Optimal Portfolios

Question

In modern portfolio theory, one famous problem is the Markowitz mean variance optimal portfolio, defined by solving

$$\underset{\mathbf{w}}{\mbox{min}\,\,}\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}$$

subject to $\mathbf{w}^{T}\mathbf{1}=1$ and $\mathbf{w}^{T}\boldsymbol{\mu}=\eta$.

Another example that I've seen in lectures is the Minimum Variance Portfolio which is the same as above except the condition $\mathbf{w}^{T}\boldsymbol{\mu}=\eta$ is dropped.

I was wondering, there are surely lots of other similar sorts of optimisation problems similar to these. For example,

• imposing each entry of $\mathbf{w}$ is >0 -- to avoid short shelling
• imposing each entry of $\mathbf{w}$ is < $\alpha$ to avoid putting too much weight into one stock

My question is as follows: is there a convenient list of these sorts of optimisation problems, and their solutions?

Answer 1

As a practitioner, I have worked on the following

• Maximize Yield/OAS for a Fixed Income Portfolio keeping the Rates Duration (Key Rate Durations) and Spread duration in a constrained range . There are other constraints such as

  1. No short selling
  2. Max amount you can buy is X% of Max outstanding amount in market
  3. Maximum exposure to a perticular country , issuer, Sector , currency etc is constrained
  4. Maximum portfolio turnover is within a certain limit.
  5. Transaction Cost (Defined as function of DV01 Bid-Offer Spread) is within a range

• Instead of the objective function being Yeild/OAS or any other measure of return we can also try minimize functions such as RWA(Risk Weighted Assets) , Basel 3 Capital required etc. These problems have similar set of constraints as the previous one.

• I am attempting to solve a dynamic optimization exercise where we would have re-balancing based on a simulated environment of rates , inflation , fx etc.

Most of these are not purely Markowitch type and I end up using Linear / Quadratic programming based on the use case.

Hope this helps you in some small way.

Answer 2

One more kind of problem in your basket: $$ \max_w \left(w^T \mu -q \cdot w^T \Sigma w\right) $$ where $q\geq 0$ is a risk-aversion parameter. In case $q\to\infty$ you are extremely risk-averse, and you minimize the variance without caring about the mean. If $q =0 $ you are risk-neutral, and you're only interested in maximizing the mean. You can put all possible linear constraints on top: $Aw \leq b$, $A'w = b'$ etc. and they will all fall into the class of quadratic optimization problems, that are very-well studied in math - in particular, most of them won't have nice analytic formulas for $w$, but can compute it numerically rather fast.