Improved Risk Reporting with Factor-Based Diversification Measures

Improved Risk Reporting with Factor-Based Diversification Measures — February 2014

2. Portfolio Diversification Measures

the portfolio return r P as the sum of N uncorrelated factors F 1 , ..., F N (see also Deguest et al. (2013) or Meucci et al. (2013)). We thus have: , where r denotes the vector of returns of the original constituents, and r F the vector of uncorrelated factors’ returns. The main challenge with this approach is to turn correlated asset returns into uncorrelated factor returns. The factor returns can typically be expressed as a linear transformation of the original returns: r F = A'r for some well chosen transformation A guaranteeing that the covariance matrix of the factors Σ F = A' Σ A is a diagonal matrix. A is an N × N transition matrix from the assets to the factors and it is therefore critical that it be invertible since F = A −1 . Then, we define the contribution of each factor to the overall variance of the portfolio as:

α ≥ 0, α ≠ 1.

(2.4)

This family of measures, indexed with the free parameter α , takes into account not only the number of available assets but also the correlation properties between them. More specifically, a constituent that is highly positively correlated with all the other constituents will tend to have a higher contribution to the variance (considering long-only portfolio for simplicity), leading to a lower effective number of correlated bets since most of the portfolio risk is concentrated in that constituent. However, the measure of the contribution of each constituent to the variance is somehow arbitrary in how the overlapping correlated terms are affected to the various constituents. Indeed, the approach consists in adding to the variance of the position in constituent k , equal to , which is unambiguously related to constituent k , all covariance terms emanating from the overlap between constituent k and the other constituents, and which could also be affected to the other constituents. More generally, having dollars (eggs) evenly spread across bets (baskets) is not necessarily a sufficient condition for proper diversification if the bets are correlated (baskets are tied together). For these reasons, we now turn to the analysis of uncorrelated bets. 2.2.2 Measures of the Effective Number of Uncorrelated Bets (ENB) To alleviate the concern over an arbitrary affectation of the correlated components to the various portfolio constituents, Meucci (2009a) proposes to decompose

6 - One can easily show that factor risk parity portfolios give to each factor a weight that is given by the ratio of the inverse of its volatility with the harmonic average of the volatilities.

which leads to the following percentage contributions for each factor:

, where

The portfolios named factor risk parity portfolios in Deguest et al. (2013) are built such that the contribution p k of each factor to the variance are all equal. 6 Mimicking the definition of ENC and ENCB, we now introduce the effective number of “uncorrelated” bets of portfolio , or ENB in short, as:

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