Correlation of portfolio of two strategies

Question

I have two trading strategies, both having a correlation of 0.5 to indicator 'i'. If I take a portfolio of these two strategies, what will be the correlation of this portfolio with the indicator 'i'.

Answer 1

Let $\vec{a}$, $\vec{b}$ and $\vec{s}$ be the returns of the two strategies and the SP500, respectively.

Given that

$$\frac{\vec{a}\cdot \vec{s} }{as}= 0.5, \>\>\>\frac{\vec{b}\cdot \vec{s} }{bs}= 0.5$$

the correlation of the combined strategy with the SP500 is

$$\frac{ \vec{a}\cdot \vec{s}+\vec{b}\cdot \vec{s} }{ |\vec{a}+\vec{b}|s} =\frac{0.5as+0.5bs}{|\vec{a}+\vec{b}|s}=0.5\frac{a+b}{|\vec{a}+\vec{b}|}$$

Assume that the two trading strategies are not correlated and have same expected returns ($a=b$). Then, the correction of the combined strategy with the SP500 becomes

$$0.5\frac{a+b}{\sqrt{a^2+b^2}}=0.5\frac{2}{\sqrt 2}=0.5\sqrt 2$$

Answer 2

The only possible answer is "It depends". It depends on:

• What is your mathematical model for the strategies and the S&P?
• What do you mean by "combine strategies"?
• And of course it will depend on your specific strategies.

My answer is concerned with the most simple interpretation. I assume the two strategies and the S&P are vectors in $\mathbf{R}^n$ and "combining" means adding the cash flows. Without loss of generality I assume all vectors as zero mean and unit variance. Furthermore $n>1$ since otherwise correlation 0.5 were not possible. Under these assumptions every correlation between 0.5 and one is possible.

The case 0.5 is quite trivial: If both strategies are equal, their sum will again have correlation 0.5.

In case you have only two linearly independent strategies (i.e. $n=2$) the only other correlation possible is 1. This can by visualised by a sketch in the plane. Call the S&P $S$ and your strategies $u$ and $v$. Assume that $S$ is the canonical unit vector $(1, 0)$. Then $u$ and $v$ must be the unit vectors making an angle $\frac{\pi}{3}$ with $S$. Their coordinates are $(\cos \frac{\pi}{3}, \sin \frac{\pi}{3})$ and $(\cos \frac{\pi}{3}, -\sin \frac{\pi}{3})$. Since $\cos \frac{\pi}{3}=0.5$ their correlation with $S$ is 0.5 and their sum is equal to $S$. Hence the correlation is 1.

In higher dimensions the set of all unit vectors with correlation 0.5 to $S$ is a hyper-sphere (in $n=3$ a circle) around the point $(1,0,\ldots,0)$. Each pair of different strategies is defined by a pair of points on this circle. If the points are close to each other the correlation of the sum will be close to but above 0.5, if they are opposite to each other the correlation will be 1. You can do more precise calculations simply in the plane containing $u$ and $v$.