example, the contribution of Asset 2 to the total risk of the portfolio is 6.93 % × 35 % = 2.43 %. These figures were calculated with the help of Equation 2.12. Let's consider another example to see how the equation can be applied.
Suppose the weights of three asset classes in the portfolio displayed in Exhibit 2.3 are changed to 50 %, 40 %, and 10 % for Assets 1, 2, and 3, respectively. What will be the risk contribution of Asset 1? First, we need to calculate the standard deviation of the portfolio:
Next, the contribution of Asset 1 can be calculated using Equation 2.12 and the information provided by Exhibit 2.3. The correlation between Asset 1 and the portfolio is 0.9586, and its standard deviation was reported in Exhibit 2.3 to be 8.83 %.15
It can be seen that 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).
Here, RPt 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:
Here,
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
