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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'.

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    $\begingroup$ 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%? $\endgroup$ – develarist Dec 6 '19 at 11:59
  • $\begingroup$ Yes. Let's say that the indicator is a benchmark portfolio. And, let's assume that the signals are equally weighted. Thanks. $\endgroup$ – SHUBHAM BHAT Dec 6 '19 at 16:34
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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:

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

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