Short Course at Your Company
We have found that giving the short course in
a company can be an effective way to get a team understanding of the concepts
of investment science. This facilitates the acceptance of the methods
as the best way to view project design and other decisions with major
financial impact. If you are interested in having the short course in
your company, contact me.
Powerful Project Pricing
Pricing of a project fundamentally
rests on evaluation of the cash flows that the project will generate.
If these cash flows are known with certainty at the time of project conception
the value of the project can be determined by a discounted cash flow analysis.
But, the cash flow stream of most projects cannot be determined with certainty
ahead of time. Future cash flows are uncertain and this is why a different
pricing theory is needed.
Modifications of discounted cash
flow analysis are available, including the popular Capital Asset Pricing
Model (CAPM), which is based on the relation of the cash flows of the
project to an overall market index. The relation is summarized by the
parameter "beta" that is familiar to those in the finance field.
Although the CAPM is a benchmark for many theoretical discussions, its
use as a method for assigning value has been somewhat limited because
it is not easy to determine the value of beta for a project.
A more popular (and simpler)
method is the method known as comparables, which is intuitive, relatively
simple, and fairly reliable. In this method an asset's value is estimated
by comparing it with similar assets with known prices. For example, when
attempting to determine the value of a house, one generally looks at recent
sales of similar houses in the same neighborhood. Likewise, when evaluating
the value of a potential company acquisition, a firm finds comparable
companies that have recently been purchased. Indeed, the concept of using
comparables is one of the oldest and most accepted methods of price determination.
There appears to be a distinct
difference between the highly analytical method of the traditional CAPM
and the looser, more intuitive, method of comparables. However, our recent
work in investment science has forged a simple link between the two, making
the CAPM value easier and more intuitive, and making the method of comparables
more robust. This new method, called the correlation pricing formula,
is described below.
Correlation Pricing Formula
Suppose that an asset has its
payoff in exactly one year. The payoff is denoted y but it is uncertain.
It can be characterized by its expected (or mean) value E(y) and
its standard deviation
. A first approximation to the value at the current time of a payoff y
one year later is v = E(y)/R where R is the
total return on a risk free asset (such as a Tbill). This merely discounts
the mean value by the risk free rate.
To account for the fact that
y is uncertain and therefore risky, some people recommend discounting
by a higher rate, and often they suggest that the greater the uncertainty
(the ), the
greater the discount rate. However, the CAPM formula does not increase
the discount rate, it decreases the numerator. The exact formula is
In this formula M is the
payoff of a special "market" portfolio. Normally, it is taken
to be the payoff of a market index such as the S&P 500. The term cov(y,M)
is the covariance of y with M,
is the current price of the market portfolio, and is
the variance of M.
*
The important point of the CAPM is that the economic value of y
does not depend on the standard deviation but
rather on "beta" defined as
In particular, we note that if
y is independent of the market portfolio M, then =
0 and we find the simple formula v = E(y)/R.
In practice, it is difficult
to find for project
investments. For example, it would be difficult to evaluate a house by
determining its beta on the basis of the next year's value and some general
housing index. Likewise, it would difficult to determine the value of
a start up company by finding its beta relative to the S&P 500.
The correlation pricing formula
(CPF) has exactly the same form as the CAPM but with one critical difference.
The formula is
where now C is the payoff
of an existing priced (or marketed) asset most correlated with y.
For example, in pricing a house, C might be the payoff of the house
next door which sold last week. If we thought the two houses were virtually
identical, then y =C and the formula would reduce to .
If we thought that our house was about halfway in between two other houses
in terms of its value characteristics, then the price would be the average
of the prices of these two other houses.
Sometimes a more complex analysis
is required. For example, if we value a project by its cash flow, we need
to relate the cash flow of the project to that of the most correlated
priced asset (such as a company that has similar projects) or to an appropriate
combination of such assets. All of this is likely to be easier and more
precise than attempting to relate the new house or project to a general
market index.
The surprising and important
result is that this correlation pricing formula gives precisely the same
result as would the CAPM formula if both were computed exactly. That is,
if the exact beta of our project were known and entered into the CAPM
formula, and if the most correlated asset were known and related to the
project to find the CPF price, then both would agree. Hence, the correlation
pricing formula is not arbitrary. It is a solid link between the theoretical
CAPM and the intuitive and easily implemented method of comparables. The
method can be easily extended to the case where cash flows occur at several
periods.
Example. Your company has value
equal to
Value = Debt + Equity.
Suppose that debt represents
a fraction a of the value and equity represents a fraction 1  a.
Then the value V can be written as
to reflect the division into
debt and equity. One year from now the value will be
where R is the total risk
free return (which increases the debt) and S is the (random) return
on the equity which is the return on the stock.
Your company keeps a fraction of
its value in cash and invests a fraction 1 
in projects. Hence the value next year is
where P is the (random)
return on projects.
By equating the two expressions
for V' we find
and hence
.
Now suppose that you have a project
that is much like every other project in the firm; that is, its random
payoff Q is similar in character to other projects. In that case,
you can take the stock of your company to be a most correlated marketed
asset. The payoff of your project is of the form Q = qP and thus
the value of your project is according to the CPF equal to
Solving for q we have
Hence you may find the value
of your (typical) project by discounting its expected value by
As a special case, suppose that
a = =
0, which means that the company has no debt and all value is invested
in projects. In that case the discount factor in the denominator is E(S),
which means that you discount by the rate of growth of the stock. This
is, of course, exactly the principle of how market risk is discounted,
for in this special case the project is exactly copying the market in
terms of risk.
As another special case, suppose
that = 0 but
a is not zero. This means there is debt but all value is invested
in projects. In this case, the discount factor in the denominator is ,
which is termed the weighted average cost of capital WACC, reflecting
the value equation
A most interesting case is when
> a
which means that your company has more cash than debt. Lots of cash and
no debt is often considered the sign of a healthy company, and maybe it
is. But in the single period framework of this analysis, it means that
projects must be discounted more heavily than the WACC, and even more
heavily than the stock if there is no debt. The relatively small proportion
of projects must carry a heavy load, exhibiting high returns, to enable
the company's stock to grow appropriately when burdened by cash that earns
a low return.
Of course, if your project is
not of the same character as other projects you can modify this analysis
to find the appropriate discount factor.
Reference: Luenberger, David G. "A Correlation
Pricing Formula", Journal of Economic Dynamics and Control, 26, (2002),
11131126. v
