11A.1Estimation Errorand Denominator-Based Biases in Present Value

Estimates

Assume that the true cost of capital is rand the estimated cost of capital is re˜,the true cost

of capital plus an error where the expected error E(e˜)0.Then, if the expected cash flow is

˜

E(C),the expected present value estimate for the cash flow is

E(˜)

C

E(PV)E1re˜

This is approximately equal to the sum of the true present value

˜

E(C)

1r

and the product of the discounted expected cash flow and the discounted variance of the dis-

count rate estimation error

E(˜)var(e˜)

C

1r 1r

which generates an expected present value estimate of

˜˜)

E(C)var(e

E(PV)1

1r 1r

E˜

(C)

,

Since the first factor in brackets, is the true present value and the second factor,

1r

var(˜)

e

,

1is greater than 1, the expected present value estimate is larger than the true present

1r

value.

¹For example, data on the market value of debt are not readily available.

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Chapter 11

Investing in Risky Projects

419

The upward bias in the present value estimate arises because the denominator of a ratio is

estimated with error and the numerator tends to be positive. An upward adjustment to the dis-

count rate can eliminate the upward bias in the present value estimate. The degree of the adjust-

ment depends on one’s (admittedly ballpark) estimate for var(˜),the estimation error variance

e

of the cost of capital.

The recommended bias correction is to increase the estimated discount rate by an amount

equal to the estimated variance of the error divided by 1 plus the estimated discount rate;that

var(˜)

e

is, .The estimate of present value of the cash flow would then be

1r˜

e

E˜

(C)

var(˜)

e

1re˜

1re˜

Example 11A.1 illustrates the procedure.

Example 11A.1:Obtaining Unbiased PVs from an Unbiased Cost of Capital

Estimate

In Example 11.2, the three comparison firms for Marriott’s restaurant division were used to

generate 10.55 percent as the division’s cost of capital.Assume that this cost of capital esti-

mate has a standard deviation of 3 percent per year.What is the best discount rate to use

for a six-month cash flow for Marriott’s restaurant division?

Answer:If the standard deviation is .03 per year, the variance is .0009.Hence, it is appro-

priate to increase the estimated discount rate from 10.55 percent to

.09%

10.63%10.55%

1.1055

For reasons that will be explained in the next subsection, one should implement this upward

adjustment to the cost of capital adjustment only for short-horizon cash flows.