EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

allows them to quantitatively examine the returns of any risk parity strategy to determine whether their growth (equity) and inflation (bond) risk are indeed in parity. They consider a portfolio made of the S&P 500 index and the 10-year Treasury US bond, and they find that achieving parity in risk exposures has a positive impact on the Sharpe ratio, especially during bear period. They also show that asset-based risk parity portfolios can often concentrate too much in just one component of risk exposures, particularly equity risk, in contrast to factor-based risk parity which allows a more robust risk diversification. To illustrate this decomposition, we consider the same mutual funds as in Section 1.2.1, and we plot the contributions of the Carhart factors to their volatility in Figure 2. Most of the ex-post risk of the long-only funds in the Figure 2 comes from their market exposure and from specific risk. This specific risk was not rewarded in

this sample, as appears from the negative alphas in Figure 1, while market risk was well rewarded over the period. The long/ short fund has the highest specific variance and the lowest market risk but this specific risk is not rewarded for this fund over the period while the market factor obtains the highest reward. Common intuition and portfolio theory both suggest that the degree of diversification of a portfolio is a key driver of its ability to generate attractive risk-adjusted performance across various market conditions. Carli, Deguest and Martellini (2014) argue that balanced factor contributions to portfolio risk lead to higher performance in the long run (due to higher performance during bear market). Hence, fund managers could manage a better risk factor diversification with more balanced factor exposures. For this aim, they need to analyse and measure with accuracy their fund risk factor exposures.

Figure 2: Risk (Volatility) Decomposition Using Euler 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. Returns are monthly. We regress for each mutual fund their excess returns on factor returns for the period 2001-2015. We measure historical factor covariances matrix over the same period and use the formula (1.8).

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