11.6Estimating Beta from Scenarios: The Certainty Equivalent Method

Hypothetical examples like Example 11.6 can illustrate the problem of using scenarios

with the risk-adjusted discount rate method. In analyzing a real-world project, however,

financial managers face a significant challenge whenever projects are mutually exclu-

sive and it is difficult to identify a comparison tracking portfolio. Managers do not

know the true PVunless they know the beta computed using a base cost that makes

the project a zero-NPVinvestment. However, they cannot know this true project return

beta unless they know the true PV.This section suggests a way out of this quandary.

The method introduced is also applicable in cases where the present value is not of the

same sign as the expected cash flow.

Defining the Certainty Equivalent Method

As noted earlier, the certainty equivalent method is closely related to the risk-adjusted

discount rate method. However, instead of discounting expected cash flows at risk-

adjusted discount rates, certainty equivalent cash flows are discounted using risk-free

interest rates.

To understand what a certainty equivalent cash flow is, consider a project that pays

off either $100, $200, or $300 next year, depending on the state of the economy. If

these three states of the economy are equally likely, the expected cash flow is $200.

Because of risk aversion, however, a project that paid $200 for certain would be pre-

ferred to this project. In other words, the certainty equivalent cash flow for this hypo-

thetical project is less than $200—the project’s expected cash flow. On the other hand,

we know that this risky project is more valuable than a project with a guaranteed pay-

off of $100, the project’s lowest cash flow. The certainty equivalent cash flow is thus

some certain amount between $100 and $200 that would make the manager indifferent

between taking the certain cash flow and taking the risky cash flow.²³Specifically, the

certainty equivalent of a risky cash flow paid at future date tis the riskless cash flow

paid at date tthat has the same present value as the risky cash flow.

Finding the present value of a stream of certainty equivalent cash flows is straight-

forward. Simply discount the certainty equivalent cash flows at the relevant risk-free

rates, exactly as riskless cash flows were discounted in Chapter 10. The certainty equiv-

alent methodfirst obtains the certainty equivalent of the future cash flow. It then dis-

counts the certainty equivalent back to date 0 at the risk-free rate.

The difference between the risk-adjusted discount rate method and the certainty

equivalent method is simply a matter of where the risk adjustment occurs. Recall that

²³In

rare cases, specifically projects with negative betas, the certainty equivalent may exceed the

expected cash flow.

Grinblatt822Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw822Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

404Part IIIValuing Real Assets

the present value of a cash flow can be expressed as the ratio of the projected cash

flow to 1 plus the discount rate. With the certainty equivalent method, the numerator

of that ratio, the certainty equivalent cash flow, is adjusted for risk and discounted at

a risk-free interest rate in the ratio’s denominator. By contrast, the risk-adjusted dis-

count rate method places the expected cash flow in the numerator and discounts it in

the ratio’s denominator at a risk-adjusted interest rate.Either method is theoretically

acceptable; it is appropriate to risk-adjust in either the numerator or the denominator

of the present value expression. The preferred method depends on the practical con-

siderations emphasized throughout the chapter.