# Capitalization constraint in portfolio optimization

My problem boils down the the classical $$a \cdot w - \lambda w \Sigma w \rightarrow Max$$ under the constraints $$A \cdot w \le b,$$ where the above constraints also contain information about my benchmark weights. I also have constraints in place about minimum investment if invested and maxinal over- resp. underweights.

However due to the nature of my constraints I see a strong concentration of the resulting portfolio either on the large cap segment or on the small cap segment of the market I work on.

Is there an elegant constraint such that I can neutralize a small cap bias. Maybe an elegant way to neutralize any capitalization bias?

EDIT: we can assume that the capitalization information is contained in the BM-weights. The stocks of the largest weights are the large caps. Can we apply some entropy approach?

You can introduce equality constraints

$$a^Tw = b$$

where the matrix $a$ contains information about the capitalization of each stock, and $b$ contains information about your benchmark.

For example, say you have six stocks, of which 1, 2 and 3 are small cap and 4, 5 and 6 are large cap. You also know that your benchmark has 80% of its market value as large caps, and 20% as small caps. Then you can use

$$a = \left[ \begin{array} 11 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \\ \end{array} \right], \quad b = \left[\begin{array} 00.2 \\ 0.8 \end{array}\right]$$

this ensures that whatever weights are selected, you will match your benchmark's investment in both small and large caps, thereby neutralizing any capitalization bias relative to your benchmark.

An alternative approach is to allow $a$ to contain both positive and negative values, where large cap stocks have generally positive values and small cap stocks have generally negative values, and demand that

$$a^Tw = 0$$

The question is how to select the weights in $X$. You certainly want your benchmark weights, $w_0$, to have no capitalization bias -

$$a^Tw_0 = 0$$

Beyond that you are free to choose the weights in any fashion that you wish. One example might be to choose the weights to co-vary with some measure $v$ of market capitalization which has a roughly normal distribution, for example the logarithm of market cap -

$$v = \log V$$

Then you could solve

$$\min_a \; (a-v)^T(a-v) \; \textrm{s.t.} \; a^Tw_0 = 0$$

which gives

$$a = v - \frac{w_0^T v}{w_0^T w_0} w_0$$

Now enforcing $a^Tw=0$ in your portfolio optimization ensures that your selected weights are free of capitalization bias in the same way that the benchmark is free of capitalization bias. Of course, what you mean by "capitalization bias" is encoded in the choice of weights in $a$, and there are multiple valid ways to choose this.

• Thanks for this answer. This is the obvious way to do it - but we have to draw a line in terms of capitalization. Can't we use the concept of entropy here? I have edited the question to make this clear. – Ric Mar 7 '17 at 10:17
• Very good answer! Looks very good theoretically and I will try it out ! – Ric Mar 7 '17 at 11:55
• I wasn't clear what you meant by "using the concept of entropy" so I went a different route. Did you mean to treat the weights $w$ as probabilities, and look at the entropy $\sum_i w_i \log w_i$? – Chris Taylor Mar 7 '17 at 12:58
• Right, this is what I meant. But if we set $V_i =w_i$ and thus work with $\log(V_i) = v_i = \log(w_i)$ then your approach is pretty similar -right? – Ric Mar 8 '17 at 7:04