11.1Tracking Portfolios and Real Asset Valuation

The discounted cash flow valuation formula is founded on the tracking portfolio

approach. The formula is a statement that the market price of a combination of finan-

cial investments that track the future cash flows of the project should be the same as

the value of the project’s future cash flows.

Although this method is fairly straightforward with riskless cash flows, it is much

more difficult to apply to risky projects. Aportfolio that perfectly tracks the cash flows

of a risky project exists only in special circumstances. One case might be an oil well

whose cash flows can be perfectly tracked by a portfolio of forward contracts on oil

and some investment in risk-free bonds. Another case is a copper mine, which can be

perfectly tracked by a portfolio of copper forward contracts and a risk-free investment.⁴

In most cases, however, there will be some tracking error;that is, a difference between

the cash flows of the tracking portfolio and the cash flows of the project. The analyst

³However, one practical issue of great importance, of which the reader needs to be aware, is not

addressed in this chapter: the issue of how corporate taxes affect the valuation of projects financed with

debt. Unless the reader is valuing only equity-financed projects or projects with no corporate tax

implications, we urge the study of Chapter 13, which addresses this topic.

⁴See Chapter 12.

Grinblatt760Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw760Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

374Part IIIValuing Real Assets

who wants to value a project in these cases needs to employ a theory that describes

how to generate tracking portfolios for which the tracking error has a present value of

zero. The next section elaborates on this point.

Asset Pricing Models and the Tracking Portfolio Approach

When tracking error exists, the analyst generally must turn to asset pricing models, like

the Capital Asset Pricing Model (CAPM) and the arbitrage pricing theory (APT), to

derive a project’s present value. Specifically, imperfect tracking portfolios can be used

for valuation purposes if the tracking error has zero present value, which is the case

only when the tracking error consists entirely of unsystematic or firm-specific risks.

An Example of How to Use Tracking Portfolios forValuation.Consider the cash

flows from the following hypothetical project, which, for simplicity, we initially assume

can be perfectly tracked by a mix of the market portfolio and the risk-free asset: Faced

with the possibility of legalized onshore gambling in Louisiana, the senior management

of Hilton Hotels wants to evaluate the prospects for a hotel/casino. To simplify the

analysis, focus only on the valuation of a single risky cash flow from the casino to be

received by Hilton one year from now. Assume further that this cash flow can take on

one of only three values:

•	$12.3 million in the good state (40 percent probability)
•	$11.3 million in the average state (40 percent probability)
•	$9.3 million in the bad state (20 percent probability)

as shown in the table below. Attached to these states are the future values of the mar-

ket portfolio per dollar invested.

StateProbabilityCash Flow Next Yearof

Hilton Hotel/Casino	Market Portfolio
(in $ millions)	(per $1 invested)

Good state	.4	$12.3	$1.40
Average state	.4	11.3	1.20
Bad state	.2	9.3	.80

If the risk-free rate is 6 percent, the future cash flow of the Hilton Hotel casino can be

perfectly tracked by a $10 million investment:

•	$5 million in a risk-free asset (worth $5.3 million one year from now) and
•	$5 million in the market portfolio.

Since a portfolio of financial assets worth $10 million tracks the cash flow of the casino,

the value of the casino’s future cash flow is $10 million.

Note that if one were discounting the casino’s expected cash flow to obtain a pres-

ent value, the appropriate discount rate must equal the discount rate for the tracking

portfolio’s expected cash flow, namely the expected return of the tracking portfolio.

This is a weighted average of the expected return of the market portfolio and the risk-

free return, where the weights correspond to the respective portfolio weights on the

Grinblatt762Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw762Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

375

market ( ) and risk-free asset (1 ) in the tracking portfolio.⁵Here, it turned out that

equals 0.5.

Tracking Errorand Present Values.The perfect tracking seen above arises because

we constructed an example where the return on the market portfolio is perfectly cor-

related with the casino cash flow. Changing any one of the three casino cash flows or

three market portfolio cash flows eliminates this perfect correlation. In this case, we

would find, at best, that only imperfect tracking is possible.

With imperfect tracking, the mix of the market portfolio and the risk-free asset that

best tracks the Hilton cash flow is one that minimizes the variance of the tracking error.

Much of this chapter is devoted to illustrating how to obtain this mix. For now, sim-

ply assume that the portfolio that best tracks the casino cash flow, in the sense of min-

imizing the variance of tracking error, is one that contains $5 million of the market

portfolio and $5 million of the risk-free asset.

As noted above, since the tracking portfolio has a $10 million value, the value of

the casino cash flow also should be $10 million. As Chapter 10 illustrated, this is obvi-

ous when there is perfect tracking, given the assumption of no arbitrage. It is useful to

review why this also is the case with imperfect tracking. First, valuation requires a

tracking portfolio with the same expected future cash flow as that of the real asset it

tracks. This implies that the tracking error, measured as the difference between the

tracking portfolio’s future value and the casino’s cash flow, has an expected value of

zero. Moreover, the tracking error from a properly designed tracking portfolio repre-

sents unsystematic (or diversifiable) risk. Whenever tracking error has no systematic

(or factor) risk and has zero expected value, it has zero present value and can be

ignored. Thus, we can often use a tracking portfolio’s market value as a fair represen-

tation of what a cash flow is worth, even when it tracks the cash flow imperfectly.

In general:

Result 11.1

Whenever a tracking portfolio for the future cash flows of a project generates tracking errorwith zero systematic (or factor) risk and zero expected value, the market value of the track-ing portfolio is the present value of the project’s future cash flows.