OR Spectrum 28:289–299 (2006)
REGULAR A RT ICLE
Jörg Laitenberger .Andreas Löffler
The structure of the distributions
of cash flows and discount rates
in multiperiod valuation problems
Published online: 29 August 2005
© Springer-Verlag 2005
Abstract In capital budgeting problems future cash flows are discounted using the
expected one-period returns of the investment. In this paper we relate this approach
to the assumption that markets are free of arbitrage. Our goal is to uncover implicit
assumptions on the set of cash flow distributions that are suitable for the capital
budgeting method. Our results are twofold. First we obtain that for ...view middle of the document...
V. for financial support.
J. Laitenberger . A. Löffler (*)
Wirtschaftswissenschaftliche Fakultät, Universität Hannover,
Königsworther Platz 1, 30167 Hannover, Germany
E-mail: AL@wacc.de, JL@wacc.de
290 J. Laitenberger and A. Löffler
return of the investment. On the other hand it is well known that in arbitrage free
markets the value of a claim is given by the sum of its expected cash flows
discounted at the riskless interest rate, where the expectation is taken with respect
to the so-called risk neutral measure that is usually different from the subjective
probability measure of the investor. In this note we investigate the relationship
between the textbook approach and the assumption that markets are free of
arbitrage. Specifically, our goal is to uncover implicit assumptions on the set of
cash flow distributions on the one hand and the set of discount rates on the other
hand that are suitable to compute the net present value of an investment project.
If the discount rate is derived from an equilibrium model as the CAPM the
above problem reduces to the question under what assumptions a myopic valuation
principle can be applied. This problem was considered by Fama (1977), Myers and
Turnbull (1977), Sick (1986), Black (1988) and Franke (1984). In the context of the
CAPM the probability distributions of the one-period returns have to be normal.
The implications on the shape of the cash flow distributions are less well known.
Fama investigated the case of a single future cash flow some periods ahead.
Later he pointed out that in this case the distribution of cash flows tend to become
more and more skewed when the distributions of the one-period single returns are
roughly symmetric (see Fama 1996). Myers and Turnbull assume a specific pattern
of expectation formation that implies certain linearities in the distributions of future
cash flows. They show that in this context a constant value for the risk adjusted
discount rate is only obtained when the cash flows follow a pure random walk
process. Sick investigated additive or multiplicative cash flow processes. Black
assumed that both the cash flows of the project and the cash flows of the market
portfolio are joint normal.
In this note we take a more general view. We ask under the most general
conditions of arbitrage-free markets with rational expectations, what is the
acceptable uncertainty in the distribution of future cash flows that allows the
application of the net present value method. Specifically we tackle two problems.
On the one hand we prove that assumptions on the shape of return distributions
alone do not restrict the set of admissible cash flow distributions when no
restrictions are imposed on the shape of the distributions of the future firm values
and vice versa. Therefore the result of Fama (1996) that “distributions of payoffs
more than one period ahead are skewed right” cannot be generalized to multiperiod
budgeting problems. Our analysis...