©August 28, 2019, Christopher D. Carroll CARAPortfolio

Portfolio Choice With CARA Utility

Consider a consumer with Constant Absolute Risk Aversion utility u(c) = α1eαc, with assets aT1 who is deciding how much to invest in a risky security that will earn a normally distributed stochastic return RT ∼𝒩(R) 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!).1 ,2

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

c   =   SR    +  (a     −  S )R
 T          T      T − 1
    =   aT − 1R + (RT   −  R) S
                  ◟ ---◝◜---◞
                      ≡ΦT
where ΦT is the equity premium realized in period T. Given S and defining the expected equity premium as Φ = 𝔼 T1[RT R], the expectation as of time T 1 is:
                              − 1 − α(aT− 1R+ (RT − R)S)
𝔼T  − 1[u(cT )] =   𝔼T − 1[− α   e                   ]
                        − 1 − α(aT−1R)       − α(RT− R)S
                =   − α   e          𝔼T  − 1[e          ]           (1)
                =   − α − 1e− α(aT−1R)e− αSΦ+ (αS)2σ2∕2             (2)
                                                2 2
                =   − α − 1e− α(aT−1R)e− α(SΦ− αS σ ∕2)             (3)
and (2) follows from (1) because if z ∼𝒩zz2) then 𝔼[ez] = eΦz+σz22 . (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

      {                }
               αS2  σ2
max     S Φ −  --------
  S               2
(4)

with FOC

             2
Φ  =   αS  σ                                  (5)
         Φ
S  =    ----.                                 (6)
        ασ2

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 aT1. 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.