CORPORATE INVES1MENT AND DISCOUNTING RULES
Fischer Black
Sloan School of Management
Massachusetts Institute of Technology
February, 1983
TABLE OF CONTENTS
SUMMARY • • • • • • • • • • i
INTRODUCTION • • • • • • • • • • 1
DISCOUNT CONDITIONAL CASH FLOWS AT THE INTEREST RATE 2
GIVE ZERO VALUE TO SYHHETRIC CLAIMS Oil EXCESS RETURNS 5
DISCOUNT AFTER-TAX CASH FLOWS AT AN AFTER-TAX RATE 8
INVEST IN BONDS RATHER THAN STOCKS • • 10
USE DIFFERENTIAL CASH FLOWS • • • • 13
ASSUME llETAS ARE ZERO FOR OPTIONS • • • 15
TREAT ESTIMATED CASH FLOWS AS VERY UNCERTAIN • • 17
INVEST MORE ~IHERE THINGS HAVE GONE WELL • 19
USE MARKET PRICES WHEN AVAILABLE • • 20
HANDLE INFLATION CONSISTENTLY • • 23
USE PRO FORMA EARNINGS STATEMENTS
INSTEAD OF DISCOUNTED CASH FLOWS • 24
FOOTNOTES • • • • • • 26
REFERENCES • • • • 27
i
SUMMARY. The rules in this paper are designed to take some of the mystery and
complexity out of the process of making corporate investment decisions. For
example, there is no need to make explicit use of beta or of the discount rate
for a risky cash flow. Instead, we can estimate future cash flows under the
assumption that the return on the market portfolio is equal to interest in
every period prior to the cash flow, and discount these estimated cash flows
at the interest rate. ~e should use after-tax cash flows and an after-tax
interest rat ..... This rule suggests that investment in current coupon bonds is
a matter of indifference for a corporation, but that investment in stocks of
firms other than the corporation itself is not a desirable investment. When
the cash flows have option elements, we can take the expected value of the
truncated cash flows under the assumption that the underlying assets have zero
beta, and again discount at the interest rate. What makes this work is that
the growth rates assumed in projecting the cash flows appear again in the
discount rates for those cash flows, and cancel. Since it doesn't matter what
growth rate we use, we can make the simplest clloice. We can assume that
assets are expected to grow at the interest rate, either because the market
return is equal to interest in each period, or because the asset betas are
zero. We can then discount at the interest rate. Cash flows are hard to
estimate, so we should treat estimated cash flows as very uncertain.
Moreover, we should revise our estimates frequently as market prices and other
indicators of the future change. In general, we should invest more in areas
that have done well and less in areas that have done poorly. We should handle
inflation consistently, taking account of any effects that inflation may have
on the firm's taxes and keeping in mind that an inflation-increased interest
rate does not necessarily represent a cost that will decrease the present
value of a project. In the end, though, especially when terminal values are
important, it may turn out to be better to forget about discounted cash flow
methods entirely, and use a pro forma earnings statement approach to
evaluating corporate investments.
INTRODUCTION
CORPORATE INVESTMENT AND DISCOUNTING RULES*
t Fischer Black
Corporate investment decisions are often made by simple intuitive
methods.
For example, when a machine breaks down on your assembly line, you often
go ahead and make the investment needed to fix it. You do not spend time
on a complex analysis of the cash flows resulting from a fixed machine.
When analytic methods are used, they may take the form of a payback
analysis, a discounted cash flow analysis, or a projection of the future
earnings from a project which are then converted to an estimated value
for the project.
When the future cash flows from a project are very uncertain, as they
often are, it may not make much difference which of these methods are
used. By changing its estimated cash flows, the analyst can make the
project look good or bad under any of these methods, depending on how he
feels about it.
Thus, when analytic methods are used, it seems best to make them as
simple as possible, so that the important factors in the investment
decision will not be lost in the analysis. To this end, here is a list
of rules for evaluating corporate investments.
2
DISCOUNT CONDITIONAL CASH FLOWS AT THE INTEREST RATE
How are we to discount future cash flows from an actual or proposed
1
investment?
When the cash flows are certain, we discount them at a riskless rate with
an appropriate maturity. We can estimate the discount rate by looking at
the prices and yields of government bonds at various maturities, making
allowances for tax factors and call features.
But what if the future cash flows are uncertain? For example, suppose
that a firm is considering an investment with a cash payoff at the end of
each year that depends on the market return for that year. IVhen the
market return is 10 percent, the payoff for the year is 3150,000. IVhen
the market return is 20 percent, the payoff is $250,000. The payoff goes
up or down by 310,000 for each percentage point increase or decrease in
the market return for the year.
Each year the payoff will depend on the market return for that year but
will not depend on the market returns for prior years or on anything
else. The payoffs will continue to follow this pattern each year into
the indefinite future. Moreover, let's assume that the short-term
interest rate is constant at 10 percent per year and that the expected
return on the market is somehow known to be constant at 20 percent per
year.
What is the present value of the payoffs from this investment?
One procedure is to find the expected cash flows from the project and to
discount them at rates that depend on the betas of the cash flows and on
the expected return on the market. This procedure is correct only when
the betas of the cash flows are constant in time, and it requires that we
estimate several things that are hard to estimate. We must use the
expected cash flow, the cash flow beta, and the expected return on the
market.
3
When this procedure works, there is a simpler procedure that will also
"ork. The simpler procedure will sometimes work when the more complex
procedure does not work. The simpler procedure involves estimating the
cash flows under the assumption that the market return is equal to
interest in each period and then discounting the estimated cash flows at
the interest rate.
Let's go back to our example. Since the natural period of analysis for
this example is a year, interest for the period is equal to the interest
rate, or 10%. When the market return is equal to 10%, the payoff each
year will be $150,000. Since the interest rate is 10% each year, the
present value of an indefinite stream of payments, each equal to
$150,000, is $1.5 million. That's the present value of the investment.
We might describe the $150,000 as the certainty equivalent of the payoff
from this investment each year. Then the present value of the investment
is simply the stream of certainty equivalents discounted at the interest
rate.
Since the expected return on the market may change rapidly over time, and
since it is very hard to estimate what it is at any given point in time,
it is good that our rule does not need to make use of it. Similarly, the
beta of the project or its cash flows may be hard to estimate, and may
change over time. So it is nice to have a rule that does not require the
use of beta.
This simple discounting rule works because both the expected future cash
flows from the project and the discount rate for those cash flows depend
on the betas and the expected return on the market. They depend on these
factors in just such a way that the discounting process cancels the
effects of these factors on the expected cash flows. If a discounting
process is used that is consistent with the way in which the expected
cash flows are estimated, then it will give the same ans"er as the simple
procedure because the betas and market expected returns will cancel.
4
Normally, it seems simpler to estimate cash flows assuming that the
market return is equal to interest each period than to estimate expected
cash flows for all possible paths of the return on the market. Thus, the
simple discounting rule uses inputs that are easier to obtain than the
inputs to more complex rules.
5
GIVE ZERO VALUE TO SYHMETRIC CLAIMS ON EXCESS RETURNS
Suppose the government proposes a tax on the returns from a portfolio
that takes the following form. Each year a return on the portfolio is to
be calculated. The investor will subtract interest at the one year rate
to give the excess return on the portfolio. This excess return may be
either positive or negative. The tax is to be a constant fraction of the
excess return.
It is symmetric, because when the excess return is negative, the
government will give money back to the investor. So the government takes
some of the gains on the portfolio and gives back some of the losses.
How should an investor feel about a tax that takes this form?
Normally, an investor should be indifferent to this kind of a tax, at
least when the investor is lending or is able to obtain a secured loan at
a rate about equal to the one-year interest rate. The government is
taking some of the investor's expected return, but is also taking some of
the investor's risk. The investor can offset this by simply starting
with a larger position in the same portfolio.
For example, with taxes at a 50% rate, the investor can simply double the
size of the initial portfolio and the after-tax returns to the investor
will be the same as if there were no tax on the portfolio.
Since the investor must reduce lending or increase borrowing in order to
increase the portfolio of risky assets, it is important that the interest
rate used in the tax calculation is the interest rate the investor faces
on lending or additional borrowing. When it is, the present value of the
tax is zero.
Another way to see this is to note that an investor can create a
portfolio with cash flows equal to the tax by borrowing to buy an
appropriate amount of the original portfolio. Since the investor puts up
no equity to obtain this portfolio, the present value of the cash flows
from the portfolio must be zero.
With this kind of a tax,
government. Although it
the
has
6
investor
positive
is indifferent and so is
expected revenues, it
the
will
frequently require payments to taxpayers rather than payments from
taxpayers.
Since it has a zero present value, it cannot be used to support
government spending. It can only be used to make transfers between
people who are taking more risk and people who are taking less risk in
their investments.
For another example, suppose that an investment advisor has an agreement
under which he receives one-half of one percent per year of the value of
the portfolio being managed. Does the investment advisor have an
incentive to increase the expected value of the advisory fee by
2
increasing the beta of the portfolio?
It turns out that he does not have such an incentive, because the present
value of the advisory fee does not depend on the beta of the portfolio.
Using the simple discounting rule described above, we estimate the
advisory fees under the assumption that the return on the market is equal
to interest each year. The future value of the portfolio when the return
on the market is equal to interest each year does not depend on the beta
of the portfolio.
Thus the present value of the advisory fee does not depend on the beta of
the portfolio. A higher beta will mean a higher expected future advisory
fee, but also a higher discount rate. The effects of beta on the
expected value of the future advisory fees and on the discounts rates
will cancel, leaving a present value for the advisory fee that is
independent of beta.
Again, this assumes that the advisor is lending or can borrow at a rate
equal to the lending rate. If the advisor's borrowing rate is higher
than his lending rate, and if he wants to take on more risk, taking on
7
more risk through the portfolio he has managing may be a more efficient
way to take risk than borrowing to buy risky assets himself.
Once we see that symmetric claims on excess returns have zero present
value, we have an aiternate way to derive the simple discounting rule.
Suppose each future cash flow can be written as a portion that will occur
if the market is equal to interest each period, plus a portion that is
proportional to the excess returns on the market in one or more past
periods. The first portion can be discounted at the interest rate and
the second portion has a zero present value. Thus we can use the simple
discounting rule on cash flows like these.
A rule that is equivalent when discounting cash flows like these is to
assume that the beta of each cash flow is zero for each period. We
project the cash flows assuming zero betas, and then we discount them at
the interest rate. When the cash flows are symmetric, this will be
equivalent to estimating the cash flows under the assumption that the
market return is equal to interest each period and then discounting at
the interest rate.
Note that this means a cash flow with a positive expected value can have
a zero present value. In the tax example, the expected future taxes are
positive, but they have a zero present value. Standard discounting
procedures, because they assume a constant discount rate, would always
give positive present values for cash flows like these and thus would
give the wrong answers.
8
VISCOUNT AFTER-TAX CASH FLOWS AT AN AFTER-TAX RATE
Until now, we have ignored taxes. For any method of discounting cash
flows, it would seem natural to discount after-tax cash flows at an
3
after-tax rate, and that turns out in fact to be the right method.
The taxes that matter are corporate taxes, not personal taxes. So the
after-tax rate is the interest rate after corporate taxes. This rule
works no matter what we assume about personal taxes.
For example, we might assume that personal tax rates and corporate tax
rates are equal, and we might look at a model in which firms do not pay
dividends and there are no effective taxes on capital gains. In
addition, we might assume that bankruptcy costs and other factors
affecting optimal corporate assets and liabilities are zero.
In a model like this, capital structure will be a matter of indifference,
and the after-tax interest rate can be taken indifferently to be the rate
after corporate taxes or the rate after personal taxes. In this model,
it makes no difference whether a firm buys its own debt or not, and it
makes no difference if the firm buys debt of others. The present value
of current coupon debt is zero.
Consider, on the other hand, a model in which personal taxes are zero.
If we continue to assume that bankruptcy costs and other complicating
factors are not present, then a firm will want to have as much debt as it
can, since additional debt reduces corporate taxes without increasing
personal taxes. If the corporate tax is to be effective, we must assume
that there is a limit on the amount of debt a firm can have.
It is natural to assume this limit is related to the risk of the firm's
equity. A firm will be allowed to increase its debt until the risk of
its equity reaches a standard point. If the limit is set in this way,
then a firm that buys the debt of others will be able to increase the
amount of its own debt by an equal amount.
9
Thus, buying the debt of others will be a matter of indifference for a
firm, even though buying back its own debt and putting its total debt
below the limit would not be a matter of indifference.
In either kind of model, then, debt will have a present value of zero.
Some people say that the right way to take taxes into account is to
discount after-tax cash flows at a before-tax interest rate and then take
into account the present value of interest tax shields. While this may
give the same answer as the method we have set out if the tax shields ~~e
computed properly, it is clearly more complex to use.
So it seems that discounting at the after-tax rate is the simplest way of
handling such a cash flow. If we are using the simple discounting rule,
then we will start by estimating future after-tax cash flows assuming
that the return on the market in each period is equal to interest at the
after-tax interest rate. We will then discount these estimated cash
flows at the after-tax interest rate.
If we were to use the capital asset pricing model in discounting, we
would want to use the form of it that has the after-tax interest rate
everywhere where the before-tax interest rate might otherwise appear. If
we do that properly, we will get the same present value using the capital
asset pricing model that we get using the simple discounting rule.
10
U,VEST IN BONDS RATHER THAN STOCKS
In the last section, we considered two simple models of capital
structure. One had personal taxes at the corporate tax rate, while the
other had no personal taxes. Neither had any bankruptcy costs or related
factors affecting optimal corporate assets and liabilities.
Now let's talk about a more realistic model of capital structure. Let's
assume that the capital structure of a firm is influenced by corporate
taxes, by personal taxes, by the possibility of bankruptcy, by the way in
which incentives for managers are affected, by the degree of
diversification in the firm and the way diversification affects both
managers and lenders, and by the costs associated with diffuse ownership
of the firm's stock.
Higher corporate taxes will increase the optimum amount of debt for a
firm, while higher personal taxes will decrease the optimum amount of
debt.
Higher costs of bankruptcy and higher costs of avoiding bankruptcy will
reduce the optimal amount of debt for a firm. When we consider
management incentives, we may want to have a higher amount of debt than
we would otherwise have because managers will want to avoid bankruptcy
due to its effects on their personal wealth and reputation.
Of course, the more diversified a firm is, the more debt it can support,
at least if this diversification is through its businesses rather than
through ownership of the stock of other firms.
Diffuse ownership of the company's stock has costs because small owners
of the stock have little incentive to exert the effort and spend the
money needed to exercise responsible control over the firm. The more
debt and the less stock a firm has, the less diffuse its ownership will
be, other things equal. Thus, this factor favors a higher amount of debt
for the firm.
11
Taking all these factors into account, there will be some optimal capital
structure for the firm. In a more general context, we might say that
there is an optimal asset and liability structure for the firm.
At the optimal capital structure, small changes in the firm's debt-equity
ratio will have no effect on the value of the firm or the value of its
stock. Thus, issuing stock to buy back debt and issuing debt to buy back
stock will not affect the value of the firm or the value of the stock if
done in small amounts. We assume here that the debt bought and sold in
small amounts is of lower priority than the firm's existing debt.
If buying back small amounts of the firm's own debt is a matter of
indifference, then buying the debt of other firms will be a matter of
indifference too. This becomes a very general proposition in the current
context, and does not depend on any particular simple model of capital
structure.
If the firm is at an optimal capital structure, then current coupon bonds
will be a zero net present value investment.
Similarly, the firm will be indifferent to buying small amounts of its
own stock. Its stock will be a zero net present value investment.
However, buying the stock of other firms is worse from a tax point of
view than buying the firm's own stock. The firm will have to pay taxes on
15% of its dividend income, and it may be subject to capital gains taxes
as well. Thus, the stock of another firm will generally be a negative
net present value investment, except in cases where the firm is preparing
for a tender offer or merger proposal.
This is consistent with the fact that we see large numbers of financial
insti tutions holding debt in the form of loans, while firms other than
investment companies rarely hold the stock of other firms. Investment
companies, of course, have special tax rules that reduce or eliminate the
tax costs of holding shares of other firms.
12
Thus, any firm can be comfortable buying current coupon bonds of any
maturity but should be cautious about owning the stocks of other firms as
an investment.
13
USE DIFFERENTIAL CASH FLO\;S
One way for you to simplify the process of estimating a present value for
a project is to compare the project with some other project, either in
your firm or in another firm.
For example, if another firm in your industry is doing something
successfully, and you feel that you can do what they are doing at a lower
cost, then it seems very likely that you should do it. Your cash flows
will be higher than theirs, so if it is successful for them, it should be
successful for you too. Intuitively, the idea is to do things in which
you have an advantage over others that are doing the same thing
successfully.
Similarly, suppose that you are looking at a potential merger candidate.
If the firm has traded stock, the market price of the firm's stock
reflects investors' opinions about the cash flows of the firm in its
current form. Unless you have strong reasons for believing that
investors are mistaken, you should take the market price as a correct
estimate of the present value of the firm's cash flows as it now exists.
Presumably, the firm is a merger candidate because you believe you can
change its operations in some way, such as integrating them with your
operations, which will increase the firm's cash flows. If you find the
present value of the increased cash flows, that will suggest the amount
that you should be willing to pay as a maximum premium to acquire this
firm.
For another example, suppose you have a division that is not doing very
well and you decide you'd like to consider selling the division. The
price you get will normally depend on the present value of the division's
cash flows in the hands of the buyer.
What makes a division a good candidate for sale is that the cash flows in
the hands of someone else will be much higher than the cash flows in your
hands. You may be able to get, through the sale price, a share of the
present value of these increased cash flows.
14
However, if your firm is better able to handle the division than other
potential buyers, it doesn't make sense to sell it no matter how poorly
it is doing. It may pay to liquidate it, but if you can do better with
the division than anyone else can, it will not generally pay to sell it.
Thus, some of the factors often considered in buying and selling
divisions are not really relevant. Past growth in the division's sales
is not relevant in itself, because the price paid will reflect that past
growth and any future growth that is expected. The division' s market
share is not relevant because the higher the market share, the higher its
price will be.
Not even the skill of the managers is relevant, as a first
approximation. If they stay with the division, they will generally want
compensation reflecting their contributions. If their compensation fully
reflects their contributions, there won't be any left for you.
A decision on the optimal size of a firm is another case where the use of
differential cash flow analysis may pay. An expansion that increases the
size of the firm will generally have both positive and negative effects.
As the size of the firm increases, the differential effects of further
expansion will generally turn less positive or more negative. When the
differential effects of the expansion are zero and about to turn
negative, the firm has reached its optimal size, at least for the moment.
15
ASSUME BETAS ARE ZERO FOR OPTIONS
The simple discounting rule, as described above, says you should estimate
cash flows assuming the market return is equal to interest each period,
and then discount them all at the interest rate.
The simple discounting rule is equivalent to the following rule. Assume
the growth rates for the cash flows are what they would be if the betas
of the underlying assets were zero.
assumption and then discount at
Estimate the cash flows under this
4
the interest rate. This is
equivalent for symmetric cash flows to estimating the cash flows under
the assumption that the return on the market is equal to interest in each
period prior to the cash flow.
The simple discounting rule in its original form would not work on cash
flows that have option elements, because they are not symmetric.
For example, an option to buy the market in one year at a price that is
higher than today's price by more than the interest rate will have value
today. Under the simple discounting rule, it would appear not to have
value. If the market return is only at the interest rate, the option
will end up out of the money, so it will be worthless at year end.
What will work for this option is to assume that the expected return on
the market is equal to the interest rate, while the variance of the
market remains unchanged. We look at the possible values of the option
at the time it expires under this assumption, take the expected value of
that range of possible values, and then discount the result at the
interest rate.
Thus we find the distribution for the market return under the assumption
that the market has a zero beta, truncate that distribution, take the
expected value of the truncated distribution, and discount at the
interest rate.
16
Using this method, the total risk of the asset underlying an option will
affect the value of the option, but the beta of the asset, which will
always be taken to be zero, will not affec t the value of the opt ion.
This is consistent with what we know from option pricing theory, where
total risk matters in finding the price of an option, but beta does not
matter.
For a more down-to-earth example, assume that we are looking at a movie
production firm and the firm is thinking about the fact that a new movie,
if successful, will probably be followed by a sequel. The firm has an
option to produce the sequel if the first movie is successful.
The success of the first movie depends on economic conditions generally
and therefore will be correlated with the return on the market. In
valuing the option to make the sequel, though, we will assume that the
returns from the first movie will be independent of the market. We will
hold the total risk of the first movie the same, but will assume that its
returns are independent of the return on the market, and thus that it has
zero beta. We will find the possible future values of the option to make
the sequel under this assumption, and will then discount the expected
value of that option at the interest rate.
17
TREAT ESTIMATED CASH FLOWS AS VERY UNCERTAIN
People worry a lot about how to discount cash flows. In general, though,
uncertainty about what the cash flows will be dominates uncertainty about
how to discount them.
For example, Apple Computer recently introduced a new model called Lisa.
It has many features that were not in the company's existing computers,
including a substantial amount of built-in software and a much fancier
display unit. As a result, it is priced higher than competitive machines
such as those produced by IBM. Finally, it is aimed at the business
market, whereas Apple's previous computers were used more in the home
market than in the business market and were used more in small businesses
than in large businesses.
What will the future cash flows from this project be? The uncertainty is
enormous. It may be that large firms will snap up the computer in large
numbers and it will be a smash hit. On the other hand, it may be that
competitors will swarm in with comparable machines at lower prices, or
that no machines like Lisa will sell well.
The uncertainty is so great that it hardly seems worth making detailed
projections of cash flows. One might as well rely on intuition.
In fact, in situations like this, one of the most common uses of analytic
methods for finding present values is to justify a decision that has
already been made. Once you have decided on a discounting method, you
Can construct your cash flows so that the project seems favorable or
unfavorable depending on which way you would like the decision to go.
With a project 1 ike Lisa, who is to say your projected cash flows are
wrong?
Another common problem with cash flows is overoptimism on the part of the
people projecting the cash flows.
18
Cash flows are usually estimated by lower level managers who are familiar
with the details of the production and marketing processes that will be
used. But while those managers are in the best position to make detailed
cash flow estimates, they are also personally involved with the
decision. The initial idea for the project probably came from those same
managers, so it is likely that for the average project under consideration,
the cash flows have been overestimated. The managers have already
thrown out as unpromising projects for which they underestimated the cash
flows.
By rights, cash flow estimates should be revised frequently.
Market prices are the present values of cash flows and market prices can
change substantially from day to day, from week to week, or from month to
month. It seems logical that cash flow estimates should change by about
the same proportion that market prices change. This will give present
values which jump around the way market prices do.
If the projects being estimated are small new projects, or involve
substantial option elements, then it is possible that the cash flows for
these projects should jump around eVen more than market prices do.
One should be suspicious of cash flow estimates that remain constant from
week to week when economic conditions relevant to the project are
changing substantially from week to week.
Yet another potential problem in estimating cash flows is simple ways of
extrapolating the past into the future. It is especially dangerous to
project rapid past growth into the indefinite future. Rapid growth
always levels out and sometimes reverses itself.
In sum, there is no reliable way to estimate future cash flows. Since
estimated cash flows are so uncertain, it doesn't pay to be too precise
in the rest of the process of making decisions about corporate
investments.
19
INVEST MORE WHERE THINGS HAVE GONE WELL
Although it doesn't pay to project rapid past growth into the indefinite
future, it does pay to invest more in sectors of the economy or in parts
of your company where things have gone well in the past. When a part of
your business has been doing well, it is likely that there are many
opportunities for further investment in the business that will be
profitable.
For example, the computer program VisiCalc has been an enormous success
for personal computers. When the first versions started to be
successful, the company that produced the program developed many
variations on it, and started writing much more elaborate documentation
than had been written in the past. These investments paid off in
additional sales of VisiCalc much more than investments in a typical new
program would have paid off.
Past success can be measured in many ways: by return on book equity in
the area under consideration, or by an increase in the market price of
the company's stock if most of its business is in that area. The stock
price, of course, is looking at the future as well as at the past. So it
may be an especially good indicator of companies for which further
investments are warranted.
A company whose stock price has been rising should probably increase its
estimated cash flows for projects that it has under consideration and a
company whose stock price has been falling should probably decrease its
estimated cash flows. Sometimes, of course, an area that has been doing
poorly simply needs major investments to pull it out of its slump. More
often, though, pouring new money after old would be a bad idea.
It's always important to watch out for the influence of luck. Success
can easily be more a matter of luck than of skill in picking areas to
invest in. But when an initial investment has been successful, even if
it was lucky, it will usually still pay to make further investments in
that area in order to maximize the benefits of the original lucky
investment.
20
USE MARKET PRICES WHEN AVAILABLE
Since cash flows are so difficult to estimate, it's important to use any
tools that may help us in making such estimates. Normally, it appears
that the market price of a firm's stock reflects most or all of the
available information about the firm and its future. Thus, if we went to
estimate the value of a firm that has traded stock, it's hard to beat
just taking the current market price of all the firm's stock.
We could estimate future cash flows and try to discount them, but unless
we have special information not reflected in the stock price, it is
unlikely that we are going to do better than just taking the total value
of the firm's stock.
If we do have special information, the best way to Use it is probably to
estimate the present value of the special factors that we believe are not
reflected in the stock price, and then add those to the firm's stock
price. In doing this, though, it's important tha t we try to assure
ourselves that the special factors that we take into account are really
not known by the market. If they are known, then it may be tha t the
effect of these special factors is offset by other special factors that
we don't know about.
If we are trying to value something less than a whole firm, such as an
oil property or a piece of real estate, then it is more difficult to use
the market price of the firm's stock as a guide. However, if similar
properties have sold in the recent past, then we can look a t the
attributes of those properties, compare them with the property we are
interested in, and estimate the value that our property would have if
sold under similar circumstances.
This is probably a more reliable method than trying to project the future
cash flows of the property and discounting. Ivhen the property we want to
value differs from any property that has recently sold, however, we will
have to try to project at least the differential cash flows and discount
those.
21
If we want to estimate future interest rates, then there is no need to
ask economists. The best estimates come from the market interest rates
on government bonds of various maturities.
We can make separate estimates of tax-exempt interest rates using
tax-exempt bonds. When working with tax-exempt bonds, though, it's
important to keep in mind that the interest rates we see are interest
rates on risky bonds. Tax-exempt bonds generally have a substantial risk
of default.
If we want to estimate the future prices for agricultural commodities,
then there is no need to go to an agricultural economist. We can simply
use the futures or forward prices for those commodities.
For most applications, we can use the futures prices even if the expected
prices for the commodities are known not to be the expected values of the
futures prices. If commodities futures prices are generally higher than
the spot prices they are estimating because of some sort of risk premium,
then that same risk premium will come into a discounting process whenever
the futures prices are used as part of a procedure to estimate the
present value of the cash flows. The discounting will cancel the
premiums,
Thus, we can generally use the futures prices as if they were known
future spot prices.
Another use for market prices is in revising estimates of future cash
flows. For this we can use the level of the stock market as a whole and
the price of our own firm's stock. If we are working with a proposed
project that is more closely related to another firm's business than to
our current business, then we can use the price of that firm's stock.
Whenever these prices rise, we will increase our estimated cash flows;
and whenever these prices fall, we will reduce our estimated cash flows.
The effect of this will be that when these market prices rise, we will be
22
more likely to go ahead with any given project, and when these market
prices fall, we will be less likely to go ahead.
Firms often base their decisions to issue stock on the PiE ratio in
relation to past PiE ratios, or the ratio of price to book value, perhaps
related to past plB ratios. Since accounting figures have many arbitrary
elements in them, these comparisons may not mean much in themselves.
But the net effect of these procedures is to make it less likely that the
firm will issue stock and go ahead with certain investments when the
stock price has been falling than when it has been rising. Thus, these
procedures do seem sensible, at least when they are explained in words a
little different than the words usually used.
For a concrete example, suppose that a firm is considering a project that
is squarely within its present line of business.
It might make sense to make the estimated cash flows for this project
linear functions of the firm's stock price. In fact, if it is a small
project or a new area within the firm's business, it might make sense to
have the cash flow go up and down in percentage terms by more than the
stock price does. If the cash flows are made to depend explicitly on the
stock price in this way, then it will be easy to update the cash flow
estimates as economic conditions change.
23
illU,DLE INFLATION CONSISTENTLY
Inflation and interest rates move consistently together. On average, a
one percent increase in inflation means a one percent increase in
interest rates, and a one percent decrease in inflation means a one
percent decrease in interest rates; though at times the change in
interest rates and the change in inflation may be far apart.
The relation betwen inflation and after-tax interest rates, though, is
quite different. The after-tax interest rate seems to move less than the
inflation rate, at least in the short run.
If cash flows are estimated in constant dollars, then we should discount
using real interest rates. Real after-tax interest rates are often
negative, however, we should be discounting using a negative real
interest rate. It may be easier to project the cash flows in actual
dollars and to discount them using nominal interest rates.
Using either method, if we ignore taxes, the rate of inflation should
have no effect on the present value of the new project. Higher inflation
will mean higher cash flows and a higher discount rate, and these t'lO
effects will cancel.
When we do take taxes into account, then a change in inflation may have
some effect on the present value of the project.
For example, some people believe that with higher inflation, corporate
taxes are increased more than taxes on other kinds of capital. This
means that projected after-tax corporate cash flows should go up by less
than the rate of inflation. After-tax interest rates will also be going
up at less than the rate of inflation, so it's not clear how inflation
will affect the present value of a project.
Others, however, argue that Congress is usually quick to change tax laws
to adjust to inflation and that any effect of inflation on corporate
taxes is apt to be minor and short-lived. There is also a possibility
that Congress will change the tax structure in the future in such a way
24
as to substantially reduce taxes on capital, including corporate taxes.
In fact, some argue that recent changes in the tax law have already
reduced corporate taxes to zero or below for most new projects. The
chance that such changes in corporate taxation will be made may well be
related to the inflaiion rate.
To some extent, we can see in the structure of interest rates on both
taxable and tax-exempt bonds the likelihood tha t the market places on
changes in tax laws of this sort. If projected interest rates on
tax-exempt bonds are about equal to projected interest rates on taxable
bonds of the same quality, then the market expects that taxable bonds
will not be taxed beyond the point in time at which the projected
interest rates are equal.
As a first approximation, then, changes in expected inflation should have
no impact on estimated present values. It is not clear, as a second
approximation, how inflation will be related to present values.
Perhaps the market's reaction to changes in the rate of inflation can
guide us toward a second approximation. When the market does not react
to unexpected changes in the rate of inflation, we can stay with the
first approximation. When the market does react, we can use the strength
and direction of its reaction as a guide.
In the recent past, higher inflation has been related to lower market
values. So long as that relation persists, we should probably reduce our
5
estimated cash flows whenever the rate of inflation goes up.
25
USE PRO FORMA EARNINGS STATEMENTS INSTEAD OF DISCOUNTED CASH FLOWS
Often, when discounting estimated cash flows, we don't treat purchase of
capital equipment or plant in the same way as other cash flows. We may
smooth out these purchases so that they create a more regular pattern of
cash flows.
But this is just the kind of smoothing that we do in estimating the
earnings of a firm, or the contribution to earnings of a project. If we
are going to do this kind of smoothing, why don't we go all the way and
use all the kinds of accrual accounting that are used by accountants in
producing financial statements?
If we do this, then we can evaluate a project by estimating a normalized
earnings figure and multiplying it by an appropriate price-earnings
ratio.
We often estimate a normalized earnings figure as of some point in the
future, so we will want to discount it to the present again using the
interest rate.
For a long-lived project, discounting cash flows usually involves
stopping at some moderately distant point, estimating a residual value
and then discounting the residual value along with the other cash flows.
The residual value is often estimated using some version of the earnings
method anyway. So why don't we go all the way to a pro forma earnings
approach rather than trying to work with cash flows at all?
In practice, I believe that working wi th a project's contribution to
earnings is as common a method of project evaluation as the use of the
discounted cash flow procedure.
For most projects, a pro forma earnings statement approach may, in fact,
be easier to use than a discounted cash flow approach; though if each is
properly done, they should give the same answer. Thus we may not have to
worry about how to estimate and discount cash flows, since we may not
want to discount cash flows at all.
26
FOOTNOTES
* This is based in part on Black (1982), which contains more detailed
analysis of the "simple discounting rule." I am grateful to Richard
Ruback for extensive discussions of several of these topics.
tS10an School of Management, Massachusetts Institute of Technology.
1
Compare my "simple discounting rule" with Ross (1978). In some
cases, my rule follows from his rule.
2
This example is analyzed correctly by Margrabe (1976).
3
Ruback (1982) shows that this is the right method.
,
This rule is derived by Cox and Ross (1976).
Modigliani and Cohn (1979) argue that this negative relation between
inflation and stock prices may have been due to inconsistent handling of
inflation by investors.
27
REFERENCES
Black, Fischer. "A Simple Discounting Rule." Unpublished memorandum,
November, 1982.
Cox, John C" and Stephen A. Ross. "The Valuation of Options for
Alternative Stochastic Processes." Journal of Financial Economics 3
CJanuary/l1arch, 1976): 145-166.
Hargrabe, William. "Alternative Investment Performance Fee Arrangements
and Implications for SEC Regulatory Policy." Bell Journal of
Economics 7 (Autumn 1976): 716-718.
Uodigliani, Franco, and Richard A. Cohn. "Inflation, Rational Valuation
and the Stock J.larket." Financial Analysts Journal 35 (l1arch/ April,
1979): 24-44.
Ross, Stephen A. "A Simple Approach to the Valuation of Risky Streams."
Journal of Business 51 (July, 1978): 453-475.
Ruback, Richard S. "Calculating the Present Value of Riskless Cash
Flows." Unpublished memorandum, September, 1982.