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Macroeconomics

Portfolio Choice with CARA Utility

Authors
Affiliations
Johns Hopkins University
Econ-ARK
Johns Hopkins University
Econ-ARK

Consider a consumer with Constant Absolute Risk Aversion utility u(c)=−α−1e−αc\uFunc(\cRat) = -\CARA^{-1} e^{-\alpha \cRat_{}}, with assets aT−1\aLev_{T-1} who is deciding how much to invest in a risky security that will earn a normally distributed stochastic return factor RT∼N(R,σ2)\Risky_{T} \sim \mathcal{N}(\Risky,\sigma^{2}) versus a safe asset that will earn return factor R<R\Rfree < \Risky (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\Stocks in the risky asset, then

cT=SRT+(aT−1−S)R=aT−1R+(RT−R)S\begin{gathered}\begin{aligned} \cRat_{T} & = \Stocks \Risky_{T}+( \aLev_{T-1}-\Stocks)\Rfree \\ & = \aLev_{T-1}\Rfree + (\Risky_{T}-\Rfree)\Stocks \end{aligned}\end{gathered}
(1)

where RT−R\Risky_{T}-\Rfree is the “excess return” realized in period TT. Note that it could be negative: if the risky asset performs badly enough, shareholders lose not just the excess return they hoped to earn, but part of what they would have earned from the safe asset. Given S\Stocks, and with expected excess return ET−1[RT−R]=R−R\Ex_{T-1}[\Risky_{T}-\Rfree] = \Risky-\Rfree, expected utility as of time T−1T-1 is:

ET−1[u(cT)]=ET−1[−α−1e−α(aT−1R+(RT−R)S)]=−α−1e−α(aT−1R)ET−1[e−α(RT−R)S]=−α−1e−α(aT−1R)e−αS(R−R)+(αS)2σ2/2=−α−1e−α(aT−1R)e−α(S(R−R)−αS2σ2/2)\begin{gathered}\begin{aligned} \Ex_{T-1}[\uFunc(\cRat_{T})] & = \Ex_{T-1}[-\CARA^{-1}e^{-\alpha( \aLev_{T-1}\Rfree+(\Risky_{T}-\Rfree)\Stocks)}] \\ & = -\CARA^{-1}e^{-\alpha( \aLev_{T-1}\Rfree)} \Ex_{T-1}[e^{-\alpha(\Risky_{T}-\Rfree)\Stocks}] \\ & = -\CARA^{-1}e^{-\alpha( \aLev_{T-1}\Rfree)} e^{-\alpha\Stocks(\Risky-\Rfree)+(\alpha\Stocks)^{2}\sigma^{2}/2} \\ & = -\CARA^{-1} e^{-\alpha( \aLev_{T-1}\Rfree)} e^{-\alpha(\Stocks(\Risky-\Rfree)-\alpha\Stocks^{2}\sigma^{2}/2)} \end{aligned}\end{gathered}
(2)

and the third line follows from the second because if z∼N(μz,σz2)z \sim \mathcal{N}(\mu_{z},\sigma_{z}^{2}) then E[ez]=eμz+σz2/2.\Ex[e^{z}] = e^{\mu_{z}+\sigma_{z}^{2}/2}. (See ELogNorm).

Expected utility (2) is a negative constant times e−α(S(R−R)−αS2σ2/2)e^{-\alpha(\Stocks(\Risky-\Rfree)-\alpha\Stocks^{2}\sigma^{2}/2)}, so maximizing it means maximizing the term in parentheses:

max⁡S{S(R−R)−αS2σ22}\max_{\Stocks} \left\{ \Stocks (\Risky-\Rfree) - \frac{\alpha \Stocks^{2} \sigma^{2}}{2} \right\}
(3)

with FOC

R−R=αSσ2S=R−Rασ2.\begin{gathered}\begin{aligned} \Risky-\Rfree & = \alpha \Stocks \sigma^{2} \\ \Stocks & = \frac{\Risky-\Rfree}{\alpha \sigma^{2}}. \end{aligned}\end{gathered}
(4)

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\aLev_{T-1}. So, Warren Buffett and Homer Simpson should have the same dollar holdings of the risky asset! If Buffett is richer than Simpson, Buffett’s extra 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).

Footnotes

  1. The seminal paper examining this problem (in continuous time) was by Merton (1969); that paper also examines the case with CRRA utility and lognormal returns.

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  2. The assumption that returns are normally distributed is highly implausible. This means that with some positive probability, RT<0\Risky_{T} < 0. So, owning $1 of the risky asset in period T−1T-1 could result in negative wealth in period TT. You can lose more than everything, which is a violation of the legal principle of limited liability. (For a detailed history of limited liability, see Micklethwait & Wooldridge (2002).) Lognormally distributed returns are therefore much more plausible.

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References
  1. Merton, R. C. (1969). Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case. Review of Economics and Statistics, 51(3), 247–257. 10.2307/1926560
  2. Micklethwait, J., & Wooldridge, A. (2002). The Company: A Short History of a Revolutionary Idea. Modern Library.
Intertemporal Choice
Asset Pricing
Intertemporal Choice
Portfolio Choice with CRRA Utility (Merton-Samuelson)