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\begin{center}

 \begin{Large}

 {\bf  The  Government Risk Premium  }  \\

\end{Large}


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 \bigskip

November 12,  2008   \\

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Eric Rasmusen \\

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{\it Abstract}

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 What should the goernmet risk preimum be ?


\bigskip

\begin{small}

\noindent
  \hspace*{20pt} Dan R. and Catherine M. Dalton Professor, Department
of
Business
Economics and Public Policy, Kelley School of
Business, Indiana University.
Erasmuse@indiana.edu.   \url{http://www.rasmusen.org}. This paper:
 \url{http://www.rasmusen.org/papers/risk-rasmusen.pdf}.

\noindent
Keywords: risk premium. 

\noindent
 I thank xxx for helpful discussion.


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\noindent
  {\bf  1.   Introduction}


 

 What risk premium should the government use in deciding on projects?
First, we must narrow the scope of the question. Let's suppose the
government must decide whether to build a pollution facility that has
an immediate cost which is known and a benefit which will come in one
lump sum in one year.\footnote{A well-known problem with simply increasing the discount rate to adjust for risk is that this mixes  discounting for time and discounting for risk improperly. It leads, for example, to a   cost ten years out being discounted to a {\it smaller} value because of its risk, not a bigger value. The solution  is to convert all cash flows to certainty equivalents first and only then to do time discounting. Here I avoid the problem by having zero time discounting. }  Suppose the riskfree rate of interest is zero.
The benefit is uncertain, and is perfectly correlated, in expectation,
with increases in social wealth-- a beta of one, so to speak.\footnote{If the risk is nonsystematic, then since it is spread across all citizens, it is trivial and should be ignored. This is the well-known advice of Arrow \& Lind (19xx).} The
expected  rate of return on this project is 5\%. The expected rate of
return on a project with the same risk characteristics but in the
private sector is  6\%.\footnote{Although the return on the project is not known with certainty,   the value of 6\% for the private return is known. Thus, we are not in the situation analyzed by Weitzman (2001), who looks at what happens when the discount rate itself is uncertain.}  The government funds come from general revenue,
financed by nondistortionary lump sum taxes.  Should the government
take the project?

   One view is that the government should not apply any risk premium
at all, so it should take the project. The rationale  for this is that
people are too risk averse in their private investments. The Equity
Premium Puzzle says that if we use our standard models of risk
aversion as motivated by concave utility, the decreasing marginal
utility of consumption, the risk premium on the stock market should be
far less than it is. Rather than being 7.5\%, it should be .5\%. People
misunderstand risk. Thus, the private rate of return cannot be taken
as a guide to public investment. Perhaps we should use .5\%, but for
simplicity we might as well use 0, since our estimates of the benefits
and
riskiness of the project are going to be inaccurate anyway. (That's
not really a good argument--  it's easy enough to tack on a
requirement of a risk premium of .5\% for all  risky projects ,
ignoring
details of riskiness, and more accurate than tacking on 0\%).

Another view is that the government should reject the project, because
it should use the private sector's risk premium of 6\%. A first
rationale
for
this is that the government could take the money and invest it in the
stock market instead of the pollution project, for a higher return.
One might reply that although that is a good idea, the best thing
might be to invest in *both* the stock market and the pollution
project, investing in the stock market till  stock returns fall  to
5\%. A second rationale, however, is that the rate of  6\% represents
society's required return for risk in the private sector, and the
government should take its cue from what people do in their own
decisionmaking.

 I will  argue for a third view: the government should reject the
project, and should even reject it if the project's return were 6\%,
the same as the private sector's. The rationale is that the government
should indeed take its cue from what people do in their own
decisionmaking, but not all people are willing to accept as little as
6\% for bearing risk-- only people who are investing in the stock
market. Other people are choosing not to invest, showing that the risk
premium they require is higher. How high is hard to say.

 We need more of a framework to make this argument, though. One thing
we need is an objective function. What is the social planner trying to
do, anyway? As a first cut, let us suppose that he is choosing a risk
premium which is a weighted average of each citizen's risk premium,
for whatever weights one might choose.

 There are $n$ citizens, each with his own strictly concave utility function $u_i(c_{i1})$, where $c_{i1}$ is his consumption and  
 $i= 1, \ldots n$.     Citizen $i$ has $w_{it}$ in wealth, for $t=0, 1$. Amount $\bar{y}_{it}$ is an illiquid risky asset. He cannot sell this asset.  He chooses to put $\widetilde{x}_{it}$  of it in a liquid risky asset whose return is perfectly correlated with the illiquid risky asset and $s_{it}$ into a safe risky asset. Consumption occurs in period 1. Each citizen consumes his own wealth plus his share of government wealth, which we will assume is equal for every citizen.    Government wealth can be put into either  the safe  liquid asset or a risky project (amount $\widetilde{p}_{gt}$) which has a risk perfectly correlated with the citizens' risky asset, but has a return which is higher by the possibly negative  amount $k$. We subscript government wealth by $g$. 

 


Thus,  for $i= 1, \ldots n $,  
\begin{equation} \label{e1}
  \bar{w}_{i0}  = \bar{y}_{i0} +\widetilde{x}_{i0}+ s_{i0} 
\end{equation}
 and
\begin{equation} \label{e1}
  w_{i1}  = \bar{y}_{i1} +\widetilde{x}_{i1}+ s_{i1}, 
\end{equation}
where   for $r_s > -1 $ and $\widetilde{r}_x$ such that $E\widetilde{r}_x>r_s$, 
\begin{equation} \label{e1}
\begin{array}{ll}
    \bar{y}_{i1} &=  \bar{y}_{i0} (1+\widetilde{r}_x)\\
 \widetilde{x}_{i1} &=   \widetilde{x}_{i0} (1+\widetilde{r}_x)\\
 s_{i1} &=   s_{i0} (1+r_s)\\
\end{array}
\end{equation}


For the government, 
 \begin{equation} \label{e1}
  \bar{w}_{g0}  =  \widetilde{p}_{g0}+ s_{g0} 
\end{equation}
 and
\begin{equation} \label{e1}
  w_{g1}  =  \widetilde{p}_{g1}  + s_{g1}, 
\end{equation}
where    
\begin{equation} \label{e1}
\begin{array}{ll}
 s_{g1} &=   s_{g0} (1+r_s)\\
 \widetilde{p}_{g1} &=   \widetilde{p}_{g0} (1+r_s+k)\\
\end{array}
\end{equation}

  
 All wealth is consumed at the end of period 1. Consumption is, for $i= 1, \ldots n$,
\begin{equation} \label{e1}
\begin{array}{ll}
   c_i = y_{i1} +\widetilde{x}_{i1}+ s_{i1}  +  \widetilde{p}_{g1}+ s_{g1}\\
  & \\
  & = (1+r_s)( y_{i0} +\widetilde{x}_{i0}+  \widetilde{p}_{g0}) + k \widetilde{p}_{g0}  + r_s( s_{i1}   + s_{g1})\\
\end{array}
\end{equation}

  The order of play is: 

\noindent
 0-1. Citizens and the government  start with wealths  $\bar{w}_{i0} $ and $\bar{w}_{g0} $. \\
 0-2. Everybody simultaneously chooses a risk ratio $\frac{\widetilde{x}_{i0}} { \bar{w}_{i0}}$ or   $\frac{\widetilde{p}_{i0}} { \bar{w}_{g0}}$. \\
 1-1. The value of $\widetilde{r}_x$ is realized. \\
1-2. Citizens consume all the  wealth. 

If every citizen wanted to have more than $\bar{y}_{i0}$ at risk and $k=0$, then the government policy wouldn't matter. It would just affect the split between $p$ and $x$. 

If lending and borrowing were possible, we wouldn't   need the ``more than $\bar{y}_{i0}$'' qualifier, though we'd need a different one about aggregate private holdings of the illiquid asset.  Someone who didn't like how much the government was putting at risk could lend to someone else who didn't mind it. 

  Suppose $k<0$. Then the government should put nothing in the risky asset.  It is a bad substitute for private risk-taking. 

Suppose $k>0$. If everybody wanted to  have more than $\bar{y}_{i0}$ at risk, the government should put all its wealth into the government project, because it is a superior substitute for the risky asset. 

If some people want to have less at risk and either no borrowing is allowed or the aggregation condition fails, then we must use a social welfare function to decide what to do, or use efficiency.  If $k$ is small enough but positive, I claim the efficient thing is to invest nothing in the risky project. 


I can think about making $w0g$ endogenous. Let the government give some back right away, so people ca ninvest in the stock market if $k>0$. 

This model works very well. 

Do this for $k = 0$, $k = -\epsilon$, $k =  \epsilon$. 



  The illiquid asset
is meant to represent such things as human capital and housing. Human
capital is easiest to see as a risky illiquid asset. Housing fits the
category to the extent that a citizen chooses it because for tax or
consumption-hedge reasons but must accept the resale risk that goes
with it (see Sinai in QJE 2005,Campbell in JF 2006).

Finance theory tells us that everybody should own at least a small
amount of the risky asset, because if someone holds a small enough
amount, the second-order risk effect is tiny compared to the first-
order high-return effect. Transaction costs might prevent that. Or, it
might be that people already own so much of the illiquid asset that
they do not want to hold more risk. Which is the true reason is very
important to my argument. If it is transaction costs, then the
government might help these people by buying risk on their behalf.
Otherwise, it might hurt them.

 Most people, in reality, do not hold the liquid risky asset (if we
don't count housing as that). They are unwilling to hold it at the
market risk premium. Thus, their risk premia are higher.

 Most people don't hold much of the liquid safe asset either, though.
So it could be that they are  not risk averse-- just impatient.

Look for the efficient solution. This is NOT a matter for a social welfare function. Taxes and transfers deal with that, not choosing consumption items or risk. 

THe equity premium puzzle. What if utility is not separable across
time, and takes the simple form of U = Min(C1, C2, ...)? I guess you
wouldn't save much. Loss aversion is another possible explanation.




 


\begin{equation}label{e1}
sdf
\end{equation}


\begin{equation}label{e1}
sdf
\end{equation}


\begin{equation}label{e1}
sdf
\end{equation}



\begin{equation}label{e1}
sdf
\end{equation}


\bigskip

\noindent
{\bf 5. Closely Held Corporations}

This can be applied to closely held corporations too. What is the efficient amount of risk taking ?



\bigskip

\noindent
{\bf 9. Concluding Remarks}

 The other possiiblity is that high transaction costs mean that only rich people save in risky assets. Bank accounts need ot be in the safe asset becaue they need to be of predictable size for transactions. 


But do we really think poor people would save in risky assets if they saved? And that rich, sophisticated people are fooled about risk?


Ewijk \& Tang (2003) say that the U.S. government uses a discount rate of 7\%, France uses 8\%, Germany uses 3\%, and the Netherlands 4\%. 



\newpage




\noindent
{\bf References}

Arrow, K.J., \& Lind, R.C. (1970). Uncertainty and the evaluation of public investment decisions.
American Economic Review, 60(3), 364-378. 


Casper van Ewijk, Paul J G Tang.
  How to price the risk of public investment?
 De Economist. Leiden: Jun 2003. Vol. 151, Iss. 3; pg. 317

 Weitzman, M. (2001). Gamma discounting. American Economic Review, 91(1), 260-271.

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