aggregate level of expenses is determined and is spent among various categories. Similarly, a risk budgeting process might decide that an appropriate level of risk is 15 % and then might allocate that acceptable level of risk among the various asset classes.
2.3.1 Specifications in Risk Budgeting
Risk budgeting requires a specification of how risk is measured, but the risk budgeting approach can be used with virtually any quantitative approach. Common examples of risk measurements to use in a risk budgeting process include standard deviation of returns, standard deviation of tracking error against a benchmark, value at risk, and beta. Risk budgeting requires a clear specification of the relationship between the measured risk of the total portfolio and the measured risks of the portfolio's constituent assets. The quantitative link between the risk of the portfolio and the risks of its constituent assets permits risk to be budgeted among the available assets.
Risk budgeting does not require the specification of expected returns. However, a popular application of risk budgeting allocates a portfolio between passive investments (e.g., indexation) and active investments (e.g., alternative investments) based in part on estimates of the extent to which the active investments can be expected to have higher expected returns than the passive investments. For example, a risk-budgeting framework may be designed to guide the asset allocator into deciding how much risk out of a total risk budget of 15 % to allow for actively managed investments such as hedge funds in the pursuit of earning potential alpha. Other risk budgeting approaches may use a mean-variance framework in which the expected return of every asset is specified.
One case where expected returns are used in risk budgeting is when the asset allocator is using the standard deviation of the tracking errors between the portfolio's return and a benchmark's return. The asset allocator has to decide which managers will be allowed to deviate from a benchmark. One important factor influencing this decision is the potential alpha of the manager. For instance, an asset allocator may decide that a large-cap equity investor is not likely to generate substantial alpha; therefore, the portfolio manager will not be allocated any tracking error risk. In other words, an index fund will be used for this asset class. On the other hand, the asset allocator will be willing to spend a substantial portion of the portfolio's tracking error risk on an equity long/short manager who has the potential to generate a significant amount of alpha.
2.3.2 Implementing a Risk Budgeting Approach
The key to risk budgeting is its focus on risk allocation as the primary driver of the portfolio selection and monitoring process. Risk budgeting differs from mean-variance optimization, in which the optimizer selects assets driven by the trade-off between risk and return.
As previously mentioned, to implement and understand risk budgeting, the first step is to come up with a measure of a portfolio's total risk. The two most common measures are standard deviations of returns and value at risk. Note that under the assumption that returns are normally distributed, there is a one-to-one relationship between standard deviation and value of risk, and therefore either one could be used as a measure of total risk. In this section, standard deviation of returns will be used as the portfolio's measure of total risk.
The CAIA Level I book explains that the variance of a portfolio's return can be written as:
(2.10)
Here, wi is the weight of asset i in the portfolio, and σij is the covariance between asset i and asset j. Risk budgeting attempts to measure the contribution of each asset class to the total risk of the portfolio, typically measured by the standard deviation of returns, σP. Risk budgeting can be used to measure the risk contributions of asset classes as well as contributions of risk factors.
Having selected standard deviation to represent the total risk of a portfolio, measuring each asset class's contribution to the total risk is not complicated. In particular, the total risk can be decomposed in the following form:
(2.11)
Equation 2.11 states that the total risk of a portfolio can be decomposed into N components, with each measuring the contribution of an asset class to the total risk. The contribution of each class is measured by the asset's weight in the portfolio multiplied by the sensitivity of the portfolio's standard deviation to small changes in the weight of the asset (∂σP/∂wi). It can be seen that an asset that has a relatively large weight is likely to contribute a relatively large amount to the total risk of the portfolio. However, depending on the volatility of the asset and its correlations with other assets in the portfolio, the actual contribution of the asset could be much smaller or larger than its weight in the portfolio.
A simple analytical formula allows us to calculate the risk contribution of each asset class. The total risk of a portfolio can be decomposed into the contribution of each asset class to the total risk. Because the contribution of each asset class to the total risk was measured by (∂σP/∂wi) × wi, we can use the formula for the standard deviation of the portfolio to evaluate this contribution
where σiP is the covariance of asset i with the portfolio, and ρi is the correlation of asset i with respect to the portfolio. According to Equation 2.12, the contribution of each asset depends on its correlation with the portfolio, its own volatility, and its share in the portfolio. An increase in each of these three variables will increase the risk contribution of the asset. Note that given the definition of beta, βi = ρi × (σi/σP), the relationship expressed in Equation 2.12 can be stated in terms of betas as well.
2.3.3 A Three-Asset Example of Risk Budgeting
Consider the information in Exhibit 2.3 about a portfolio consisting of three assets.
The diagonal terms of the variance-covariance matrix represent the variances of the assets. For example, the standard deviation of Asset 1 is
Using the information provided in Exhibit 2.3, it can be seen that the mean and the standard deviation of the portfolio will be 9.20 % and 8.11 %, respectively. The risk decomposition of this portfolio will be
EXHIBIT 2.3 Properties of Three Hypothetical Assets
For 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