < 5.06% = EAR


Interest Rates Present and Future Values

Present and Future Values

Let VF (m) denote the future value of 1$ capitalized m times per yearat the interest rate R for one year:

VF (m) = (1+ 1mR)� ...� (1+

1mR) = (1+

1mR)

m

Let VP (m) denote the present value of 1$ to be received in one yeargiven that the interest rate R is capitalized m times per year:

VP (m) = (1+ 1mR)

�1 � ...� (1+ 1mR)

�1 = (1+ 1mR)

�m

Present and future values are inverse operations: VP = 1/VF


Interest Rates Continuous Capitalization

Continuous Capitalization

Frequency m VF (m) VP (m)Semester 2 (1+ R/2)2 (1+ R/2)�2

Monthly 12 (1+ R/12)12 (1+ R/12)�12

Weekly 52 (1+ R/52)52 (1+ R/52)�52

Daily 360 (1+ R/360)360 (1+ R/360)�360...

......

Continuous ∞ lim∞ VF (m) = eR lim∞ VP (m) = e�R


Interest Rates Example

Example

Suppose a bank states that the interest on 1-year deposits is 10% perannum.

I What are the values of a USD 100 investment at the end of the yearwith annual and semiannual compounding frequency?

I What are the values of this investment if the compounding frequency ismonthly, weekly, daily, and continuous?


Interest Rates Equivalent Interest Rates

Equivalent Interest Rates

Let R∞ be the continuously compounded rate and Rm denote theequivalent rate with m capitalization periods per year.

Comparing the future values VF (∞) and VF (m) shows that theseequivalent interest rates satisfy:

(1+ 1mR)

m = exp(R∞)) R∞ = m ln(1+ 1mRm)

) Rm = m�exp( 1mR∞)� 1

�These formulae are used to convert discrete capitalization interestrates into continuous capitalization interest rates and vice versa.

Example: A monthly interest rate of 5% is equivalent to acontinuously compounded rate ofR∞ = 12 ln(1+ 0.05/12) = 4.9896%


Interest Rates Spot Rates

Spot Rates

The spot rate R(t,T ) is the per annum interest rate at time t for aninvestment or a loan until the maturity date T .

For example, on December 31 2008, the US spot interest rates fordi¤erent maturities were given by:

Maturity 0.25 0.50 1 3 5 7 10 30Sport rate (%) 0.11 0.27 0.37 1 1.55 1.87 2.25 2.69

The set of spot rates T ! R(t,T ) for di¤erent maturity dates iswhat we refer to as the term structure of interest rates or yield curve.

The yield to maturity on zero-coupon bonds is sometimes also calledthe spot rate.


Interest Rates The Term Structure of Spot Rates

The Term Structure of Spot Rates

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

Maturity

Spo

t Rat

e

199020002008


Interest Rates Forward Rates

Forward Rates

The forward rate R(t, τ,T ) is the per annum interest rate at time tfor an investment or a loan that starts at time τ and lasts until timeT .

For a forward transaction to borrow money in the future:I The terms of transaction are agreed upon today at date tI The loan is received at some future date τI The repayment of the loan occurs at some future maturity date T



Forward Rates

If the forward rate for period n is fn, then fn is de�ned by theequation:

fn =(1+ yn)

n

(1+ yn�1)n�1 � 1

where yn is the yield to maturity of a zero-coupon bond with an n-periodmaturity (or equivalently the spot rate).

Example: The one and two years yield to maturities are 3% and 4%,respectively. What is the forward rate from year one to year two?

Assume continuous compounding. How do the results change?



Forward Rates

Term structure of forward rates:

YearSpot rate for ann-year investment

Forward ratefor nth year

1 3.0 �2 4.0 5.03 4.6 5.84 5.0 6.25 5.3 6.5


Interest Rates LIBOR

LIBOR

LIBOR is short for the London InterBank O¤er Rate.

Rate of interest at which a bank is prepared to make a large wholesaledeposit with other banks.

Usual quotes are 1-month, 3-month, 6-month, and 12-month in allmajor currencies.

Financial institutions can regard the LIBOR as their short-termopportunity cost of capital.

Not totally risk-free, but close to risk-free.

LIBOR rates are often used to price derivatives.


Bond Pricing Characteristics of Bonds

Characteristics of Bonds

A bond is a �nancial security in which the issuer promises to makeinterest and principal payments to the holder.

Bonds are mainly characterized by:I Maturity date - date of last promised paymentI Face, par, or principal value - promised payment at maturityI Coupon - periodical payments

Bonds may have additional characteristics: embedded options, specialtax treatments, in�ation protection, seniority, etc.


Bond Pricing Formula

Formula

The price of a bond paying coupons ct with a face value F andmaturity T is given by:

P =T

∑t=1

ct(1+rt )t

+ F(1+rT )T

As time passes, cash �ows are paid and the number of terms in thesummation decreases. This implies that ceteris paribus the couponbond price is a discontinuous function of time.

What happens to the value of a bond when interest rates increase?Decrease?


Bond Pricing Illustration

Illustration

0 1 2 3 4 595

100

105

110

Time

Bon

d P

rice


Bond Pricing Market Conventions

Market Conventions

Bond prices are quoted in percentage of their par value and roundedto 1/32. For example, a quote 98-07 means that the price is

N100 � (97+

732 ) = 0.9821875�N.

If t and T are dates it is not clear what T � t should be. The daycount convention decides upon the measurement of this di¤erence:

I 30/360 (money market)I Actual/365I Actual/Actual (for Treasuries)

Example: The di¤erence between January 4, 2007 and July 4, 2009 is2.5 in the 30/360 convention and 2.4986 in the Actual/365convention


Bond Pricing Accrued Interest

Accrued Interest

Quoted bond prices do not include the interest that accrues betweencoupon payment dates (clean price).

If a bond is purchased between coupon payments, the buyer must paythe seller for accrued interest, the prorated share of the upcomingcoupon.

The formula for the amount of accrued interest between two dates is:

AI = couponm

days since last coupondays between coupons

Relation between dirty and clean price:

Pclean(t) = Pdirty (t)� AI (t)



Accrued Interest

Example: suppose a coupon rate of 10%. Then the annual coupon is10$ and the semiannual coupon is 5$. Assume 30 days have passedsince the last coupon.

What is the clean price of the bond?

What is the accrued interest?

What is the dirty (invoice) price of the bond?



Accrued Interest

0 1 2 3 4 50

1

2

3

4

5

Time

Accr

ued

Inte

rest


Bond Pricing Clean Price

Clean Price

0 1 2 3 4 595

100

105

110

Time

Bon

d P

rice

QuotedCash


Bond Pricing Yield to Maturity

Yield to Maturity

The Yield-to-Maturity (YTM) is the interest rate that makes thepresent value of a bond�s payments equal to its price.

Example: suppose an 8% coupon, 30-year bond selling at 1, 276.76$.What is the YTM? Find the interest rate at which the present valueof the bond payments equals the bond price:

1, 276.76 =60

∑t=1

40(1+r )t +

1,000(1+r )60

Result: the yield-to-maturity is 3% per half-year

Bond yields are reported as Annual Percentage Rates (APR): theannual yield would then be 6%. The e¤ective annual yield thataccounts for compound interest is (1.03)2 � 1 = 6.09%


Bond Pricing Yield to Price relationship

Yield to Price relationship

0 5 10 15 2020

40

60

80

100

120

140

160

Interest Rate (%)

Bon

d P

rice


Bond Pricing Zero-coupon Bonds

Zero-coupon bonds

A zero-coupon bond is a bond that does not make coupon paymentsand its only cash payment is the face value of the bond on thematurity date.

Example: US Treasury Bills.

Zero-coupon bonds always trade at a discount (a price lower than theface value).

Let P denote the time 0 price of a zero bond with maturity T . Theprice P is given by:

P = F(1+rT )T



Zero-coupon bonds

The YTM of a zero-coupon bond with face value F and maturity T :

P = F(1+YTMT )T

The yield-to-maturity can be obtained in closed-form:

YTMT =� FP

�1/T � 1

Zero-coupon bonds are used to recover spot rates. Zero rates are alsooften referred to as n-year spot rates.



Examples

Assume that r = 5%. What is the price of a zero coupon bondmaturing in 20 years?

Suppose the price of a zero coupon bond with 10 years maturity is 75.What is the yield-to-maturity of this bond?


Bond Pricing Time and bond prices

Time and bond pricesPremium bonds sell above par: coupon rate is greater than YTM.Discount bonds sell below par: coupon rate is lower than YTM.


Bond Pricing Callable Bonds

Callable Bonds

The bond includes a call provision which allows the issuer to redeemthe bond for a pre-speci�ed price.

Such a bond can be thought of as a portfolio composed of anotherwise identical bond and a short position in a call option:

Callable Bond = Straight Bond - Call Option

This view implies that the market value of the callable bond is lowerthan that of the underlying straight bond.

The call feature protects the issuer against a decrease in the level ofinterest rates: if rates are too low then calling the bond allows torefund the loan at the current (better) market conditions.


Bond Pricing Callable Bonds

Callable Bonds


Bond Pricing Puttable Bond

Puttable Bond

The bond includes a put feature which allows the bond holder toforce the issuer to buy back the bond at a pre-speci�ed price.

Such a bond can be thought of as a portfolio composed of anotherwise identical bond and a long position in a put option:

Puttable Bond = Straight Bond + Put Option

The dates at which the bond can be put are �xed in advance. If thereis a single date then the option is European, otherwise it isBermudian.

The put feature protects the bond holder against an increase in thelevel of interest rates or a credit deterioration of the issuer.


Bond Pricing Puttable Bond

Puttable Bond


Bond Pricing Convertible Bond

Convertible Bond

The bond includes a feature which allows its holder to convert thebond into a pre-speci�ed number of shares of the issuer�s stock.

Such a bond can be thought of as a portfolio composed of anotherwise identical bond and a long position in an American exchangeoption for a certain number of shares of stock.

Convertibles o¤er protection against a decline in the stock price: theyearn interest even when the stock is trading down, but when thestock prices rise, the value of the convertibles also rises.

Convertible bonds typically include anti-dillution provisions whichadjust the conversion ratio in case of stock splits, stock dividends, etc.


Bond Pricing Convertible Bond

Convertible Bond


Bond Pricing CAT Bonds

Catastrophe (CAT) Bonds

Type of bond to transfer "catastrophe risk" from a �rm to the capitalmarkets.

If the insurance company who issued the CAT bond su¤ers a lossfrom a prede�ned catastrophe, then the obligation to pay interestand/or repay the principal is reduced, deferred or forgiven.

Example: the coupons of the Winterthur Insurance bond will only bepaid if during their corresponding periods no single storm or haildamages more than 6000 vehicles insured with Winterthur Insurance.

CAT bonds provide a good diversi�cation of risk since their returnsare barely correlated with those of equity and bond markets.


Credit Risk De�nition

De�nition

Credit risk: risk of default or reduction in market value caused bychanges in the credit quality of the borrower.

Pricing corporate bonds - two steps:I Estimation of cash �owsI Estimation of the discount rate

Simple representation for the discount rate:

Discount rate = Riskless rate + Risk premium


Credit Risk Rating Agencies

Rating Agencies

Rating agencies provide an indication of the credit quality of issuers:

Credit ratings are divided in two classes:I Investment gradeI Speculative grade


Credit Risk Components

Components

Components of credit risk:I Arrival risk: whether default will occur or not.I Timing risk: when default occurs.I Recovery risk: how severe are the losses / how much can be recovered.I Market risk: changes in the market value of an instrument due tochanges in credit quality or market conditions.

I Default correlation: risk of several obligors defaulting simultaneously orin sequence.


Credit Risk Determinants of Bond Safety

Bond Safety Determinants

Coverage ratios

Leverage ratios

Liquidity ratios

Pro�tability ratios

Cash-�ow-to-debt ratios

Altman0s Zscore = 3.3EBITTA

+ 99.9SalesTA

+ 0.6MV EquityBV Debt

+

1.4EarningsTA

+ 1.2Working Capital

TA


Credit Risk Pricing of Defaultable Bonds

Pricing of defaultable bonds

There are typically two modelling frameworks for credit risk:I Structural modelsI Reduced form models

In structural models, which is based on options pricing analysis, �rmsdefault when they cannot meet their �nancial obligations.

In the reduced form approach, the default probability is modeleddirectly without any explicit reference to �rm characteristics.


Investments

Session 9. The Term Structure of Interest Rates


Philip Valta

Spring 2010

Term Structure (Session 9) Investments Spring 2010 1 / 52


1 Bootstrap method2 Parametric models3 Models of the short rate


Review Term Structure of Interest Rates

De�nition

The term structure of interest rates refers to the interest rates forvarious terms to maturity embodied in the prices of default-freezero-coupon bonds.

It is the structure of interest rates for discounting cash �ows ofdi¤erent maturities.

It is also referred to as the yield curve. It is the relation betweeninterest rates and their maturities.

How do we get this curve? What does it mean?


Review Bond Price

Bond Price

The price of a coupon bond is de�ned as:

P(t) =n

∑i=1

�ci + 1fi=ng100

� 1(1+R (t ,Ti ))(

Ti�t)

=n

∑i=1

�ci + 1fi=ng100

�B(t,Ti )

Here B(t,Ti ) is the discount factor for date Ti (the price of a zerocoupon bond with that maturity date).

What exactly does this mean?I Where do we get the discount factors from?I Do we use it to obtain bond prices or do we use bond prices to obtainimplied discount factors?

I Can we deviate from this simple rule?


Review Bond Pricing

Some Answers

Answer to the chicken-and-egg second question:I It depends on the situation.I Take the price of primitive securities as given and derive implieddiscount factors or discount rates from them.

I Then use the term structure of discount rates to price other securities.This is know as relative or arbitrage pricing.

Answer to the third question:I In theory any deviation implies arbitrage opportunities.I In practice, some prices may not be exactly simultaneous, some bondsare less liquid or subject to di¤erent tax treatments than others, theremay be round-o¤ errors in the available prices...


The Bootstrap Method The Bootstrap method

De�nition

The bootstrap method is a procedure for calculating the zero-couponyield curve from market data.


The Bootstrap Method Example

Example

Consider the following three bonds:I A 10% coupon bond maturing in 1 year with price P1 = 99I A 2-year 10% coupon bond with price P2 = 97.5I A 3-year 10% coupon bond with price P3 = 96

Using this information, we can get:

P1 = 99 = 110� B(0, 1)! B(0, 1) = 0.90! R(0, 1) = 11.1%

P2 = 97.5 = 10� B(0, 1) + 110� B(0, 2)! B(0, 2) = 0.80! R(0, 2) = 11.5%

P3 = 96 = 10� B(0, 1) + 10� B(0, 2) + 110� B(0, 3)! B(0, 3) = 0.71! R(0, 3) = 11.7%


The Bootstrap Method General Formulation

General Formulation

A more elegant way of presenting this derivation:

P1= CF 1�B(0, 1)P2= CF 1�B(0, 1) +CF 2�B(0, 2)...

......

Pn= CF 1�B(0, 1) +CF 2�B(0, 2) . . . +CF n�B(0, n)

Using the following notation:I P � the vector of the bond pricesI CF � the matrix of the bonds�cash �owsI B � the (unknown) discount factors vector

We can then solve for the discount factors:

P = CF � B ! B = CF�1P


The Bootstrap Method Example

ExampleIn our former example, the matrix CF and the vector P are:

CF =

24110 0 010 110 010 10 110

35 , P =24 9997.596

35The discount factors vector is then:

B = CF�1P ! B =

24 0.900.80450.7178

35We can then recover the sport rates R(0, t) = (1/B(0, t))1/t � 1 :

R =

2411.1%11.5%11.7%

35Term Structure (Session 9) Investments Spring 2010 9 / 52

The Bootstrap Method A More Realistic Example

A More Realistic Example

We want to determine a 10-year term structure based on thefollowing information:

Maturity Coupon Bond Price1 5 100.4782 5.5 101.4123 6.25 103.5914 6 103.2755 5.75 102.5016 5.5 101.1997 4 92.2208 4.5 94.2479 6.5 107.43410 6 104.213


The Bootstrap Method A More Realistic Example

A More Realistic Example

The resulting discount factors (zeros coupon prices) and the spot ratestructure:

Maturity B(0, t) R(0, t)1 0.95693 0.0452 0.91136 0.04753 0.86507 0.04954 0.81957 0.0515 0.77609 0.0526 0.73355 0.0537 0.69202 0.0548 0.64508 0.05459 0.61763 0.05510 0.58543 0.055


The Bootstrap Method Yield Curve

Yield Curve

1 2 3 4 5 6 7 8 9 104.25

4.5

4.75

5

5.25

5.5

5.75

Maturity (Years)

Spor

t Rat

e (%

)


The Bootstrap Method Linear Interpolation

Linear Interpolation

The bootstrap method only gives us the discount factors associatedwith �nitely many dates.

I The term structure is a whole curve.I What if we need the discount factor for two and a half years?! Need to interpolate to construct the whole curve.

Interpolation: if we know the spot rates for maturities Ti andTi+1,then the spot rate for maturities θ in between is given by:

R(0, θ) = (θ�Ti )R (0,Ti+1)+(Ti+1�θ)R (0,Ti )Ti+1�Ti

This simply amounts to drawing lines between the points obtainedfrom the observed price by bootstrapping.


The Bootstrap Method Interpolated Yield Curve

Interpolated Yield Curve

1 2 3 4 5 6 7 8 9 104.25

4.5

4.75

5

5.25

5.5

5.75

Maturity (Years)

Spor

t Rat

e (%

)


The Bootstrap Method 1-Year Forward Curve

1-Year Forward Curve

1 2 3 4 5 6 7 8 9 104.25

4.5

4.75

5

5.25

5.5

5.75

6

6.25

Maturity (Years)

Forw

ard

Rat

e (%

)


The Bootstrap Method Summary

The Bootstrap Method

The bootstrap methodI Reproduces the observed market prices.I is simple to implement.I but requires the bonds to have the same coupon dates.

The linear interpolation produces non-smooth forward curves ! need�ner interpolation methods (e.g. piecewise polynomial).

Instead of re�ning our interpolation method, we will now concentrateon an alternative approach which:

I Is widely used in practice.I Allows for smooth spot and forward curves.

There is no free lunch: this indirect method will not allow us tomatch the observed market prices exactly!


Parametric Models De�nition

Parametric ModelsSelect a set of bonds paying cash �ows (CFit ) at dates (Tit ).Select a parametric model for the functional form of the discountfactors or the spot rates and generate the theoretical prices

Pk (Θ) =nk

∑i=1CFikB(0,Tik j Θ) =

nk

∑i=1CFik 1

(1+R (0,Tik jΘ))Tik

Estimate the vector Θ of parameters by solving

minΘ

K

∑k

��Pk � Pk (Θ)��2 or minΘ

K

∑k

wk��Pk � Pk (Θ)��2

Here, the weights (wk ) depend on the maturity of the spot rates andare chosen to increase the importance of short maturities.How should the functional form of the yield curve look like? Whichfactors determine the price of a bond?


Parametric Models Dimensionality

Dimensionality

The price of a coupon bond with coupons (c)ni=1 at dates (Ti )ni=1

and nominal amount of one hundred dollars is de�ned as:

P(t) =n

∑i=1

�ci + 1fi=ng100

�e�R (t ,Ti )(Ti�t)

Here R(t,Ti ) is the continuously compounded spot rate or discountrate for the maturity.

There are n risk factors in�uencing the bond price but working withso many is neither practical nor necessary.

Reduce the dimensionality of the problem by writing the discountrates as functions of 3 or 4 parameters (factors).


Parametric Models Principal Component Analysis

Principal Component Analysis (PCA)

Many correlated market variables (factors) a¤ect the yield curve.

However, empirical investigations have shown that more than 95% ofthe variance of interest rates is explained by three factors.

What are these factors?

The PCA is a popular way to study the deformations of the termstructure.

The PCA takes historical data on movements in the market variablesand attempts to de�ne a set of components or factors that explainsthe movements.

PCA tries to explain the behavior of observed variables using asmaller set of independent and unobserved factors.


Parametric Models Principal Component Analysis

Example

US Treasury zero-coupon rates, sampled daily, from 11/1985 to04/2009, with maturities from 1 to 30 years.

The observations are variations in the continuously compounded spotrates for a set of constant maturities.

Result: the �rst three principal components explain 98.6% of thevariance.


Parametric Models Principal Components

Principal Components

Computing the principal components (eigenvectors) for the short tomedium part of the curve we obtain that the three principalcomponents are given by:

p1 =

0BBBBBBBBBBBBBB@

0.2840.3100.3190.3240.3260.3260.3240.3200.3160.311

1CCCCCCCCCCCCCCAp2 =

0BBBBBBBBBBBBBB@

0.5980.4200.2560.114�0.009�0.115�0.203�0.276�0.334�0.381

1CCCCCCCCCCCCCCAp3 =

0BBBBBBBBBBBBBB@

0.664�0.102�0.359�0.374�0.281�0.146�0.0040.1270.2390.329

1CCCCCCCCCCCCCCA







1 2 3 4 5 6� 30% 86.79 7.71 4.13 0.95 0.34 0.07Cumul. 86.79 94.50 98.63 99.59 99.93 100

The �rst PC is roughly �at and in�uences all rates in the same way.This �rst factor is identi�ed as the level of the term structure.

The second PC is decreasing and thus in�uences the two ends of thecurve in opposite directions. This second factor is identi�ed withvariations in the slope of the term structure.

The third PC is hump-shaped and thus has a di¤erent impact on thevarious segments of the yield curve. This third factor identi�es thecurvature of the term structure.

Example in class.


Parametric Models Nelson-Siegel

Nelson-Siegel

In the Nelson-Siegel model, the continuously compounded sport rate isgiven by:

R(0,T ) = β0 + β11�exp(�T /τ)

T /τ + β21�(1+T /τ) exp(�T /τ)

T /τ

where

τ is a scale parameter

β0 is a level parameter (the long rate)β1 is a slope parameter (the spread short term/long term)β2 is a curvature parameter



Nelson-Siegel

To investigate the in�uence of the slope and curvature parameters ofthe Nelson-Siegel model, we perform the following experiment:

1 Start with a set of base case parameters

F τ = 3.33F β0 = 0.07F β1 = �0.02F β2 = 0.01%

2 Adjust the slope and curvature parameters

F The parameter β1 varies by ∆ 2 f�6, ...,�1, 1, ..., 6g%F The parameter β2 varies by ∆ 2 f�6, ...,�1, 1, ..., 6g%


Parametric Models Initial Curve

Initial Curve

0 5 10 15 20 25 305

5.5

6

6.5

7

Maturity (Years)

Spor

t Rat

e (%

)


Parametric Models Variation in Slope

Variation in Slope

0 5 10 15 20 25 303

5

7

9

11

Maturity (Years)

Spor

t Rat

e (%

)


Parametric Models Variation in Curvature

Variation in Curvature

0 5 10 15 20 25 304

5

6

7

8

Maturity (Years)

Spor

t Rat

e (%

)



Nelson-Siegel

Changes in the parameters induce non uniform changes in thediscount rates which in turn imply changes in the bond prices.

Thanks to the functional from of the model, we can easily computethe sensitivities of R(0,T ) to changes in any of the parameters.

This approach is consistent with the principal component analysis ofthe term structure since the NS parameters can be interpreted aslevel, slope and curvature.

The same principles can be applied to any of the parametric modelsof the term structure.



Nelson-Siegel

Let θ be a �xed maturity date. The sensitivity of the associateddiscount rate to variations in the parameters are given by

∂R (0,T )∂β0

= F0(θ) = 1∂R (0,T )

∂β1= F1(θ) =

1�exp(�θ/τ)θ/τ

∂R (0,T )∂β2

= F2(θ) =1�(1+θ/τ) exp(�θ/τ)

θ/τ

The bond price is given by a sum of exponentials involving thediscount rates associated with each of the coupon dates.

The sensitivities of the bond price to variations in the parameters canbe then computed from those of the discount rates.


Parametric Models Nelson-Siegel Factor Loadings

Nelson-Siegel Factor Loadings

0 5 10 15 20 25 300

0.10.20.30.40.50.60.70.80.9

1

Maturity

Fact

or L

oadi

ng

β0

β1

β2


Parametric Models Extended Nelson-Siegel

Extended Nelson-Siegel

Svensson has proposed an extension of the Nelson-Sielgel modelwhere the spot interest rate for the maturity date θ is given by:

R(0, θ) = β0 + β11�exp(�θ/τ)

θ/τ + β21�(1+θ/τ) exp(�θ/τ)

θ/τ

+β31�(1+θ/ρ) exp(�θ/ρ)

θ/ρ

τ, β0, β1 and β2 are as in the original N-S model.

ρ and β3 are scale and level parameters which add more �exibility inthe short term end of the yield curve.


Parametric Models Extended Nelson-Siegel

Extended Nelson-Siegel

0 5 10 15 20 25 304

5

6

7

8

Maturity (Years)

Spor

t Rat

e (%

)


Parametric Models Parametric Models: Pros and Cons

Parametric Models: Pros and Cons

Widely used in practice.

Related to some popular dynamic models of the term structure.

One key advantage is that they are parsimoniousI Do not involve many parametersI Robustness and stability

One drawback is their lack of �exibility: cannot account for all theshapes of the yield curve that are observed in practice.

Alternative methods: Spline ModelsI Less parsimoniousI More �exible and better for pricingI Various shapes: Cubic splines, Exponential splines,...


Parametric Models Summary

Selecting a Basket of Bonds

To build a coherent set of instruments from which to construct theterm structure, one must select instruments which:

1 Are denominated in the same currency2 Are of the same credit quality3 Are comparable in risk4 Are not concerned with pricing errors or liquidity5 Do not have option like features

Other �xed income securities and contracts may be used:I Swap ratesI Futures contracts


Models of the Short Rate Type of Models

Models of the Short Rate

The term structure of interest rate is characterized by one singlevariable: the short rate.

Assumes that the short rate contains all the information about theterm structure that is relevant for pricing and hedging interest ratesrelated contracts.

These term structure models can be categorized as follows:I Equilibrium models: start with assumptions about economic variables(consumption, production) and derive the process for the short rate.

I Arbitrage models: models designed to be exactly consistent withtoday�s term structure of interest rates.


Models of the Short Rate Basic setup

Basic setup

Assume the risk-neutral framework.

The short rate, r , at time t is the rate that applies to anin�nitesimally short period of time at time t.

Example: in a very short time period between t and t + dt, investorsearn r (t) dt.

Interest rate claims depend only on the process followed by r .

The risk-neutral process for the short rate is described by an Itôprocess of the form:

dr = m(r , t)dt + s(r , t)dW

Here, we haveI m(r , t) the instantaneous driftI s(r , t) the instantaneous volatilityI W is a Wiener process


Models of the Short Rate Basic setup

Basic setup

In this setup, we can compute the discounted bond prices:

P(t,T ) = E[ exp(�R Tt r(s)ds)]

If R(t,T ) is the continuously compounded interest rate at time t fora term T � t, then P(t,T ) = e�R (t ,T )(T�t), so that:

R(t,T ) = � 1T�t lnP(t,T )

These equations enable the term structure of interest rates at anygiven time to be obtained form the value of r at that time and therisk-neutral process for r .


Models of the Short Rate Mean Reversion

Mean Reversion

Reversion Level

High interest ratehas negative trend

T ime

Interest rate

Low interest ratehas positive trend


Models of the Short Rate The Vasicek Model

The Vasicek Model

In the Vasicek model, the risk-neutral process for r is:

dr = a(b� r)dt + σdW

Here, the coe¢ cients a, b and σ are constantI b is the long term rateI a is the speed of mean-reversionI σ is the instantaneous volatility

Model properties:I Incorporates mean-reversionI All rates move in the same direction but not necessarily in the samemagnitude.

I The interest rate can be negative!



The Vasicek ModelVasicek shows that the price at time t of a zero-coupon bond thatpays 1$ at time T is:

P(t,T ) = A(t,T )e�B (t ,T )r (t)

In this equation, r(t) is the value of r at time t and:

B(t,T ) = 1�e�a(T�t)a

A(t,T ) = exp[ (B (t ,T )�T+t)(a2b�σ2/2)

a2 � σ2B (t ,T )2

4a ]

The spot rate is de�ned as:

R(t,T ) = � 1T�t lnA(t,T ) +

1T�t lnB(t,T )r(t)

The entire term structure can be determined as a function of r(t)once a, b and σ are chosen.



The Vasicek Model

The short rate dynamics are:

drt = a(b� rt )dt + σdWt

It can be veri�ed using Itô�s formula that:

rt = e�athr0 +

R t0 abe

audu + σR t0 e

audWu

iBased on this solution, we can derive µt � E [rt ] and σ2t � V [rt ] :

µt = e�at [r0 + b(eat � 1)]

σ2t = σ2�1�e�2at2a

�Therefore, we have that rt � N(µt , σ2t )



The Vasicek ModelPossible shapes of the term structure

Maturity

Spot

Rat

e

Maturity

Spot

Rat

e

Maturity

Spot

Rat

e


Models of the Short Rate The Cox, Ingersoll, and Ross Model

The Cox, Ingersoll, and Ross Model

In the CIR model, the risk-neutral process for r is:

dr = a(b� r)dt + σprdW

In this setup, we haveI The same mean-reverting drift as in VasicekI But the volatility of the change in the short rate is proportional to r

Model properties:I Rates are always non-negative


Models of the Short Rate The Cox, Ingersoll, and Ross Model

The Cox, Ingersoll, and Ross Model

In the CIR model the price at time t of a zero-coupon bond that pays1$ at time T is:

P(t,T ) = A(t,T )e�B (t ,T )r (t)

In this equation, r(t) is the value of r at time t and:

B(t,T ) = 2(eγ(T�t)�1)(γ+a)(eγ(T�t)�1)+2γ

A(t,T ) =h

2γe (a+γ)(T�t)/2

(γ+a)(eγ(T�t)�1)+2γ

i2ab/σ2

Here we have γ =pa2 + 2σ2


Models of the Short Rate The Ho-Lee Model

The Ho-Lee Model

In the Ho-Lee model, the risk-neutral process for r is:

dr = θ(t)dt + σdW

θ(t) = Ft (0, t) + σ2t

The price at time t of a zero-coupon bond that pays 1$ at time T is:

P(t,T ) = A(t,T )e�B (t ,T )r (t)

Here F (0, t) denotes the instantaneous forward rate for maturity tand the subscript t denotes a partial derivative with respect to t. Theexpression for A(t,T ) is:

lnA(t,T ) = ln P (0,T )P (0,t) + (T � t)F (0, t)�12σ2t(T � t)2


Models of the Short Rate The Hull-White Model

The Hull-White ModelIn the Hull-White model, the risk-neutral process for r is:

dr = [θ(t)� ar ] dt + σdW

θ(t) = Ft (0, t) + aF (0, t) + σ2

2a (1� e�2at )

The price at time t of a zero-coupon bond that pays 1$ at time T is:

P(t,T ) = A(t,T )e�B (t ,T )r (t)

Here F (0, t) denotes the instantaneous forward rate for maturity t.The expression for A(t,T ) and B(t,T ) are:

B(t,T ) = 1�e�a(T�t)a

lnA(t,T ) = ln P (0,T )P (0,t) + B(t,T )F (0, t)�14a3 σ2(e�aT � e�at )2(e2at � 1)


Models of the Short Rate Term Structure Models

Term Structure Models

Merton dr = bdt + σdWVasicek dr = a (b� r) dt + σdWDothan dr = brdt + σrdWCIR dr = a (b� r) dt + σ

prdW

Ho and Lee dr = θ(t)dt + σdWHull and White I dr = (θ(t)� ar) dt + σdWHull and White II dr = (θ(t)� ar) dt + σ

prdW

Black and Karasinski dr = (θ(t)� a log r) dt + σdW


Models of the Short Rate Options on Discount Bonds

Options on Discount Bonds

For the Vasicek model, the price at time zero of a call/put optionthat matures at time T on a zero coupon bond maturing at time s is:

Call = LP(0, s)N(h)�KP(0,T )N(h� σP )

Put = KP(0,T )N(�h+ σP )� LP(0, s)N(�h)

Here, L is the principal of the bond, K is the strike price and

h = 1σPln LP (0,s)KP (0,T ) +

σP2

σvP =σa [1� e�a(s�T )]

q1�e�2aT2a

σhlP = σ[s � T ]pT

Here σvP holds for the Vasicek and the Hull and White models and σhlPholds for Ho-Lee model.


Models of the Short Rate Options on Coupon Bonds

Options on Coupon Bonds

An option on a coupon-bearing bond is expressed as a sum of aoptions on zero-coupon bonds. The procedure is as follows:

1 Calculate r�, the critical value of r for which the price of the couponbearing bond equals the strike price of the option on the bond atoption maturity.

2 Calculate the prices of options on the zero-coupon bonds that comprisethe coupon-bearing bond. Set the strike price of each option equal tothe value the corresponding zero-coupon bond will have at time Twhen r = r�.

3 Set the price of the option on the coupon bond equal to the sum of theprices on the options on zero-coupon bonds calculated in step 2.




Example: suppose that a = 0.05, b = 0.08, and σ = 0.015 in theVasicek model with r0 = 6%. Calculate the price of a 2.1-yearEuropean call option on a bond that will mature in three years.Suppose that the bond pays a coupon of 5% semiannually. Theprincipal of the bond is 100 and the strike price of the option is 99.(The strike price is the cash price, not the quoted price, that will bepaid for the bond)




Stage 1:

2.5A(2.1, 2.5)e�B (2.1,2.5)r�+ 102.5A(2.1, 3)e�B (2.1,3)r

�= 99

! r � = 0.066

Stage 2:

2.5A(2.1, 2.5)e�B (2.1,2.5)�0.066 = 2.43473102.5A(2.1, 3)e�B (2.1,3)�0.066 = 96.56438

The call option on the coupon bond can be decomposed into a call optionwith a strike price of 2.43473 on a bond that pays o¤ 2.5 at time 2.5 yearsand a call option with a strike price of 96.56438 on a bond that pays o¤102.5 at time 3.0 years.Stage 3: The value of the �rst option is 0.009085 and the value of thesecond options is 0.815238.


Investments

Session 10. Managing Bond Portfolios


Philip Valta

Spring 2010

Bond Portfolios (Session 10) Investments Spring 2010 1 / 54


Duration

Convexity

Managing Bond Portfolios

Immunization


Duration Yield to Price Relationship

Yield to Price Relationship

0 5 10 15 2020

40

60

80

100

120

140

160

Yield to Maturity (%)

Bond

Pric

e


Duration Interest Rate Risk

Interest Rate Risk

Interest rates can �uctuate substantially.

As a results, bondholders experience capital gains and losses.

Thus, even though coupon and principal payments are guaranteed(Treasuries), �xed income investments are risky.

Why?

Why do bond prices respond to interest rate �uctuations?


Duration Change in Bond Price as a Function of Change in YTM

Change of Bond Price as a Function of Change in YTM


Duration Bond Price as a Function of YTM

Bond Price as a Function of YTM

Bond prices and yields are inversely related: as yields increase, bondprices fall; as yields fall, bond prices rise.

An increase in a bond�s yield to maturity results in a smaller pricechange than a decrease in yield of equal magnitude.

Prices of long-term bonds tend to be more sensitive to interest ratechanges than prices of short-term bonds.


Duration Bond Price as a Function of YTM

Bond Price as a Function of YTM

The sensitivity of bond prices to changes in yields increases at adecreasing rate as maturity increases. In other words, interest raterisk is less than proportional to bond maturity.

Interest rate risk is inversely related to bonds coupon rate. Prices oflow-coupon bonds are more sensitive to changes in interest rates thanprices of high-coupon bonds.

The sensitivity of a bond�s price to a change in its yield is inverselyrelated to the yield to maturity at which the bond currently is selling.


Duration Changes in Interest Rates

Changes in Interest Rates

We have seen how to use interest rates to calculate the value of azero coupon bond, coupon bond, etc.

Since interest rates vary over time, it is important to know whathappens to the value of a bond when interest rates change.Suppose you are managing a USD 100 Mio. bond portfolio. Howdoes the value of your portfolio change when interest rates increaseby one basis point? Two possibilities:

1 Recompute the value of your portfolio using the new interest rate.2 Short cut: if changes in interest rates are not too large, use durationand convexity.


Duration Duration

Duration

If changes in y are not too large, we can use a Taylor expansion toapproximate the change in the price of a bond.

As the degree of the Taylor series rises, it approaches the correctfunction.

Suppose a bond has an initial value of P0 and the initial yield is y0.The yield then changes to y1 = y0 +4y . The new price of the bondcan be written as

P1 = P0 +dPdy4y + 1

2d2Pdy2

(4y) + ...

For �xed-income instruments, the derivatives dPdy andd 2Pdy 2 are so

important that they have been given special names.


Duration Duration

Duration

The duration of a bond measures the sensitivity of a bond price tochanges in interest rates.

The duration is also a measure of how long on average the holder ofthe bond has to wait before receiving cash payments.

Duration assume a �at term structure.

Let P(y) denote the price of a bond with a yield to maturity y andcash-�ow Ct at time t

The price of a bond is given by the usual P(y) = ∑Tt=1

Ct(1+y )t

. Then,

we have:

P0(y) = �

T

∑t=1

tCt(1+y )t+1

= � 11+y

T

∑t=1

tCt(1+y )t


Duration Duration

Duration

Therefore, we have:

P(y + ∆y) = P(y)� 11+y

T

∑t=1

tCt(1+y )t

∆y

The change in the bond price (the return on the bond caused by ashift in interest rates) is given by:

P (y+∆y )�P (y )P (y ) = �

∑Tt=1

tCt(1+y )t

P (y )

�∆y1+y

�

D =∑Tt=1

tCt(1+y )t

P (y ) is called the Macaulay Duration of the bond.


Duration Duration

Duration

The duration is a weighted average of the times when payments aremade, with the weight applied to time t being equal to the proportionof the bond�s total present value provided by the cash �ow at time t:

D =∑Tt=1

tCt(1+y )t

P (y ) =T

∑t=1t(Ct/(1+y )

t

P (y ) ) =T

∑t=1twt

Thus the bond price change due to a small change in interest ratescan be approximated by:

dPP = �D dy

1+y

D� = D/(1+ y) is called the Modi�ed Duration.


Duration Example

Example

Consider a 10-year 6% (annually) coupon bond with yield to maturityof 5%. We can calculate the duration as follows:

Date Cash Flow Weight (2)� (3)1 6 0.0530467 0.053047

2 6 0.0505207 0.101041

3 6 0.0481149 0.144345

4 6 0.0458238 0.183285

5 6 0.0436417 0.218208

6 6 0.0415635 0.249381

7 6 0.0395843 0.277090

8 6 0.0376993 0.301595

9 6 0.0359041 0.323137

10 106 0.6041010 6.041010

Total 1.000 7.892150


Duration Example

Example

Suppose interest rates increase by ∆y = 0.75%. By how much wouldthe bond price change?

We have:

∆PP = �D ∆y

1+y = �7.890.00751+0.05 = �5.6373%


Duration Properties of Duration

Properties of Duration

1 The duration of a zero-coupon bond equals its time to maturity.2 The bond�s duration is lower when the coupon is higher.3 The bond�s duration generally increases with time to maturity.Duration always increases with maturity for bonds selling at par or ata premium.

4 The duration of a coupon bond is higher when the bond�s yield tomaturity is lower.

5 The duration of a perpetuity is D = (1+ y)/y .




Maturity Coupon YTM Price DurationBond 1 1 7 6 100.94 1.00Bond 2 1 6 6 100.00 1.00Bond 3 5 7 6 104.21 4.40Bond 4 5 6 6 100.00 4.47Bond 5 10 4 6 85.48 8.28Bond 6 10 8 6 114.72 7.45Bond 7 20 4 6 77.06 13.22Bond 8 20 8 7 110.59 11.05Bond 9 50 6 6 100.00 16.71Bond 10 50 0 6 5.43 50.00





Duration Linearity of Duration

Linearity of Duration

Consider a portfolio of K bonds with annual payments eachrepresenting a fraction πk of the total value. The duration of theportfolio is:

DΠ = �(1+ y)Π0(y )

Π(y ) =K

∑k=1

πkDk

Here Π denotes the total value of the portfolio and Dk denotes theduration of the k-th bond in the portfolio.

! The duration of a portfolio is the weighted average of the duration ofthe securities in the portfolio.


Convexity Convexity

ConvexityConsider a 30-year bond with 8% coupon and initial YTM = 8%


Convexity Convexity

Convexity

Modi�ed duration is the appropriate measure of interest rate risk.

Duration only measures the �rst-order (linear) e¤ect. But therelationship between bond prices and yields is not linear.

Duration is good only for small changes in yields. When there arelarge changes in yields, duration is not a su¢ cient measure of interestrate exposure.

It might therefore be necessary to take into account the curvature(the second-order quadratic term) of the price-yield relation.

Notice that duration always understates the value of the bond.


Convexity Convexity

Convexity

By Taylor�s theorem we have:

P(y + ∆y) = P(y) + P0(y)∆y + 1

2P00(y) (∆y)2

The price of a bond is given by the usual P(y) = ∑Tt=1

Ct(1+y )t

. Then,

we have:

P0(y) = �

T

∑t=1

tCt(1+y )t+1

= � 11+y

T

∑t=1

tCt(1+y )t

P00(y) =

T

∑t=1

t(t+1)Ct(1+y )t+2

= 1(1+y )2

T

∑t=1

t(t+1)Ct(1+y )t


Convexity Convexity

Convexity

Therefore, we have:

P(y + ∆y) = P(y)� 11+y

T

∑t=1

tCt(1+y )t

∆y + 12

1(1+y )2

T

∑t=1

t(t+1)Ct(1+y )t

(∆y)2

The change in the bond price (the return on the bond caused by ashift in interest rates) is given by:

P (y+∆y )�P (y )P (y ) = �

∑Tt=1

tCt(1+y )t

P (y )

�∆y1+y

�+ 1

2

∑Tt=1

t(t+1)Ct(1+y )t

P (y )

�∆y1+y

�2P (y+∆y )�P (y )

P (y ) = �D�

∆y1+y

�+ 1

2C�

∆y1+y

�2Here D denotes the duration and C the convexity of the bond.


Convexity Example

Example

Consider a 10-year 6% (annually) coupon bond with yield to maturityof 5%

Suppose interest rates increase by ∆y = 0.75%. By how much wouldthe bond price change?

We have:

∆PP = �D ∆y

1+y +12C�

∆y1+y

�2!

∆PP = �7.89 0.00751+0.05 +

1279.57

� 0.00751+0.05

�2= �5.43%


Convexity Properties of Convexity

Properties of Convexity

Maturity Coupon YTM Price ConvexityBond 1 1 7 6 100.94 1.78Bond 2 1 6 6 100.00 1.78Bond 3 5 7 6 104.21 22.47Bond 4 5 6 6 100.00 22.92Bond 5 10 4 6 85.48 75.89Bond 6 10 8 6 114.72 65.17Bond 7 20 4 6 77.06 211.53Bond 8 20 8 7 110.59 157.93Bond 9 50 6 6 100.00 440.04Bond 10 50 0 6 5.43 2269.50




Convexity of a portfolio:

CΠ =K

∑k=1

πkCk

Here, πk and Ck denote respectively the weight of bond k in the totalvalue of the portfolio and the convexity of bond k.





Convexity Example

Example

Consider a 10-year zero coupon bond with a yield of 6% (semiannual)and present value of USD 55.368.

What is the bond�s duration?

What is the bond�s modi�ed duration?

What is the bond�s convexity?

Suppose the yield goes up to 7%.I How good is the linear approximation?I How good is the linear and convexity approximation?

Do investors like convexity?


Convexity Callable Bonds

Duration and Convexity of Callable Bonds


Convexity Analogy

An Analogy

Are there related concepts to duration and convexity when we talkabout stocks and options?

How does the relation between the price of a call option and the priceof the underlying stock look like?

Delta: The rate of change of the option price with respect to theunderlying stock.

Gamma: The rate of change of delta with respect to the price of theunderlying stock.

Dynamic delta hedging.


Managing Bond Portfolios Managing Bond Portfolios

Managing Bond Portfolios

Active Bond ManagementI Interest rate forecastingI Identi�cation of relative mis-pricing

Passive Bond ManagementI Bond index fundsI Cash �ow matchingI Immunization of interest rate risk

1 Net worth immunizationDuration of assets = Duration of liabilities

2 Target date immunizationHolding period matches duration


Managing Bond Portfolios Example

Example 1

Example: consider an insurance company that issues a guaranteedinvestment contract (GIC) for 10, 000. The GIC has a 5 year maturityand a guaranteed interest rate of 8%. Then the insurance company isobliged to pay 10, 000� (1.08)5 = 14, 693.28 in 5 years.Suppose that the insurance company chooses to fund its obligationwith 10, 000 of 8% annual coupon bonds, selling at par value with 6years to maturity.

! As long as the market interest rate stays at 8% the company has fullyfunded the obligation, as the present value of the obligation exactlyequals the value of the bonds.

Can the bond generate enough income to pay o¤ the obligation 5years from now regardless of interest rates movements?



Example 1



Example 1

Fixed income investors face two type of risks:I Price riskI Reinvestment risk

Increases/decreases in interest rates cause capital losses/gains but atthe same time increase/decrease the rate at which reinvested incomewill grow.

For a horizon equal to the portfolio�s duration, price risk andreinvestment risk exactly cancel out.Example: if interest rates fall to 7%, the total funds will accumulateto 14, 694.05$ providing a small surplus of .77$. If rates increase to9% the fund accumulates to 14, 696.02$, providing a small surplus of2.74$.



Example 2

An insurance company must make a payment of 19, 478$ in 7 years.The interest rate is 10%, so the present value of the obligation is10, 000. The portfolio manager wishes to fund the obligation using3-year zero-coupon bonds and perpetuities paying annual coupons.How can the manager immunize the obligation?



Example 2

Immunization requires that the duration of the portfolio of assetsequals the duration of the liability. We then proceed in 3 steps:

1 Calculate the duration of liability: 7 years2 Calculate the duration of asset portfolio: the duration of the portfolio isthe weighted average of durations of each component asset, withweights proportional to the funds placed in each asset. Here wehave: DZC = 3 years and DP = (1+ y)/y = 11 years and theportfolio duration is: DA = w � 3+ (1� w)� 11

3 Find the asset mix that sets the duration of assets equal to the 7-yearduration of liabilities. This requires to solve:w � 3+ (1� w)� 11 = 7! w = 1/2



Example 2

Suppose that 1 year has passed and that interest rates are still at10%. Is the position still fully funded? Is it immunized?

The PV of the obligation will have grown to 11, 000$. The manager�sfunds also have grown to 11, 000$: value of zero-coupon bonds goesfrom 5,000 to 5,500 with passage of time and the perpetuity has paid500$ of coupon and remains worth 5, 000$ ! the obligation is stillfunded.

The portfolio weights must be changed and mustsatisfy: DA = w � 2+ (1� w)� 11 = 6! w = 5/9Immunization based on duration is a dynamic strategy. As timepasses the duration and time to maturity changes ! need torebalance the immunized portfolio!


Managing Bond Portfolios Comments

Comments

Duration and convexity are build on restrictive assumptionsI The yield curve is �atI Term structure only a¤ected by parallel shifts

F All bonds have the same yield to maturityF Risk on the general level of interest rates

Bad news: not only is the term structure not �at, but it also changesshape through time!

Solutions:I Principal Component Analysis: sheds light on the dynamics of the yieldcurve

I Application of general immunization theory


Application of Immunization Theory An Application of Immunization Theory

Idea

Apply an immunization theory that allows for arbitrary changes in thespot rate structure.

These changes include parallel shifts but also changes in the curvatureof the term structure.

Illustration with a numerical example of how to immunize a portfolioto a radical change in the term structure.


Application of Immunization Theory Application

Initial spot rate curve - continuously compounded

0 5 10 15 200.04

0.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06Initial spot rate curve

Maturi ty (years)

Spot

rate



Assumptions

Suppose that you want to invest today a value of $100 for seven years.

Investment horizon is therefore seven years.

No AAA zero-coupon bond that would guarantee you a terminal valueof $100 � e(0.0528�7) = $144.72 in seven years.However, there are 4 bonds a, b, c , and d . Their characteristics aresummarized in the next table.

Assume for simplicity that they pay annual coupons.

Can an appropriate portfolio using bonds a, b, c , and d guarantee theterminal value of $144.72 in seven years?



Main features of bonds

Coupon Maturity (years) Par value Initial valueBond a 4 7 100 92.18556Bond b 4.75 8 100 95.42947Bond c 7 15 100 111.0813Bond d 8 20 100 123.5471



Question

What should your investments in these bonds be, such that your $100will be transformed into that terminal value of $144.72, even if thereis a dramatic shift in the term structure just after you bought thebonds?

For example, the steep spot curve from before turns clockwise tobecome horizontal at a given level, for instance 5.5%.

How should you constitute your bond portfolio in order to secure avalue extremely close to $144.72 in seven years?



New spot rate curve

0 5 10 15 200.04

0.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06Initial and new spot rate curve

Maturi ty (years)

Spot

rate



De�nitions

Denote the amounts invested in bonds a, b, c , and d with na, nb , nc ,and nd .

Denote Hk the investors horizon (or duration) to the power of k.

The moment of order k of bond l , mlk , is the weighted average of thekth power of its times of payments, the weights being the shares ofthe bond�s cash �ows in the initial bond value:

mlk =N

∑t=1tkclte�s(t)t

B l0

where s(t) is the spot rate for maturity t.



The Moment of Order k of a Bond Portfolio

The moment of order k of a bond portfolio is the weighted average ofthe kth power of its times of payments, the weights being the sharesof the portfolio�s cash �ows in the initial portfolio value.

mPk =L

∑l=1

nlB l0P0

N

∑t=1tkclte�s(t)t

B l0

=L

∑l=1

nlB l0P0

mlk

The moment of order 0 of a portfolio (or a bond) is one, since it isthe weighted average of 1�s.

The moment of order 1 of a portfolio (or a bond) is its duration.



A General Immunization Theorem

TheoremSuppose that the spot rate structure can be expanded into a Taylor seriesof order m� 1 and that it undergoes a variation. Then a su¢ cientcondition for a bond portfolio to be immunized against such a variation isthe following:

1 Any moment of order k (k = 0, 1, ..., 2m� 1) of the bond portfolio isequal to the kth power of the investor�s horizon H.

2 The moment of order 2m is equal to the 2mth power of H plus apositive arbitrary constant.



System of Equations

This result leads to the following system of four linear equations withfour unknowns na, nb , nc , and nd :

naB a0P0ma0 +

nbB b0P0mb0 +

ncB c0P0mc0 +

ndB d0P0md0 = H

0

naB a0P0ma1 +

nbB b0P0mb2 +

ncB c0P0mc3 +

ndB d0P0md4 = H

1

naB a0P0ma2 +

nbB b0P0mb2 +

ncB c0P0mc2 +

ndB d0P0md2 = H

2

naB a0P0ma3 +

nbB b0P0mb3 +

ncB c0P0mc3 +

ndB d0P0md3 = H

3



System of Equations

For simplicity de�neI n � the row vector of unknowns (na, nb , nc , nd )

I m � the square matrix of terms Bl0P0mlk , for l = a, b, c , d , and

k = 0, 1, 2, 3I h � the vector of horizons to the powers of 0, 1, 2, 3

The system can now be written:

nm = h

and solved for n:n = m�1h



Solution

n = m�1h

,2664nanbncnd

3775 =266443.6218 -11.4694 0.81844 -0.0188-48.7516 13.4968 -1.0056 0.023710.2440 -3.3070 0.3078 -0.0088-3.2934 1.1062 -0.1107 0.0036

377526641749343

3775



Solution

The solution n is then:I na = �2.99784I nb = 4.57423I nc = �0.82821I nd = 0.257722

We go long the bonds b and d , and short the bonds a and c .



Solution

Suppose now that at time ε, immediately after the purchase ofportfolio (na, nb , nc , nd ), the initial spot structure that was increasingturns clockwise to become horizontal at level 5.5% per year(continuously compounded).

What happens to the value of our portfolio under the initial and newstructure?

What is the value of the portfolio in 7 years under the initial and newstructure?

How e¢ cient is the immunization strategy?



Solution

# of bonds na nb nc ndValue at t0 92.19 95.43 111.08 123.55Value in portfolio -276.36 436.52 -92 31.84Total = 100Value at tε 90.65 94.31 113.37 127.68Value in portfolio -271.76 431.39 -93.90 32.91Total = 98.637Value of portfolio in 7 yearsInitial term structure 100e(0.0528�7) = 144.72New term structure 98.637e(0.055�7) = 144.96



More changes

Suppose that the term structure experiences an even more radicalchange.

From the initial term structure, make it turn clockwise to tenhorizontal structures, from 1% to 10%.



More changes

New spot rate structure Portfolio value in 7 years0.01 144.72410.02 144.74480.03 144.7910.04 144.85240.05 145.92180.06 145.99520.07 145.07080.08 145.1490.09 145.23210.10 145.3237


Investments

Session 11. Interest Rate DerivativesEPFL - Master in Financial Engineering

Philip Valta

Spring 2010

Interest Rate Derivatives (Session 11) Investments Spring 2010 1 / 53


Forward contracts

Future contracts

Interest Rate Swaps

Interest Rate DerivativesI Bond OptionsI Interest Rate Caps and FloorsI European Swap Options


Forward Contracts De�nition

Forward Contracts - De�nition

A forward contract is an agreement to buy or sell an asset at a certainfuture time for a certain price.

By contrast, a spot contract is an agreement to buy or sell an assettoday.

Forward contracts are popular in the foreign exchange market.

They are widely used for hedging purposes.



Forward Contracts - De�nition

Suppose a Swiss �rm expects to receive a payment of 1 million Euroin one month.

The �rm is exposed to the risk that the Euro depreciates (Swiss francappreciates).

In order to hedge this risk, the �rm can sell 1 million Euro today at aforward price.



Payo¤

Profit

Price of Underlyingat Maturity, STK

Profit

Price of Underlyingat Maturity, ST

Profit

Price of Underlyingat Maturity, STK


Forward Contracts Pricing

Pricing of Forward Contracts

Consider a long forward contract to purchase a non-dividend payingstock in 3 months. Assume the current stock price is $40 and the3-month risk-free rate is 5% per annum.

Suppose that the forward price is relatively high at $43.

Suppose that the forward price is relatively low at $39.

Under what circumstances do arbitrage opportunities not exist?




Consider a forward contract on an underlying investment asset withprice S0 that provides no income, T is the time to maturity, and r isthe risk-free rate.

F0 is the forward price given by

F0 = S0erT

If F0 > S0erT , arbitrageurs buy the asset and short forward contractson the asset.

If F0 < S0erT , they can short the asset and enter into long forwardcontracts on it.

In the example above: S0 = 40, r = 0.05, T = 0.25, so thatF0 = 40e0.05�0.25 = $40.50.




With dividends (known yields), interest payments, carrying costs, etc.the price generalizes to

F0 = S0e(r+v�c�d )T

v denotes the annualized, continuously compounded storage costs ordepreciation,

c the convenience yield (leasing, foreign currency interest rate), and

d cash payouts (dividends)



Example 1

Consider a 4-month forward contract to by a zero-coupon bond thatwill mature 1 year from today (this means that the bond will have 8months to go when the forward contract matures). The current priceof the bond is $930. Assume that the 4-month continuouslycompounded risk-free rate is 6% per annum. What is the negotiateddelivery price of the forward contract today?



Example 2

Consider a forward contract on March 1 where a French �rm agreesto purchase CHF 100�000 on June 1. The CHF is sold spot for Euro0.70. The Swiss annual interest rate is 4%, and the Euro area interestrate is 10%. What is the no arbitrage price for such a contract?



Value of Forward Contract

What is the value of a forward contract at the time it is �rst enteredinto?

Denote f the value of a forward contract today.

Suppose K is the delivery price for a contract that was negotiatedsome time ago. F0 is the forward price what would be applicable if wenegotiated the contract today.

As time passes, K stays the same, but F0 changes and f becomeseither positive or negative.

f = (K � F0) e�rT



Value of Forward Contract - Example

A long forward contract on a non-dividend paying stock was enteredinto some time ago. It currently has 6 months to maturity. Therisk-free rate of interest (with continuous compounding) is 10% perannum, the stock price is $25, and the delivery price is $24. What isthe value of the forward contract?


Futures Contracts De�nition

Futures contracts - De�nition

Like a forward contract, a futures contract is an agreement betweentwo parties to buy or sell an asset at a certain time in the future for acertain price.

Unlike forward contracts, futures contracts are normally traded on anexchange.

Exchange speci�es certain standardized features of the contract.

Exchange provides mechanism that gives the two parties a guaranteethat the contract will be honored.

Underlying comprise a wide range of commodities and �nancial assets:I pork bellies, live cattle, sugar, wool, lumber, copper, aluminium, gold,tin, stock indices, currencies, Treasury bonds, etc.


Futures Contracts De�nition

Forwards and Futures - Di¤erences

Forwards FuturesPrivate contract between two parties Exchange tradedNon-standard contract Standard contractUsually one speci�ed delivery date Range of delivery datesSettled at maturity Settled dailyUsually delivery or �nal cash settlement Usually closed out before maturity


Futures Contracts Interest Rate Futures

Treasury Bond Futures

Treasury bond futures are traded on the Chicago Board of Trade(CBOT).

They are quoted the same way as the Treasury bond prices.

Suppose the settlement price of a contract is 110� 03, or 100 332 .One contract involves the delivery of $100000 face value of the bond.

A $1 change in the quoted futures price would lead to a $1000change in the value of the futures contract.

Delivery can take place at any time during the delivery month.



Treasury Bond Futures

Treasury Bond Futures contracts allow the party with the shortposition to choose to deliver any bond that has a maturity of morethan 15 years and that is not callable within 15 years.

When a particular bond is delivered, the conversion factor de�nesthe price received by the party with the short position.

The quoted price applicable to the delivery is the product of theconversion factor and the most recent settlement price.

Take into account the accrued interest.

Example.



Cheapest-to-deliver Bond

At any given time during the delivery month, there are many bondsthat can be delivered in the CBOT Treasury bond futures contract.

These vary widely regarding coupon and maturity.

The party with the short position can choose which of the availablebonds is "cheapest" to deliver.

The party with the short position receives

(Settlement price� Conversion factor)+ Accrued interest

and the cost of purchasing a bond is

Quoted bond price+ Accrued interest

and the cheapest-to-deliver bond is

Quoted bond price� (Settlement price� Conversion factor)



Cheapest-to-deliver Bond - Example

The party with the short position has decided to deliver and is tryingto choose between the three bonds in the table below. Assume thatthe most recent settlement price is 93� 08, or 93.25. Which bond ischeapest to deliver?

Bond Quoted bond price Conversion factor1 99.50 1.03822 143.50 1.51883 119.75 1.2615



Pricing

An exact theoretical futures price for the Treasury Bond contract isdi¢ cult to determine.

If the cheapest-to-deliver bond and the delivery date are known, thenthe futures contract is like a futures contract on a security providingthe holder with known income.

The futures price F0 relates to the spot price S0 by

F0 = (S0 � I ) erT

I is the present value of the coupons during the life of the futurescontract, T is the time until the future contract matures, and r is therisk-free interest rate applicable to a time period of length T .



Pricing - Example

Suppose that the cheapest-to-deliver bond is a 12% coupon bondwith conversion factor 1.400. The delivery will take place in 270 days.Coupons are paid semiannually on the bond.

The last coupon date was 60 days ago, the next coupon date is in 122days, and the coupon date thereafter is in 305 days.

The term structure is �at and the continuously compounded rate ofinterest is 10% per annum.

The current quoted bond price is $120.

What is the quoted futures price?



Eurodollar Futures

The most popular interest rate futures contract in the US is the3-month Eurodollar futures contract traded on the ChicagoMercantile Exchange (CME).

A Eurodollar is a dollar deposited in a US or foreign bank outside theUS.

The Eurodollar interest rate is the rate of interest earned onEurodollars deposited by one bank with another bank.

Eurodollar futures contracts allow an investor to lock in an interestrate on $1 million for a future 3-month period.

The 3-month period to which the interest rate applies starts on thethird Wednesday of the delivery month.

Delivery months are March, June, September, and December for upto 10 years into the future.



Eurodollar Futures

Suppose a March 2005 Eurodollar futures contract has a settlementprice of 97.63. The contract ends on the third Wednesday of thedelivery month, Wednesday, March 16.

On March 16, 2005, the settlement price is set equal to 100� R,where R is the actual 3-month Eurodollar interest rate on that day,expressed with quarterly compounding and an actual/360 day countconvention.

Contracts are designed so that a 1 basis point move in the futuresquote corresponds to a gain or loss of $25 per contract.



Eurodollar Futures - Example

On February 4, 2004, an investor wants to lock in the interest ratethat will be earned on $5 million for 3 months starting on March 16,2005. The investor goes long �ve March05 Eurodollar future contractsat 97.63. On March 16, 2005, the 3-month LIBOR interest rate is2%, so that the �nal settlement price proves to be 98.00 (100� R).The investor gains 5� 25� (9800� 9763) = $4625 on the longfutures position.

The interest earned on the $5 million for 3 months at 2% is5000000� 0.25� 0.02 = $25000.The gain on the futures contract is therefore $29625. This is theinterest that would have been earned if the interest rate had been2.37%, i.e. locking in an interest rate equal to 2.37%.



Duration-based hedging strategies

Suppose that you want to hedge a bond portfolio or a money marketsecurity using an interest rate futures contract.

FC : price for the interest rate futures contractDF : is the duration of the underlying asset at the maturity of thefutures contract.

P : is the forward value of the portfolio being hedged at the maturityof the hedge.

DP : is the duration of the portfolio at the maturity of the hedge.The number of contracts required to hedge against an uncertain 4yis

N� =PDPFCDF



Duration-based hedging strategies - Example

It is August 2 and a fund manager with $10 million invested ingovernment bonds is concerned that interest rates are expected to behighly volatile over the next three months. The fund manager decidesto use the December T-bond futures contracts to hedge the value ofthe portfolio. The current futures price is 93� 02, and each contractis for the delivery of $100000 face value of bonds.Suppose the duration of the bond portfolio in 3 months will be 6.80years. The cheapest-to-deliver bond in the T-bond contract isexpected to be a 2-year 12% per annum coupon bond. The yield onthis bond is currently 8.80% per annum, and the duration will be 9.20years at maturity of the futures contract.

Should the fund manager go long or short the futures contracts?

How many contracts?


Swaps De�nition

Swaps - De�nition

A swap is an agreement to exchange cash �ows at speci�ed futuretimes according to speci�ed rules.

Usually, the calculation of the cash �ows involves the future value ofan interest rate, an exchange rate, or other market variables.

A forward contract can be viewed as a simple example of a swap.

Swaps typically lead to cash �ow exchanges taking place on severalfuture dates.

The most common type of swap is a "plain vanilla" interest rate swap.


Swaps De�nition

De�nition

A company agrees to pay cash �ows equal to interest at apredetermined �xed rate on a notional principal for a number of years.

In return, it receives interest at a �oating rate on the same notionalprincipal for the same period of time.

A swap can be seen as the exchange of a �xed-rate bond for a�oating-rate bond, or vice versa.

The �oating rate in most interest rate swap agreements is the LIBOR.

For instance, a 5-year bond may have a speci�ed rate of interest of6-month LIBOR plus 0.5% per annum.

The interest on a �oating-rate bond is generally set at the beginningof the period to which it will apply.


Swaps Example

Example of Interest Rate Swap

Consider a 3-year swap initiated on March 5, 2004, between Microsoftand Intel.

Microsoft agrees to pay to Intel an interest rate of 5% per annum ona notional principal of $100 million.

Intel agrees to pay to Microsoft the 6-month LIBOR rate on the samenotional principal.

Microsoft is the �xed-rate payer.

Intel is the �oating-rate payer.

Payments are exchanged every 6 months and the 5% interest rate isquoted with semiannual compounding.


Swaps Example


Millions of DollarsLIBOR FLOATING FIXED Net

Date Rate Cash Flow Cash Flow Cash FlowMar.5, 2004 4.2%

Sept. 5, 2004 4.8% +2.10 –2.50 –0.40Mar.5, 2005 5.3% +2.40 –2.50 –0.10

Sept. 5, 2005 5.5% +2.65 –2.50 +0.15Mar.5, 2006 5.6% +2.75 –2.50 +0.25

Sept. 5, 2006 5.9% +2.80 –2.50 +0.30Mar.5, 2007 6.4% +2.95 –2.50 +0.45

Millions of DollarsLIBOR FLOATING FIXED Net

Date Rate Cash Flow Cash Flow Cash FlowMar.5, 2004 4.2%

Sept. 5, 2004 4.8% +2.10 –2.50 –0.40Mar.5, 2005 5.3% +2.40 –2.50 –0.10

Sept. 5, 2005 5.5% +2.65 –2.50 +0.15Mar.5, 2006 5.6% +2.75 –2.50 +0.25

Sept. 5, 2006 5.9% +2.80 –2.50 +0.30Mar.5, 2007 6.4% +2.95 –2.50 +0.45


Swaps Example


Convert a liability from �oating-rate to �xed-rate.I Microsoft has arranged to borrow USD 100 Mio. at LIBOR plus 10basis points.

Microsoft now pays LIBOR plus 0.1% to outside lenders, receivesLIBOR from swap, and pays 5% under terms of swap.

These three sets of cash �ows net out to an interest rate payment of5.1%.


Swaps Example


Convert a liability from �xed-rate to �oating-rate.I Intel has a 3-year USD 100 Mio. loan outstanding on which it pays5.2%.

Intel now pays 5.2% to outside lenders, pays LIBOR under terms ofswap, and receives 5% from swap.

These three sets of cash �ows net out to an interest rate payment ofLIBOR plus 0.2%.


Swaps Swaps

Financial Intermediary

Usually, two non-�nancial companies such as Intel and Microsoft donot get in touch directly to arrange a swap.

They each deal with a �nancial intermediary such as a bank orother �nancial institution.

Swaps are structured in such a way so that the �nancial institutionearns about 3 or 4 basis points on a pair of o¤setting transactions.

In practice, unlikely that two companies contact �nancial institutionat the same time and want to take exactly opposite swap positions.

Large �nancial institutions act as market maker.

Enter swaps without having an o¤setting swap with another counterparty and hedge the risk.


Swaps Swaps

Swap - An illustration

LIBOR+0.1%Intel Financial

InstitutionMicrosoft

LIBOR LIBOR

4.985% 5.015%5.2%

LIBOR+0.1%Intel Financial

InstitutionMicrosoft

LIBOR LIBOR

4.985% 5.015%5.2%


Swaps Swaps

Market Makers

In practice, it is unlikely that two companies contact �nancialinstitution at the same time and want to take exactly opposite swappositions.

Large �nancial institutions act as market maker.

They enter swaps without having an o¤setting swap with anothercounter party.

Market makers must quantify and hedge the risk.


Swaps Swaps

Market Makers

This table shows bid and o¤er �xed rates in the swap market andswap rates (percent per annum).

Maturity (years) Bid O¤er Swap rate2 6.03 6.06 6.0453 6.21 6.24 6.2254 6.35 6.39 6.3705 6.47 6.51 6.4907 6.65 6.68 6.66510 6.83 6.87 6.850


Swaps Swaps

Why Swaps?

Suppose that two companies, AAACorp and BBBCorp, both wish toborrow $10 million for 5 years and have been o¤ered these rates:

Fixed FloatingAAACorp 4.0% 6�monthLIBOR + 0.3%BBBCorp 5.2% 6�monthLIBOR + 1.0%

Assume that BBBCorp wants to borrow at a �xed rate of interest,and AAACorp wants to borrow at a �oating rate of interest linked tothe 6-month LIBOR.

Because BBBCorp has a worse credit rating, it pays higher interest in�xed and �oating markets.

Suppose a �nancial institution acting as a intermediary earns a spreadof 4 basis points per year.


Swaps Swaps

Swap Valuation

Value of interest rate swap is zero, or close to zero, when �rstinitiated.

After in existence for some time, its value may be positive or negative.

From the point of view of the �oating-rate payer, a swap can beregarded as a long position in a �xed-rate bond and a short positionin a �oating-rate bond:

Vswap = B�x � B�

Vswap is the value of the swap, B� is the value of the �oating-ratebond, B�x is the value of the �xed-rate bond.

Conversely, from the point of view of �xed-rate payer, a swap is along position in a �oating-rate bond and a short position in a �xedrate bond:

Vswap = B� � B�x


Interest Rate Derivatives Introduction

Interest Rate Derivatives

Interest rate derivatives are instruments whose payo¤s depend insome way on the level of the interest rate.

The volume of trading in interest rate derivatives has increasedsigni�cantly over the past 20 to 30 years.

Interest rate derivatives are more di¢ cult to value than equityderivatives:

I Behavior of interest rate more complicated than that of a stock price oran exchange rate

I For valuation purposes, it may be necessary to develop a modeldescribing the behavior of the entire zero-coupon yield curve.

I The volatilities of di¤erent points on the yield curve are di¤erent.I Interest rates are used for discounting as well as for de�ning the payo¤from the derivative.


Interest Rate Derivatives Black�s Model

Black�s Model

Consider a European call option on a variable whose value is V .

De�ne:I T : Time to maturity of the optionI F : Forward price of V for a contract with maturity TI F0 : Value of F at time zeroI K : Strike price of the optionI P (t,T ) : Price at time t of a zero-coupon bond paying $1 at time TI VT : Value of V at time TI σ : Volatility of F



Black�s Model

Assume ln(VT ) is normal with mean F0 and standard deviation σpT .

Discount the expected payo¤ at the T�year rate.The payo¤ from the option at time T is: max(VT �K , 0).The expected payo¤ from the option is: E (VT )N (d1)�KN (d2) .E (VT ) is the expected value of VT and

d1 =ln�E (VT )K

�+ σ2T

2

σpT

and d2 = d1 � σpT



Black�s Model

Assume E (VT ) = F0. Then the values of call and put options are:

c = P (0,T ) [F0N (d1)�KN (d2)]

p = P (0,T ) [KN (�d2)� F0N (�d1)]

d1 =ln�F0K

�+ σ2T

2

σpT



Interest Rate Derivatives Bond Options

Bond Options

A bond option is an option to buy or sell a particular bond by aparticular date for a particular price.

Many bond options trade in the over-the-counter market and have aEuropean options character.

Bond options are also frequently embedded in bonds when they areissued to make them more attractive to either the issuer or potentialpurchaser.

Callable bond.

Puttable bond.



Bond Options

Set F0 equal to the forward bond price FB . Set σ equal to the forwardbond price volatility σB .

B0 is the bond price at time zero and I is the present value of thecoupons that will be paid during the life of the option.

The equations for pricing a European bond option are:

c = P (0,T ) [FBN (d1)�KN (d2)]

p = P (0,T ) [KN (�d2)� FBN (�d1)]

d1 =ln�FBK

�+

σ2BT2

σBpT

and d2 = d1 � σBpT

FB =B0 � IP (0,T )



Bond Options - Example

Consider a 10�month European call option on a 9.75�year bondwith a face value of $1000. Suppose that the current cash bond priceis $960, the strike price is $1000, the 10-month risk-free interest rateis 10% per annum, and the volatility of the forward bond price in 10months is 9% per annum.

The coupon is 10%, paid semiannually, and coupon payments of $50are expected in 3 months and 9 months.

Suppose that the 3-month and 9-month risk-free rates are 9.0% and9.5% per annum, respectively.

What is the price of the call option if the strike price is the cash pricethat would be paid for the bond on exercise?

What�s the price if the strike price is the quoted price?



Yield Volatilities

The volatilities quoted for bond options are often yield volatilitiesrather than price volatilities.

The market uses the duration concept to convert a quoted yieldvolatility into a price volatility.

The volatility of the forward bond price σB can be approximatelyrelated to the volatility of the forward bond yield σy by

σB = D�y0σy

D� is the modi�ed duration of the bond, and y0 is the initial value ofyF .


Interest Rate Derivatives Interest Rate Caps and Floors

Interest Rate Caps

Consider a �oating-rate note where the interest is reset periodicallyequal to LIBOR. The time between resets is known as tenor.Suppose it is 3 month.

The interest rate on the note for the �rst 3 months is the initial3-month LIBOR rate; the interest rate for the next 3 months is setequal to the 3-month LIBOR rate prevailing in the market at the3-month point; and so on.

An interest rate cap provides insurance against the rate of interest onthe �oating-rate note rising above a certain level. This level is thecap rate.



Interest Rate Caps

Assume a principal amount of $10 million, the tenor is 3 months, thelife of the cap is 3 years, and the cap rate is 4%.

Suppose that on a particular reset date the 3-month LIBOR interestrate is 5%.

The cap provides a payo¤ of $25000 in this case.



Interest Rate Caps

Consider a cap with a total life of T , a principal of L, and a cap rateof RK .

Suppose that the reset dates are t1, t2, ..., tn+1 = T . De�ne Rk as theinterest rate for the period between time tk and tk+1 observed at timetk (1 6 k 6 n).The payo¤ of the cap at time tk+1 (k = 1, 2, ...n) is

Lδk max (Rk � RK , 0)

δk = tk+1 � tk . Rk and RK are expressed with a compoundingfrequency equal to the frequency of resets.

This equation is a call option on the LIBOR rate observed at time tkwith the payo¤ occurring at time tk+1.

The cap is a portfolio of n such options. The options underlying thecap are caplets.



Interest Rate Caps

If the rate Rk is assumed to be lognormal with volatility σk , the valueof a caplet and �oorlet are

caplet = LδkP (0, tk+1) [FkN (d1)� RKN (d2)]

�oorlet = LδkP (0, tk+1) [RKN (�d2)� FkN (�d1)]

d1 =ln�FkRK

�+

σ2k tk2

σkptk

and d2 = d1 � σkptk



Interest Rate Caps - Example

Consider a contract that caps the LIBOR interest rate on $10000 at8% per annum (with quarterly compounding) for 3 months starting in1 year.

Suppose that the LIBOR curve is �at at 7% per annum with quarterlycompounding and the volatility of the 3-month forward rateunderlying the caplet is 20% per annum.

The continuously compounded zero rate for all maturities is 6.9394%.

What is the price of the caplet?


Interest Rate Derivatives Swap Options

Swaptions

Swap options, or swaptions, are options on interest rate swaps and areanother popular type of interest rate option.

They give the holder the right to enter into a certain interest rateswap at a certain time in the future.

Consider a company that knows that in 6 months it will enter into a5-year �oating-rate loan agreement and knows that it will wish toswap the �oating interest payments for �xed interest payments.

At a cost, the company could enter into a swaption giving it the rightto receive 6-month LIBOR and pay a certain �xed rate of interest (say8%) for a �ve year period starting in 6 months.

Swaptions provide companies with a guarantee that the �xed rate ofinterest will not exceed some level.



Swaptions

Suppose that there are m payments per year under the swap and thatthe notional principal is L.

The value of a swaption that gives the holder the right to pay a �xedrate of sK is

swaption = LA [s0N (d1)� sKN (d2)]

A =1m

mn

∑i=1P (0,Ti )

d1 =ln�s0sK

�+ σ2T

2

σpT


A de�nes the value of a contract that pays 1m at times

Ti (1 6 i 6 mn), s0 is the forward swap rate at time zero, and σ isthe volatility of the forward swap rate.



Swaptions - Example

Suppose that the LIBOR yield curve is �at at 6% per annum withcontinuous compounding.

Consider a swaption that gives the holder the right to pay 6.2% in a3-year swap starting in 5 years.

The volatility of the forward swap rate is 20%.

Payments are made semiannually and the principal is $100.

What is the value of the swaption?


Investments

Session 12. Credit RiskEPFL - Master in Financial Engineering

Philip Valta

Spring 2010

Credit Risk (Session 12) Investments Spring 2010 1 / 44


Credit risk:I Default probabilitiesI Recovery rates

Extract default probabilities from bond prices

Contingent claim models:I Merton modelI KMV modelI Leland model


Credit Risk Introduction

Credit Risk

Credit risk arises from the possibility that borrowers andcounterparties in transactions may default.

Credit rating agencies (S&P, Moody�s, and Fitch) provide ratingsdescribing the creditworthiness of corporate (sovereign) bonds.

For instance, on April 27 2010, S&P reduced its credit rating forGreece to BB+, junk status.


Credit Risk Default Probabilities

Historical default probabilities - Moody�s , 1970-2003

Term 1 2 3 4 5 7 10 20Aaa 0.00 0.00 0.00 0.04 0.12 0.29 0.62 1.55Aa 0.02 0.03 0.06 0.15 0.24 0.43 0.68 2.70A 0.02 0.09 0.23 0.38 0.54 0.91 1.59 5.24Baa 0.20 0.57 1.03 1.62 2.16 3.24 5.10 12.59Ba 1.26 3.48 6.00 8.59 11.17 15.44 21.01 38.56B 6.21 13.76 20.65 26.66 31.99 40.79 50.02 60.73Caa 23.65 37.20 48.02 55.56 60.83 69.36 77.91 80.23


Credit Risk Default Probabilities

Credit Risk

The previous table shows the default experience through time ofcompanies that started with a certain credit rating.

For example, a bond issue with an initial credit rating of Baa has a0.20% change of defaulting by the end of the �rst year, a 0.57%chance of defaulting by the end of the second year, and so on.

The probability that a bond initially rated Baa will default during thesecond year of its life is 0.57� 0.20 = 0.37%.For investment grade bonds, the probability of default in a year tendsto be an increasing function of time.For bonds with poor credit ratings, the probability of default is oftena decreasing function of time.


Credit Risk Default Intensities

Default Intensities

The probability of a Caa bond defaulting during the third year is48.02� 37.20 = 10.82%.This is the bond�s unconditional default probability (as seen attime 0).

The probability that the Caa-rated bond will survive until the end ofyear 2 is 100� 37.20 = 62.80%.The probability that it will default during the third year conditionalon no earlier default is 0.1082/0.6280=17.23%.

This is the bond�s default intensity or hazard rate.



Intensity Modeling

Models of default probabilities and timing are often based on thenotion of an arrival intensity of default.

The simplest version of such a model de�nes default as the �rstarrival time τ of a Poisson process with some constant mean arrivalrate, called intensity, often denoted λ.

In this setup we have:I The probability of survival for t years is p(t) = e�λt .I The probability of default by time t is Q (t) = 1� e�λt .I The expected time to default is 1/λ.

Example: for a constant default intensity λ = 0.04 the probability ofdefault within 1 year is 3.9% and the expected default time is 25years.



Intensity Modeling

The classic Poisson-arrival model is based on the notion ofindependence of arrival risk over time.

It is implausible to assume that the default intensity λ is constantover time ! simple extension of the Poisson process is to allow fordeterministically time-varying intensities.

In such a model, the probability of survival at time t given survival attime t � 1 is:

p(t � 1, t) = e�λ(t)

The probability of survival at time t seen from time 0 is then:

p(t) = e�[λ(1)+...+λ(t)]

Deterministic variation in intensities implies that the only informationrelevant to default risk that arises over time is the mere fact ofsurvival until that date.


Credit Risk Recovery Rates

Historical Recovery Rates on Corporate Bonds - Moody�s,1981-2003

The recovery rate for a bond is normally de�ned as the bond�s marketvalue immediately after a default, as a percent of its face value.

Debt Class Average Recovery Rate (%)Senior secured 51.6Senior unsecured 36.1Senior subordinated 32.5Subordinated 31.1Junior subordinated 24.5


Extract Default Probabilities from Bond Prices Default Probabilities

Default Probabilities from Bond Prices - Approximation

The probability of default can be estimated from the prices of bondsit has issued.

Assumption: the only reason a corporate bond sells for less than asimilar risk-free bond is the possibility of default (what aboutliquidity?).

The probability of a default per year conditional on no earlier default is

h =s

1� R

h is the default intensity per year, s is the spread of the corporatebond yield over the risk-free rate, and R is the expected recovery rate.

Example: Consider a bond yields 200 basis points above a similarrisk-free bond and that the expected recovery rate in default is 40%.



Default Probabilities from Bond Prices

Consider a bond that lasts 5 years, provides an annual coupon of 6%(paid semiannually), and has a yield of 7% (continuouscompounding). The price is 95.34.

Consider a similar risk-free bond with yield 5% (continuouscompounding). The price is 104.09.

The expected loss from default over the 5-year life of the bond is104.09� 95.34 = $8.75.Suppose the probability of default per year is Q (constant). Therisk-free rate is 5% for all maturities.

Default can happen at times 0.5, 1.5, 2.5, 3.5, and 4.5 years(immediately before coupon payment dates).

What is the probability of default Q per year?




Time Recov Rf value ($) LGD ($) Disc. factor PV exp. loss ($)0.5 40 106.73 66.73 0.9753 65.08Q1.5 40 105.97 65.97 0.9277 61.20Q2.5 40 105.17 65.17 0.8825 57.52Q3.5 40 104.34 64.34 0.8395 54.01Q4.5 40 103.46 63.46 0.7985 50.67QTotal 288.48Q




Assume that defaults take place more frequently.

Assume a constant default intensity or a particular pattern for thevariation of default probabilities with time.

Use bonds with various maturities to estimate several parametersdescribing the term structure of default probabilities (bootstrapprocedure).


Contingent Claim Models The Merton Model

Contingent Claim Models

Instead of relying on the company�s credit rating, use equity prices toestimate the probability of default.

Contingent claim models allow to value the components of a �rm�sliability mix.

Assume certain �rm value dynamics and consider equity and debt ascontingent claims.

Based on option pricing techniques.

Seminal work by Black and Scholes (1973) and Merton (1974).

These models provide an underlying economic context for the event ofdefault.



The Merton Model: Assumptions

Markets are frictionless ! no transaction costs, no taxes, nobankruptcy costs.

Agents are price takers ! trading in assets has no e¤ect on prices.

Borrowing and lending can be done at the same riskless rate r andthe money-market account evolves as:

dBt = rBtdt

The value of �rm�s assets Vt follows a Geometric Brownian Motion:

dVt = µVtdt + σVtdWt

µ is the instantaneous expected rate of return, σ is the volatility ofthe underlying assets, and Wt is a standard Wiener process.



The Merton Model: Assumptions

The value of the �rm�s assets is �nanced by equity, E , and onerepresentative zero-coupon noncallable debt contract, D, maturing attime T with face value F :

Vt = Et +Dt

Managers act to maximize shareholder wealth (no agency con�ict).

The debt contract is �xed with the initial hypothesis that the �rm isnot already at default.

There are neither payouts, nor issues of any type of security duringthe life of the debt contract ! default can only happen at maturity.

The absolute priority rule is enforced.

Remark: we are in the Modigliani-Miller framework ! the value ofthe �rm does not depend on the capital structure!



The Merton Model: Payo¤s

The �rm has thus two classes of securities: equity and a singlehomogenous class of zero-coupon discount bonds.

Default is simply de�ned as the event when the face value payment isnot met. This yields the following payo¤s:

ST = max(VT � F , 0)DT = min(F ,VT ) = F �max(F � VT , 0)

The value of equity is a call option with strike price the face value ofdebt.

The value of debt is a portfolio of risk-free debt and a short positionin a put with strike price the face value of debt.



The Merton Model

The value of equity is given by the fundamental PDE:

0 = ∂E (V )∂t + 1

2σ2V 2 ∂2E (V )∂V 2 + rV ∂E (V )

∂V � rE (V )

subject to boundary conditions

Et (V )/V � 1Et (0) = 0ET (V ) = max(VT � F , 0)



The Merton Model

The solution of the PDE is the Black, Scholes, and Merton�s result:

Et (V ,T , σ, r ,F ) = VtN(d1)� Fe�r (T�t)N(d2)

where N(.) is the cumulative normal distribution function and

d1 =ln(Vt/F )+(r+σ2/2)(T�t)

σpT�t

d2 = d1 � σpT � t



The Merton Model

The value of debt is given by the fundamental PDE:

0 = ∂D (V )∂t + 1

2σ2V 2 ∂2D (V )∂V 2 + rV ∂D (V )

∂V � rD(V )

subject to

Dt (V )/V � 1Dt (0) = 0DT (V ) = min(VT ,F )



The Merton Model

The value of debt is given by:

Dt (V ) = Vt � Et= Vt � VtN(d1) + Fe�r (T�t)N(d2)= VtN(�d1) + Fe�r (T�t)N(d2)

or alternatively:

Dt (V ) = Fe�r (T�t) � European Put= Fe�r (T�t) � [�VtN(�d1) + Fe�r (T�t)N(�d2)]= VtN(�d1) + Fe�r (T�t)N(d2)



The Merton Model

In this setup, we can obtain analytical expressions for the yield tomaturity, the credit spread, the default probability, and the discountedexpected recovery value.

The yield to maturity, y , of a discount bond in a continuous timeframework is given by:

Dt (V ) = Fe�y (T�t) ! yt = � 1T�t ln(Dt/F )

The credit spread, cst = yt � r , is given by:

cst = � 1T�t ln (Dt/F )� r



The Merton ModelThe risk neutral default probability corresponds to N(�d2).The value of debt can be re-expressed as:

Dt (V ) = Fe�r (T�t) �N(�d2)hFe�r (T�t) � N (�d1)

N (�d2)Vti

where the fraction N(�d1)/N(�d2) is the expected discountedrecovery rate and h

Fe�r (T�t) � N (�d1)N (�d2)Vt

iis the expected discounted shortfall.The expression for debt can be reinterpreted as

Dt (V ) = Fe�r (T�t) �DP � EDLGD

where DP and EDLGD stand for the default probability and theexpected discounted loss given default.



Subordinated Debt

Suppose the �rm has two debt issues outstanding with equal maturityT . The �rst debt contract is a senior debt with face value FS . Thesecond debt contract is a junior debt with face value FJ .

At the maturity date, the payo¤s of the �rm�s securities are:

Claim VT < FS FS � VT � FS + FJ VT > FS + FJSenior Debt VT FS FSJunior Debt 0 VT � FS FJEquity 0 0 VT � FS � FJ

In addition the total cash �ow to debtholders is VT if VT � FS + FJand FS + FJ otherwise.



Subordinated Debt

We know that the value of senior debt is:

DS = VtN(�d1(FS )) + FS e�r (T�t)N(d2(FS ))

with

d1(FS ) =ln(Vt/FS )+(r+σ2/2)(T�t)

σpT�t , d2(FS ) = d1 � σ

pT � t

Using the total cash �ow to debtholders, the total value of debt is:

D = VtN(�d1(FS + FJ )) + (FS + FJ ) e�r (T�t)N(d2(FS + FJ ))

Then the value of junior debt is DJ = D �DS .



Subordinated Debt

Subordinated debt has many characteristics which are quite di¤erentfrom those normally associated with bonds.

While senior bonds are always a concave function of V , the juniorbonds are initially a convex function of V , becoming a concavefunction for larger values of V .

Unlike senior debt, junior debt can be an increasing function ofvolatility and debtholders as a group may thus have con�ictinginterests.



The Merton Model

The Merton model can be extended in several directions:I Stochastic interest ratesI Jumps in asset valueI Discrete coupon paymentsI Continuous coupon paymentsI Default barriersI Perpetual debtI Dynamic capital structure


Contingent Claim Models The KMV Model

The KMV Model

The KMV model is based on the dynamics of the asset value of theissuer



The KMV Model

The KMV model is best applied to large publicly traded companies,where the value of equity is determined by the stock market.

The derivation of actual probabilities of default proceeds in threestages:

1 Estimation of the market value and volatility of �rm�s assets.2 Calculation of the distance-to-default which measures the default risk.3 Mapping of the distance-to-default to actual probabilities of defaultusing a default database.



The KMV Model

The process Vt and the volatility σ are unobserved.

We can recover these values based on market data for equity.

Assume that the �rm�s equity has the following dynamics:

dEt = rEtdt + σEEtdWt

The volatility of equity and the volatility of assets are related by:

σE = σV∂Et∂Vt

VtEt

Also, the theoretical and observed values of equity should match:

Et (Vt ) = Et

We obtain thus two equations for the two unknowns, Vt and σ.



The KMV Model

Using a large dataset, KMV observes that �rms default when theasset value reaches a level that is between the value of total liabilitiesand the value of short term debt.

The default point (DPT ) is therefore set to be the sum of short termdebt and half of long term debt.

The distance to default is de�ned as the number of standarddeviations between the mean of the distribution of the asset valueand the default point:

DD = E (V )�DPTσ

Finally, the DD is mapped to an actual default probability based onhistorical information.



The KMV Model

Consider the following information

Current market value of assets V = 1000Expected growth of asset per annum 20%Expected asset value in one year V � 1.2 = 1200Annualized asset volatility 50%Default point 1000

We can then compute that DD = (1200� 1000)/50 = 4Assume that among a population of 5000 �rms with DD = 4 at onepoint in time, 20 defaulted one year later. Then we have:

EDF = 20/5000 = 0.4%


Contingent Claim Models The Leland Model

The Leland ModelLeland (1994) proposes a structural model with the followingassumptions:

I Assets are continuously traded in arbitrage-free and complete markets.I The default-free term structure is �at with an instantaneous risklessrate r at which investors may lend and borrow freely.

I Management acts in the best interest of shareholders.I Debt is perpetual with coupon payment c .I Corporate taxes are paid at a rate τ on operating income.I There are proportional liquidation costs α of the value of remainingassets.

I Absolute priority is enforced.

The value of the �rm, under the risk-neutral measure Q, evolvesaccording to the process:

dVt = (r � δ)Vtdt + σVtdWt

Here, δ is the �rm�s payout rate, σ is the constant volatility of assetreturns, and Wt is a standard Wiener process.



The Leland Model

Consider a claim that pays a non negative coupon c per instant whenthe �rm is solvent and denote its value by F (V , t).

When the �rm �nances the net cost of the coupon by issuingadditional equity, the value of this asset satis�es:

12σ2V 2FVV (V , t) + (r � δ)VFV (V , t) + Ft (V , t) + c = rF (V , t)

Here, the boundary conditions are determined by payments atmaturity and by payments in bankruptcy, if bankruptcy occurs beforematurity.

When the security does not have an explicit time dependency, wehave:

12σ2V 2FVV (V ) + (r � δ)VFV (V ) + c = rF (V )



The Leland Model

The general solution to the PDE is given by:

F (V ) = AV β+ + BV β� + cr

where A and B are constant and β+ and β� are the positive andnegative solutions to the quadratic equation

12σ2β(β� 1) + (r � δ)β� r = 0

Finally, we obtain:

β� =12 �

r�δσ2�q� 1

2 �r�δσ2

�2+ 2r

σ2

and A and B are determined by boundary conditions.



The Leland Model

Debt promises a perpetual coupon payment c , whose level remainsconstant unless the �rm declares bankruptcy.

De�ne the value of debt by D(V ).

Let VB denote the level of asset value at which bankruptcy isdeclared.

At default, debtholders get (1� α)VB .

The boundary conditions for the debt value are as follows:

limV!∞ D(V ) = cr

D(V )jV=VB = (1� α)VB



The Leland Model

Using these boundary conditions we get:

A = 0

B =�(1� α)VB � c

r

� � 1VB

�β�

Thus the value of debt is given by:

D(V ) = cr +

�(1� α)VB � c

r

� � VVB

�β�



The Leland Model

Debt issuance a¤ects �rm value in two waysI It reduces �rm value because of bankruptcy costsI It increases �rm value due to the tax deductibility of interest debtpayments

We want to solve for the value of bankruptcy costs. Bankruptcy costsare αVB at V = VB . Denote by BC (V ) the current value. Thecurrent value satis�es:

12σ2V 2BCVV (V ) + (r � δ)VBCV (V ) = rBC (V )

The boundary conditions are:

limV!∞ BC (V ) = 0

BC (V )jV=VB = αVB



The Leland Model

Solving for the value of bankruptcy costs yields:

BC (V ) = αVB�VVB

�β�

Tax bene�ts resemble a security that pays a constant coupon equal tothe tax-sheltering value of interest payments (τc) as long as the �rmis solvent and nothing in bankruptcy. Using the same methodology weget:

TS(V ) = τcr

�1�

�VVB

�β��



The Leland Model

The total value of the �rm, v(V ) re�ects the three terms: the �rm�sasset value, the tax bene�ts of debt, and the bankruptcy costs. Weobtain:

v(V ) = V + τcr

�1�

�VVB

�β�� αVB

�VVB

�β�

The value of equity is the total value of the �rm minus the value ofdebt:

E (V ) = V � (1�τ)cr

�1�

�VVB

�β�� VB

�VVB

�β�



The Leland Model

If the �rm is not otherwise constrained by covenants, bankruptcyoccurs only when the �rm cannot meet the required coupon paymentsby issuing additional equity. This is the case when the equity valuefalls to zero.

When VB can be chosen by shareholders and the priority rule isenforced, the threshold selected by shareholders satis�es:

∂E (V )∂V

��V=VB

= 0

Finally, the solution is:

VB =β�

β��1(1�τ)cr



The Leland Model

The asset value at which bankruptcy occursI Is proportional to the coupon cI Is independent of the current asset value VI Decreases as the corporate tax rate increasesI Is independent of bankruptcy costsI Decreases with σ

Plugging this expression into the expression for �rm value and solvingfor the value maximizing c yields:

c� = rVβ��1

β�

h1� β�

1�(1�τ)(1�α)τ

i1/β�



The Leland Model

Some quantities of interest:

yield spread: y = c/D � rleverage: L = D(V )/v(V )

A base case scenario with V = 100, r = 6%, δ = 5.5%, σ = 30%,τ = 15%, α = 40% generates the following results:

leverage ratio: L(V ) = 56.80%yield spread: y(V ) = 102 basis points

Model shortcomings:I Leverage is too high compared with the leverage ratio of the medianS&P500 �rm (which is around 20%)

I Credit spreads are too low given the high leverage ratios predicted bythe model



The Leland Model: Extensions

The Leland model can be extended in order to address the empiricalshortcomings in various directions:

1 Strategic default2 Finite debt maturity3 Dynamic capital structure4 Agency con�icts5 Imperfect accounting information6 etc.


Investments

Session 13. Credit DerivativesEPFL - Master in Financial Engineering

Philip Valta

Spring 2010

Credit Derivatives (Session 13) Investments Spring 2010 1 / 31


Credit default swaps

Total return swaps

Asset backed securities


Credit Derivatives Introduction

Credit Derivatives

Credit derivatives are contracts where the payo¤ depends on thecredit-worthiness of one or more companies or countries.

Credit derivatives allow companies to trade credit risks in much thesame way that they trade market risks.

Allow active management of portfolios of credit risk.

Banks have been the biggest buyers of credit protection and insurancecompanies have been the biggest sellers.



Credit Derivatives

Credit derivatives can be categorized as:I Single name: credit default swap or CDSI Multiname: collateralized debt obligation or CDO

In July 2007, investors lost con�dence in the subprime mortgagemarket in the US: the interest in multiname structured productsdeclined but single name credit derivatives continue to be activelytraded.



Who Bears the Credit Risk?

Traditionally banks have been in the business of making loans andthen bearing the credit risk that the borrower will default.

Banks have been reluctant to keep loans on their balance sheets.

E¤ect of capital requirements and regulation.

During the 1990s banks created asset-backed securities to pass loanson to investors.

Banks have been net buyers of credit protection while insurancecompanies have been net sellers: insurance companies are notregulated in the same way as banks are.


Credit Derivatives Credit Default Swaps

Credit Default Swaps

The most popular credit derivative is a credit default swap (CDS).This contract provides insurance against the risk of a default by aparticular company.

The buyer of the insurance obtains the right to sell bonds issued bythe company for their face value when a credit event occurs.

The seller of the insurance agrees to buy the bonds for their facevalue when a credit event occurs.

The terms of the contract include:I The reference entityI Credit eventI Notional principalI Payments are made in arrearsI There is either physical delivery or cash payment



Credit Default SwapsExample: two parties enter into a 5 year credit default swap onMarch 1, 2010 with notional amount $100 million. The buyer agreesto pay 90 basis points annually for protection against default by thereference entity.

If default occurs, and the reference bond is worth $35 per $100 offace value, the payment would be $65.

DefaultProtectionBuyer, A

DefaultProtectionSeller, B

90 bps per year

Payoff if there is a default byreference entity=100(1R)

DefaultProtectionBuyer, A

DefaultProtectionSeller, B

90 bps per year

Payoff if there is a default byreference entity=100(1R)

The total amount paid per year, as a percent of the notional principal,to buy protection is known as the CDS spread.




A key aspect of a CDS is the de�nition of default.

In contracts on European reference entities restructuring is typicallyincluded as a credit event, whereas in contracts on North Americanreference entities it is not.

CDS spreads vs bond yields.

The cheapest to deliver bond.



Credit Default Swaps: Valuation

CDS spreads on individual reference entities can be calculated fromprobability default estimates.

Suppose that the probability of a reference entity defaulting during ayear conditional on no earlier default is 2%

The probability of default during the �rst year is 0.02 and theprobability that the reference entity will survive until the end of the�rst year is 0.98

The probability of default during the second year is0.02� 0.98 = 0.0196 and the probability of survival until the end ofthe second year is 0.98� 0.98 = 0.9604The probability of default during the third year is0.02� 0.9604 = 0.0192, and so on...




Time Default probability Survival probability1 0.0200 0.98002 0.0196 0.96043 0.0192 0.94124 0.0188 0.92245 0.0184 0.9039




We will assume that default always happens halfway through a yearand that payments on the credit default swap are made once a year,at the end of each year.

Assume that the risk-free (LIBOR) interest rate is 5% per annumwith continuous compounding and the recovery rate is 40%.

There are three parts to the calculation:I Calculation of the present value of expected payments.I Calculation of the present value of expected payo¤.I Calculation of the present value of accrual payment.




This table shows the calculation of the present value of the expectedpayment made on the CDS assuming that payments are made at therate s per year and the notional principal is $1.For example, there is a 0.9412 probability that the third payment of sis made and the present value of the expected payment is0.9412� s � e�0.05�3 = 0.8101s

TimeProba.survival

Expectedpayment

Discountfactor

PV of exp.payment

1 0.9800 0.9800s 0.9512 0.9322s2 0.9604 0.9604s 0.9048 0.8690s3 0.9412 0.9412s 0.8607 0.8101s4 0.9224 0.9224s 0.8187 0.7552s5 0.9039 0.9039s 0.7788 0.7040s

Total 4.0704s



Credit Default Swaps: ValuationThis table shows the calculation of the present value of the expectedpayo¤ assuming a notional principal of $1We assume that default always happens halfway through a year andthat the recovery rate is 40%.For example, there is a 0.0192 probability of a payo¤ halfway throughthe third year and the present value of the expected payo¤ is:0.0192� 0.6� 1� e�0.05�2.5 = 0.0102.

TimeProba.default

Recoveryrate

Expectedpayo¤

Discountfactor

PV exp.payo¤

1 0.0200 0.4 0.0120 0.9753 0.01172 0.0196 0.4 0.0118 0.9277 0.01093 0.0192 0.4 0.0115 0.8825 0.01024 0.0188 0.4 0.0113 0.8395 0.00955 0.0184 0.4 0.0111 0.7985 0.0088

Total 0.0511Credit Derivatives (Session 13) Investments Spring 2010 13 / 31


Credit Default Swaps: ValuationThis table shows the calculation of the present value of accrualpayment in the event of default.For example, there is a 0.0192 probability that there will be a �nalaccrual payment halfway through the third year. The accrual paymentis 0.5s.The present value of the expected accrual payment at this time is0.0192� 0.5s � e�0.05�2.5 = 0.0085s.

TimeProba.default

Expectedaccrual payment

Discountfactor

PV exp. accpayment

1 0.0200 0.0100s 0.9753 0.0097s2 0.0196 0.0098s 0.9277 0.0091s3 0.0192 0.0096s 0.8825 0.0085s4 0.0188 0.0094s 0.8395 0.0079s5 0.0184 0.0092s 0.7985 0.0074s

Total 0.0426sCredit Derivatives (Session 13) Investments Spring 2010 14 / 31



The present value of the expected payment is:

4.0704s + 0.0426s = 4.1130s

The present value of the expected payo¤ is 0.0511. Equating the twogives:

4.1130s = 0.0511! s = 0.0124

We have found that the 5-year CDS spread should be 0.0124 of theprincipal or 124 basis points per year.

A more realistic valuation setup would comprise:I More frequent payment datesI Possibility of default to happen more frequently than once a year



Credit Indices

Participants in credit markets have developed indices to track creditdefault swap spreads.

Two important standard portfolios are:I CDX NA IG, a portfolio of 125 investment grade companies in NorthAmerica

I iTraxx Europe, a portfolio of 125 investment grade names in Europe



Credit Indices

Example: suppose that the 5-year CDX NA IG index is quoted by amarket maker as bid 65bps and o¤er 66bps. This means that a tradercan buy CDS protection on all 125 companies in the index for 66bpsper company. Suppose a trader wants $800, 000 of protection on eachcompany. The total cost is 0.0066� 800, 000� 125 = 660, 000 peryear. The trader can also sell $800, 000 of protection on each of the125 companies for a total of $650, 000 per annum. When a companydefaults, the protection buyer receives the usual CDS payo¤ and theannual payment is reduced by 660, 000/125 = $5, 280.




There are many credit products based on the CDS:I Binary CDSI CDS forwardsI CDS optionsI Basket credit default swaps (kth-to-default CDS)




Important di¤erence between credit default swaps and otherover-the-counter derivatives.

Most derivatives depend on the interest rate, exchange rate, equityand commodity prices etc. There is no reason to assume that any onemarket participant has better information than any other marketparticipant about these variables.

CDS depend on the probability that a particular company will defaultduring a particular period of time.

Some market participants may have more information to estimate thisprobability than others, i.e. a �nancial institution.

There is an asymmetric information problem.Is the decision by a risk manager to buy protection against the defaultby a company based on special information about this company?



The Future of the CDS Market

The credit default swap market survived the credit crunch of 2007.

Credit default swaps have become important tools for managingcredit risk.

A �nancial company can reduce exposure to particular companies bybuying protection.

Use CDS to diversify credit risk.


Credit Derivatives Total Return Swaps

Total Return Swaps

The total return swap is an agreement to exchange the total returnon a bond (or any portfolio of assets) for LIBOR plus a spread.

The total return includes coupons, interest, and the gain or loss onthe asset over the life of the swap.

Example: a 5-year total return swap with a notional principal of $100millions to exchange the total return on a corporate bond for LIBORplus 25 basis points.


Credit Derivatives Total Return Swaps

Total Return Swaps

At the end of the life of the swap there is a payment re�ecting thechange in value of the bond.

Example: if the bond increases in value by 10%, the payer is requiredto pay $10 millions, similarly, if the bond decreases in value by 15%,the receiver is required to pay $15 millions.The spread received by the payer depends on the credit quality of thereceiver, of the bond issuer, and the correlation between the two.


Credit Derivatives Asset Backed Securities

Asset Backed Securities

An asset-backed-security (ABS) is a security created from aportfolio of loans, bonds, credit card receivables, mortgages, autoloans, aircraft leases, or other �nancial assets.

Example: consider a bank that has made a large number of autoloans. The loans would typically be classi�ed according to the creditquality of the borrower, as prime, nonprime, and subprime. Supposethat there are 10, 000 non prime loans. Rather than keeping theseassets on its balance sheet, the bank might decide to sell them to aspecial purpose vehicle (SPV). Then the SPV issues securities thatare backed by the cash �ows of the loans and proceeds to sell thesecurities to investors. The investors return depends solely on thecash �ows from the loans. The bank earns a fee.







Typically, the senior tranche is rated AAA.

The mezzanine tranche might be rated BBB.

The equity tranche is usually not rated and is sometimes retained bythe creator of the ABS.

An asset-backed security is therefore a way of taking a portfolio ofrisky loans with a principal of $100 millions and creating from it $75millions of AAA-rated debt.



The Creation of an ABS CDO

Dealers have been very creative (too creative?)

Because mezzanine tranches are di¢ cult to sell, dealers have put themezzanine tranches from, for example, 20 di¤erent ABS into a newABS.

This product is known as an ABS CDO (Collateralized Debt

Obligation).

Then they "convinced" rating agencies to assign a AAA rating to themost senior tranche for the new structure.

Con�ict of interest of rating agencies.







The AAA rating for the senior tranche of the ABS CDO is reasonableif the losses experienced by di¤erent mezzanine tranches areindependent of each other.

If all of the mezzanine tranches are likely to experience a high lossrate at the same time, the AAA rated trance becomes quite risky!

Investors who bought AAA rated tranches that were created fromBBB rated mezzanine tranches that were in turn created fromsubprime mortgages found that their investments were steeplydowngraded by rating agencies!



Synthetic CDOs

The structure we have seen is know as a cash CDO.

Note that a long position in a corporate bond has essentially the samecredit risk as a short position in the corresponding CDS.

Alternative way for creating a CDO: form a portfolio of shortpositions of CDS then pass the credit risk on to tranches.

A CDO created in this way is know as a synthetic CDO.

Example:I Tranche 1 is responsible for the �rst $5m of losses. As compensation,it earns 15% on the remaining Tranche 1 principal.

I Tranche 2 is responsible for the next $20m of losses. As compensation,it earns 100bps on the remaining Tranche 2 principal.

I Tranche 3 is responsible for all losses in excess of $25m. Ascompensation, it earns 10bps on the remaining Tranche 3 principal.



Single Tranche Trading

Credit indices are used to de�ne standard CDO tranches.

The trading of these standard tranches is known as single tranchetrading.

A single tranche trading is an agreement where one side agrees to sellprotection against losses on a tranche and the other side agrees tobuy the protection.

The tranche is not part of a synthetic CDO that someone has createdbut cash �ows are calculated in the same way as if it were part ofsuch a synthetic CDO.


Credit Derivatives Asset Back Securities

Single Tranche Trading

Example: Five-year CDX NA IG and iTraxx Europe tranches onMarch 28, 2007. Quotes are in basis points except for 0-3% tranche,where the quote indicates the percent of the tranche principal thatmust be paid up front in addition to 500bps per year

CDX NA IG

Tranche 0-3% 3-7% 7-10% 10-15% 15-30% 30-100%

Quote 26.85% 103.8 20.3 10.3 4.3 2.0

iTraxx Europe

Tranche 0-3% 3-6% 6-9% 9-12% 12-22% 22-100%

Quote 11.25% 57.7 14.4 6.4 2.6 1.2


Investments

Session 14. Alternative InvestmentsEPFL - Master in Financial Engineering

Philip Valta

Spring 2010

Alternative Investments (Session 14) Investments Spring 2010 1 / 21

Outline

Managing portfolios

Overview of asset classes

Hedge Funds

Structured Products


Alternative Investments Managing Portfolios

Process of Portfolio Management

Making investment decisions

Constraints

Asset allocation

Managing portfolios of individual investors

Pension funds


Alternative Investments Overview of Asset Classes

Asset Classes

Private EquityI Capital to �rms not quoted on a stock market

Venture CapitalI Investments in start-up or early stage companies with high growthpotential

Real EstateI Direct investments, REITS, real estate derivatives

Exotic ProductsI Weather derivatives, ...

Commodities

Hedge Funds

Other Structured Products


Alternative Investments Hedge Funds

Hedge Funds

What is a hedge fund?

Surprisingly, hedge funds have no legal de�nition!"a mutual fund that employs leverage and uses various techniques ofhedging", (George Soros)

"all investment funds with an absolute return goal", (TASS)

"a multitude of skill-based investment strategies with a broad rangeof risk and return objectives. A common element is the use ofinvestment and risk management skills to seek positive returnsregardless of market directions", (Goldman Sachs)



Hedge Funds: Characteristics I

Hedge funds are actively managedI Returns not derived from passive long positionI Manager skill in identifying opportunities

Hedge funds target absolute returnsI No benchmarks

Hedge funds have �exible investment policiesI Assets, strategies, marketsI Short-selling, leverage, derivatives

Hedge funds have limited liquidityI Lock-up periods, exit fees



Hedge Funds: Characteristics II

Hedge funds typically have two type of feesI Management fee (as percent of assets under management): 1%� 2%I Performance fees: 20%

Hedge funds are partners, not employeesI Personal commitment

Hedge funds have limited transparencyI Legal requirement or necessity?

Hedge funds exhibit a low correlation with traditional investment

Hedge fund investorsI Institutional investors, high net worth individuals, fund of funds, butalso pension funds



Hedge Funds vs. Traditional Investments

Hedge Funds Traditional InvestmentsLong and short Long onlyOpportunistic exposure Fully investedMay use leverage No leverageAbsolute performance Relative performancePerformance driven by α Performance driven by βFlexible investment approach BenchmarkLow correlation with market High correlation with market



Hedge Funds: Some Strategies

Long-short EquityI Sector or geographic funds

Market neutral

Directional strategiesI Global macro, market trend, emerging markets

Event-drivenI Distressed securities, merger arbitrage

Relative valueI Convertible, �xed income, statistical arbitrage

Commodities (CTAs)

Multi-strategy

Fund of funds (diversi�cation but double fees)



Fund Selection



Hedge Funds - Other Issues

Measuring performance

Measuring risk

Performance attribution

Asset allocation with hedge funds

Growing assets under management and new entrants may be reducinghedge fund returns.

Potential problems:I High leverageI Limited transparencyI Limited regulationI LiquidityI Survivorship bias



Hedge Funds: Returns

2008 2007 2006

HFRX Equity Market Neutral �1.16% 3.11% 4.76%HFRX Energy/Basic Materials �29.42% 14.78% 12.44%HFRX Technology/Healthcare �8.40% 18.17% 13.22%HFRX Event Driven �22.11% 4.88% 10.32%HFRX Distressed Securities �30.69% 3.99% 9.56%HFRX Merger Arbitrage 3.69% 4.85% 10.73%HFRX Macro 5.61% 3.19% 5.61%HFRX Commodity 14.73% 13.42% 15.56%HFRX Russia/Eastern Europe 16.39% 39.99% -

HFRX Brazil 11.32% 23.00% -

HFRX BRIC 34.62% 42.46% -

HFRX China 44.94% 41.99% -

HFRX India 62.24% 45.79% -



Hedge Funds

Some open questions:I Can hedge funds deliver α?I Are hedge funds providing insurance (selling puts)?I Are hedge funds providing liquidity?I Are hedge funds the "bad" for �nancial markets?

The future of hedge funds:I Surviving?I Regulation?I Consolidation


Alternative Investments Structured Products

Structured Products

Structured products are investment products available to the publicwhose repayment value derives from the development of one orseveral underlying assets.

I stocks, interest rates, currencies, commodities such as gold, crude oil,copper, sugar etc.

Structured products are a combination of a traditional investment(e.g. bond) and a derivative �nancial instrument.

A suitable product can be created for virtually every derivativestrategy.

Legally, structured products are obligations for whose ful�llment theissuer is liable with all of its assets.



Writing a Covered Call

Profit

STK



Protective Put

ST

K



Bull Spread using Calls

K1 K2

Profit

ST



Butter�y Spread using Calls

K1 K3

Profit

STK2



Straddle

Profit

STK



Strangle

K1 K2

Profit

ST



Structured Products - Categorization

Capital protection

Yield enhancement

Participation

Leverage without knock-out

Leverage with knock-out

etc.

http://www.svsp-verband.ch/home/index.aspx?lang=en


Documents

Session1. Introduction - Daniel Andrei · 2013-02-06 · Course Overview Course Essentials Pedagogical Material Classnotes Textbooks I Bodie,KaneandMarcus,Investments I Hull,Options,Futures,