\documentclass[12pt,epsf]{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\parskip 10pt
\input{tcilatex}
\begin{document}
\baselineskip 16pt
\parindent 24pt
\parskip 10pt
\titlepage
\vspace*{12pt}
\begin{center}
{\large \textbf{Risk-Adjusted Discounting of Negative Cash Flows } \\[0pt]
}
July 18, 2003 \\[0pt]
\bigskip
Eric Rasmusen \\[0pt]
\bigskip
\textit{Abstract}
\end{center}
Be careful in choosing a risk-adjusted discount rate for negative cash flows. This paper is really just in the early notes stage right now.
\noindent
{\small \hspace*{20pt} Indiana University Foundation Professor,
Department of Business Economics and Public Policy,
Kelley School of
Business,Indiana University, BU 456, 1309 E. 10th Street,
Bloomington, Indiana, 47405-1701.
Office: (812) 855-9219. Fax: 812-855-3354.
Erasmuse@indiana.edu.\newline
Php.indiana.edu/$\sim$erasmuse,
Php.indiana.edu/$\sim$erasmuse/papers/negvalue\_rasmusen.pdf. }
\noindent {\small I thank ddd and Laurent Booth for their comments.
}
\newpage
\noindent
\textbf{I. Introduction}
Risk-adjusted interest rate versus the other method of converting riskless first, then discounting to present.
See what Dirk and what SHokcley have to say.
Relate to Real Options Literature.
Examples of what the textbooks say.
The risk-adjusted discount rate method is clearly wrong. BUt it has
survived, no doubt because it usually is good enough, given the
uncertainty over the numbers being discounted. Robichek and Myers
(1966) pointed out that it assumes that risk increases over time in a
particular way. To be more correct, a separate risk-adjusted discount
rate should be used for each period's cash flow. Or, each period's
cash flow should be put into present value, and then adjusted for risk
using the appropriate risk premium (note that these are two distinct
methods). One of these is known as the certainty equivalent approach.
Yet if the risk-adjusted discount rate approach is wrong, it is still
used, because of its convenience. It thereofre is useful to explore
conditions under which it is more reliable adn conditions under which
it is less reliable, even if it is never perfect and sometimes grossly
wrong.
I would like to point out a different problem, and one with perhaps
even greater potential to lead to wrong decisions--- even grossly
wrong decisions. The problem will immediately become apparent with an
example:
There are two states of the world, Good and Bad.
A company can pay \$100,000 now for product redesign which will
increase its sales by 10\%, an expected value of \$5,000/year forever.
This \$5,000 is risky, however, and is actually \$7000 in the good
state and \$3000 in the bad state. Thus, the cash flows of the project
are:
-100, +5, +5, +5...
A company can settle a lawsuit in two ways; with a one-time payment of
\$100,000 now or with
10\% of its sales forever, an expected value of
\$5,000/year forever. This \$5,000 is risky, however, and is actually
\$7000 in the good state and \$3000 in the bad state. Thus, the cash
flows of the second alterntaive are
+100, -5, -5, -5...
A company can pay \$100,000 now to avoid losing workers if the bad
state occurs. This is worth \$3000 in the bad state, and \$7000 in
the....
I am all confused now. I clearly need these examples. What kind of
losses are greater in the bad state than in the good state?
Or is that a separate issue?
\noindent
{\bf Another approach}
Suppose you have the risky cash flow of either 10 or 0 five years
from now, where the risk is market risk, and 2\% is the riskfree
discount rate. The present value of that cash flow is less than 5 for
two reasons: 1. it is in the future, and 2. it is risky. So you
discount by more than 2\%.
Suppose you have a risky cash flow of either 0 or -10 in five years,
where the risk is market risk. If you were risk-neutral, you would
discount by 2\%. If you are risk averse, you should discount by LESS
than 2\%, not more. The point of the discounting is to see what present
payment would make you indifferent between the present payment and the
risky cash flow.
If there is a mixture of positive and negative cash flows, I think
what matters is the possible net outcomes. What are the possible end
states?
But Craig Holden suggests turning the insurance problem into a
combination of two things. Start out owning a -10 loss for sure. Then
think about investing in a +10 risky asset. That is just a positive
investment problem.
How is insurnace differnet form this? I will pay X to avoid a 50%
chance of a \$100 loss.
Instead, I start by owing \$100. But I can pay Y to get a 50% chance of
getting \$100.
Alternately: I start at zero. I can pay Z to get a 50\% chance of a
$100 gain.
Will Y=X? Yes. Will Z=X? No. I start at a different wealth.
\noindent
{\bf Estimating Betas}
What is the beta of costs? Probably near zero, or negative.
\bigskip
\noindent
\textbf{IV. Concluding Remarks}
sdfsdfsdfsdf
\newpage
\bigskip \noindent \textbf{References}\newline
Booth, Laurent (1982) "Correct Procedures for Discounting Risky Cash Outflows," Journal of
Financial and Quantitative Analysis, (June 1982).
Capital Budgeting
Financial Appraisal of Investment Projects
Don Dayananda, Richard Irons, Steve Harrison, John Herbohn, Patrick
Rowland
Conceptual Problems in the Use of Risk-Adjusted Discount Rates (in
Notes)
Alexander A. Robichek; Stewart C. Myers
The Journal of Finance, Vol. 21, No. 4. (Dec., 1966), pp. 727-730.
Valuing Risky Projects: Option Pricing Theory and Decision Analysis
James E. Smith; Robert F. Nau
Management Science, Vol. 41, No. 5. (May, 1995), pp. 795-816.
Discounting Under Uncertainty
Eugene F. Fama
The Journal of Business, Vol. 69, No. 4. (Oct., 1996), pp. 415-428
Shockley papers
\end{document}