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that allocations are made to other asset classes (e.g., U.S. equity). Effectively, hedge funds are substituted for exposure to other asset classes.

Alternatively, investors can treat the hedge fund portfolio as a substitute for other active strategies (e.g. active U.S. large-cap equity or active U.S. fixed income). Suppose an investor wanted to substitute a hedge fund portfolio for a traditional active manager, say an active U.S. large-cap equity manager. If the hedge fund manager equitizes a portion of the cash (e.g., by purchasing futures contracts), and invests the rest in the specific hedge fund portfolio, the investor now has a portfolio that can be compared with a traditional active manager. This strategy is called a "portable alpha" strategy.

For our purposes, we'll assume that investors are substituting away from equity and fixed income and into hedge funds. The basic principles that are described in this case can be easily applied to analyze portable alpha strategies.

Let's look at the case where an investor decides to make outright allocations to hedge funds. In this case, the investor must consider the volatility of a hedge fund portfolio and its correlation with other asset classes. For discussion purposes, we'll assume that the hedge fund portfolio is the equal risk portfolio discussed earlier (i.e., Portfolio B). This portfolio has a volatility of 5.2 percent and a correlation with U.S. equity of 0.51.

An investor who chooses to make an outright allocation to hedge funds must also choose how to fund the allocation. That is, the investor must choose which asset class (or combination of asset classes) the hedge fund program substitutes for in the overall portfolio. In our simple example, there are three natural alternatives: (1) the investor can scale all other assets down proportionately; (2) the investor can substitute away from equity holdings and into the hedge fund portfolio; (3) the investor can substitute away from bonds and into hedge funds.

The impact on total portfolio volatility of each funding alternative is summarized in Figure 26.3. The chart plots alternative allocations to hedge funds and the resulting portfolio volatility for each of the three funding methods.

What happens when we substitute out of equity and into hedge funds? In our

Allocation to Hedge Funds

Funded from Equities

Funded from Bonds

### Funded Pro Rata

FIGURE 26.3 Portfolio Volatility and Hedge Fund Allocations example, hedge fund volatility declines almost linearly. The principal reason for this is that we are effectively substituting an asset with low volatility (the hedge fund portfolio) for one with higher volatility (the equity portfolio or the total portfolio). In addition, the hedge fund portfolio is not perfectly correlated with the equity portfolio. Both of these effects mean that substituting into the hedge fund portfolio reduces total portfolio volatility. Clearly, if the hedge fund portfolio is riskier or more highly correlated with equity market returns, then total portfolio volatility will not be reduced by as much, or even at all, when we substitute into hedge funds.

Suppose, though, that an investor wanted to add hedge funds to the portfolio, but didn't want a change in total portfolio risk. Since the hedge fund portfolio (Portfolio B) in our hypothetical example has a bondlike volatility, the investor might substitute hedge funds for fixed income. For example, in Figure 26.3 allocations to hedge funds funded through reductions in fixed income leave the total portfolio volatility more or less unchanged. Again, this result depends on the structure of the hedge fund portfolio and our assumptions on hedge funds volatility and correlation. If the hedge fund portfolio is skewed toward higher-volatility strategies or strategies that are more highly correlated with equity markets (e.g., equity long/short), then total portfolio volatility will increase if the hedge fund allocation is funded out of fixed income.

The analysis of the impact on total portfolio volatility is important to investors for two reasons. First, it reinforces the point that investors should analyze the characteristics of their hedge fund portfolios prior to investing. The second reason Figure 26.3 is important is because it provides investors with an easy decision rule: How they fund the hedge fund allocation depends in part on how much risk they would like to take in the overall portfolio.

In addition to analyzing the impact on total portfolio volatility, investors should consider the impact of each funding alternative on the marginal contribution to total portfolio risk. Figure 26.4 illustrates this point by showing the mar

FIGURE 26.4 Hedge Fund Contribution to Risk

Allocation to Hedge Funds

Funded from Equities —■— Funded from Bonds —A— Funded Pro Rata

FIGURE 26.4 Hedge Fund Contribution to Risk ginal contribution to risk (expressed in percentage terms) for each hedge fund allocation and under each scenario.

The important feature of Figure 26.4 is the illustration that the marginal impact on portfolio risk from hedge fund allocations can be quite small.1 In this example, even a 20 percent allocation to hedge funds contributes less than 10 percent of the total portfolio risk at the margin, irrespective of which funding choice is made. Of course, this conclusion depends on the actual structure of the hedge fund portfolio. If the hedge fund portfolio were concentrated in a highly volatile sector (e.g., equity long/short), then we would anticipate a more significant marginal contribution to total portfolio risk at each hedge fund allocation.

Figures 26.3 and 26.4 suggest that a hedge fund program can be designed to have a modest impact on total portfolio volatility and the distribution of portfolio risk. What about the returns associated with hedge fund allocations?

Rather than focus on projecting future returns to hedge funds on the basis of historical averages, our preferred approach is to find the implied excess returns (i.e., returns over cash rates) associated with alternative allocations.2 Implied returns are the returns that are implied by the optimality of the portfolio structure under the assumed correlation and volatility structure of all the asset classes in the portfolio. The results are shown in Figure 26.5, again using the same equal risk weight hedge fund portfolio. In keeping with the analysis of Figures 26.3 and 26.4, we also show the impact of alternative funding scenarios.

What is striking about the numbers in Figure 26.5 is how low the implied premiums actually are. For instance, the implied return for a 10 percent allocation to hedge funds is around 107 basis points, irrespective of which funding choice is used. In fact, the choice of how the hedge fund allocation is funded really begins to matter only at more significant hedge fund allocations.

For example, suppose that an investor allocated 25 percent to hedge funds. If the hedge fund allocation is made out of equities, then the implied hedge fund return is around 127 basis points. On the other hand, if the hedge fund portfolio is made out of bonds, then the implied return is 14 basis points lower. Similar to Figures 26.3 and 26.4, the relationship between the implied returns and the hedge fund allocation will depend on the actual structure of the hedge fund portfolio: A more volatile hedge fund portfolio (e.g., one that is concentrated in equity long/short managers) will have a higher implied return. Alternatively, a hedge fund portfolio that is not especially highly correlated with the other assets (e.g., concentrated in commodities futures trading) will have a lower implied return at every allocation.

The implied returns shown in Figure 26.5 are best interpreted as hurdle rates. In other words, they are the minimum returns required by the investor to hold the hedge fund allocation and all other asset classes in the indicated proportions. Of course, higher returns on hedge funds would be preferred (and perhaps even expected). In

1The marginal contribution to risk from a hedge fund depends on its weighting in the portfolio, its level of volatility, and its correlations with the other assets in the portfolio.

2Implied returns can be found for any set of portfolio weights as R = XQX. In this equation, R is a vector of asset returns, Q a covariance matrix of asset returns, X a vector of portfolio weights, and X a risk aversion parameter.

Allocation to Hedge Funds

Funded from Equities j Funded from Bonds ^ Funded Pro Rata FIGURE 26.5 Implied Hedge Fund Hurdle Rates some senses, then, it is reasonable for an investor to ask whether a particular implementation of a hedge fund program can achieve these hurdle rates.

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