Exposure
In the process of asset allocation, whether strategic or tactical, the investor may wish to limit the exposure of the portfolio to certain sources of risk (and potentially some returns) other than or in addition to the risk of the overall market. For example, the investor may wish to cap the exposure of the portfolio to changes in the price of oil. As long as an observable factor representing the source of risk exists, the constraints on factor exposures can be incorporated into the mean-variance approach with little difficulty.
The first step is to estimate the factor exposures of the assets that are being considered for inclusion in the portfolio. To do this, a linear regression of the following form is run:
(2.6)
Here, Rit is the rate of return on asset i; ai and bi are the intercept and the slope of the regression, respectively; Ft is the risk factor that is of interest; and ϵit is the residual. For example, Ft might be the percentage change in the price of oil. In this regression, bi measures the factor exposure of asset i. Since the factor exposure of the portfolio is a weighted average of the individual asset's exposures, the factor exposure of the portfolio can be expressed as
(2.7)
In this case, the constraint is that the total factor exposure of a potential portfolio must be less than or equal to the target,
2.2.3 Adjustment of the Mean-Variance Approach for Estimation Risk
The inputs to the portfolio allocation process are unknown and have to be estimated. Further, accurate estimates of some inputs such as expected returns are more difficult to obtain. As seen in Chapter 1, to estimate the mean return on an asset class, a long price history is generally beneficial. This is particularly troublesome when considering alternative asset classes. Most of the funds that provide access to alternative assets have a relatively short history. As a result, the level of confidence in the estimated expected means and alphas of the funds is low. Therefore, it would be preferable if the portfolio optimization problem would be able to account for the uncertainty associated with the estimated input values. In particular, it seems reasonable to adjust the optimization process to reduce allocations to those asset classes that are subject to significant estimation risk (e.g., because a short return history is available).
Suppose an investor is considering an allocation to a relatively new hedge fund strategy for which only five years of monthly data are available (60 observations). Suppose the monthly mean and the standard deviation of an index representing the strategy are estimated to be 0.83 % and 2.89 %, respectively (equivalent to 10 % annually for both). The estimated mean is a random variable, as one is likely to obtain different estimates when using different samples of returns, and we know from the CAIA Level I book that the standard error of the estimate mean will be
The constant 1.96 is obtained from the fact that Pr(z > 1.95) ≈ 2.5 %, where z is a standard normal random variable. Given the large estimation error of this strategy, the true annual return is estimated to be 95 % likely to lie between 1.2 % and 18.7 %, a range found by multiplying the monthly returns by 12. It would be a mistake to use 0.83 % (10 % annualized) as the input, as if it were the true expected return. Another hedge fund strategy with 20 years of data and an estimated mean of 0.83 % should be preferred to the new strategy due to its reduced estimation risk compared to funds with short return histories. How can estimation risk be incorporated into the mean-variance optimization problem?
Since estimation risk is an important consideration, dealing with it has a long history. Although a full discussion of its different treatments is beyond the scope of this book, robust optimization can be discussed as one particular approach. Robust optimization selects final solutions that incorporate estimation error directly into the modeling process. A simple version is presented here.
Suppose there are N assets with estimates of their expected returns,
That is, the analyst can be 95 % confident that the error in the estimate of the mean return does not exceed ϵi in absolute terms. If returns are normally distributed, then the estimation error term (ϵi) can be expressed as
The process of incorporating estimation risk into the optimization is to select the optimal weights using estimated expected means that are reduced to the lower value of the selected confidence interval. Including the lower value of the confidence interval displayed in Equation 2.8 in the optimization problem, the problem will simplify into a regular mean-variance optimization by replacing
That is, the estimated expected mean returns are replaced according to Equation 2.9, and then the optimization proceeds in the normal way, as if there were no estimation risks.14 It is important to notice the impact of the estimation risk on the optimal allocation. The estimated expected returns on assets are reduced by the size of the estimation error. That is, the larger the assumed estimation error, the lower the mean that is used to perform the optimization. Consequently, those asset classes for which the portfolio manager has the least estimation confidence will be most penalized and will receive lower allocations than would have been received if the estimation risk were ignored.
For example, suppose the mean return from a new strategy is estimated to be 1 % per month. Assume that given the volatility of returns and the number of observations, the 95 % confidence interval for the estimated mean is 1 % ± 1.2 %. The input to the optimization based on estimation risk is simple. Instead of using 1 % as the expected mean return of the strategy, we should use 1 % − 1.2 % = −0.2 %. That is, in the presence of estimation risk, the optimization constructs the optimal portfolio as
