In this section the term “diversification” is applied to the portfolio risk itself. The portfolio risk can in principle be captured by either a volatility-based measure (i.e., the portfolio’s standard deviation) or a downside-based measure (i.e., the conditional value at risk (CVaR) or expected shortfall). Hence, two approaches will be presented here. Heuristically these approaches are motivated by the empirical observation that the risk contributions are a good predictor for actual portfolio losses, and hence, by diversifying directly on these contributions, portfolio losses can potentially be limited compared to an allocation that witnesses a high risk concentration on one or a few portfolio constituents (see Qian 2005).
In the first approach, an asset allocation is sought such that the contributions of the portfolio’s constituents contribute the same share to the overall portfolio volatility. In this case, the equal-weight allocation is now applied to the assets’ risk contributions, leading to the term “equal-risk contribution” (ERC). Diversification is therefore defined and achieved by a weight vector that is characterized by a least-concentrated portfolio allocation with respect to the risk contributions of its constituents. It was introduced in the literature by Qian (2005, 2006, 2011) and the properties of this portfolio optimization approach were analyzed in Maillard
k k et al. (2009, 2010). Zhu et al. (2010) showed how this approach can be adapted
when the risk contributions are constrained or budgeted. Spinu (2013) introduced risk-parity portfolio optimizations as a convex program formulation with predefined risk contributions. Hereby, the ERC portfolio is entailed as a special case. Last but not least, the monograph by Roncalli (2013) is an encompassing treatment of risk-parity and risk-budgeting portfolio optimization techniques.
In the second approach, the risk contributions of the portfolio constituents are mea- sured by the portfolio’s downside risk, such as the CVaR. The term “diversification”
is now related to the notion that the downside risk contributions of the assets con- tained in a portfolio are either bounded by an upper threshold or evenly spread out between the constituents. This approach was introduced by Boudt et al. (2010, 2011) and Ardia et al. (2011b), based on results in Boudt et al. (2007, 2008) and Peterson and Boudt (2008).
In the following paragraphs, ERC portfolios are more formally presented and then the focus is shifted toward the second kind of portfolio optimization which limits the contributions to a downside risk measure. Recall from Section 4.3 the definition of the risk contribution of an asset, which is restated in more general terms below:
CiM𝝎∈Ω =𝜔i𝜕M𝝎∈Ω
𝜕𝜔i
, (11.7)
whereM𝝎∈Ωdenotes a linear homogeneous risk measure and𝜔iis the weight of the ith asset. As such, the risk measureM𝝎∈Ω can be the portfolio’s standard deviation, the value at risk, or the expected shortfall. All of these measures have in common the above characteristic and hence, by Euler’s homogeneity theorem, the total portfolio risk is equal to the sum of the risk contributions as defined in (11.7). These contribu- tions can also be expressed as percentage figures by dividing the risk contributions by the value ofM𝝎∈Ω:
%CiM𝝎∈Ω = CiM𝝎∈Ω
M𝝎∈Ω ×100. (11.8)
In the empirical application, the marginal contributions will be provided in this notation.
If one inserts the formula for the portfolio standard deviation𝜎(𝝎) =√
𝝎′Σ𝝎for M𝝎∈Ω, where𝝎is the(N×1)weight vector andΣdenotes the variance-covariance matrix of asset returns with off-diagonal elements 𝜎ij and with𝜎2i theith element on its main diagonal (i.e., the variance of the returns for theith asset), then partial derivatives in the above equation are given by
𝜕𝜎(𝝎)
𝜕𝜔i = 𝜔i𝜎i2+∑N i≠j𝜔j𝜎ij
𝜎(𝝎) . (11.9)
Thesei=1,…Npartial derivatives are proportional to theith row of(Σ𝜔)iand hence the problem for an ERC portfolio with a long-only and a budget constraint can be
k k stated as
PERC∶ 𝜔i(Σ𝜔)i=𝜔j(Σ𝜔)j∀i,j,
0≤𝜔i≤1fori=1,ã ã ã,N, (11.10) 𝝎′i=1,
whereiis an(N×1)vector of 1s. A solution to this problem can be found numerically by minimizing the standard deviation of the risk contributions. The optimal ERC so- lution is valid if the value of the objective function is equal to zero, which is only the case when all risk contributions are equal. A closed-form solution can only be de- rived under the assumption that all asset pairs share the same correlation coefficient.
Under this assumption, optimal weights are determined by the ratio of the inverse volatility of theith asset and the average of the inverse asset volatilities. Assets with a more volatile return stream are penalized by a lower weight in an ERC portfolio (see Maillard et al. 2010). It was further shown by these authors that with respect to the portfolio’s standard deviation, the ERC solution takes an intermediate position be- tween the solution of a GMV and an equal-weighted portfolio. They also showed that under the assumption of constant correlations, and if the Sharpe ratios of the assets are all identical, then the ERC solution coincides with that of the tangency portfolio.
As mentioned in Section 11.1, Spinu (2013) cast the problem of risk-parity opti- mizations as a convex program. The objective is then given as
F(x) = 1 2𝝎′Σ𝝎−
∑N
i=1
bilog(𝜔i), (11.11)
wherebiis the marginal risk contribution of theith asset. Ifbi=1∕Nfori=1,…,N in (11.11) then an ERC portfolio optimization results. Thus, non-negativity con- straints for the weights must be included in the convex program definition.
If one inserts the portfolio’s CVaR instead of its standard deviation as a measure of risk in (11.7), one obtains for the marginal CVaR contribution of theith asset to the portfolio’s CVaR,
CiCVaR𝝎∈Ω,𝛼 =𝜔i𝜕CVaR𝝎∈Ω,𝛼
𝜕𝜔i
. (11.12)
Obviously, by employing the CVaR or any other quantile-based risk measure, all portfolio optimizations will be dependent on the prespecified nuisance parameter𝛼, which is the confidence level pertinent to the downside risk. The derivation of the partial derivatives in (11.12) depends on the distribution model assumed, but can be provided fairly easily in the case of elliptical distributions. As such the CVaR either based on the normal assumption or modified by the Cornish–Fisher extension is assumed in practice (see Section 4.3). If one assumes that the returns follow a multivariate normal distribution, then the CVaR of a portfolio allocation is given by
CVaR𝝎∈Ω,𝛼= −𝝎′𝝁+√
𝝎′Σ𝝎′𝜙(z𝛼)
𝛼 , (11.13)
k k where𝝁denotes the(N×1)vector of expected returns,𝜙the standard normal den-
sity function, andz𝛼the𝛼-quantile of the standard normal distribution. The marginal contribution of theith asset is then given by
CiCVaR𝝎∈Ω,𝛼=𝜔i
[
𝜇i+ (Σ𝝎)i
√𝝎′Σ𝝎′ 𝜙(z𝛼)
𝛼 ]
. (11.14)
It was shown in Scaillet (2002) that the CVaR contributions are equal to the loss contributions for portfolio losses that exceed the portfolio’s VaR at the confidence level𝛼. Incidentally, if one further assumes that the assets’ returns behave like a ran- dom walk and hence the best forecast for the expected returns is 𝝁=𝟎, then the portfolio allocation obtained by equating all marginal CVaR contributions will be identical to an ERC allocation, regardless of the specified confidence level. This is because the first summand drops out of the objective function and𝜙(z𝛼)∕𝛼is a con- stant which does not affect the portfolio composition. However, (11.14) could be fruitfully employed in portfolio optimizations when either placing an upper bound on the marginal CVaR contributions or demanding a well-diversified allocation with respect to the constituents’ downside risk contributions. The latter objective can be achieved by minimizing the maximum marginal CVaR contribution:
C𝝎∈Ω,𝛼 =maxCiCVaR𝝎∈Ω,𝛼, (11.15) as proposed in Boudt et al. (2011). The authors termed this approach “minimum CVaR concentration” (MCC), as it turns out that by using this objective in prac- tice one ordinarily obtains a more balanced solution—not equal—with respect to the marginal contributions, and the CVaR level itself is reasonably low, compared to a solution where the risk contributions are budgeted (BCC). Instead of minimizing the maximum marginal contribution to CVaR, one could directly utilize a measure of divergence instead.