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{\large {\sc ``Why  Firms Reduce Business Risk Revisited: Projects with Risky Cash Flows Are Harder To Evaluate? ''   
}} 

February 28, 2020

%IDEA: Evaluation noise rather than cash-flow noise. 

 


   Eric   Rasmusen  

{\it Abstract}

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\begin{small}
  Firms  care a lot about ``risk management'': the practice of hedging risks---  whether they are correlated with market risk or not.  The standard reasons why widely held corporations might be averse to idiosyncratic risk are  principal-agent problems, bankruptcy costs, external finance, and tax convexity. This paper offers a different reason:    risk makes business decisions more difficult.  We must distinguish, however, between  risk over the expected value of profits  (``value risk'') and risk over the volatility of cash flows (``cash-flow noise'').   Value risk is good    because an unprofitable policy can be abandoned (option value).  Cash-flow noise   is bad  because it makes learning when to abandon   more difficult.
   This distinction is unrelated to Knightian risk   or  ambiguity aversion, and  it  matters even if the firm's agents are risk neutral. 
 
 
 
 
 \noindent
Rasmusen: 
Professor, Department of Business Economics and Public Policy, Kelley
School
of Business, Indiana University. 1309 E. 10th Street,
Bloomington,
Indiana, 47405-1701. (812) 855-9219.  
\href{mailto:erasmuse@indiana.edu}{ erasmuse@indiana.edu}, \url{http://www.rasmusen.org}.   
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{\small
  \noindent This paper:
\url{http://rasmusen.org/papers/risk_aversion.pdf}. }

{\small 
 \noindent
  Keywords: idiosyncratic risk, risk aversion,   real options,   capital budgeting, risk management      \\
JEL Codes:  D21, D25, D81, G32, L21.  }


 {\small    I would like to thank Philip Bromiley,  Brett Green,  David Hirshleifer, Robert Pindyck, and Wolfgang Kursten  for helpful comments,  and  Aaron Kolb for   exceptionally helpful input.    } 

 
  
 

\end{small}




\newpage

 


\noindent
{\sc 1. Introduction}

  

 Risk is commonly divided into two categories: systematic and idiosyncratic.  A company's systematic risk is   the part of its risk  correlated with the returns an  investor could earn by holding other assets in the economy; its idiosyncratic risk,  the part which is not. Under the CAPM and other theories of asset pricing  a publicly traded corporation should be averse to systematic but not   idiosyncratic risk   because its shareholders can diversify away idiosyncratic risk by holding a variety of assets.  Thus, firms should not take idiosyncratic risk into consideration.
 
  Yet it seems   firms do   avoid idiosyncratic risk in both investment   (Panousi \& Papanikolaou, [2012])  and operating decisions ((Amit \& Wernerfelt [1990], Chatterjee, Lubatkin \& Schulze [1999]).   They  manage enterprise risk (Bromiley, McShane,   Nair  \& Rustambeko [2015]),  hedge against movements in commodity prices and currencies (Stulz [1996], Smith [2008]),   smooth accounting  earnings  (Burgstahler \& Dichev [1997]), and  buy insurance (Harrington, Niehaus \& Risko [2002]).  There  may even be a ``diversification discount'': investors prefer firms with diversified lines of business (Villalonga [2004], though   see also Laevena \& Levine  [2007]). 
 
 
     The most common explanation for aversion to idiosyncratic risk is that to incentivize managers  the firm must link  pay to  firm-specific performance.  As a result,  managers   wish to reduce risk  regardless of shareholder desires. Even  if  shareholders can control the managers they will need to design complex contracts and pay  extra for   bearing risk (Smith [2008]  Panousi \& Papanikolaou, [2012],  Glover \& Levine [2017]).  A second  explanation    is bankruptcy costs. Bankruptcy arises  even from idiosyncratic risk  and creates not just   direct costs in reorganization but indirect costs because of anticipatory moves  of   lenders, suppliers, and customers  (Stulz [1996], Kuerstein \& Linde 2017]).  A third  explanation   is that external finance  is more costly than retained earnings as a source of capital. If retained earnings are  variable, the firm will sometimes be short of capital and sometimes have more than it has use for (Froot, Scharfstein \& Stein [1993]).  A final explanation is  convex taxation, which makes it advantageous to smooth   profits (Nance, Smith \& Smithson [1993]).   

 

     This article  offers a different  explanation, one that does not seem to have been discussed despite its  common sense: risk's effect on the quality of  decisions. If the firm encounters more noise in evaluating whether an activity is profitable or not, it will make worse decisions, so it chooses to  avoid risk.
     
       The value of risk has indeed been much studied in the context of decisions, in the theory of real options, but with the opposite conclusion: risk is good, because    option value increases with  volatility.  If an activity might have either high or low returns, the firm has the option to abandon  if the return turns out to be low.  This   suggests that corporations should  be risk-loving, not risk-averse.  Real options have  been the topic of much study, some focussed on  adjustment costs and competitive conditions (Hartman [1972], Abel [1983], Kim \& Kung [2017]), some on valuation  (Dixit \& Pindyck [1994], Trigeorgis [1996], Lambrecht [2017]), and  some on management implications    (Trigeorgis \& Reuer [2017]). 
       
       The  downside--- this paper's point---  is that  volatility makes learning   more difficult.   Idiosyncratic risk  hurts learning even more than systematic risk, since it cannot be backed out from the market return.  Thus, two contrary forces are at work in volatility:  the advantage of   a more valuable option, and the disadvantage of not knowing what to do with it.   I propose 
distinguishing between  ``value risk'' and ``cash-flow noise''. Value risk increases  with divergence in   possible expected values.  Cash-flow noise increases with   the variability of   cash flows,  for given expected value.  Value risk has positive option value because the firm has the option to discontinue  the policy  if it   learns the  expected value is low. Cash-flow noise hurts option value because
it hinders the firm from knowing whether to exercise the option.  Thus, when firms avoid projects whose idiosyncratic risk consists of cash-flow noise, they are being rational even if their investors are risk-neutral. 
        
\newpage
\noindent
 {\sc 2. Distinguishing Value Risk   from   Cash-flow Noise}
      

There are two reasons one might say, ``This project is risky.'' The first   is that one doesn't  know   whether the project is profitable.   The second   is that even though   profitability is known,   the  cash flow   varies over time.  
Profitability is akin to Aristotle's ``substance'',   a data-generating process,  with  cash flows as  the data (Aristotle's ``accidents'').   We will   call  dispersion in possible expected values   ``value risk'', and    dispersion  in cash flows around the  expected value  ``cash-flow noise''.  

 More formally,  let     activity  A have  the  realized cash flows  $X^A_{it}$,   $t= 1, \ldots T$    given by  the    stationary stochastic process   $X^A_i$, $i= 1, \ldots  N^A$   with probabilities $F^A_i$ given by   distribution $F^A$. An example  for  $t= 1, \ldots 10$,  is that  either  $X_t = X^A_{1t} = 10 $ with probability  .7 and 0 with probability .3 for all $t$,   or  $X_t = X^A_{2t} = -100$ with probability  .6 and 0 with probability .4 for all $t$, where $F^A_1= .9$ and   $F^A_2  = .1$. Then $E(X_1^A) = 7$, $E(X_2^A) = -60$,  and $E(X^A) = .9(7) + .1 (-60)= 0.3$. 


Following the standard partial ordering of riskiness in Rothschild \& Stiglitz (1970),  we will say that when  $E(X^A) =  E(X^B)$,  activity A has  {\it greater  value risk}  than B if any agent with strictly concave utility would prefer     $\{E(X_i^B) , F^B \}  $  to $  \{E(X^A_i), F^A \} $.  We will say that   A has {\it  greater  cash flow noise} than B if any agent with strictly concave utility would prefer $\{X^B_i- E(X^B_i), F^B_i\}$, $i= 1, \ldots  N^B$ to $\{X^A_i-E (X^A_i), F^A_i\}$, $i= 1, \ldots  N^A$.  

Confusing the two kinds of risk  is easy because we model them the same way.  Suppose  project X  has  70\% chance   of yielding  a cash flow of +1 and    30\% chance of 0  in any particular year, whereas 
project Y  has 
 70\% chance  of yielding a steady cash flow of +1  and   30\% chance of 0.  They each have an annual expected value of .7, but  project X has zero value risk and project Y has zero cash flow noise. 
   
 

 \bigskip
 
    
     \noindent
{\bf Example 1:   A Firm Choosing Which Product to Sell } \\    
       A firm is considering  which of  several  new brands of soap to introduce for a three-year product cycle, after which  the brand will be replaced by something else.    Assume zero time discounting. The firm, which  is either ``well-suited''   or ``ill-suited''  to sell this particular product  with probability .5 of each,  does not know its ability.  Market conditions will be ``normal'' in some years and ``difficult'' in others.  Product A is the safest,  with no risk whatsoever.  Product B's value is unknown, but    its cash flow will not depend on market conditions. Product C is  the most complicated, yielding  positive  profit  in normal years  and   losses in difficult years. 
       Specifically: 
   
Product A yields a yearly cash flow of 0, regardless of market conditions or whether the firm is well-suited.  
        
 
Product B yields a  yearly cash flow of   +1     if the firm is well-suited and -1 if  ill-suited, regardless of market conditions.   Product B's   expected cash flow is also 0. 

Product C yields a   cash flow of  +2 in a normal year, which has 70\% probability. If the firm is well-suited,  it   yields a cash flow of  -4/3 in a difficult year, for an  expected yearly cash flow of   +1 over all years.  If the firm is ill-suited,  it yields a cash flow of  -8 in a difficult year,  for an expected   flow  of  -1.  Product C's overall expected cash flow is also 0. 

     If the firm chooses Product A, it has no decisions to make  and the product's lifetime value is 0.
     
      If the firm chooses Product B, it 
   will immediately discover if it is ill-suited  because the cash flow will be -1 the first year,  and  it will  cancel the product.   The expected three-year profit  consists of a 50\% chance of -1+0+0 and a 50\% chance of +1+1+1, for an expected profit of +1,  even though if the firm refused to cancel  the expected value would be 0.
 
  If the firm   chooses Product C,  the  best strategy  is to keep selling  until a year in which the cash flow is -8, which would make plain that   it is ill-suited to sell soap, and then  cancel. If the  firm is ill-suited, its probability of   selling  for one year would   be    .30, .21 for two years,   and .49 for three years,    (which includes probability .34  of never discovering  the truth),    for an expected profit of  -2.19.   If the firm is well-suited,  its expected three-year profit  is +3.  Overall,   expected profit is  .405.\footnote{$.405=  .5[ .3(-8) + .21 (2-8) + .147 (4-8) + .343(6) ] + .5 [3] $.}  Product C is  better than Product A, because it   has option value, but worse than B,  because learning  C's value is more difficult.\footnote{ 
   Whether the firm can switch products in the  middle of the three-year product life cycle does not matter.   Even with switching, if product C is chosen and the firm then discovers it was ill-suited the first year, C would be discontinued, but  B  would be no more profitable for the ill-suited firm.    }
   
 In Example 1,    value risk  
  is desirable. Product B's uncertainty makes it superior to Product A--- but only because the firm both  (a) learns, and (b)  has the option to cancel the product.  Product C has even more uncertainty than B, but   the wrong kind: cash-flow noise. That uncertainty does not affect option value directly, because  the firm has the same option to cancel an unprofitable product. Indirectly,  however,  cash-flow noise makes learning more expensive: it costs $.5(-1)$   to learn that product B is unprofitable, compared to $.5(-2.19)$ for product C.
  
    Example 1 is reminiscient of   ambiguity aversion,    the   Ellsberg Paradox's idea that even if two lotteries have the same risk, if one of them is a ``lottery of lotteries'' people will avoid it  (Ellsberg [1961], Ju \& Miao [2012]).   Product C   is indeed a lottery of lotteries,  a 50-50 gamble between two different 70-30 lotteries rather than a  50-50 gamble between  two integers,  like product B. As with the Ellsberg Paradox, easier decisionmaking is preferred.      An executive viewing an   MBA's powerpoint proposal to undertake product C
 might well worry about  the cognitive cost of calculating the value of complex lotteries,   the   likelihood of making a mistake,  and the danger of being  fooled. Here,  what is driving his distaste for C is that it requires more 
   expensive experimentation.   This will be apparent in the next section's model, in which the two alternatives are equally simple, with cash flow distributions that differ only in  a single    parameter. 
   
    
\newpage
 \noindent
 {\sc 3. A  Continuous-Time Model }
 
  This section's model will use stochastic calculus, a tool  often   used to look at adoption and abandonment decisions (see Pindyck [1993],  Dixit \& Pindyck 1994, p. 345],  Bernardo \& Chowdhry [2002],  Decamps , Mariotti  \& Villeneuve [2006],   Kwon \& Lippman [2011]). The model  makes no claims to technical novelty; indeed, it could be considered  a special case of    Lippman \& Ryan (2003), despite its different look.\footnote{Lippman \& Ryan (2003) uses  ${\mu}(H)$ and $-{\mu}(L)$ rather than ${\pi}$ and $-{\pi}$. Imposing symmetry makes  my propositions    simpler and stronger (because less general), and  allows value risk to be parameterized by  ${\pi}$. An even earlier and more general analysis of the project  stopping problem is Karatzas (1984).  } It will, however, be useful for thinking about the difference between value risk and cash-flow noise. 
  
  
  Consider  a  single  project  whose  type $\theta\in \{H,L\}$  is initially unknown, drawn from a prior distribution with $\Pr(\theta=H)=p_0\in (0,1)$. The project delivers a stochastic cash flow  $X_t$ according to   a continuous-time diffusion  process:
\begin{equation} 
 dX_t=\mu(\theta) dt+\sigma dW_t,
\end{equation}
 where $W_t$ is a standard Brownian motion process for positive mean $  \mu(\theta)$ and standard deviation $\sigma$. Let  the possible  means of the process be $\mu(H)={\pi}$  and  $\mu(L)= -{\pi}$. The firm discounts cash flows at rate $r$.   Its only decision, which it makes every instant,    is whether  to continue the project or abandon it. 

Here  the parameter $ {\mu}(\theta)$ represents the  amount of  value risk and $\sigma$ the  amount of cash-flow noise.
 As time passes, the firm continuously updates its posterior belief $p_t$ about $ \mu(\theta)$  using Bayes's rule.   Rather than analyzing the  cash flow path, $X_t$,  it is most convenient   (see section 2.3 of 
Daley  \& Green [2012]  or section A1 of Kolb  [2019])
  to analyze the belief path, $p_t$,   converting to   log-likelihood  beliefs, $Z_t$ that update linearly and vary on $[-\infty, +\infty]$ as    $p_t$ varies  on [0,1]:
 \begin{equation} 
Z_t \equiv\ln  \left(\frac{p_t}{1-p_t} \right)
 \end{equation}
 
     Using  filtration results  standard in the literature (e.g., section 3 of Bolton \& Harris [1999]),   the law of motion for the posterior belief conditional on  the type being $\theta\in
      \{H,L\}$ is 
\begin{align}
Z_t^H& =\frac{\phi^2}{2} dt+\phi dW_t\\
Z_t^L& =-\frac{\phi^2}{2} dt+\phi dW_t,
\end{align}
where $\phi \equiv \frac{\mu(H)-{\mu}(L)}{\sigma}=\frac{2\pi}{\sigma}$. 

The firm's optimal strategy is to 
abandon the project whenever the belief $Z_t$ falls to some lower threshold $z^*$, as stated in the following proposition. 

\fbox{\parbox{\textwidth}{     
\begin{proposition}\label{prop:optimalpolicy}
The firm's optimal abandonment threshold is
\begin{equation} \label{e-threshold}
z^*=\ln \left[\frac{ \pi^2+r \sigma^2- \pi \sqrt{ \pi ^2+2r\sigma^2}}{r\sigma^2}\right]<0,
\end{equation}
 and its  value function is 
\begin{equation} \label{e-value}
 V(z)=\begin{cases} \left(  \frac{ e^{z }-1 }{e^z+1} \right)  \frac{\pi}{r}  + \left( \left(   \frac{1 - e^{z^*} }{e^z+1}  \right) e^{m(z-z^*)}\right)  \frac{\pi}{r}\qquad &\text{if } z\geq z^*\\
0\qquad &\textit{otherwise,}\end{cases}
\end{equation}
where $m=\frac{1}{2}-\sqrt{\frac{1}{4}+\frac{r\sigma^2}{2\pi^2}}<0$.
\end{proposition}
}}

 \begin{proof}[Proof] 
   
By Ito's Lemma, since the expected flow return is   $ p_t \pi + (1-p_t)(-\pi)= \left( \frac{e^z-1}{e^z+1} \right) \pi$,  the value function prior to abandonment satisfies  
 \begin{equation}\label{e0}
 r V(z)= \left(  \frac{e^z-1}{e^z+1}\right)\pi + \left(  \frac{e^z-1}{e^z+1}\right) \frac{2\pi^2}{\sigma^2}V'(z)+ \left( \frac{2\pi^2}{\sigma^2}  \right) V''(z).
\end{equation}

This differential  equation has solutions of the form 
 \begin{equation}\label{e0}
 V(z) =    \left( \frac{e^z-1}{e^z+1}\right) \frac{\pi}{r} +  \left(  \frac{e^{mz}}{e^z+1}\right)   C_1  +  \left(  \frac{e^{(1-m)z}}{e^z+1}\right) C_2,
 \end{equation}
  where $m$ is defined in the proposition and we need to find $C_1$ and $C_2$. We know that the value  is  $\frac{\pi}{r}$  if the project type is high, which is  belief   $p_t=1$   (and thus     $Z_t = ln \left(  \frac{p_t}{1-p_t}\right) \to \infty$), so it must be true that     $\lim_{z\to+\infty} V(z)=\frac{\pi}{r}$ and we can conclude  that  $C_2=0$. Pinning down $z^*$ and $ C_1 = \frac{\pi}{r} \left(1-e^{z^*}\right)e^{-m z^*} $ then follows from the standard value matching and smooth pasting conditions $V(z^*)  = 0$ and $V'(z^*)=0$  (Dixit [1993]).  Simplifying the second term   yields (\ref{e-value}).
 \end{proof}
 
Figures 1 and 2 show   one realization of  stochastic  process paths over time when $r=.12$, the project is unprofitable ($\mu(\theta)=-15$),   and the belief starts at 40\% probability the project is profitable.\footnote{The Python 3 code is available at \url{http://rasmusen.org/papers/risk_aversion.py}.}  The red   paths are for $\sigma =10$ and  the blue  paths
  for $\sigma = 30$, using the same discrete approximation realization of  the  Weiner process $dW_t$ for both. The optimal thresholds  $p^*(\sigma)$ are .02 and .14, which are  below .50 because of option value. With low volatility, the belief that the project is  high must fall all the way to 2\% before abandonment, but the information is so good that this happens quickly, in period 299,   after losses have only accumulated to  11.   With  high volatility, the belief  only needs to fall to 35\%, but the information is poor enough that this happens   in period 800,  after losses have  accumulated to 42.  
  
  
 \newpage
 \begin{center}
 {\sc Figure 1: \\
   Cash Flows   for High  and Low  Volatility }\\
\vspace{6pt}
\includegraphics[width=5in]{typical_pathx.png}\\

{\sc Figure 2: \\
   Beliefs for for High   and Low   Volatility }\\
\vspace{6pt}
\includegraphics[width=5in]{typical_pathp.png}\\
\end{center}


 
 Returning to  the optimal value function, equation (\ref{e-value}),  note that it is composed of two terms. The first term represents the expected value of running the project forever, $ (2p_t-1)  \left( \frac{\pi}{r}  \right) $, since $ \frac{e^{z_t}-1}{e^{z_t}+1}  = 2p_t-1$.   That term is positive if $p_1>0$ and negative if $p_t<0$.  
   The second term represents the gain from the ability to abandon the project after poor returns.  It is always positive, because $z^*<0$ so $1 - e^{z^*}>0$. If $z_t = z^*$, then 
   $  \frac{1 - e^{z^*} }{e^{z^*+1}}  =  2p^*-1$   and    $e^{m(z_t-z^*)} = 1$, so the second term's magnitude is exactly equal to the first term's.  The value to running the project another instant is $rV(z_t)$, so when $z_t = z^*$, the positive learning flow from continuing is exactly cancelled by the negative cash flow. This, of course, is  what we used in solving the $V(z)$ function, because for  any lower $z_t$ the option value would not be worth the cash flow loss. The next proposition shows how value risk and cash-flow noise affect the tradeoffs. 
  
\fbox{\parbox{\textwidth}{  
\begin{proposition}\label{prop:compstatics}
As  cash-flow noise $\sigma$ rises, the abandonment threshold $v^*$  increases and the project's value falls. As value risk $\pi$ rises, the abandonment threshold  decreases and the project's value rises. 
\end{proposition}}}

 
 \begin{proof}[Proof.]
 In what follows, we use $ C_1 = \frac{\pi}{r} \left(1-e^{z^*}\right)e^{-m z^*} $   from   Proposition \ref{prop:optimalpolicy}'s proof  so   we can write the positive part of the  value function as  
\begin{equation}
  V^+(z)= \left( \frac{e^z-1}{e^z+1} \right)  \frac{\pi}{r}+   \left(  \frac{e^{mz}}{e^z+1}\right) C_1
  \end{equation} 
   We will define $F(z)$ to be the function which extends  $V^+(z)$ over the entire real line.  From Proposition 1's proof, $V(z^*)=0$ and $V'(z^*)=0$, and $V(z)>0$ for large enough $z$,  so since 
  $F(z)(e^z+1)$ is an exponential polynomial  with at most  two sign changes    it follows that $F(z)$   is nonnegative with  a single minimum at $z = z^*$. 

First, consider the effect of an increase from $\sigma$ to $\tilde{\sigma}$ and let tilde denote the solution components for  $\tilde{\sigma}$.  We claim that $F(z)$ and $\tilde{F}(z)$ can intersect at most once, since $\Delta(z) =(e^z+1)[\tilde{F}(z)-F(z)]=\tilde{C}_1 e^{\tilde{m}z}-C_1  e^{m z}$, where the expression in brackets is an exponential polynomial with exactly one sign change. Now $\frac{\partial z^*}{\partial \sigma}=\frac{2\pi}{\sigma\sqrt{\pi^2+2 r\sigma^2}}>0$, so $\tilde{z}^*>z^*$.  The minimum of $\tilde{F}(z)$, but not of $ F(z)$,   is at $\tilde{z}^*$, so $\tilde{F}(\tilde{z}^*)=0,  F (\tilde{z}^*)>0   $, and      $\Delta(z^*)<0. $ Since $\Delta(z^*) >0$ by the same kind of argument,  there exists a root   $\hat{z}\in (z^*,\tilde{z^*})$ of $\Delta(z)$  where  $\Delta(\hat{z})=0$  and   $\tilde{F}$ intersects $F$ from above. Since that is the only sign change $\Delta(z)$ can have, it follows that 
 that $ F(z)> \tilde{F}(z)$   for $z> \hat{z} <  \tilde{z}^*$.  Recall that $V(z)$    is identical to   $F(z)$ except for equalling zero on $z < z^*$. Hence,  $ \tilde{V}(z)=0$      for all $z< \hat{z} <\tilde{z}^*$, so $V(z) >  \tilde{V}(z)$ for all $z > z^*$ (and $V(z)= \tilde{V}(z)=0$ for $z  \leq  z^*$).  
 

Next, consider the effect of an increase from $\pi$ to $\tilde{\pi}$, and use tilde to denote solution components for $\tilde{\pi}$. We have $\Delta(z) =(e^z+1)[\tilde{F}(z)-F(z)]=\frac{\tilde{m}-m}{r}(e^z-1)+\tilde{C}_1  e^{\tilde{m}z}-C_1  e^{mz}$. Now $\frac{\partial z^*}{\partial \pi}=-\frac{2}{\sqrt{\pi^2+2 r\sigma^2}}$, so $\tilde{z}^*<z^*$, and thus $\Delta$ has a root in $(\tilde{z}^*,z^*)$ where $\Delta$ crosses 0 from below. We claim that any root above $\tilde{z}^*$ must cross from below, and thus there is only one root above $\tilde{z}^*$. Note that if $z$ is a root, then  $\tilde{F}(z)=F(z)$,  so     $\Delta'(z)=\Delta'(z)-\tilde{m} \Delta(z)=\frac{\tilde{m}-m}{r}\left[e^z-\tilde{m}(e^z-1)\right]+C_1  e^{mz}(\tilde{m}-m).$  It is easy to verify that $\tilde{z}^*=\ln \frac{-\tilde{m}}{1-\tilde{m}}$, so the first term is positive for all $z>\tilde{z}^*$. The second term is always positive because $m$ increases in $\pi$, so the claim holds. It follows that for all $z\geq z^*$, we have $\Delta(z)=\tilde{F}(z)-F(z)>0$. Moreover, for $z\leq \tilde{z}^*$, $\tilde{V}(z)=V(z)=0$ and for $z\in (\tilde{z}^*,z^*)$, $\tilde{V}(z)>0=V(z^*)$. We conclude that $\tilde{V}(z)\geq V(z)$ for all $z\in \mathbb{R}$, with strict inequality for all $z>\tilde{z}^*$.
\end{proof} 
  
     The  firm's belief about project type changes faster if $\pi$ is big and $\sigma$ is small. This is equivalent to faster learning as news comes in from the  cash flows.  A bigger $\sigma$ means bigger movements in cash flows, a more variable cash flow process, but when converted to  belief space  it means smaller movements in beliefs, because a given change in cash flow is less communicative of whether the state is high or low. 
As the cash-flow noise $\sigma$ increases, the signal-to-noise ratio of the cash flow signal decreases, and the firm learns more slowly about the project's type. This reduces the option value to continuing the project to try to learn more, which is  the second term in the value function. When the option value falls,  that makes the firm worse off and   causes it to abandon projects sooner. Thus, the firm should be averse to cash-flow noise because it makes decisionmaking harder, whether or not its managers and shareholders are  risk-averse  in their utility functions. 


  On the other hand,  when   value risk  increases  the spread between the expected returns $- \pi$  and $\pi$ of low and high projects,  the firm benefits twofold. First, the upside potential  $\pi/r$ from running a good project forever increases, while the downside potential is still bounded below at zero, as the firm can always abandon it. This is the standard result that increasing variance increases an option's value.  This first benefit would apply even if the firm's learning was instantaneous  as  in Example 1.    The second benefit is that  the signal-to-noise ratio increases and the firm can more quickly learn about the project's type. The firm can  respond    with  more aggressive experimentation. Its  abandonment threshold is lower because it is willing to continue the project  even under a pessimistic belief  because quicker learning means quicker abandonment if the project type is indeed low. 
  
 
    We have described the model as one of project choice, but it could apply to any of a business's activities. In this setting of normally distributed risk, we can also interpret a higher value of  $\sigma$ as   the result of adding normally distributed zero-mean noise to an existing activity, since adding an independent  zero mean normal variable to an existing normal variable   increases the variance while leaving normality and the mean unchanged. In particular, one could view accounting systems as attempts to reduce cash-flow noise. In practice, standard business earnings are not cash flows, which are higher-variance measures of profitability, 
    but flows of accruals and depreciation, which require careful definition intended to balance current economic profits against variability  across time that makes determining profitability more difficult. Thus, we can  view accounting imperfections as a source of risk in ``cash flows''.  Deficiencies in accounting quality would often merely be idiosyncratic risk, but it is risk that hurts the quality of investment decisions. 
 
     \bigskip
 \noindent
  {\sc 4. Two Distractions:  Knightian Uncertainty    and the Principal-Agent Problem  }
  
    
      Splitting risk into two types   brings to mind   Knight's distinction between risk and uncertainty (Knight [1921]).  What  Knight was getting at remains controversial (see Leroy \& Singell [1987]), but one interpretation is that  Knightian risk is unpredictability with known probabilities in repeated situations, the kind of risk that is insurable,  whereas  Knightian uncertainty is unpredictability  in one-time situations  where the probabilities are more subjective. 
 In Example 1, the cash-flow noise was about whether the market was normal or difficult, something objective  that would seem to be Knightian risk.  The value risk 
 was that  the firm might be  either well-suited or  ill-suited, something ``unknowable''  even though  the firm will have subjective probabilities, and would thus be Knightian uncertainty. Similarly, in the continuous-time model, the cash flows were  Knightian risk,  but the expected value was either high or low, but unknown: Knightian uncertainty. Example 2 will illustrate that this is not a reliable equivalence. 
   
   \bigskip
    \noindent
{\bf Example  2: Hedging Foreign Exchange Risk} \\   
      An American firm has set up a London office for  a three-year period. The new office is a ``good'' or ``bad'' idea with equal probability.  The office  buys and sells  in both dollars and  pounds, so it is unclear how the exchange rate affects its profitability. If the office is a good idea, though, the profit will be +1 if the exchange rate moves beneficially, which has probability .70, and 0 if it moves harmfully,   an expected annual profit of +.7. If the office is a bad idea, the profits will be 0  or -1,  an expected profit of -.3. The firm's optimal strategy is to open the office,  and shut it down if and only if its profit is seen to be -1 in some year.   If the idea is a good one,  the expected profit will then be  2.1.  If the idea is a bad one, the office will   have 30\% probability  of closing after one year, 21\% after two years, 14.7\%  after three years due to losses in the third year, and 34.3\% of operating profitably for three years and closing down only because the three-year period is over.  Its expected profit if the idea is bad is -.657. Overall, balancing the probability the idea is good with that it is bad, the expected profit is .7215.\footnote{  Another strategy is to close down the London office the first year if     -1  is observed,  or the second year  if   0 is observed in the first year and   -1 or 0     in the second.  That would have expected profit of -.51  if the  idea is bad, of   -.51 and 1.547 if  it is good, an average of .5185. }
      
   Now suppose the firm instructs the office to hedge its foreign exchange risk. If the office is a good idea  and hedging is costless, its profit each year is +.7, while if it is a bad idea, its profit each year is -.3. The firm will know to shut down   the first year if its profit is -.3, so the expected profit over the three years is .5(-.3) + .5 (.7+.7+.7) = .90.  This is higher than .7215, so the hedging helps, and the firm would be willing to  pay as much as .1785 in transactions costs  to be able to hedge.
   
 

   
   
As in  Example 2 and the continuous-time model, the exchange rate risk is insurable and objective,  so it would seem that  again  cash-flow noise corresponds to Knightian risk. Whether the London office is a good idea or not appears to be subjective and uninsurable, so value risk would correspond to Knightian uncertainty. Knightian uncertainty would thus be good for profit, and Knightian risk would be bad. 
 
   But Example 2 can be interpreted to give the opposite results.  Suppose the firm has extensive experience setting up foreign offices, and has found that in exactly 50\% of them the idea is a good one. Then the value risk is Knightian risk. Moreover,  while exchange rate risk is insurable, the big problem is that the firm doesn't know enough about the inner workings of the office to know whether it benefits from a strong dollar or a weak one. Thus,  the cash-flow noise is Knightian uncertainty.   The questions of whether (a)  there is  opportunity for learning because of value risk, and (b)  whether learning will be difficult because of cash-flow noise  are   separate from whether the probabilities are subjective or objective. 
   
   \bigskip
  
  
    For our last example, we will return to the principal-agent problem. I have emphasized   that option learning is an alternative to managerial risk aversion as an explanation for corporate aversion to idiosyncratic risk. It is also, however, a supplement to the principal-agent   explanation,  a supplement that applies even if the agent is risk-neutral.  That noise worsens a principal-agent problem is an old idea. Holmstrom's classic 1979 ``sufficient statistic'' paper notes how the informativeness of a signal is hurt by additional risk, and the negative welfare effect shows up naturally in many models,  despite the lack of attention to the  implications for  corporate decisions. (An exception is D. Hirshleifer \& Suh [1992], in which a firm avoids high-variance projects because of the difficulty of incentivizing the manager.) Our next example will be of the problem of incentivizing a risk-neutral manager, to show that option learning does not depend on anyone  having concave preferences. 
     
 
     
    
    \noindent
     {\bf Example 3:  The Principal-Agent Problem}.  Management is considering one of two protocols for managing customers. Under Protocol X, the worker dealing with a customer reads from a script.  If  the worker exerts high effort, the customer is satisfied, a value of +10 for the firm. If he exerts low effort, the customer is dissatisfied, a value of -10. Under Protocol Y, the worker is trained   to try to  adapt to the particular customer.  If  the worker exerts high effort, the customer is  highly satisfied with probability .5, a value of +40 for the firm and dissatisfied with probability .5,  a value of -20, which comes to an expected value of +10. If he exerts low effort, the customer is  satisfied with probability 1/6 and dissatisfied with probability 5/6, which comes to an expected value of -10.  The workers are risk-neutral, with the utility function $w-e$ for wage $w$ and effort $e=0$ or 3, and their expected utility must equal at least 1 to take the job. They cannot be paid  a negative wage. 
     
    First let us look at the optimal contract for Protocol X.  If the company pays the worker 4 if the customer is satisfied and  0 otherwise, the worker would have utility of 1 ($= 4-3$) from high effort and 0 from low effort  and so would choose high effort.    With the contract $(w_{sat}=4, w_{diss}=0)$, the firm   ends up obtaining   +10 from the customer and paying  4 to the worker.  
    
      Protocol Y takes a bit more consideration. 
      The participation constraint is  $E(w -e|high \; effort) = .5w_{sat} + .5 w_{diss} -3 \geq  1$. The incentive compatibility constraint is $E(w -e|high \; effort) \geq E(w -e|low \; effort)$, which is $.5w_{sat} + .5 w_{diss} -3 \geq (1/6)w_{sat} + (5/6) w_{diss} -0$. Because the wage cannot be negative, the biggest gap between $w_{sat}$ and $w_{diss}$ is  if $w_{diss}=0$, so the optimal contract will choose that.   The participation constraint  then    requires  $.5w_{sat} + .5(0)   -3 \geq  1$:     the worker is willing to take the job  if the contract is $(w_{sat} =8, w_{diss} =0) $.  But the  incentive compatibility constraint tells us that $.5 w_{sat} + .5(0) - 3 \geq (1/6)w_{sat} + (5/6)(0) -0$:  the lowest $w_{sat}$ that will induce high effort is $w_{sat} =9$.  As a result,  the firm will wish to use an ``efficiency wage'', a contract that gives the worker more than his reservation utility because the firm needs to be generous to induce   effort (Yellen [1984], Rasmusen [2007 ch. 8]).   With the contract $(w_{sat}=9, w_{diss}=0)$, the firm would end up obtaining an expected value of +10 from the customer and paying an expected wage of 4.5 to the worker.   So Protocol Y is worse, with a profit of 5.5 instead of 6.0. 
      
     What Example  3 illustrates is that even if  both employer and employees are risk-neutral, risk in cash flows   creates contracting costs. The basic idea in principal-agent models is that if agents  are risk-averse  then when output is noisy   the optimal contract balances the incentivizing effect of  incentive pay against  the  extra risk that requires a higher expected wage.    In Example 3,   the agent needs no  compensation for bearing risk,  since he is risk-neutral, but noisy output requires   divergence in pay to incentivize effort. When pay is bounded below by zero, the  firm needs to increase the upper bound  and thus the expected value, of pay. 
    
       
      
       
       \bigskip
  
 \noindent
  {\sc 5. Concluding Remarks}
  
  What's wrong with risk?  Our usual answer is  concavity of utility:  sometimes you have too much, sometimes too little.   We focus so much on utility  that we neglect another answer: Risk makes it hard to  figure out what to do. 
 On retiring to the family farm, my parents hung up a sign on the front porch saying   ``Simplify''.  Executives  live  a frenzied existence.  Much of their job is to solve unexpected problems. leaving less time than they would like for evaluating   people and projects.  They want simplicity, and  if they can get a steady bottom-line, whether positive or negative, that  simplifies  their decisionmaking. 
   
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  \noindent
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