Alternative Investments. Black Keith H.

while the weight of Asset 1 in the portfolio is 50 %, close to 52 % (4.32 %/8.21 %) of the total risk comes from Asset 1.

      Example: The correlation of Asset 3 with the previous portfolio is 0.1734 and its standard deviation is 20 %. What is the contribution of Asset 3 to the total risk of the portfolio?

      In this case, Asset 3 is 10 % of the portfolio, but it contributes only 0.35 % to the total risk, or only 4.2 % (0.35 %/8.21 %) of the total risk is due to Asset 3.

      To summarize, the total risk of a portfolio can be decomposed so that the contribution of each asset class to total risk is measured. This decomposition increases the portfolio manager's understanding of how the portfolio is likely to react to changes in each asset class. In addition, once the risk budget of each asset class is measured, the portfolio manager may consider changing the allocation so that each asset class's contribution does not exceed some predetermined risk budget. Finally, the mean-variance optimization problem discussed earlier in this chapter can be adjusted to incorporate constraints related to the risk budget.

      2.3.4 Applying Risk Budgeting Using Factors

      The preceding discussion focused on risk budgets associated with asset classes. The total risk of a portfolio was decomposed into the contribution of each asset class to that total risk. It is possible to use the same approach to decompose the total risk of a portfolio by measuring the contribution of each risk factor to the total risk. A number of macroeconomic and financial factors can affect the performance of a portfolio. An asset owner may already have significant exposures to some of these factors. For example, the manufacturer of a product that is sold in foreign markets may have significant exposures to currency risk. That may lead the asset owner to instruct the portfolio manager to measure and then limit the exposure of the portfolio to currency risk. On the other hand, the same asset owner may wish to measure and adjust the risk exposure of the portfolio to the interest rate factor because it needs the assets to fund a liability that is interest rate sensitive. Here we discuss how the contribution of factor volatilities to total risk of a portfolio can be measured.

      To see this, suppose there are two risk factors, F₁ and F₂, that are the major drivers of the portfolio's return. Their degree of importance can be measured by regressing the portfolio's rate of return on these two factors (e.g., one factor could be changes in the credit spread and the other could be changes in the price of oil).

(2.13)

      Here, R_Pt is the rate of return on the portfolio; a, b₁, and b₂ are the estimated parameters of the regression model; and ϵt is the residual part of the regression that represents the part of the return that cannot be explained by the two factors. The total risk of the portfolio can now be decomposed into the contributions of each risk factor:

(2.14)

Here,

,

, and ρ_ϵ are the correlations of the two factors and the residual risk with the portfolio's return, respectively. Each of the first two terms that appear on the right-hand side of Equation 2.14 represents the contribution of a factor to the total risk of the portfolio. The last term represents the contribution of unknown sources of risk.

For example, suppose the correlation between changes in the oil price and the return on the portfolio displayed in Exhibit 2.3 is 0.31. The standard deviation of changes in the oil price is estimated to be 20 %, and the factor loading of the portfolio on oil (i.e., the coefficient of oil in Equation 2.13) is 0.757. What is the risk contribution of oil to the total risk of the portfolio?

      Therefore, 4.69 % of the total risk of 8.11 % of the portfolio described in Exhibit 2.3 can be contributed to the volatility in oil prices. It can be seen how the risk budget associated with risk factors can be helpful in understanding the risk profile of a portfolio. In addition, limits on risk budgets associated with risk factors can be incorporated into the mean-variance optimization. In fact, we saw a version of this earlier in the chapter when we discussed how limits on factor exposures can be added to the mean-variance model.

      So far, the measure of total risk has focused on standard deviation. Value at risk (VaR) and conditional value at risk (CVaR) can also be used to measure total risk. Much of the discussion presented here can be represented using these two measures of total risk with minimal change. For further discussion of these measures of total risk and risk budgeting, see Pearson (2002).

      2.4 Risk Parity

      While risk budgeting is mostly concerned with measuring the exposure of a portfolio, the risk parity approach uses risk budgeting results and attempts to create portfolios that equally weight the risk contributions of each asset or asset class in a portfolio. In risk parity, a portfolio allocation model is constructed based entirely on the risk contribution of each asset to the total risk of the portfolio, with no consideration of expected return of each asset class. Specifically, and as is detailed later, the risk-parity approach recommends that the allocation to each asset class should be set so that each asset class has the same marginal contribution to the total risk of the portfolio. The result is that the allocation to each asset tends to be inversely proportional to its risk once the diversification effect is taken into account. In risk parity, if equity is viewed as an asset that contributes more risk to the portfolio than bonds do, then less of the portfolio should be allocated to equities than to bonds.

      2.4.1 Three Steps in Implementing the Risk Parity Approach

      There are three steps in implementing the risk parity approach:

      First, similar to risk budgeting, risk parity requires a definition of the total risk of a portfolio. Risk parity does not impose a uniform measure of total risk. However, total risk is typically measured by the standard deviation of the rate of return on the portfolio. Alternatively, one could use value at risk (VaR) or some other measure of risk. The advantage of using VaR as a measure of total risk is that one can incorporate skewness and kurtosis into the measure of total risk. For the purpose of this discussion, standard deviation is used as a measure of total risk.

      Second, risk parity requires a method to measure the marginal risk contribution of each asset class to the total risk of the portfolio. The marginal risk contribution of an asset to the total risk of a portfolio indicates the rate at which an additional unit of that asset would cause the portfolio's total risk to rise. The marginal risk contribution of an asset depends on the composition of the portfolio. For example, adding a hedge fund to an otherwise diversified portfolio may contribute little or no risk, since the hedge fund may offer substantial diversification benefits. However, as the hedge fund's allocation to the portfolio is increased, the effect of additional allocations of hedge funds (i.e., the marginal contribution) also increases. Accordingly, at high levels of allocation to hedge funds, additional allocations may increase risk substantially, as the portfolio becomes concentrated in hedge funds rather than diversified. The measurement of risk contributions was discussed in the previous section.

      Finally, portfolio weights are determined for all available assets. The weights are typically computed using a trial-and-error process until the marginal contributions from all assets to the total risk of the portfolio are equal. The previous section showed that the marginal contribution of an asset class to the total risk of a portfolio is given by:

(2.15)

      Therefore, the risk parity approach to asset allocation seeks a portfolio