Portfolio Weight Constraints in Dynamic Asset Allocation
Summary
The document presents a question about translating a multi-period asset allocation problem into a dynamic programming formulation. Wealth evolves through investment returns applied to current wealth plus income, while the investor chooses portfolio fractions at each period and seeks to maximize expected utility. The discussion asks what the constraint that portfolio weights sum to one means, and where additional linear equality and lower and upper bound constraints come from.
The accepted response clarifies only the latter constraints: the matrix equality and bounds implement restrictions on portfolio weights. The response says those restrictions were not specified in the cited book passage and were found in a related presentation. The document does not resolve the notation question about the vector of ones, nor does it explain how to choose particular bounds or equality restrictions. It is therefore a limited clarification of how allocation constraints can be represented inside a dynamic optimization model, rather than a full treatment of dynamic programming or portfolio construction.
Key ideas
- The model chooses asset allocation fractions over multiple periods to maximize expected utility of wealth.
- Wealth at the next period depends on current wealth plus income and portfolio returns.
- The sum-to-one equality expresses a fully allocated portfolio under the stated setup.
- Linear equalities and lower and upper bounds encode additional restrictions on portfolio weights.
- The response clarifies the weight constraints but leaves the notation question unresolved.
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Full text
# Dynamic asset allocation strategies using a stochastic dynamic programming approach
# 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.
## Answer by apitsch (score 0, accepted)
https://quant.stackexchange.com/a/40968
Ok. After my first question was solved in the comments, I've found the information I needed for the second question in a keynote on Dynamic Asset Allocation by Mr. Infanger on page 19.
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.