Читать книгу Alternative Investments - Black Keith H. - Страница 52

This section addresses the central point of the risk parity approach: how to determine portfolio weights. Equations 2.11 and 2.15 demonstrate that, in all cases, the total risk of a portfolio may be expressed as the sum of the marginal contributions of the portfolio's constituent assets. The risk parity approach is the simple prescription that the portfolio's weights should be selected such that the marginal contribution of each asset is equal. Thus, to create a portfolio of N assets using the risk parity approach, the weights need to be adjusted until the marginal contribution to risk for each asset in the portfolio is equal to (1/N) times the total risk of the portfolio. The portfolio weights that equalize all the marginal contributions to risk can be easily found using a trial-and-error approach or an optimization package such as Microsoft Excel's Solver.

Consider the information for three asset classes in Exhibit 2.4.

The data from Exhibit 2.4 can be used to generate portfolio weights using the risk parity approach. A trial-and-error search can lead to the risk parity solutions depicted in Exhibit 2.5. For comparison purposes, the weights associated with other approaches are presented as well.

EXHIBIT 2.4 Properties of Three Asset Classes

Source: Bloomberg, HFRI, and authors' calculations.

EXHIBIT 2.5 Portfolio Weights and Their Properties

Source: Bloomberg, HFRI, and authors' calculations.

There are three different portfolios in Exhibit 2.5. The first one is constructed using no optimization or risk parity. The risk parity portfolio is constructed to equalize the risk contributions of the three asset classes. The minimum volatility portfolio is constructed using mean-variance optimization, in which the goal is to use positive weights to create a portfolio with minimum standard deviation regardless of the mean. It can be seen that the risk parity portfolio allocates relatively high weights to bonds and hedge funds. The minimum volatility portfolio has no allocation to equities. The risk contributions of the three asset classes for each of the three portfolios are presented in Exhibit 2.6.

As expected, in the risk parity portfolio, each asset contributes the same marginal risk (0.53 %), which is 33.3 % of the resulting portfolio's total risk. The process is iterative, because the total risk of the portfolio changes as the allocations are changed.

Note that in this example risk parity requires a substantial allocation to fixed income. This is because the fixed-income investment exhibits lower total risk; therefore, more of the portfolio can be allocated to fixed income while keeping its marginal contribution to the portfolio's total risk the same as the marginal contributions of equities and hedge funds. The risk parity approach typically prescribes a low-risk portfolio by overweighting low-risk assets relative to the market portfolio.

EXHIBIT 2.6 Risk Contributions of the Three Asset Classes

* Because of rounding error the columns do not add up to the total.

Source: Bloomberg, HFRI, and authors' calculations.