SAAM Algorithm

Math Forms

After having another attention to the (*), we know that when $\mu$ and $\Sigma$ are unknown and treated as state variables whose uncertainties are specified by their posterior distributions given the observations $r_1,r_2,\cdots,r_n$ in a Bayesian framework. Note that if the prior distribution puts all its mass at$(\mu_0,\Sigma_0)$, then the minimization problem (*) reduces to Markowitz’s portfolio optimization problem that assumes $\mu_0$ and $\Sigma_0$ are given. The Lagrange multiplier $\lambda$ in (*) can be regarded as the investor’s risk-aversion index when the variance is used to measure risk.

The problem (*) is not a standard stochastic optimization problem because of $[E(\omega^T r_{n+1})]^2$ ,which is included in $Var(\omega^T r_{n+1})=E[(\omega^T r_{n+1})^2]-[E(\omega^T r_{n+1})]^2$. The method hereof solving (*) is to convert it to a standard stochastic control problem by using an additional parameter. Note that $E(W)-\lambda Var(W) = h(EW,EW^2)$, where $h(x,y) = x+\lambda x^2-\lambda y$. Let $W_B =\omega_B^T r_{n+1}$ and $\eta =1+2\lambda E(W_B)$, where $\omega_B$ is the Bayes weight vector. Then

$$\begin{equation} 0  \geq   h(EW,EW^2)-h(EW_B,EW_B^2) \\ =\eta\{E(W)-E(W_B)\} +\lambda \{E(W_B^2)-E(W^2)\} +\lambda \{E(W)-E(W_B)\}^2 \\ \geq \{\lambda E(W_B^2)-\eta E(W_B)\}-  \{\lambda E(W^2)-\eta E(W)\} \end{equation}$$

Since $\eta =1+2\lambda E(W_B)$ and the target is $max\{E(\omega^T r_{n+1}）-\lambda Var(\omega^T r_{n+1})\}$, so we have an equivalence problem as follows:

$\underset{\eta}{max} \{E[\omega^T(\eta) r_{n+1}]-\lambda Var[\omega^T(\eta) r_{n+1}]\}$

, $\omega(\eta)$ is the solution of the stochastic optimization problem

$\omega(\eta) =\underset{\omega}{argmin} \{\lambda E[(\omega^T r_{n+1})^2]-\eta E(\omega^T r_{n+1})\}$