Instruments

Consider a two-factor model, where the two factors are interest rate movements and changes

in inflation.Assume that the Noah’s Bagel Company has a future cash flow with factor betas

of 2.5 on the interest rate factor and 4.5 on the inflation factor.Noah’s would like to elimi-

nate its sensitivity to both factors by acquiring financial securities, yet it does not wish to use

its own cash to do this.

Noah’s contacts its investment bank and learns that it can (1) enter into a five-year inter-

est rate swap contract, (2) purchase 30-year government bonds, and (3) acquire a sizable

chunk of shares in DESPERATE, Inc.Investment 1, the swap contract, has no up-front cost

and, per contract, has a factor equation for its future value described by

˜5 5˜ 3˜

CFF

1inflationinflation

Investment 2, the 30-year government bond, per million dollars invested, has a factor equa-

tion for its future cash flow of

˜10 5˜ 1˜

CFF

2inflationinflation

Investment 3 is the stock of DESPERATE, Inc., which, per million dollars invested, has a

factor equation for its end-of-period value of

˜01˜1˜

CFF

3inflationinflation

Design a proper hedge against inflation and interest rate movements in this environment.

Answer:To design a future cash flow that is insensitive to inflation or interest rate move-

ments, we need to find a costless portfolio of financial investments with an interest rate sen-

sitivity of 2.5 and an inflation sensitivity of 4.5.The three investments are denoted by

Grinblatt1625Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1625Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

807

xthe number of contracts in the costless swap investment

1

xmillions of dollars in the 30-year government bonds

2

xmillions of dollars in DESPERATE, Inc.

3

The portfolio of these three investments has a cost of 0x1,000,000x1,000,000x.Its

123

sensitivity to the interest rate factor is

5x 5x1x

123

Its sensitivity to the inflation factor is

3x 1x1x.

123

Therefore, the hedge portfolio is found by simultaneously solving

0x1,000,000x1,000,000x0

123

5x 5x1x 2.5

123

3x 1x1x 4.5

123

Since the first equation says x x, one can substitute for xin the other two equations,

323

implying

5x 6x 2.5

12

and

3x 2x4.5

12

Multiplying the first equation (of the two immediately above) by .6 and adding it to the sec-

ond equation yields

1.6x 3

2

or

x1.875

2

Plugging this back into either of the above equations yields x2.75.Thus, the solution is

1

1.	Buy 2.75 swap contracts, representing $2.75 million in notional amount
2.	Short $1.875 million in 30-year government bonds
3.	Buy $1.875 million of DESPERATE, Inc.

22.9	Hedging with Regression

Regression provides a shortcut method for estimating a hedge ratio that minimizes risk

exposure. By regressing the cash flow that one is trying to hedge against the value of

the financial instrument used in the hedge, one derives a beta coefficient that deter-

mines the quantity of the hedging instrument that minimizes variance.¹⁴

¹⁴Chapter

4 notes the conditions required for variance minimization and for the portfolio variance

formulas used here.

Grinblatt1627Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1627Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

808Part VIRisk Management

Hedging a Cash Flow with a Single Financial Instrument

Consider a future cash flow ˜, which is random, and a financial instrument with a

C

future value of ˜, also random, per unit bought. The variance of the combination of

P

the cash flow and the short position in the financial instrument is

var(˜2˜) 2cov(˜, ˜)	(22.4)
C)var(PCP

Hedge Ratios from Covariance Properties.Because covariance can be interpreted

as the marginal variance, the combination of the cash flow and the hedge instrument

that minimizes variance cannot have either a positive marginal variance or a nega-

tive marginal variance with the financial instrument. If it has a positive marginal

variance, then a small reduction in the holdings of the financial instrument in the

combination reduces variance. If it has a negative marginal variance, then a small

increase in the holdings of the financial instrument reduces the variance of the com-

bination. Only when the financial instrument has zero covariance with the combina-

tion of the financial instrument and the cash flow will the variance be minimized.

The covariance of the combination with the financial instrument’s payoff is

cov(˜ P˜, P˜)cov(C˜, P˜) cov(P˜, P˜)cov(C˜, P˜) var(P˜)

C

This covariance is zero when

cov(˜, P˜)var(P˜)

C

or when the number of units of the financial instrument that are sold, b, satisfies

cov(˜, P˜)

C

var(˜)

P

To show the same result with calculus, set the derivative with respect to bof the

variance expression, equation (22.4), to zero and solve for b. The resulting derivative

2var(˜) 2cov(C˜, P˜)

P

is zero when

˜˜

cov(C, P)

˜

var(P)

This is the same result that we achieved more intuitively above.

Note that the that gives the minimum variance hedge is the regression coeffi-

cient (see Chapter 5). This suggests that the minimum variance hedge ratio can be found

by using historical data to estimate regression coefficients. Example 22.17 illustrates

how to use regression to determine hedge ratios after estimating slope coefficients from

historical data.¹⁵

Example 22.17:Using Regression to Determine the Hedge Ratio

Consider the problem of using one month crude oil futures to hedge a heating-oil contract

to deliver 2.5 million barrels of heating oil one year from now at a prespecified price.His-

torical data suggest that for every $1.00 change in the one-month futures price of crude oil,

¹⁵˜

Alternatively, managerial forecasts of the cash flows contingent on different values of Pcould be

used to estimate the slope coefficient.

Grinblatt1629Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1629Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

809

the “year ahead”heating-oil prices change by $0.75.How many futures contracts should the

heating-oil company use to hedge the 2.5 million barrel heating-oil obligation?

Answer:Since the regression coefficient is .75, and there are 2.5 million barrels for deliv-

ery, one should buy contracts to receive 1.875 million barrels (2.5 million .75 barrels)

of crude oil.

There are many things that can be learned from regression estimates. For example,

commodities tend to have a term structure of volatilities much like the term structure

of interest rates in Chapter 10. There are often regular patterns to these volatilities. The

ten-year maturity oil forward price for instance, when regressed against the one-year

maturity oil forward price has a regression coefficient of about .5, indicating that long-

maturing oil forward prices are less volatile than short-maturing forwards. This reflects

the fact that when oil prices have historically been high, they tend to decline, and when

low, they tend to rise.

Cross Hedging.Regression techniques also apply to cross-hedging, which is hedg-

ing across different commodities. Crude oil and heating oil are similar but not identi-

cal. Hence, it is possible to find the best hedge of heating oil using crude oil futures,

or to hedge the value of a lemon crop with orange juice futures, or whatever com-

modity one selects as the financial instrument for hedging.

Basis Risk.In addition, the regression method automatically takes account of basis

risk in coming up with the variance-minimizing hedge. Generally, the more basis risk

there is in the financial instrument, the smaller the hedge ratio.

Hedging with Multiple Regression

It is also possible to use two or more financial instruments to hedge a risk. For exam-

ple, Neuberger (1999), studying long-dated oil price obligations from 1986 to 1994,

used an elaboration of the regression technique to suggest that such obligations are best

hedged, per barrel, by selling futures contracts to deliver 1.839 barrels of oil seven

months ahead and buying contracts to deliver 0.84 barrels of oil six months ahead. He

found that using futures contracts of multiple maturities in this way improves the vari-

ance reduction of the hedge dramatically.

Example 22.18 illustrates how to use multiple regression to determine hedge ratios

for multiple hedging instruments.

Example 22.18:Using Multiple Regression to Determine the Hedge Ratios

Suppose we have estimated a regression equation for General Motors (GM) in which the

left-hand variable is the quarterly change in GM’s profits in millions of dollars and the right-

hand variables are the quarterly change in the three-month ¥/US$ exchange rate futures

price (currently 120) and the quarterly change in the three-month S&P futures price (cur-

rently 1400).The coefficient on the first right-hand variable is 70, and the coefficient on the

second variable is 2.Interpret these coefficients in terms of risk exposures and discuss their

implications for the minimum variance hedge ratio.

Answer:The coefficient of 70 on the exchange rate implies that (holding the S&P 500

fixed) for each yen increase in the yen/dollar exchange rate (for example, the dollar increases

in value from 120 to 121 yen per US$), GM’s quarterly profits increase by $70 million, on

average.The coefficient of 2 on the S&P 500 implies that (holding the ¥/US$ exchange rate

fixed) for each one-unit increase in the S&P 500 futures (for example, the three-month S&P

Grinblatt1631Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1631Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

810Part VIRisk Management

500 futures moves from 1400 to 1401), GM’s quarterly profits are expected to increase by

$2 million.To hedge this risk with the two futures instruments, GM should enter into futures

contracts to sell 70 million yen, as well as sell futures contracts on 2 million units of the S&P

500 index.