©August 28, 2019, Christopher D. Carroll CARAPortfolio

Portfolio Choice With CARA Utility

Consider a consumer with Constant Absolute Risk Aversion utility
u(c) = −α^{−1}e^{−αc}, with assets a_{T−1} who is deciding how much to invest in a risky
security that will earn a normally distributed stochastic return R_{T} ∼𝒩(R,σ)
versus a safe asset that will earn return R < R (the risky asset gets
a bold font because you must be a bold person to invest in a risky
asset!). ^{,}

Consumption in the last period of life will be the entire amount of resources. If
the consumer invests an absolute amount of money $S in the risky asset, then

where Φ_{T} is the equity premium realized in period T. Given S and deﬁning the
expected equity premium as Φ = 𝔼 _{T−1}[R_{T} −R], the expectation as of time
T − 1 is: and (2) follows from (1) because if z ∼𝒩(Φ_{z},σ_{z}^{2}) then 𝔼[e^{z}] = e^{Φz+σz2∕2
}. (See
MathFacts [ELogNorm]).
Because (3) is negative, the optimal S will be the one that yields the
largest negative exponent on e, which occurs at the value of S given
by

| (4) |

with FOC

This yields the intuitive result that the greater is risk aversion or the greater is
the risk, the less the consumer wants to invest in the risky asset, while the
greater is the expected excess return, the more the consumer wants to invest.
Note, however, that the model implausibly says that the dollar amount invested
in the risky asset does not depend on the total dollar amount of resources a_{T−1}.
So, Warren Buﬀett and Homer Simpson should have the exact same dollar
holdings of the risky asset! If Buﬀett is richer than Simpson, Buﬀett’s excess
wealth is held in the safe form. Not very plausible. (That is why models with
CARA utility are increasingly unfashionable in the economics and ﬁnance
literatures).

### References

Merton, Robert C. (1969): “Lifetime Portfolio Selection under
Uncertainty: The Continuous Time Case,” Review of Economics and
Statistics, 51, 247–257.

Micklethwait, John, and Adrian Wooldridge (2002): The
Company: A Short History of a Revolutionary Idea. Modern Library.