5.4.1 Bayesian Investor’s Asset Allocation

5.4.1 Bayesian Investor’s Asset Allocation

To illustrate Bayesian decision making in the context of a simple prediction model, we use the example provided by Kandel and Stambaugh (1996) of a single-period investment decision. Consider an investor who, at time t, can allocate a fraction of his wealth, $\omega ,$ to stocks which pay continuously compounded returns $y_{t+1}$ or hold riskfree T-bills which for simplicity we assume pay zero returns. For simplicity assume that the investor’s initial wealth at time t is $W_t=1$ . Then the future wealth at time $t+1,W_{t+1}$ , is given by

$$W_{t+1}=1+\omega \left[\exp (y_{t+1})-1\right].$$

The investor is assumed to have logarithmic utility over future wealth,

$$U(W_{t+1})=\ln (W_{t+1}),$$

and maximizes expected utility conditional on current information, $Z_t=$ $\{z_1,\dots ,z_t\}$ :

$$\underset{0<\omega <1}{\max }\int U(W_{t+1}{)}_{PY}(y_{t+1}|Z_t)d{y}_{t+1}.$$

To choose asset holdings, the investor therefore requires the conditional density of $y_{t+1},p_Y(y_{t+1}|Z_t)$ . This is assumed to be known up to a set of unknown parameters, θ.

The predictive density for $y_{t+1}$ can be obtained by integrating the return distribution over the conditional density of θ , $p(\theta |Z_t)$ :

$$p(y_{t+1}|Z_t)={\int }_{\theta }{p}_Y(y_{t+1}|\theta ,Z_t)p(\theta |Z_t)d\theta ,$$

where, from Bayes’ theorem, the posterior density of θ conditional on $Z_t$ is $\begin{array}{r}p(\theta |Z_t)\propto \end{array}$ $p(\theta )p(Z_t|\theta ),p(\theta )$ is the prior on $\theta ,$ and $p(Z_t|\theta )$ is the p.d.f. for the data given the parameters.

As an illustration, consider the binary case where stock returns take two possible values: $y_{t+1}=+20\%$ or $y_{t+1}=-20\%$ . Suppose the information consists of a sequence of i.i.d. binomial state variables, $s_t,$ that take the value $s_t=1$ (state 1) or $s_t=2$ (state 2): $Z_t=\left\{s_t,\{s_{i-1},r_i{\}}_{i=1}^t\right\}$ . The model is parameterized through the probability that the high return state will occur next period given that state i is observed at time $t,i.e.,\theta =\{{\theta }_1,{\theta }_2\}$ , where ${\theta }_i=P(y_{t+1}=20|s_t=i)$

Following Kandel and Stambaugh (1996), assume that the investor’s prior joint distribution is the product of two independent distributions for ${\theta }_1$ and ${\theta }_2$ each of which is given by

$$\hat{p}({\theta }_i)=\frac{[{\theta }_i(1-{\theta }_i){]}^{c-1}}{B(c,c)},i=1,2,\text{\hspace{1em}(5.13)}$$

where $c\ge 1$ , and $B(\cdot )$ is the beta function. Hence, the prior depends on a single parameter, c.

Suppose the investor observes the following sample information presented in a $(2\times 2)$ contingency table.

$$\begin{array}{rl}\frac{|\mathrm{State}1(t-1)\mathrm{State}2(t-1)}{8}& \\ & y_t=20\%\frac{\mathrm{State}3}{5}\\ & y_t=-20\%\end{array}$$

If the economy has been in state i in $T_i$ months during the observed sample, the probability that the high return state is realized in $n_i$ of those months has a binomial likelihood function:

$$\begin{array}{r}p(n_i|{\theta }_i,T_i)=(\genfrac{}{}{0ex}{}{T_i}{n_i}){\theta }_i^{n_i}(1-{\theta }_i{)}^{T_i-n_i},i=1,2.\end{array}\text{\hspace{1em}(5.14)}$$

Combining the likelihood function in (5.14) with the prior in (5.13), the posterior for ${\theta }_i$ can be shown to follow a beta distribution:

$$\begin{array}{l}p({\theta }_i|Z_t)=p({\theta }_i|n_i,T_i)\\ =\frac{{\theta }_i^{n_i+c-1}(1-{\theta }_i{)}^{T_i-n_i+c-1}}{B((n_i+c),(T_i-n_i+c))},i=1,2.\end{array}$$

Letting $\hat{\mathbf{p}}$ be the probability of observing the high return state conditional on $Z_t.$ Kandel and Stambaugh (1996) show that, from properties of the beta distribution,

$$\hat{p}=\frac{n_j+c}{T_j+2c}.$$

Moreover, the expected stock return can be calculated from

$${\hat{y}}_{t+1}=E\left[y_{t+1}|Z_t\right]=0.2\hat{p}-0.2(1-\hat{p}).$$

The investor’s constrained optimization problem therefore becomes

$$\underset{0<\omega <1}{\max }(\hat{p}\ln (1+0.2\omega )+(1-\hat{p})\ln (1-0.2\omega )).$$

This has solution

$${\omega }^{\ast }=\{\begin{array}{ll}0& \mathrm{if}\left(2\hat{p}-1\right)<0,\\ (2\hat{p}-1)/0.2& \mathrm{if}0<2\hat{p}-1<0.2,\\ 1& \mathrm{if}\left(2\hat{p}-1\right)\ge \mathrm{0.2.}\end{array}$$

Suppose the signal $s_t=2$ is observed at time T . Since $n_2=8,T_2=16$ , then $\hat{p}=0.5$ independent of which value c takes in the prior. In this case the conditional risk premium on stocks equals ${}^{0,}$ and so risk-averse investors choose not to hold any stocks.

Next, suppose $s_t=1$ . Using that $n_1=11,T_1=16,\hat{p}=(11+c)/(16+2c)$ , we have the following.

Even though it cannot be rejected from a statistical point of view that there is little information in receiving the signal $s_t=1$ (under the null that ${\theta }_1=0.5,$ the probability of drawing $n_1\ge 11{\stackrel{\ll }{\mathrm{up}}}^{\gg }$ states in $T_1=24$ draws exceeds 10%), Bayesian investors change their asset allocation decisions quite strongly based on the observed signal, with the allocation to stocks varying from 54 to 100% in the above example, depending on the strength of the investors’ prior beliefs.