How Strategy Correlation Determines Portfolio Correlation to a Benchmark

Summary

The document explains why knowing each of two strategies’ correlations with a benchmark does not by itself determine the correlation of their combined portfolio. It derives the portfolio’s covariance with the benchmark as the weighted sum of each strategy’s covariance, then converts that covariance into correlation by dividing by the portfolio and benchmark standard deviations.

For an equally weighted portfolio whose two strategies each have correlation 0.5 with the benchmark, the resulting correlation depends on the strategies’ individual volatilities and on the volatility of their combined returns. Thus the portfolio correlation need not equal the shared individual correlation; diversification between the strategies changes portfolio volatility and therefore the result. The derivation assumes the stated equal weights and equal benchmark correlations. It does not provide a numeric answer because the strategy volatilities and their covariance with each other are not specified.

Key ideas

• Portfolio covariance with a benchmark is the weighted sum of the component strategies’ covariances with it.
• Portfolio correlation also depends on the portfolio’s standard deviation.
• Equal component correlations do not determine the combined portfolio correlation by themselves.
• The strategies’ volatilities and their relationship to each other are needed to calculate the result.

Tags

# Correlation of a portfolio of trading strategies to a benchmark


# Correlation of a portfolio of trading strategies to a benchmark












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'.

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/50140

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:

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.