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II. Market Risk Module

II. 1. A. INTRODUCTION TO VAR 3

II. 1. B. PUTTING VAR TO WORK 9

II. 2. MECHANICS OF FUTURES MARKETS 13

II. 2. A. HEDGING STRATEGIES USING FUTURES 16

II. 2. B. DETERMINATION OF FORWARD &FUTURES PRICES 19

II. 2. C. INTEREST RATES 26

II. 2. D. SWAPS 35

II. 2. (NA). MECHANICS OF OPTION MARKETS 43

II. 2. E. PROPERTIES OF STOCK OPTIONS 45

II. 2. F. TRADING STRATEGIES INVOLVING OPTIONS 49

II. 2. G. BINOMIAL TREES 53

II. 2. H. THE BLACK–SCHOLES MODEL 57

II. 2. I. THE GREEK LETTERS 65

II. 2. J. VOLATILITY SMILES 72

II. 2. K. EXOTIC OPTIONS 75

II. 3. A. VAR METHODS 85

II. 3. B. VAR MAPPING 87

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II. 3. C. STRESS TESTING 94

II. 4. COMMODITY FORWARDS & FUTURES 98

II. 5. A. MARKET RISK 104

II. 5. B. FOREIGN EXCHANGE RISK 110

II. 6. A. FIRM-WIDE RISK MANAGEMENT 114

II. 6. B. CASH FLOW EXPOSURES 119

II. 6. C. THE DEMAND AND SUPPLY FOR DERIVATIVE PRODUCTS 125

II. 7. A. BOND PRICES, DISCOUNT FACTORS, AND ARBITRAGE 129

II. 7. B. BOND PRICES, SPOT RATES & FORWARD RATES 132

II. 7. C. YIELD-TO-MATURITY (YTM) 136

II. 7. D. GENERALIZATIONS AND CURVE FITTING 140

II. 7. E. ONE-FACTOR MEASURES OF PRICE SENSITIVITY 142

II. 7. F. MEASURES OF PRICE SENSITIVITY BASED ON PARALLEL YIELD SHIFTS 150

II. 7. G. KEY RATE AND BUCKET EXPOSURES 153

II. 7. H. THE SCIENCE OF TERM STRUCTURE MODELS 158

II. 7. I. MORTGAGE-BACKED SECURITIES 162

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II. 1. A. Introduction to VaR LO 7.1: Discuss reasons for the widespread adoption of VaR as a measure of risk.

LO 7.2: Define value at risk and calculate VaR for a single asset on both a dollar and percentage basis.

LO 7.3: Convert a daily VaR measure into a weekly, monthly, or annual VaR measure.

LO 7.4: Discuss assumptions underlying VaR calculations.

LO 7.5: Explain why it is best to use continuously compounded rates of return when calculating VaR.

LO 7.6: Calculate portfolio VaR and describe the primary factors that affect portfolio risk.

Background on value at risk (VaR)

Why it Became Popular

LO 7.1 Discuss reasons for the widespread adoption of VAR as a measure of risk

The capital asset pricing model (CAPM) is popular but controversial. CAPM divides (decomposes)

risk into systemic (market) risk and residual (company-specific) risk. CAPM quantifies risk as beta

(), but beta is controversial.

Reasons for popular adoption of VAR include:

The traditional approach has been the capital asset pricing model (CAPM), where beta is the risk metric. However, beta has a “tenuous connection” to actual returns. Further, as a one-factor model, CAPM is viewed as too simplistic by many practitioners

JP Morgan created an “open architecture metric” (i.e., not proprietary) called RiskMetrics

Bank for International Settlements (BIS) in 1998 started to allow banks to use internal models such as VaR in order to calculate their capital requirements

JP Morgan later said about the introduction of RiskMetrics in 1994, “we took the

bold step of revealing the internal risk management methodology…and a free data

set…At the time, there was little standardization in the marketplace”.

What is VAR?

LO 7.2 Define value at risk (VaR)…

VaR answers a risk measurement question (not a risk management question): “how much can we

lose in a given time frame, with a specified confidence level?”

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VaR is a statistical or probabilistic approach. VaR gives the worst expected loss given some

confidence level. It does not give the worst-case scenario.

The VaR question needs two specifications to be asked: a time horizon and a

confidence level. You can look at the same portfolio and ask, for example, “What is

the VaR with 95% confidence over one year?” or “What is the VAR with 99%

confidence over one day?”

A shortcoming of VaR is that it gives no information about loss in excess of VaR.

For example, if the VaR at 95% is a loss of $1 million dollars, this gives no

information about the distribution of losses in excess of $1 million; e.g., are there

outliers at $2 million? This is the domain of extreme value theory (EVT).

Assumptions behind VAR LO 7.4 Discuss assumptions underlying VaR calculations

The following assumptions underlie the VaR calculation:

Stationarity: the (shape of the) probability distribution is constant over time

Random walk: tomorrow’s outcome is independent of today’s outcome

Non-negative: requirement: assets cannot have negative value

Time consistent: what is true for a single period is true for multiple periods; e.g., assumptions about a single week can be extended to a year

Normal: expected returns follow a normal distribution

Normality is an especially dubious assumption; many research studies have proven that asset

returns are not normally distributed.

Calculate value at risk (VaR) LO 7.2 (continued) …and calculate VaR for a single asset on both a dollar and percentage basis

Assume that daily returns for the S&P index are normally distributed with an average (expected)

return of zero (0%) and a standard deviation of 1% (100 basis points). Further assume the portfolio

value is $1 million.

In order to calculate VaR, we need to specify a percentile. The most common percentiles are the

1st percentile (i.e., corresponds to 99% level of confidence) and 5th percentile (corresponds to 95%

confidence level).

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Dollar VaR with x% confidence

For a single asset over a single period, dollar value at risk (VaR) is the product of asset value,

volatility (standard deviation) and the critical-z. The critical-z depends on the confidence level.

$VaR = dollar value($) critical-z

Dollar VaR with 95% confidence

For a 5% VAR (i.e., the worst loss we can expect in 95% of our cases), the z value is -1.645 and the

dollar VAR is given by:

$VaR (95%) = $1 million 1% (-1.645) =-16,450

The 5% dollar value at risk (the “dollar VAR”) is therefore $16,450.

Dollar VaR with 99% confidence

For a 1% VAR (i.e., the worst loss we can expect in 99% of our cases), the z value is -2.326 and the

dollar VAR is given by:

$VaR (99%) = $1 million 1% (-2.326) =-$23,260

The 1% “dollar VAR” is therefore $23,260.

Percentage VaR

We can also compute the “percentage VAR.” If we instead want to compute VAR on a

percentage basis, then for a 5% VAR it will be given by:

%VaR (95%) = 1% (-1.645) =-1.645%

For a 1% VAR, the z value is -2.326 and the percentage VAR is given by:

%VaR (99%) = 1% (-2.326) =-2.326%

In these examples, we used the negative (-) sign because the z-value is negative or

“to the left of the mean.” The text produces a positive value and then refers to the

positive value of the loss; e.g., “a 1% chance of a loss greater than $23,260 or

2.326%.” They have the same meaning.

VAR is about the worst expected loss with some degree of confidence but it is not

the absolute worst case loss. We typically specify a 95% confidence level (which

corresponds to a 95% one-tailed confidence interval) or a 99% confidence level.

The 95% confidence level corresponds to a 5% significance level (1 – 95%) and the

99% confidence level corresponds to a 1% significance level (1-99%).

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Calculating Rates of Return

LO 7.5 Explain why it is best to use continuously compounded rates of return when calculating VAR

Rate of return can be calculated in absolute, simple or continuous terms. Continuous is best with

the exception of interest rate-related variables

Absolute change (today’s price – yesterday’s price): violates the stationarity requirement.

Simple change ([today price – yesterday’s price] [yesterday’s price]): satisfies stationarity requirement, but does not comply with time consistency requirement.

Continuous compounded return is best because it satisfies the time consistency requirement: the two-period return is the sum of two single period returns. The sum of two random variables that are jointly distributed is itself (i.e., the sum) normally distributed.

The exception is interest rate-related variables (e.g., credit spreads, zero coupon spot rates):

absolute changes should be used for these variables.

VAR Scales with the Square Root of Time

LO 7.3 Convert a daily VAR measure into a weekly, monthly, or annual VAR measure

We use the “square root of time” rule. Assume that X daily is the daily VAR. Then:

daily

daily

daily

Weekly VaR =VaR 5

Monthly VaR =VaR 20

Annual VaR =VaR 250

The above assumes 20 trading days in a month and 250 trading days in a year; the 250 is not

magic. Some would instead use 252. What’s important is, how many daily periods are we trading

over? If Monday is a holiday next week, for example, such that next week will have four trading

days, then next week’s VAR is scaled by multiplying by the square root of four.

In general terms, the VAR for the J-Period return is given by the 1-period VAR multiplied by the

square root of J.

Portfolio VAR

Diversification and VAR

When assets are combined into a portfolio, total risk is less than the sum of the individual

(component asset) risks. Risk is not naively additive; diversification provides some (offsetting)

risk reduction benefit.

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LO 7.6 Calculate portfolio VAR…

If a portfolio has two assets, the variance is given by:

2 2 2 2 21 2 1,2(1 ) 2 (1 )p w w w w <II. 1. 1>

Where (w) is weight of the first asset, (1-w) is the weight of the second asset (by definition since

we only have two assets) and σ1,2 is the covariance between the assets. The portfolio variance can

then be factored into the VAR calculation directly.

Memorize the portfolio variance for a two-asset portfolio. This is a very common

test question. You should know that the covariance is a product of [correlation]

[standard deviation of 1st asset] [standard deviation of 2nd asset].

Let’s do an example. Assume Asset X has a standard deviation of 4% and Asset Y

has a standard deviation of 6%. The correlation coefficient (r) between X and Y is

0.2. Finally, assume our portfolio is evenly weighted between the two assets (50%,

50%).

The portfolio’s standard deviation is given by:

2 2 2 2 21 2 1,2

2 2 2 2

(1 ) 2 (1 )

(0.5) (0.04) (0.5) (0.06) 2(0.5)(0.5)(0.2)(0.04)(0.06)

0.00154

0.00154 0.03924 3.92%

p w w w w

Notice we were not given the covariance directly, so we replaced the covariance with the product

of: correlation coefficient (0.2) standard deviation (0.04) standard deviation (0.06). The

covariance therefore equals 0.00048, so we could have plugged that into the equation directly:

2 2 2 2 21 2 1,2

2 2 2 2

(1 ) 2 (1 )

(0.5) (0.04) (0.5) (0.06) 2(0.5)(0.5)

0.00154

0.00154 0

(0.0

.039

004

24 3 %

8

.92

)

p w w w w

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Decomposing Risk into Systemic and Residual Risk

Assume all assets have the same standard deviation (sigma) and correlation across assets (rho) is

the same. Portfolio standard deviation is given by:

p

1 ( 1)N

N N

Then, for a large portfolio of uncorrelated assets (rho = 0), the portfolio’s standard deviation tends

toward zero.

2

plim lim 0N N N

What Impacts Portfolio Volatility? LO 7.6 (continued) …and describe the primary factors that affect portfolio risk

Higher portfolio volatility implies greater value at risk (VAR). Consider the following directional

impacts:

Directional Impacts Factor Impact on Portfolio volatility

Higher variance Higher (direct function) Greater asset concentration Higher More equally weighted assets Lower Lower correlation Lower Higher systematic risk Higher Higher idiosyncratic risk Irrelevant

Discuss the role of correlation in the downfall of Long-Term Capital Management (LTCM) (Orange = not a learning outcome. Optional but recommended)

Long-term Capital Management employed several seemingly diverse hedge fund strategies

including mortgage-backed securities, foreign bonds, global swap spreads, and hedged corporate

bonds. Given the diversity of strategies, the assumption was that asset correlations were low or

non-existent. But when Russia defaulted on sovereign debt obligations in August 1998, a panic

rippled through all of the asset classes. Correlations spiked, which of course the models did not

predict.

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II. 1. B. Putting VAR to Work LO 7.7: Differentiate between linear and non-linear derivatives.

LO 7.8: Describe the calculation of VAR for a linear derivative.

LO 7.9: Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives.

LO 7.10: Discuss why the Taylor approximation is ineffective for certain types of securities.

LO 7.11: Explain the differences between the delta-normal and full-revaluation methods for measuring the risk of non-linear derivatives.

LO 7.12: Describe the structured Monte Carlo approach to measuring VAR, and identify the advantages and disadvantages of the SMC approach.

LO 7.13: Discuss the implications of correlation breakdown for scenario analysis.

LO 7.14: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach.

LO 7.15: Describe the worst case scenario measure as an extension to VAR.

Derivatives LO 7. 7 Differentiate between linear and non-linear derivatives

Derivatives are either linear or nonlinear. If the delta is constant, the derivative is linear. If the

delta is variable (i.e., changes), the derivative is nonlinear.

Linear derivative. Price of derivative = linear function of underlying asset. For example, a futures contract on S&P 500 index is approximately linear

Non-linear derivative. Price of derivative = non-linear function of underlying asset. For example, a stock option is non-linear.

All assets are locally linear. Use an option as an example. The option is convex in the value of

the underlying. The delta is the slope of the tangent line. For small changes, the delta is

approximately constant.. For large changes, it is not. What is the fix? A Taylor Series

approximation provides a correction.

Delta

Delta is the rate of change of the derivative “with respect to” (i.e., divided by) the rate of change of

the underlying asset:

ΔPrice of DerivativeDelta =

ΔPrice of Underlying Asset

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To illustrate this key idea, consider a stock option (c) on an underlying stock (S). The change in

option price given by c and a change in the stock price is given by S. The delta is given by:

stock optionDeltac

S

LO 7.8 Describe the calculation of VaR for a linear derivative

If the derivative is linear (or approximately linear), VAR is delta () multiplied by the underlying

risk factor:

Linear Derivative Underlying Risk FactorVaR VaR <II. 1. 2>

For example, if the contract multiple for a futures index is $250 (i.e., the futures contract is worth

$250 multiplied by the index value), then the VAR of the futures contract is given by:

VaR $250 VaRFuture Index

LO 7.9 Explain how the addition of second-order terms through the Taylor approximation improves

the estimate of VAR for non-linear derivatives

The linear approach above is problematic for non-linear derivatives because of the curvature (or

convexity) of the curve-relationship. The Taylor approximation is a mathematical extension of the

linear relationship that helps to account for the curvature. The Taylor approximation is given by:

20 0 0 0 0( ) ( ) ( )( ) 1 2 ( )( )f x f x f x x x f x x x

The first term is the linear approximation:

0 0 0( ) ( )( ): first term, the linear approximationf x f x x x

The second term is (effectively) the adjustment for the convexity:

20 01 2 ( )( ) : second term, the convexity correctionf x x x

LO 7.10 Discuss why the Taylor approximation is ineffective for certain types of securities;

The Taylor approximation is not helpful where the derivative exhibits extreme non-linearities.

This includes mortgage-backed securities (MBS); i.e., fixed income securities with embedded

options.

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Measuring the Risk of nonlinear securities LO 7.11 Explain the differences between the delta-normal and full-revaluation methods for measuring the risk of non-linear derivatives;

There are two approaches: full revaluation and delta-normal.

Full Revaluation

Every security in the portfolio is re-priced. Full revaluation is accurate but computationally

burdensome.

Delta-Normal

A linear approximation is created. This linear approximation is an imperfect proxy for the

portfolio. This approach is computationally easy but may be less accurate. The delta-normal

approach (generally) does not work for portfolios of nonlinear securities.

Structured Monte Carlo LO 7.12 Describe the structured Monte Carlo (SMC) approach to measuring VAR, and identify the advantages and disadvantages of the SMC approach;

In the structured Monte Carlo (SMC) approach, we simulate a large number (e.g., thousands) of

asset distributions and re-order the outcomes to determine percentile VARs.

The key advantage of structured Monte Carlo: we can generate correlated scenarios based on a

statistical distribution. The key disadvantages are: simulations may not be representative of future

outcomes; do not handle correlation breakdown in extreme situations.

Advantage Disadvantage

Structured Monte

Carlo

Able to generate

correlated scenarios based

on a statistical

distribution

Generated scenarios may

not be relevant going

forward

Scenario Analysis (Stress Testing) LO 7.13 Discuss the implications of correlation breakdown for scenario analysis;

The problem with the SMC approach is that the covariance matrix is meant to be “typical;” but

severe stress events wreak havoc on the correlation matrix. That is correlation breakdown.

Scenarios can attempt to incorporate correlation breakdowns. One approach is to stress test

(simulate) the correlation matrix. This is easier said than done; e.g., the variance-covariance

matrix needs to be invertible.

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LO 7.14 Describe the primary approaches to stress testing and the advantages and disadvantages of each approach;

The common practice is to provide two independent sections to the risk report: (i) a VAR-based

risk report and (ii) a stress testing-based risk report. The VAR-based analysis includes a detailed

top-down identification of the relevant risk generators for the trading portfolio. The stress

testing-based analysis typically proceeds in one of two ways: (i) it examines a series of historical

stress events and (ii) it analyzes a list of predetermined stress scenarios.

In regard to stressing historical events, this can be informative about portfolio weaknesses. The

analysis of predetermined (standard) scenarios can be good at highlighting weaknesses relative to

standard risk factors (e.g., interest rate factors). However, the analyzing pre-prescribed scenarios

may create false red flags.

The problem with historical stress testing is that it could miss altogether important risk sources

(i.e., because they happened not to arise in historical events).

Advantage Disadvantage

Stress Testing Can illuminate riskiness of

portfolio to risk factors

Can specifically focus on

the tails (extreme losses)

Complements VaR

May generate unwarranted

red flags

Highly subjective (can be

hard to imagine

catastrophes)

Worst-Case Scenario (WCS) LO 7.15 Describe the worst case scenario (WCS) measure as an extension to VAR.

The worst case scenario measure asks, what is the worst loss that can happen over a period of

time? Compare this to VAR, which asks, what is the worst expected loss with 95% or 99%

confidence? The probability of a “worst loss” is certain (100%); the issue is its location.

As an extension to VAR, there are three points regarding the WCS:

1. The WCS assumes the firm increases its level of investment when gains are realized; i.e., that the firm is “capital efficient.”

2. The effects of time-varying volatility are ignored

3. There is still the extreme tail issue: it is still possible to underestimate the likelihood of extreme left-tail losses

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II. 2. Mechanics of Futures Markets LO 27.5: Distinguish between a long futures position and a short futures position.

LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.

LO 27.7: Describe the marking-to-market procedure, the initial margin, and the maintenance margin.

LO 27.8: Compute the variation margin.

LO 27.9: Explain the role of the clearinghouse.

Futures Contract Specifications LO 27.6 Describe the characteristics of a futures contract…

A futures contract is a standardized contract traded on a futures exchange to buy or sell an

underlying asset at a delivery date at a pre-set futures price. Its characteristics include:

An (underlying) asset

A Treasury bond futures contract is made on the underlying U.S. Treasury with maturity of at

least 15 years and not callable within 15 years (15 years ≤ T bond).

A Treasury note futures contract is made on the underlying U.S. Treasury with maturity of at least

6.5 years but not greater than 10 years (6.5 ≤ T note ≤ 10 years).

When the asset is a commodity (e.g., cotton, orange juice), the exchange specifies a grade

(quality).

Contract Size

Contract size varies by the type of futures contract:

Treasury bond futures: contract size is a face value of $100,000

S&P 500 futures contract is index $250 (multiplier of 250X)

NASDAQ futures contract is index $100 (multiplier of 100X)

Recently, “mini contracts” have been introduced: These have multipliers of 50X for

the S&P and 20X for the NASDAQ. In other words, each contract is one-fifth the

price in order to attract smaller investors.

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Delivery Arrangements

The exchange specifies delivery location.

Delivery Months

The exchange must specify the delivery month; this can be the entire month or a sub-period of

the month.

Futures Positions LO 27.5 Distinguish between a long futures position and a short futures position

LO 27.6 (continued) …and explain how futures positions are settled

A long-futures position agrees to buy in the future and a short-futures position agrees to sell in the

future. The price mechanism maintains a balance between buyers and sellers. For example, if

there are more buyers than sellers, the price increases until new sellers enter the futures market.

Most futures contracts do not lead to delivery, because most trades “close out” their positions

before delivery. Closing out a position means entering into the opposite type of trade from the

original.

Exchanges and Regulation Chicago Board of Trade (CBOT, www.cbot.com) Chicago Mercantile Exchange (CME, www.cme.com) London International Financial Futures and Options Exchange (www.liffe.com) Eurex (www.eurexchange.com) Regulation: Commodity Futures Trading Commission (CFTC, www.cftc.gov)

Operations of Margins LO 27.7 Describe the marking to market procedure, the initial margin, and the maintenance margin

LO 27.8 Compute the variation margin

When an investor enters into a futures contract, the broker requires an initial margin deposit into

the margin account. At the end of each trading day, the margin account is marked-to-market. If

the account balance falls below the maintenance margin (i.e., typically lower than the initial

margin), a margin call requires the investors to “top up” the account back to the initial margin

amount.

Margin account: Broker requires deposit.

Initial margin: Must be deposited when contract is initiated.

Mark-to-market: At the end of each trading day, margin account is adjusted to reflect gains or losses.

15 / 166

Maintenance margin: Investor can withdraw funds in the margin account in excess of the initial margin. A maintenance margin guarantees that the balance in the margin account never gets negative (the maintenance margin is lower than the initial margin).

Margin call: When the balance in the margin account falls below the maintenance margin, broker executes a margin call. The next day, the investor needs to “top up” the margin account back to the initial margin level.

Variation margin: Extra funds deposited by the investor after receiving a margin call.

There is only a variation margin if and when there is a margin call.

Variation margin = initial margin – margin account balance

The maintenance margin is a trigger level—once triggered, the investor must “top

up” to the initial margin, which is greater than the maintenance level.

LO 27.9 Explain the role of the clearinghouse

The exchange clearinghouse is a division of the exchange (e.g., the CME Clearing House is a

division of the Chicago Mercantile Exchange) or an independent company. The clearinghouse

serves as a guarantor, ensuring that the obligations of all trades are met.

Types of Orders

Market order: Execute the trade immediately at the best price available.

Limit order: This order specifies a price (e.g., buy at $30 or less)—but with no guarantee of execution.

Stop order: (aka., stop-loss order) An order to execute a buy/sell when a specified price is reached.

Stop-limit: Requires two specified prices, a stop and a limit price. Once the stop-limit price is reached, it becomes a limit order at the limit price.

Market-if-touched: Becomes a market order once specified price is achieved.

Discretionary (aka., market-not-held order): A market order, but the broker is given the discretion to delay the order in an attempt to get a better price.

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II. 2. A. Hedging Strategies Using Futures LO 28.1: Differentiate between a short hedge and a long hedge, and identify situations where

each is appropriate.

LO 28.2: Define and calculate the basis.

LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.

LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.

LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.

LO 28.6: Identify situations when a rolling hedge is appropriate, and discuss the risks of such a strategy.

Short and long hedges LO 28.1: Differentiate between a short hedge and a long hedge, and identify situations where each is appropriate.

A short forward (or futures) hedge is an agreement to sell in the future and is appropriate

when the hedger already owns the asset. The classic example is a farmer who wants to lock in a

sales price for his/her crop, and therefore protect him/herself against a price decline.

A long forward (or futures) hedge is an agreement to buy in the future and is appropriate

when the hedger does not currently own the asset but expects to purchase in the future. An

example is an airline which depends on jet fuel and enters into a forward or futures contract (a

long hedge) in order to protect itself from exposure to high oil prices.

Basis and basis risk LO 28.2 Define and calculate the basis

Remember that the basis itself converges to zero over time, as the spot price converges toward the

future price.

Basis = Spot Price Hedged Asset – Futures Price Futures Contract = S0 – F0

LO 28.3 Define the types of basis risk and explain how they arise in futures hedging

When the spot price increases by more than the futures price, the basis increases and this is said

to be a “strengthening of the basis” (and when unexpected, this strengthening is favorable for a

short hedge and unfavorable for a long hedge).

When the futures price increases by more than the spot price, the basis declines and this is said to

be a “weakening of the basis” (and when unexpected, this weakening is favorable for a long hedge

and unfavorable for a short hedge).

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Optimal Hedge Ratio LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.

If the spot and future positions are perfectly correlated, then a 1:1 hedge ratio results in a perfect

hedge. However, this is not typically the case. The optimal hedge ratio (a.k.a., minimum variance

hedge ratio) is the ratio of futures position relative to the spot position that minimizes the

variance of the position. Where is the correlation and is the standard deviation, the optimal

hedge ratio is given by:

* S

F

h [II.1.1]

For example, if the volatility of the spot price is 20%, the volatility of the futures price is 10%, and

their correlation is 0.4, then

20%* (0.4) 0.8

10%h

And the number of futures contracts is given by N* when NA is the size of the position being

hedged and QF is the size of one futures contract:

*

* A

F

h NN

Q [II.1.2]

Stock Index Futures LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.

Given a portfolio beta (), the current value of the portfolio (P), and the value of stocks

underlying one futures contract (A), the number of stock index futures contracts (i.e., which

minimizes the portfolio variance) is given by:

*P

NA

[II.1.3]

By extension, when the goal is to shift portfolio beta from () to a target beta (*), the number of

contracts required is given by:

( * )P

NA

[II.1.4]

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Assume the following: a portfolio value of $10 million with a beta of 1.2. Further, assume the

S&P 500 Index value is 1500 (one futures contract is for delivery of $250 multiplied by the index).

$10 million* (1.2) 32

(1500)(250)

PN

A

The hedge trade is short 32 futures contracts.

The above essentially changes the beta to zero. Now assume that we want to change the beta of

the portfolio to 2.0.

$10 million( * ) (2 0.8) 21.33

(1500)(250)

PN

A

The hedge trade here is to enter into a long position on 21.33 futures contracts. Note we could

have used (beta minus target beta) in which case the result would be negative (-) 21.33. But in

either case, we must buy (go long) futures contracts because we are increasing the beta. If we are

reducing the beta, then we short futures.

Rolling the Hedge Forward LO 28.6: Identify situations when a rolling hedge is appropriate, and discuss the risks of such a strategy.

When the delivery date of the futures contract occurs prior to the expiration date of the hedge,

the hedger can roll forward the hedge: close out a futures contract and take the same position on

a new futures contract with a later delivery date.

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II. 2. B. Determination of Forward &Futures Prices LO 27.1: State and explain the cost-of-carry model for forward prices using both assets that have

interim cash flows and assets that do not have interim cash flows.

LO 27.2: Compute the forward price given both the price of the underlying and the appropriate carrying costs of the underlying.

LO 27.3: Calculate the value of a forward contract.

LO 27.4: Describe the differences between forward and futures contracts.

Forward and Futures Contracts LO 27.4 Describe the differences between forward and futures contracts

While both forwards and futures are agreements to buy or sell an asset in the future (at a

specified price), a forward contract is traded over-the-counter and the forward is not

standardized. The futures contract is traded on an exchange, standardized (often highly

standardized) and typically closed out before maturity.

Forward vs. Futures Contracts

Forward Futures

o Trade over-the-counter o Trade on an exchange

o Not standardized o Standardized contracts

o One specified delivery date o Range of delivery dates

o Settled at the end of a contract o Settled daily

o Delivery or final cash settlement

usually occurs

o Contract usually closed out

prior to maturity

Notations

The following notations apply to forward contracts:

T: Time until delivery date in a forward/futures contract (in years)

S0: Price of the underlying asset (spot price)

F0: Today’s forward or futures price

K: Delivery price

r: Risk-free rate—annual rate but expressed with continuous compounding

rf: Foreign risk-free interest rate

I: Present value of income received from asset (in dollar terms)

q: Dividend yield rate (in percentage terms; e.g., 2% dividend yield)

U, u: Storage cost. U = dollar cost and u = cost in % terms

y: convenience yield

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Cost of Carry Model LO 27.1 State and explain the cost-of-carry model for forward prices using both assets that have interim cash flows and assets that do not have interim cash flows

LO 27.2 Compute the forward price given both the price of the underlying and the appropriate carrying costs of the underlying

The cost-of-carry model sets a futures price as a function of the spot price: the futures price (F)

equals the spot price (S0) compounded at the interest rate (r, required to finance the asset) plus

the storage cost of the asset less any income earned on the asset.

For a non-dividend-paying investment asset (i.e., an asset which has no storage cost) the cost

of carry model says the futures price is given by:

0 0 0 0cT rTF S e F S e <II. 2. 5>

If the asset provides interim cash flows (e.g., a stock that pays dividends), then let (I) equal

the present value of the cash flows received and the cost-of-carry model is then given by:

0 0( ) rTF S I e <II. 2. 6>

If the asset has a storage cost and produces a convenience yield, the cost-of-carry model

expands to:

( )0 0

r u y TF S e <II. 2. 7>

Where r is the risk-free rate, u is the storage cost as a constant percentage, and y is the

convenience yield.

Forward Prices

The equations for forward prices are essentially similar to futures prices. The generalized forward

price (F0) is given by:

0 0rTF S e

<II. 2. 8>

If the asset provides income with a present value equal to (I), the forward price is:

0 0( ) rTF S I e <II. 2. 9>