©August 28, 2019, Christopher D. Carroll CARAPortfolio
Consider a consumer with Constant Absolute Risk Aversion utility u(c) = −α−1e−αc, with assets aT−1 who is deciding how much to invest in a risky security that will earn a normally distributed stochastic return RT ∼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!).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
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
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 aT−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).