# Dynamic asset allocation strategies using a stochastic dynamic programming approach

I am currently reading Gerd Infanger's Chapter 5 on "Dynamic asset allocation strategies using a stochastic dynamic programming approach" in the Handbook of Asset and Liability Management edited by S.A. Zenios and W.T. Ziemba.

On page 215, Infanger describes how one can write a multi-period portfolio selection problem as a dynamic programming problem. However, I don't understand two things in his formulation/conversion.

So, given that

• $W_t$ is the investor's wealth at time $t$,
• $s_t$ is the investor's income at time $t$,
• $u(W_t)$ is the investor's utility of wealth at time $t$,
• $x_t$ is the vector of fractions invested in each asset class at time $t$,
• $R_t$ is the vector of asset returns at time $t$, and
• that $T$ is the terminal period,

the multi-period investment problem can be written as:

max $E(u(W_t))$

subject to

$e^Tx_t = 1 \qquad t =0, T-1,$

$W_{t+1} = R_t x_t (W_t + s_t) \qquad t = 0, ..., T-1$

Now in the dynamic programming formulation that is supposed to be:

$u_t(W_t) =$ max $E(u_{t+1}((W_t + s_t)R_t x_t)$

subject to

$e^Tx_t = 1$

$Ax_t = b \qquad l \leq x_t \leq u$

where $u_T(W_T) = u(W)$.

Now my questions are:

1. What is $e^Tx_t = 1$ supposed to mean? If it requires the portfolio weights to sum up to one, why not use $\mathbf{1}^Tx_t$?

2. Where does the condition $Ax_t = b \qquad l \leq x_t \leq u$ in the dynamic programming formulation come from? What is it saying?

Any help is greatly appreciated.

• $e$ (einheit in German) is the same vector that you call $\mathbf{1}$ ("ones" in English), just a different notation. – Alex C Jul 26 '18 at 15:25
• Thanks for that clarification. I wasn't familiar with the $e$ notation. – apitsch Jul 26 '18 at 15:36

The condition $Ax_t = b \qquad l \leq x_t \leq u$ implements bounds on the portfolio weights. Unfortunately, that was not specified in the book itself.