EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

Figure 1: Performance Decomposition of Selected Mutual Funds with the Carhart Model (2001-2015) Mutual fund returns are downloaded from Datastream and factor returns are from Ken French's library. All returns are monthly. We regress for each mutual fund their excess returns on factor returns for the period 2001-2015 and use ordinary least square (OLS) to estimate the factor exposures. We measure historical risk premia over the same period and use the formula (1.5).

1.2.2 Volatility Decomposition Returns are linear functions of factors exposures making the expected return decomposition straightforward. Because the portfolio risk is non-linear in factor exposures, volatility decomposition is less immediate and we explore several methods to handle this issue. In what follows, we apply these methods to the decomposition of “absolute risk”, that is volatility, and to “relative risk”, defined as the tracking error with respect to a benchmark. 1.2.2.1 Absolute Versus Relative Risk Volatility We recall that multi-factor models for returns on portfolio have the general form:

We make the following assumptions: 1. The factor realisations vector, F t , is stationary with unconditional moments

2. Error terms, ε t

, are uncorrelated with

each of the common factors, F k,t , for all k and t .

3. Error terms ε t

are serially uncorrelated for t = s = 0, otherwise

With these assumptions, we can write the variance of the portfolio return as: (1.6)

With the variance fraction explained by the factor:

The coefficient of determination from our model is the ratio of systematic variation to the total return variation:

• R t is the return in excess of the risk-free rate on portfolio in time period t ( t = 1,..., T ), • B k is the factor loading or factor beta for the portfolio on the k -th factor, and B is the vector of beta.

In the multi-factor framework, the systematic risk depends not only on the

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