2.5.2 Portfolio Choice under Mean–Variance Utility

2.5.2 Portfolio Choice under Mean–Variance Utility

As an illustration of the relationship between economic utility and predictability, consider the single-period portfolio choice problem for an investor who can either hold T-bills which, for simplicity we assume pay a zero risk-free rate, or stocks which pay an excess return over the T-bill rate of ${\gamma }_{t+1}$ . Assuming that the investor has initial wealth $W_t=1$ , and letting ${\omega }_t$ be the portion of the investor’s portfolio held in stocks at time t, future wealth at time $t+1,W_{t+1}$ , is given by

$$W_{t+1}={\omega }_t{y}_{t+1}.\text{\hspace{1em}(2.39)}$$

The portion of wealth held in stocks, ${\omega }_t$ , is the investor’s choice variable. To analyze this decision we need to specify the investor’s utility as well as how accurately $y_{t+1}$ can be predicted.

Suppose the investor has mean–variance utility over future wealth and maximizes expected utility:

$$E\left[U(W_{t+1})|Z_t\right]=E\left[W_{t+1}|Z_t\right]-\frac{a}{2}\mathrm{Var}(W_{t+1}|Z_t),\text{\hspace{1em}(2.40)}$$

where a captures the investor’s risk aversion. The quantities $E\left[{\mathbf{W}}_{t+1}|Z_t\right]$ and $\mathrm{Var}(W_{t+1}|Z_t)$ are the conditional mean and variance of $W_{t+1}$ given information at time $t,Z_t=\{z_1,...,z_t\}$ . Under this utility function, the investor’s expected utility increases in expected returns but decreases in the amount of risk, as measured by the conditional variance of wealth. We can think of expected loss as the negative of (2.40).

Following Campbell and Thompson (2008), consider the following datagenerating process for excess returns on stocks:

$$y_{t+1}=\mu +z_t+{\epsilon }_{t+1},\text{\hspace{1em}(2.41)}$$

where $z_t\sim (0,{\sigma }_z^2),{\epsilon }_{t+1}\sim (0,{\sigma }_{\epsilon }^2)$ and $\mathrm{Cov}(z_t,{\epsilon }_{t+1})=0$ . Here $z_t$ represents a potentially predictable return component which may be known at time t, while ${\epsilon }_{t+1}$ is an unpredictable shock to returns. For an investor without information on $z_t,$ the expected value of $y_{t+1}$ is ${\mu }_{:}$ , while the variance of $y_{t+1}$ is $({\sigma }_z^2+{\sigma }_{\epsilon }^2)$ , and so

$${\omega }_t^{\ast }=\underset{{\omega }_t}{\mathrm{argmax}}\left\{{\omega }_t\mu -\frac{a}{2}{\omega }_t^2({\sigma }_z^2+{\sigma }_{\epsilon }^2)\right\},$$

which implies the following optimal holdings of stocks:

$${\omega }_t^{\ast }=\frac{\mu }{a({\sigma }_z^2+{\sigma }_{\epsilon }^2)}.\text{\hspace{1em}(2.42)}$$

Given this weight on stocks, the average (unconditional expectation) of the excess return on the uninformed investor’s stock holdings becomes

$$E\left[{\omega }_t^{\ast }{y}_{t+1}\right]=E\left[\frac{\mu (\mu +z_t+{\epsilon }_{t+1})}{a({\sigma }_z^2+{\sigma }_{\epsilon }^2)}\right]=\frac{{\mu }^2}{a({\sigma }_z^2+{\sigma }_{\epsilon }^2)}=\frac{S^2}{a}.\text{\hspace{1em}(2.43)}$$

where $S=\mu /\sqrt{{\sigma }_z^2+{\sigma }_{\epsilon }^2}$ is the unconditional Sharpe ratio, i.e., the expected excess return per unit of risk (volatility). Similarly, $\mathrm{Var}[{\hat{\omega }}_t^{\ast }{y}_{t+1}]={\mu }^2/[a^2(\widehat{{\sigma }_z^2}+{\sigma }_{\epsilon }^2)]$ ], and so the expected utility, evaluated at the optimal stock holdings, ${\omega }_t^{\ast }$ , is

$$E\left[U(W_{t+1}(f_t^{\ast }))\right]=\frac{{\mu }^2}{2a({\sigma }_z^2+{\sigma }_{\epsilon }^2)}=\frac{S^2}{2a}.\text{\hspace{1em}(2.44)}$$

Turning to the informed case where the investor exploits the predictable component in stock returns, $z_t,$ the conditional expectation and variance of future wealth are $E\left[W_{t+1}|Z_t\right]={\omega }_t(\mu +z_t)$ and $\mathrm{Var}(W_{t+1}\overline{|}{Z}_t)={\omega }_t^2{\sigma }_{\epsilon }^2$ , respectively, and so this investor’s optimal stock holding is

$${\tilde{\omega }}_t^{\ast }(z_t)=\frac{\mu +z_t}{a{\sigma }_{\epsilon }^2}.\text{\hspace{1em}(2.45)}$$

This investor’s expected average excess return becomes

$$E\left[\frac{(\mu +z_t)(\mu +z_t+{\epsilon }_{t+1})}{a{\sigma }_{\epsilon }^2}\right]=\frac{{\mu }^2+{\sigma }_z^2}{a{\sigma }_{\epsilon }^2}.\text{\hspace{1em}(2.46)}$$

Noting that the predictive $R^2$ in (2.41) is given by $R^2={\sigma }_z^2/({\sigma }_z^2+{\sigma }_{\epsilon }^2)$ , the informed investor’s average (unconditionally expected) excess return in (2.46) can be written as

$$E\left[{\tilde{\omega }}_t^{\ast }(z_t){\gamma }_{t+1}\right]=\frac{S^2+R^2}{a(1-R^2)}.\text{\hspace{1em}(2.47)}$$

Comparing expected returns under the unconditional forecast in (2.43) to the expected return under the conditional forecast in (2.47), the proportional increase in the investor’s expected excess returns is

$$\frac{E\left[{\tilde{\omega }}_t^{\ast }(z_t)y_{t+1}\right]}{E\left[{\omega }_t^{\ast }{y}_{t+1}\right]}=\frac{1+(R^2/S^2)}{1-R^2},\text{\hspace{1em}(2.48)}$$

whereas the simple return difference (2.47)–(2.43) amounts to $R^2(1+S^2)/$ $(a(1-R^2))$ .

Empirical work indicates that predictive return regressions have an $R^2$ close to $0,$ so the ratio in (2.48) is close to $(1+R^2/S^2)$ , suggesting that the magnitude of the predictive $R^2$ should be evaluated relative to the squared Sharpe ratio. Campbell and Thompson (2008) use historical data to estimate a squared monthly Sharpe ratio of 0.012 or 1.2%. Hence, even a monthly $R^2$ of “only” 0.5% would increase the average portfolio excess return by a factor 0.5/1.2, i.e., by roughly 40%. Given their historical data, this corresponds to an increase in the expected portfolio return of approximately 1.7% per annum assuming a risk aversion coefficient of $a=3$ . Even small $R^2$ -values can thus make a considerable difference to portfolio performance in this case. Moreover, the predictive $R^2$ can be used as a measure of the expected return gains arising from predictability.

Mean–variance investors are concerned with expected utility rather than expected returns. For uninformed investors their expected utility is given by (2.44). For informed investors, using the optimal stock holdings in (2.45), we have

$$\begin{array}{rl}& E\left[{\tilde{\omega }}_t^{\ast }(z_t)y_{t+1}|Z_t\right]=E\left[\frac{(\mu +z_t)(\mu +z_t+{\epsilon }_{t+1})}{a{\sigma }_{\epsilon }^2}|Z_t\right]=\frac{(\mu +z_t{)}^2}{a{\sigma }_{\epsilon }^2},\\ & \mathrm{Var}[{\tilde{\omega }}_t^{\ast }(z_t)y_{t+1}|Z_t]=\frac{(\mu +z_t{)}^2{\sigma }_{\epsilon }^2}{a^2{\sigma }_{\epsilon }^4}=\frac{(\mu +z_t{)}^2}{a^2{\sigma }_{\epsilon }^2},\end{array}$$

so that

$$E\left[U(W_{t+1}({\tilde{\omega }}_t^{\ast }(z_t)))|Z_t\right]=\frac{(\mu +z_t{)}^2}{a{\sigma }_{\epsilon }^2}-\frac{a}{2}\frac{(\mu +z_t{)}^2}{a^2{\sigma }_{\epsilon }^2}=\frac{(\mu +z_t{)}^2}{2a{\sigma }_{\epsilon }^2}.$$

The average (unconditional expectation) value of this expression is

$$E\left[E\left[U(W_{t+1}({\tilde{\omega }}_t^{\ast }(z_t)))|Z_t\right]\right]=\frac{{\mu }^2+{\sigma }_z^2}{2a{\sigma }_{\epsilon }^2}.\text{\hspace{1em}(2.49)}$$

Comparing (2.49) to (2.44), it is clear that the two are identical only when ${\sigma }_z^2=0;$ otherwise $(2.49)>(2.44)$ , and the increase in expected utility due to using the predictor variable is given by

$$\mathrm{CER}=\frac{{\sigma }_z^2}{2a{\sigma }_{\epsilon }^2}=\frac{R^2}{2a(1-R^2)}.$$

This is the certainty equivalent return (CER), i.e., the additional guaranteed return which, if paid to uninformed investors, would equate their expected utility with that of investors with access to the predictor variable. Using the earlier empirical numbers, for $R^2=0.005$ and $a=3$ , this amounts to an annualized certainty equivalent return of about 1%.