Computational Finance 1/34 Panos Parpas Asset Pricing Models 381 Computational Finance Imperial College London

Computational Finance 1/34

Panos Parpas

Asset Pricing Models

381 Computational Finance

Imperial College

London


Problem Types in Investment Science

Determining

correct, arbitrage free price of an asset:

price of a bond, a stock

the best action in an investment situation:

how to find the best portfolio –

how to devise the optimal strategy for managing an investment

Single period Markowitz model


Topics Covered

The Capital Asset Pricing Model (CAPM)

Single and Multi Factor Models

CAPM as a Factor Model

The Arbitrage Pricing Theory (APT)


M-V model

investor chooses portfolios on the efficient frontier –

deciding if given portfolio is on efficient frontier or not

no guarantee that a portfolio that was efficient ex ante

will be efficient ex post

statistical considerations regarding time period over

which to estimate & which assets to include are non-trivial

not mention implications of m-v optimisation on asset

pricing

CAPM describes MV portfolios and provides asset pricingCAPM describes MV portfolios and provides asset pricing


CAPM: Capital Asset Pricing Model

developed by Sharpe, Lintner and Mossin

single period asset pricing model

determines correct price of a risky asset within

the mean-variance framework

highlights the difference between systematic &

specific risk


Assumptions

All investors

– are mean variance optimisers –portfolios on efficient frontierare mean variance optimisers –portfolios on efficient frontier

– plan their investments over a single period of timeplan their investments over a single period of time

– use the same probability distribution of asset returns: the same use the same probability distribution of asset returns: the same

mean, variance, & covariance of asset returnsmean, variance, & covariance of asset returns

– borrow and lend at the risk free rate borrow and lend at the risk free rate

– are price-takers: investors’ purchases & sales do NOT influence are price-takers: investors’ purchases & sales do NOT influence

price of an asset price of an asset

– There is no transaction costs and taxesThere is no transaction costs and taxes


Market Portfolio

Everyone purchases single fund of risky asset, borrows (lends) at risk-free rate.

Form a portfolio that is a mix of risk free asset and single risky fund

Mix of the risky asset with risk free asset will vary across individuals according to their individual tastes for risk

Seek to avoid risk – Seek to avoid risk – have high percentage of the risk free asset in their portfoliohave high percentage of the risk free asset in their portfolio More aggressive to risk – More aggressive to risk – have a high percentage of the risky assethave a high percentage of the risky asset

What is the fund that everyone purchases?This fund is Market Portfolio and defined as summation of

all assets – total invested wealth on risky assets

An asset weight in market portfolio is the proportion of that asset’s total An asset weight in market portfolio is the proportion of that asset’s total capital value to total market capital value capital value to total market capital value – capitalization weights– capitalization weights


The Capital Market Line (CML)

Consider single efficient fund of risky assets (market portfolio) and a risk free asset (a

bond matures at the end of investment horizon):

If a risk free asset does not exist, investor would take positions at various points on

the efficient frontier. Otherwise, efficient set consists of straight line called CML.

Pricing Line: prices are adjusted so that efficient assets fall on this line

CML describes all possible mean-variance efficient portfolios that are a combination

of the risk free asset and market portfolio

Investors take positions on CML by

– buying risk free asset (between buying risk free asset (between MM and and rrff)) or or

– selling risk free assetselling risk free asset (beyond point (beyond point MM) ) and and

– holding the same portfolio of risky assetsholding the same portfolio of risky assets


The Capital Market Line

Equation describes all portfolios on CML

CML relates the expected rate of return of an efficient portfolio to its

standard deviation

The slope the CML is called the price of RISK!

– How much expected rate of return of a portfolio must increase if the risk of the portfolio How much expected rate of return of a portfolio must increase if the risk of the portfolio

increases by one unit?increases by one unit?

M

fMf

rrrrrE

)(

Expected Value of market rate of return

Standard Deviation of market rate of return


The Pricing Model

How does the expected rate of return of an individual asset relate to

its individual risk?

If the market portfolio M is efficient, then the expected return of an

asset i satisfies

The beta of an asset (risk premium):

fMifi

fMifi

rrrr

rrErrE

][][

2M

iMi


expected excess rate of return of an asset is proportional to the expected excess rate of return of the market portfolio: proportional factor is the beta of asset..

Amount that rate of return is expected to exceed risk free rate is proportional the amount that market portfolio return is expected to exceed risk free rate

fMifi rrrr

describes relationship between risk and expected return of asset

The Pricing Model


Beta of an Asset

beta of an asset measures the risk of the asset with

respect to the market portfolio M.

high beta assets earn higher average return in

equilibrium because of

beta of market portfolio: average risk of all assets

fMi rr

1)(

),(2

2

M

M

M

MMM rVar

rrCov


The Beta of Portfolio

If the betas of the individual assets are known,

then the beta of the portfolio is

This can be shown by using

rate of return of the portfolio

covariance

n

iiip w

1

n

iiip rwr

1

n

iMiiMp rrwrr

1

),cov(),cov(


Systematic and Specific Risk

CAPM divides total risk of holding risky assets into two parts:

systematic (risk of holding the market portfolio) and specific risk

Consider the random rate of return of an asset i:

Take expected value and the correlation of the rate of return with rM

The total risk of holding risky asset i is

ifMifi errrr )(

0),cov( and 0)( Mii reeE

risk specific

2

risk ystematic

22

risk total

2 ie

s

Mii


Summary: CAPM

The capital market line: expected rate of return of an efficient

portfolio to its standard deviation

The pricing model: expected rate of return of an individual asset

to its risk

The risk of holding an asset i is

M

fMf

rrrr

2

whereM

iMifMifi rrrr

risk specific

2

risk ystematic

22

risk total

2

s


Beta of the Market

Average risk of all assets is 1 (beta of the market portfolio)

Beta of market portfolio is used as a reference point to measure risk of other assets.

– Assets or portfolios with betas greater than 1 are above average risk: tend to move more Assets or portfolios with betas greater than 1 are above average risk: tend to move more

than market. than market. Example:Example:

If risk free rate is 5% per year and market rises by 10 %, then assets with a beta of 2 will If risk free rate is 5% per year and market rises by 10 %, then assets with a beta of 2 will

tend to increase by 15%. tend to increase by 15%.

If market falls by 10%, then assets with a beta of 2 will tend to fall by 25% on average. If market falls by 10%, then assets with a beta of 2 will tend to fall by 25% on average.

– Assets or portfolios with betas less than 1 are of below average risk: tend to move less Assets or portfolios with betas less than 1 are of below average risk: tend to move less

than market. than market.

rr

fr fr

),cov( Mrr2M 1

Capital Market Line Security Market LineM M

M

fMf

rrrr

2M

iMi

fMifi rrrr


CAPM as a Pricing Formula

CAPM is a pricing model.

standard CAPM formula only holds expected rates of return

suppose an asset is purchased at price P and later sold at price S.

rate of return is substituted in CAPM formula

CAPMin asset of Price 1

formula CAPM

return of Rate

fMf

fMf

rrr

SP

rrrP

PSP

PSr


Discounting Formula in CAPM

CAPMin asset an of Price 1 fMf rrr

SP

rate interest adjusted-risk

factor discount case, randomthe In

factor discount case,tic deterministhe In

)(1

1

1

1

fMf

f

rrr

r


Single-Factor Model

Consider n assets with rates of return ri for i=1,2,…,n and one factor f

which is a random quantity such as inflation, interest rate

Assume that the rates of return and single factor are linearly related.

Errors 1.1. have zero meanhave zero mean

2.2. are uncorrelated with the factorare uncorrelated with the factor

3.3. are uncorrelated with each otherare uncorrelated with each other

niefbar iiii ,,2,1 RandomConstantConstant

Intercept Factor Loadings

Error

0ieE

0ifeE

jieeE ij ,0


Multi-Factor Model

Single factor model is extended to have more than one factor. For two factors f1 and f2 the model can be written as

For k number factors

niefbfbar iiiii ,,2,12211 RandomConstantConstantConstant

niefbar i

k

jjjiii ,,2,1

1


How to Select Factors?

Factors are external to securities: consumer price index, unemployment rateconsumer price index, unemployment rate

Factors are extracted from known information about security returns:

the rate of return on the market portfoliothe rate of return on the market portfolio

Firm characteristics: price earning ratio, dividend payout ratioprice earning ratio, dividend payout ratio

How to select factors: It is part science and part art!Statistical approach – Statistical approach – principal component analysis Economical approach – Economical approach – its beta, inflation rate, interest rate, industrial production etc.


The CAPM as a Factor Model

Special case of a single-factor model f = rM

iiiM

Mii

MiMi

fMiifi

ifMii

ifMifiifi

iMiii

iiii

br

rrb

rbrr

rrEbrrE

errb

errbrbarr

erbar

efbar

0 and ]var[

],cov[

]var[],cov[

)][(][

)(

)()1(


The CAPM as a Factor Model: Example

Single Index Model applied to Lloyds

-0.1

-0.05

0

0.05

0.1

0.15

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

Market Returns

Re

turn

s o

n L

lloyd

s

slope = Beta = 1.77

intercept = alpha = 0 (approx)

•Single factor model equation defines a linear fit to data

• Imagine several independent observations of the rate of return and factor• Straight line defined by single factor model equation is fitted through these

points such that average value of errors is zero. • Error is measured by the vertical distance from a point to the line


Arbitrage: “The law of one price”

Arbitrage relies on a fundamental principle of finance : the law of one price

sayssays – – security must have the same price regardless of security must have the same price regardless of

the means of creating that securitythe means of creating that security..

implies – implies – if the payoff of a security can be synthetically if the payoff of a security can be synthetically

created by a package of other securities, the price of the created by a package of other securities, the price of the

package and the price of the security whose payoff package and the price of the security whose payoff

replicates must be equal.replicates must be equal.


Arbitrage – Example

How can you produce an arbitrage opportunity involving securities A,

B,C?

Replicating Portfolio: – combine securities combine securities AA and and BB in such a way that in such a way that – replicate the payoffs of security replicate the payoffs of security CC in either state in either state

Let wA and wB be proportions of security A and B in portfolio

Security PricePayoff in State 1

Payoff in State 2

A £70 £50 £100

B £60 £30 £120

C £80 £38 £112


Example Continued

Payoff of the portfolio

Create a portfolio consisting of A and B that will reproduce the payoff of C regardless of the state that occurs one year from now.

Solving equation system, weights are found wB = 0.6 and wA = 0.4

An arbitrage opportunity will exist if the cost of this portfolio is different than the cost of security C.

Cost of the portfolio is 0.4 x £70 + 0.6 x £60 = £64 - price of security C is £80. The “synthetic” security is cheap relative to security C.

BA

BA

ww

w w

120100 :2 statein

3050 :1 statein

112120100

383050

BA

BA

ww

w w


Example – Continued

Security Investment State 1 State 2

A -400000 5714 x 50 = 285700 5714 x 100 = 571400

B -600000 10000 x 30 = 300000 10000 x 120 =1200000

C 1000000 12500 x 38 = -475000 12500 x 112 = -1400000

Total £0 £110,700 £371,400

The outcome of forming an arbitrage portfolio of £1m

Riskless arbitrage profit is obtained by “buying A and B” in these proportions and “shorting” security C. Suppose you have £1m capital to construct this arbitrage portfolio.

Investing £400k in A £400k £70 = 5714 shares

Investing £600k in B £600k £60 = 10,000 shares

Shorting £1m in C £1m £80 = 12,500 shares


The Arbitrage Pricing Theory

CAPM is criticised for two assumptions: The investors are mean-variance optimizersThe investors are mean-variance optimizers

The model is single-periodThe model is single-period

Stephen Ross developed an alternative model based purely on

arbitrage arguments

Published Paper:

““The Arbitrage Pricing Theory of Capital Asset Pricing”, The Arbitrage Pricing Theory of Capital Asset Pricing”,

Journal of Economic Theory, Dec 1976.Journal of Economic Theory, Dec 1976.


APT versus CAPM

APT is a more general approach to asset pricing than CAPM. CAPM considers variances and covariance's as possible measures of risk while APT allows for a number of risk factors. APT postulates that a security’s expected return is influenced by a variety of factors, as opposed to just the single market index of CAPM APT in contrast states that return on a security is linearly related to “factors”. APT does not specify what factors are, but assumes that the relationship between security returns and factors is linear.


Simple Version of APT

Consider a single factor model.

Assume that the model holds exactly; no error

The uncertainty comes from the factor f

APT says that ai and bi are related if there

is no arbitrage

nifbar iii ,,2,1for


Derivation of APT

Choose another asset j such thatForm a portfolio from asset i and j with weights of w and (1-w)

Choose w so that the coefficient of factor is zero; so

ji bb

fbwwbawwar jijip ])1([)1(

ji

ij

ij

jip

jiij

j

bb

ba

bb

bar

bwwbbb

bw

0)1( and


Derivation of APT

cba

cb

a

b

a

b

a

bb

ba

bb

ba

ii

i

i

i

i

j

j

ji

ij

ij

ji

0

0

00

0

ai and bi are not independent


Arbitrage Pricing Formula

Once constants are known, the expected rate of return of an asset i is determined by the factor loading.

The expected rate of return of asset i

CAPM?

10

0

0

][

][

][][

i

i

ii

iii

iii

b

cfEb

fEbcb

fEbarE

fbar


CAPM as a consequence of APT

The factor is the rate of return on the market

APT is identical to the CAPM with

fMifi

ii

fM

f

M

rrEbrrE

brE

rrE

r

rf

][][

][

][

10

1

0

iib

Documents

Computational Finance 1/34 Panos Parpas Asset Pricing Models 381 Computational Finance Imperial College London