Alternative Investments. Black Keith H.

inclusion in the portfolio. To do this, a linear regression of the following form is run:

      (2.6)

      Here, R_it is the rate of return on asset i; a_i and b_i are the intercept and the slope of the regression, respectively; F_t is the risk factor that is of interest; and ϵit is the residual. For example, F_t might be the percentage change in the price of oil. In this regression, b_i 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

. The following constraint can now be added to the mean-variance optimization of Equation 2.4:

      (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,

, in order for the portfolio to be a feasible solution. The impact of imposing this constraint is to reduce allocations to those asset classes that have large exposures to the source of risk being considered. It is important to note that since short sales are not allowed, it may not be feasible to create a portfolio with the desired level of factor exposure. For example, suppose an investor decides to have a negative exposure of the portfolio to the equity risk of the overall equity market. It may not be possible to create a negative equity beta portfolio when short positions are not permitted.

      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 Level I book discussed how confidence intervals can be calculated once the standard error of an estimate is known. In the present case, the 95 % confidence interval for the estimated mean is approximately:

      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,

for i = 1, … N. Similar to the previous example, suppose we have some ideas about the accuracy of our estimates. In particular, if

is the true value of the mean return, then a 95 % confidence in the estimate can be expressed as:

(2.8)

      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

for a 95 % confidence level, where σi is the standard deviation of the asset return, and T is the number of observations.

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

, the estimated mean return of asset i, with

:

(2.9)

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 if there were no estimation risk and with the adjusted expected rate of return on the asset being equal to − 0.2 %. Since short sales are ruled out, the optimal allocation to this strategy may turn out to be zero if it does not provide enough diversification benefits. It is important to stress that the expected returns on all assets will likely need to be reduced somewhat, because it is reasonable to assume that no expected return can be estimated without any error. So the net impact is not going to be limited to a reduction in allocation to some specific assets, but rather the entire allocation to risky assets may be reduced.

      2.3 Risk Budgeting

      Risk budgeting refers to a broad spectrum of approaches to portfolio construction and maintenance that emphasize the selection of a targeted amount of risk and the allocation of that aggregate portfolio risk to various categories of risk. A risk budget is analogous to an ordinary