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{\large \textbf{Risk-Adjusted Discounting of Negative Cash Flows } \\[0pt]

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July 18, 2003 \\[0pt]



\bigskip



Eric Rasmusen \\[0pt]



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\textit{Abstract}

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 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.

  }



 



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\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. 

 

 

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\noindent 

\textbf{IV. Concluding Remarks}



sdfsdfsdfsdf





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\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



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