## Tuesday, April 25, 2017

### Alternative portfolio mean-variance formulations

In (1) and (2) a portfolio optimization model was used as an example to illustrate a linear reformulation of a QP problem. The problem was stated as:
 \bbox[lightcyan,10px,border:3px solid darkblue]{\begin{align} \min \>& \sum_t w_t^2\\ & w_t = \sum_i r’_{i,t} x_i\\ & \sum_i \mu_i x_i \ge R\\ & \sum_i x_i = 1\\ &x_i\ge 0\end{align}}
Here we denote:
• $$x_i$$ are the optimal (no short) positions (these are the decision variables),
• $$R$$ is the minimum required portfolio return,
• $$r’_{i,t}=r’_{i,t}-\mu_i$$ are the mean-adjusted (historic) returns
• $$w_t$$ are intermediate variables
The more familiar mean-variance model is:
 \bbox[lightcyan,10px,border:3px solid darkblue]{\begin{align} \min&\>x^TQx = \sum_{i,j} q_{i,j} x_i x_j \\ & \sum_i \mu_i x_i \ge R\\ & \sum_i x_i = 1\\ &x_i\ge 0\end{align}}
Here $$Q$$ is the variance-covariance matrix.
These models are essentially the same. Using the fact how $$Q$$ could have been calculated:
 $q_{i,j} = \frac{1}{T} \sum_t (r_{i,t}-\mu_i)(r_{j,t}-\mu_j)$
we can write:
 \begin{align} \sum_{i,j} q_{i,j} x_i x_j &= \frac{1}{T}\sum_{i,j}\sum_t \left(r_{i,t}-\mu_i\right)\left(r_{j,t}-\mu_j\right) x_i x_j\\ &= \frac{1}{T} \sum_t \left(\sum_i r’_{i,t} x_i\right)\left(\sum_j r’_{j,t} x_j\right)\\ &=\frac{1}{T} \sum_t w_t^2 \end{align}
I dropped the factor $$\frac{1}{T}$$ from the model.

This means:
1. These models are indeed identical
2. We have implicitly proven that the covariance matrix is positive semi-definite as $$w_t^2\ge 0$$
3. This formulation can be use to derive a linear alternative by replacing the objective $$\min \sum_t w_t^2$$ by $$\min \sum_t |w_t|$$ (just a different norm).
4. The simpler quadratic form $$\sum_t w_t^2$$ is often easier to solve
5. When $$N>T$$ (number of instruments is larger than the number of time periods – not so unusual as older returns are less interesting), we have less data in the model: $$Q$$ is $$N \times N$$ while $$r’_{i,t}$$ is $$N \times T$$. In this case we “invent” data when forming the covariance matrix.
Note that nowadays it is popular to solve these type of portfolio models as a conic problem (see (3)). Some solvers convert QP problems (and quadratically constrained problems) automatically in an equivalent conic form.
##### References
1. QP as LP: piecewise linear functions, http://yetanothermathprogrammingconsultant.blogspot.com/2017/04/qp-as-lp-piecewise-linear-functions.html
2. QP as LP: cutting planes, http://yetanothermathprogrammingconsultant.blogspot.com/2017/04/qp-as-lp-cutting-planes.html
3. Erling D. Andersen, Joachim Dahl and Henrik A. Friberg, Markowitz portfolio optimization using MOSEK. MOSEK Technical report: TR-2009-2, http://docs.mosek.com/whitepapers/portfolio.pdf