EDHEC-Risk Institute October 2016

Multi-Dimensional Risk and Performance Analysis for Equity Portfolios — October 2016

1. Literature and Practice Reviews

We can define the relative contribution of one risk factor to the risk of the portfolio as the fraction of variance explained by the factor in the linear regression:

If F 1

and F 2

are orthogonal, we verify:

This example with K = 2 can be generalised. For j ≥ 1, we let j ( i ) denote the set of permutations of j factors chosen among 1 , ..., K and distinct of i (no repetition is allowed, and the order matters), and 0( i ) denotes the singleton that contains only the empty set.

To see how to build a measure of relative importance, we first consider a simple example with two risk factors. There are two ways of decomposing R2 (1,2):

and

Assume that R2 (1: K ) is not zero, we define the relative risk contribution of factor k as:

In the two decompositions above, one clearly sees that the order of introduction of a given factor matters in its contribution to the total coefficient of determination. In other words, the coefficient of determination of adding factor 2 to factor 1, denoted R2(1, 2 | 1 ), is different from the coefficient of determination of only factor 2, denoted R2(2) , that is unless the two factors are uncorrelated. One way to get rid of this dependency consists in decomposing the R2 as:

Then we have that 1) The relatives risk contributions are nonnegative; 2) The relatives risk contributions of the various factor sums up to 1: 3) If factor 1 ,…, K are orthogonal to each other, then the relative contribution of factor k is given by: In the aim to access factor relative contributions, with a four-factor model, we would need to run 8 regressions for each factor. Hence it would give us 32 regressions to perform. For each regression we would obtain the observable marginal increase of the coefficient of determination by adding the new factor needed to compute relative risk contribution of each factor. 1.3 Other Factor Models Used in Practice The Fama-French and the Carhart models are at the intersection between academic and industry practices. They are commonly

In this context, one may define the relative importance of factor 1 and 2 as the average contribution of the factor under consideration over all possible permutations of the set of existing factor, or, in this example:

And

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