Aug 16

Value at Risk (VaR). 2007 FRM, Part 12. Diversification

by David Harper, CFA, FRM, CIPM

FRM | Risk |

 diversifyIntro

Learning Outcome

  • LO 9.4: Calculate the standard deviation and VAR of an equally weighted portfolio of assets whose returns all have the same standard deviation and where the correlations of the returns are all equal for each pair of assets.

Add uncorrelated assets to diversify

This learning outcome is a classic application of modern portfolio theory (MPT). The assumptions are unrealistic: a portfolio where all positions have the same standard deviation and equivalent correlations/covariances. We only need three assumptions:

  • sigma is (each) position's volatility,  
  • N is the number of positions, and
  • rho is the correlation between the returns of each pair of assets

In which case portfolio standard deviation is given by:

diversifiedport

 

...or sometimes you see the equivalent portfolio variance:

diversifiedport_alt

These are the same. Regarding the last term, note that covariance (COV) is equal to the product of: (correlation)(volatility)(volatility).

The EditGrid spreadsheet below contains a familiar matrix of [number of positions] versus [correlation coefficients] for a given standard deviation (e.g., 20%). You can see on the graph why most people tend to suggest diversification is achieved with a limited number of positions (e.g., 30) as the lines tend to appear asymptotic somewhere around these levels.

EditGrid Spreadsheet by bt/frm2007.

Comments

  1. Parimal 10 Sep 2007

    Very helpful in understanding the MVAR and IVAR calculation.

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