## Wednesday, September 16, 2015

### Scaling by correlation matrix

We analyze the effect of scaling a signal by the inverse of correlation matrix here. We start by assuming that the two assets $A_1$ and $A_2$ have unit variance. This reduces the co-variance matrix to correlation matrix. We assume a simple correlation matrix of the form $$\begin{bmatrix} 1 & c \\ c & 1 \end{bmatrix}.$$ Now let's say we have generated a signal of $\mu_1$ and $\mu_2$ for the two assets before scaling. This means that the unscaled portfolio can be written as $$\mu_1 A_1 + \mu_2 A_2.$$ Now the inverse of the correlation matrix is $$\frac{1}{1-c^2}\begin{bmatrix} 1 & -c \\ -c & 1\end{bmatrix}.$$ This makes the scaled signal ($\Sigma^{-1}\mu$) $$\frac{\mu_1-c\mu_2}{1-c^2}A_1+\frac{\mu_2-c\mu_1}{1-c^2}A_2.$$ We can see that based on the 'original signal' ($\mu_1$ and $\mu_2$) and the correlation value ($c$) the 'scaled signal' is altered. Another way to look at the 'scaled signal' is to write the portfolio as $$\mu_1\left[\frac{1}{1-c^2}A_1-\frac{c}{1-c^2}A_2\right] + \mu_2\left[\frac{1}{1-c^2}A_1-\frac{c}{1-c^2}A_2\right].$$ This is another way of saying that we trade the same original signal but replace the assets $A_1$ and $A_2$ with the spreads $\left[\frac{1}{1-c^2}A_1-\frac{c}{1-c^2}A_2\right]$ and $\left[\frac{1}{1-c^2}A_2-\frac{c}{1-c^2}A_1\right]$. In the table below we look at this 'spread' for different values of correlation coefficient $c$.  We also see the 'altered' signal value for the assets $A_1$ and $A_2$.
$$\begin{array}{c|cc|cc} c & \text{\mu_1} & \text{\mu_2} &A_1 & A_2 \\ \hline +0.9 & 5.3A_1-4.7A_2 & 5.3A_2-4.7A_1 & 5.3\mu_1-4.7\mu_2 & 5.3\mu_2-4.7\mu_1 \\ +0.5 & 1.3A_1-0.7A_2 & 1.3A_2-0.7A_1 & 1.3\mu_1-0.7\mu_2 & 1.3\mu_2-0.7\mu_1 \\ +0.1 & 1.0A_1-0.1A_2 & 1.0A_2-0.1A_1 & 1.0\mu_1-0.1\mu_2& 1.0\mu-0.1\mu \\ 0.0 & A_1 & A_2 & \mu_1 & \mu_2\\ -0.1 & 1.0A_1+0.1A_2 & 1.0A_2+0.1A_1 & 1.0\mu_1+0.1\mu_2 & 1.0\mu_2+0.1\mu_1 \\ -0.5 & 1.3A_1+0.7A_2 & 1.3A_2+0.7A_1 & 1.3\mu_1+0.7\mu_2 & 1.3\mu_2+0.7\mu_1 \\ -0.9 & 5.3A_1+4.7A_2 & 5.3A_2+4.7A_1 & 5.3\mu_1+4.7\mu_2 & 5.3\mu_2+4.7\mu_1 \end{array}$$
For the case of high absolute correlations, till $\mu_1$ and $\mu_2$ are comparable the total portfolio values are within limits. But if $\mu_1$ and $\mu_2$ differ substantially huge positive and negative positions can be created, which may be undesirable. This is a likely scenario as signals are based on recent updated information while the correlations rely on slow window.

What if we add a third asset $A_3$ with signal $\mu_3$ which is uncorrelated to the first two assets? We have the correlation matrix as $$\begin{bmatrix} 1 & c & 0 \\ c & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},$$ the inverse of this matrix is $$\frac{1}{1-c^2}\begin{bmatrix}1 & -c & 0\\ -c & 1 & 0 \\ 0 & 0 & 1-c^2\end{bmatrix}.$$ This results in the following 'altered' portfolio $$\frac{\mu_1-c\mu_2}{1-c^2}A_1+\frac{\mu_2-c\mu_1}{1-c^2}A_2+\mu_3A_3.$$ This shows that the signal of the uncorrelated asset is not changed.