Portfolio with lots of subportfolios

Question

An account manager has $N$ distinct, equally-sized pots of money, which will be used to make $N$ distinct subportfolios, each of which is drawn from a slightly different (but potentially overlapping) set of potential assets.

The rates of return from all of the assets in the whole problem are contained in the vector $\textbf{R}$:

$\textbf{R}= \begin{pmatrix} R_1 \\ R_2 \\ \vdots \\ R_I \end{pmatrix}$

The rates of return for each asset are normally distributed random variables.

Each subportfolio $n=1,2,...,N$ allocates investment weights, represented in the vector $x_n$, to the different assets in its particular set of $K$ potential assets, which are found at scattered positions inside the vector $\textbf{R}$. (The 'investable universe' for each subportfolio is not random. The assets available to each subportfolio are very specific, but they are not necessarily clustered in the same region of the vector $\textbf{R}$.)

$\textbf{x}_n= \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_K \end{pmatrix}$

The weights within each subportfolio must sum to 1. i.e., where $\textbf{1}$ is a K-by-1 vector of 1s:

$\textbf{x}_n\textbf{1}=x_1+x_2+...+x_K=1$

The rates of returns from the specific assets belonging to subportfolio $n$ could be listed in another vector, $\textbf{R}_n$.

A variance-covariance matrix $\textbf{C}$ contains all the variances associated with each asset and the covariances in returns between them. The manager is interested only in the expected return and variance of the whole portfolio. If some subportfolios suffer for the greater good of the whole portfolio, so be it.

If you could put all the subportfolio vectors $\textbf{x}_n$ into a single vector of weights $\textbf{X}$ (containing all the investments made across the whole suite of subportfolios), the variance of the whole portfolio would, I presume, be $\sigma^2_p=\textbf{X}'\textbf{C}\textbf{X}$.

I would like to know how you would work out the optimization problem for the account manager as to how to distribute its investment within each subportfolio, given the covariances etc between the different assets. Putting all their investments on the asset with the single highest expected return within each subportfolio is presumably a bad move if these assets are all very highly correlated!

Would you simply set this up as a Lagrange multiplier problem with loads of constraints, each one specifying the 'investable universe' available to each subportfolio? I'm unsure where to begin, so any advice about this kind of question would be really appreciated. Thanks!

Answer 1

One way to this is the following (you can code all these constraints if you use the right software, I am doing such things using mathematica)

1. You define $w_{i,j}$ which is the weight of asset $j$ in subportfolio $i$, furthermore you define $w =(w_j)_{j=1}^{\text{no of assets}}$ the total weight of the portfolio in asset $j$.
2. the objects for the optimization then are the total portfolio variance $w^T \Sigma w $ and the total expected return $w^T \mu$.
3. you define a bunch of constraints. The ones that you need and the universe constraints $$ w_{i,j} = 0 $$ if asset $j$ is not allowed in subportfolio $i$, the total weight constraint for each asset $j$: $$ \sum_{i=1}^{\text{no of portfolios}} w_{i,j} - w_j = 0 $$ which is just a linear constraint. The budget for each subportfolio $i$ $$ \sum_{j=1}^{\text{no of assets}} w_{i,j} = \text{total fraction of subportfolio}. $$

Thus you have a lot of varables: number assets times number subportfolio + number assets. Only the total weights in the assets enter the objective function. Everything else is done in clever constraints.