# Correlation of a portfolio of trading strategies to a benchmark [closed]

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

• by indicator do you mean a benchmark portfolio? and how do you plan to weight the two trading strategies within the single portfolio, 50% and 50%? – develarist Dec 6 '19 at 11:59
• Yes. Let's say that the indicator is a benchmark portfolio. And, let's assume that the signals are equally weighted. Thanks. – SHUBHAM BHAT Dec 6 '19 at 16:34

This is best seen via properties of covariance and the relationship between covariance and correlation. Let's represent the two strategies by X and Y, and the benchmark by I.

Letting C represent the covariance between the two arguments of $$C\left[A,B \right]$$, we have for the equally weighted portfolio of X and Y:

$$C\left[ 0.5X+0.5Y,I\right]=0.5C\left[ X,I\right]+0.5C\left[Y,I\right]$$

Now make use of the relationship between covariance and correlation $$C\left[ X,Y\right]=\rho \sigma_x \sigma_y$$

$$\rho\left[ 0.5X+0.5Y,I\right]0.5 \sigma_{x+y}\sigma_i=0.5 \rho\left[ X,I\right] \sigma_x \sigma_i+0.5\rho\left[Y,I\right] \sigma_y \sigma_i$$

Cancelling $$0.5\, \sigma_i$$, this simplifies:

$$\rho\left[ 0.5X+0.5Y,I\right] \sigma_{x+y}= \rho\left[ X,I\right] \sigma_x +\rho\left[Y,I\right] \sigma_y$$

Now you want to assume that both strategies have the same correlation with I (0.5), so let's represent this by $$\rho=\rho\left[ X,I\right]=\rho\left[ Y,I\right]$$:

$$\rho\left[ 0.5X+0.5Y,I\right]\sigma_{x+y}= \rho \sigma_x + \rho \sigma_y$$

And it follows that:

$$\rho\left[ 0.5X+0.5Y,I\right]=\rho \frac{\sigma_x +\sigma_y }{\sigma_{x+y}}$$