22.10Minimum Variance Portfolios and Mean-Variance Analysis

So far, we have assumed that the real investments of the corporation are fixed and

have examined positions in derivative securities that minimize the corporation’s risk.

There are cases, however, where the corporation can alter the scale of its real invest-

ments as well as its derivatives position. In these cases, the variance-minimizing

hedge ratio can differ from the hedge ratio identified with regression analysis. As we

discuss below, the regression approach provides the variance-minimizing hedge ratio

in instances where the hedging instrument is a contract, like a forward or a futures

contract, that has zero value. However, when the hedging instrument is costly and the

real investment can be scaled up or down in size, the relative weights in the minimum

varianceportfolio differ from the hedge ratio identified with regression analysis.

Hedging to Arrive at the Minimum Variance Portfolio

As Chapter 4 shows, the minimum variance portfolio of a set of stocks is the portfo-

lio with a return that has the same covariance with the returns of each of its compo-

nent stocks. This idea is easily extended to hedging a real asset with either a costly or

a costless financial instrument that is used for hedging. In the case of a costless finan-

cial instrument, like a swap or a forward contract, the hedge ratio provided by the min-

imum variance portfolio from mean-variance analysis is identical to the hedge ratio

from regression. That is, hedging with regression can be viewed as a special case of

minimizing variance hedging when the hedging instrument is costless.

There are situations, however, where the financial instrument is costly and the cor-

poration has flexibility in the size of its real asset. For example, a corporation that

wishes to combine the purchase of oil refineries with oil-linked bonds would minimize

the variance of that combination by forming a portfolio that has the same covariance

with the return of oil refineries and the return of oil-linked bonds. The associated

hedgeratio would be appropriate in cases where the size of the oil refinery position in

the portfolio is variable. Similarly, an airplane manufacturer of jet airplanes who wishes

to hedge the sales price of these airplanes by acquiring stock in an airline company

would use the same approach. As we show in Example 22.19 below, these minimum

variance combinations will not generally be the same as the hedge positions obtained

with regression analysis.

Example 22.19:Minimum Variance Hedge Ratios from Mean-Variance

Analysis

Assume that Boeing is thinking about building a factory to manufacture its new Boeing 777

line.The company has $1 billion to spend on the factory or on financial investments.The

margin per 777 plane, that is, the selling price less variable costs, is an uncertain cash flow

that will be captured five years from now.The fixed costs of building the factory, a cash out-

flow that occurs today, are proportionate to the output of the factory.With such a scalable

production technology, the factory size required to produce 100Q planes costs $Q billion.

This means that Boeing can purchase a factory for $1 billion that will produce 100 planes

five years from now.Alternatively, it can build a factory that will produce twice as many planes

Grinblatt1633Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1633Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

811

(200) at twice the fixed costs ($2 billion), half as many planes (50) at half the fixed costs

($500 million), 1 plane for $10 million in fixed costs, etc.After analyzing the historical sales

margin of a Boeing 777 plane against the return of AMR stock (the parent of American Air-

lines), Boeing’s analysts find that the regression coefficient is $20 million.This means that

when AMR’s stock price increases by 1 percent, the sales margin of each Boeing 777

declines by $200,000 on average.Assuming that each plane’s sales margin five years from

now has a standard deviation of $200,000 and the five-year variance of AMR’s stock return

is .75, identify the minimum variance portfolio combination of AMR stock and the factory,

and interpret the result.

Answer:The covariance between AMR’s (decimal) stock return and the Boeing 777 plane

sales margin per dollar invested is

$20 million

1.5 var (AMR return)( 2) (.75)

$10 million

The variance of the sales price per $1 invested in the factory is

4^$20 million 100Q ²

$Q billion

The covariance of a minimum variance portfolio with the fraction xinvested in the factory

and the fraction 1 xin AMR stock has xsatisfying the equation that the covariance of the

portfolio with the factory return and the AMR return is the same;that is

4x 1.5(1 x) 1.5x .75(1 x)

This is solved by x.29, and it implies that if Boeing has $1 billion to spend and the fac-

tory is scalable, the minimum variance portfolio combination is achieved by making a fac-

tory that produces 29 planes at a cost of $290 million and buying $710 million of AMR stock.

The hedging solution in Example 22.19 differs from the regression solution. The

regression solution has Boeing buy $2Q billion of AMR stock for each $Q billion spent

on the factory (as a fixed cost) and results in a 2-to-1 hedge ratio instead of the 2.45-

to-1 hedge ratio (710/290) of Example 22.19. The former ratio produces a cash flow

that has no correlation with AMR’s stock return and makes any further reduction in

variance impossible. This would be an appropriate hedge ratio if capital expenditures

were not constrained by the investment in the hedging instrument. When costless finan-

cial instruments for hedging are not available, as in the case of Boeing’s factory, man-

agers need to think about downsizing the project as a hedging vehicle rather than using

regression blindly. However, if the size of the position in the real asset is not alterable,

then the hedge ratio provided by the mean-variance method is inappropriate.

Hedging to Arrive at the Tangency Portfolio

Managers implement real investments whenever they have higher mean returns than

the financial instruments that track them. In the case of Boeing, $10 million in pro-

duction costs per plane seems inexpensive compared to the cost of the AMR stock that

tracks the future of a 777 series aircraft, which is usually obtained by analyzing a

regression coefficient like the 2 that would be computed in Example 22.19. This is

not an argument for regression-based hedge ratios as much as it is an argument for

bringing mean returns into the picture. As in investment theory (see Chapter 5), man-

agers trade off mean and variance in their project and hedging decisions. In many

instances, the proper hedging goal should be the capital market line on the mean-

standard deviation diagram instead of some minimum variance criterion.

Grinblatt1635Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1635Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

812Part VIRisk Management

Example 22.20 illustrates how to achieve a hedged position on the capital market line.

Example 22.20:Tangency Portfolios from Mean-Variance Analysis

Given the data in Example 22.19:4, 1.5, and .75, where ˜repre-

P

CCCPPP

sents the AMR investment and Crepresents a Boeing 777 factory, identify the tangency

portfolio mix of factory and AMR stock if the expected sales margin of the 100 planes is

$5.25billion, the expected return of AMR stock over five years is 75 percent, and the risk-

free rate over five years is 25 percent (or about 5 percent per year).

Answer:Following the analysis in Chapter 5, the equations that solve for the tangency

portfolio satisfy the property that the ratio of the covariances of the investment pair are equal

to the ratio of the expected excess returns

4x 1.5 (1 x)5.25 1 .25

1.5x.75 (1 x).75 .25

This is solved by x.32.Thus, if Boeing has $1 billion to spend and the factory is scala-

ble, the mean-variance efficient combination of factory and AMR stock is a $320 million fac-

tory that produces 32 planes and a purchase of $680 million of AMR stock.

The tangency portfolio hedge ratiois based on the idea that capital is fixed, the

financial instruments for hedging are given (for example, AMR stock), and the firm

has a trade-off between mean and variance where it wishes to mix the financial instru-

ment and the real investment in proportions that maximize the ratio of the expected

excess return of the portfolio to its standard deviation.

The discussion in this section is summarized as follows:

Result	22.8	Regression coefficients represent the hedge ratios that minimize variance given no capital
		expenditures constraints and no constraints on the use of costless financial instruments for
		hedging. Given flexibility in the scale of a real investment project, a cost to the hedging
		financial instrument, and a capital expenditure constraint, the techniques of mean-variance
		analysis for finding the efficient portfolio or the global minimum variance portfolio may be
		more appropriate for finding a hedge ratio.