4343 words - 18 pages

OR Spectrum 28:289–299 (2006)

DOI 10.1007/s00291-005-0001-8

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...

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