Bayesian methods have long played a role in finance and asset allocation since the seminal work of de Finetti (1941) and Markowitz (2006). In this paper, we show how the principle of maximum expected utility (MEU) (Ramsey, 1926, Savage, 1956, Bernardo and Smith, 2000) together with Stein's lemma for stochastic volatility distributions (Gron, Jorgensen and Polson, 2011) solves for the optimal asset allocation. Stein's lemma provides the solution to the first order condition that accompanies MEU. The optimal asset allocation problem couched in equilibrium then leads to models such as the Capital Asset Pricing Model (CAPM) or Merton's inter-temporal asset pricing model (ICAPM).
We consider an investor who wishes to invest in the risky asset in order to maximize the expected utility of her resulting wealth. Under logarithmic utility, this leads to the famous Kelly criterion which maximizes the expected long-run growth rate of the risky asset. We review the link between the Kelly rule (Kelly, 1956) and the Merton optimal asset allocation (Merton, 1969). We illustrate their implementation for a discrete binary setting and for the standard historical returns on the S&P500. Under a CRRA utility we use Stein's lemma to derive fractional versions of the Kelly rules where the amount allocated is normalized by the investor's relative risk aversion.
The rest of the paper is as follows. Section 2 reviews the impact of Bayesian thinking in models of finance: asset pricing equilibrium models; how agents learn from prices; properties of returns data including predictability and stochastic volatility. Section 3 views asset allocation from a Bayesian decision theoretic perspective (see, for example, Bernardo and Smith, 2000). Section 3 studies maximization of the expected long-run growth rate and derives the classic Kelly and Merton allocation rules. Section 4 describes methods for estimating this long-run growth rate. Section 5 describes estimation methods for long-run asset allocation. Section 6 considers extensions to Bayesian dynamic learning (Bellman and Kalaba, 1956) and time-varying investment opportunity sets (Ferguson and Gilstein, 1985).
When investors are faced with a return distribution that is an exchangeable Beta-Binomial process, the effect of dynamic learning makes investors willing to invest a small amount of capital to current returns that have a negative expectation even though they are averse to risk. This is due to the fact that they might learn that the investment opportunity set improves in the future and this is taken account of in the Bayesian MEU solution. Finally, Section 7 concludes.