Introduction

Portfolio insurance is a technique that allows the investor to participate in the upside potential of a risky portfolio (called the target asset), while reducing or eliminating the downside risk. Typically, this means guaranteeing a specified minimum return over a given investment horizon. The cost of the insurance is paid in terms of a return differential by which the investment return lags behind the performance of the target asset.

The target asset itself can be a portfolio containing a number of investment assets. For instance, the target portfolio can be a balanced fund consisting of equities, bonds, and mortgages. The allocation of funds within this portfolio can be fixed, or can vary according to a passive or active strategy. The insurance then applies to the total fund, and depends only on its total performance rather than on the performance of the individual components. Although the insured portfolio consists of multiple assets, from the viewpoint of the insurance strategy it is a single target asset.

In this chapter, we address a different type of multiple asset strategy, called the best return strategy. Instead of ensuring the performance of a combination of assets, this strategy assures that the return on the total investment will be that of the best performance of the individual assets, less the known cost of the strategy. Thus, if the individual assets are stocks, bonds, and mortgages, and if stocks happen to perform the best of the three, the return on the strategy will be that of the stock portion, less the known cost. If bonds perform better over the investment horizon than stocks and mortgages, the investment return will equal the return on bonds, less cost. If mortgages do better than stocks or bonds, the investor will realize the return, after costs, of the mortgage portfolio on his total investment.

Moreover, if one of the assets has a fixed return over the investment horizon, this strategy guarantees a specified minimum return in addition to assuring the best of the remaining asset returns. This minimum guaranteed return is the return on such safety asset less the known costs. Thus, if the assets include a pure discount bond maturing at the end of the investment horizon in addition to stocks, bonds, and mortgages, the strategy will yield a specified minimum return even if stocks, bonds, and mortgages all perform poorly.

There is no restriction on the number of assets used in the best return strategy. It can be applied to individual securities (such as getting the best of several individual stock returns), to portfolios (for example, assuring that the return achieved by the best—or luckiest—of several portfolio managers is realized on the total plan), or to whole markets (for instance, obtaining the best performing of several international equity markets). The cost of the strategy, of course, increases with the number of assets involved. In addition to the number of assets, the costs depend on the riskiness of the assets (the riskier the assets, the higher the costs), on the correlations among the assets (the higher correlated they are, the lower the costs), and on the length of the investment horizon (the costs per year decrease with increasing horizon length).

The strategy is implemented by a dynamic allocation of the investment funds among the several assets. The proportions of the total investment allocated to the individual assets are continuously monitored and adjusted, depending on their performance to date, and on the time remaining to the horizon date.

The Objective

The best-return strategy is a generalization of portfolio insurance to multiple assets. To understand the connection, consider the objective of portfolio insurance. The objective function can be written in the following form:

where is the total investment return, is the target asset return, is the assured minimum return, and c is the insurance cost. (Throughout this chapter, all returns are assumed to be expressed in terms of annual, continuously compounded rates.)

Portfolio insurance is nothing other than getting the better of two asset returns. Indeed, when the insurance plan is implemented by means of dynamic asset allocation, it is necessary to utilize a second asset that has a fixed return over the insurance horizon, such as a pure discount bond with no default risk. The difference between the second asset return and the assured minimum return can be viewed as a second insurance cost, assigned to this second asset. The objective function can then be written as

where are returns on the two assets and are the corresponding insurance costs. A minimum return guarantee is just a special case when one of the two assets has a fixed terminal value. (In a world of changing interest rates, however, that asset is also risky, since its value fluctuates during the horizon.)

The two insurance costs, (both of which must be positive) are subject to a pricing equation that determines one of them if a value for the other is selected. This equation reduces to a version of the Black-Scholes (1973) option pricing formula if one of the assets is a bond and interest rates are deterministic and constant, and to Merton's (1973) extension of that formula if one asset is a pure discount bond and the variability of interest rates is independent of their level. Margrabe (1978) has provided a formula for two risky assets. In general, the insurance costs will depend on the risk structure of the two assets throughout the horizon (i.e., their instantaneous covariance matrix as a function of time and state variables) and the length of the horizon.

It is a natural generalization of the two-asset case to postulate an objective function of the form

where is the number of assets, are their returns, and are the corresponding insurance costs. This objective is to get the best of multiple risky asset returns, less the cost of insurance. This constitutes the goal of the best-return strategy.

The Costs

The valuation formula from which the costs of the best-return strategy are calculated is a single equation for the n costs, so that of the costs can be independently chosen (subject to feasibility constraints) and the remaining one is then determined. Alternatively, relationships can be imposed on the costs (such as that they be all equal) to determine their values.

The formula depends on the number of assets, the risk structure of the n-dimensional stochastic process that describes their behavior over the horizon, and the horizon length. For diffusion processes, the valuation formula involves -dimensional cumulative normal distribution functions with covariance matrices that are transformations of the n-dimensional instantaneous covariance matrix of the assets, integrated over the horizon. In addition to the two-asset formulas of Black and Scholes, Merton, and Margrabe mentioned earlier, the other result previously available is Stulz's (1982) formula for two risky and one riskless asset.

The values of the costs are determined by the investor's preferences, much like in the case of portfolio insurance. In that special case of two assets, the investor chooses a tradeoff between the minimum guaranteed return (which is the fixed return of the safety asset less the cost attributed to the safety asset), and the return differential between the target asset return and the total investment return (which is the cost attributed to the target asset). Thus, portfolio insurance strategies can guarantee a relatively high minimum return at a high cost of the insurance, or a lower minimum return at a more modest insurance cost.

In the best return strategy, one alternative is to choose the costs to be all equal,

In that case, the objective function of the strategy has a particularly simple form

where the common value c of the n costs is determined from the valuation formula. This case, which will be called uniform cost allocation, assigns the costs equally to all of the assets included in the objective.

As an example, consider n assets whose stochastic behavior is described by a logarithmic Wiener process. Let the instantaneous covariance matrix be specified by standard deviations all equal to 15 percent annual, with correlations among the assets all equal to 0.4, and assume a five-year horizon. Table 26.1 lists the value c of the uniform costs, in annual percent, as a function of the number of assets.

These costs are the price to pay for getting the best out of a number of asset returns. Suppose that the values of the parameters chosen for the example are descriptive of the international equity markets. It is possible to implement a strategy whose realized return is equal to the highest of six separate national stock markets, over a five-year period, less 6.2 percent annual. It goes without saying that no prediction is needed as to which of these markets will have the highest return, or, for that matter, what are the expected returns of each.

Table 26.1 is an extract from Table 26.2, which lists the values of the uniform costs as a function of the number of assets and of the correlation among them, assumed to be the same between any pair. The investment horizon is taken to be five years, and the standard deviations of the individual asset returns are assumed to be all equal to 15 percent per year. It can be seen that the costs decrease drastically with an increase in the correlation among the assets. The uniform costs under the same assumptions but for a one-year horizon are given in Table 26.3.

Nonuniform Costs

The costs of the strategy do not have to be made equal. It is possible to choose them in such a way that a disproportionate part of the burden is borne by those assets in which the investor has a secondary interest. For instance, consider the case of four assets with standard deviations of 15.7 percent, 11.7 percent, 15.2 percent, and 1.3 percent, and a correlation matrix as shown in Table 26.4. These values correspond to historical estimates of volatilities and correlations (over the period January 1979 to December 1980) for S&P 500 stock index, Lehman government/corporate bond index, GNMA index, and Treasury bill index. For a five-year horizon, the uniform costs are 3.8 percent annually. This means that the best return strategy applied to these four assets assures the investor a return equal to the highest of the realized annual returns over the five-year period of stocks, bonds, mortgages, and cash, less 3.8 percent. For instance, if stocks turned out to do the best of these four assets with an annual return of 20 percent, the investor would realize 16.2 percent annually over the horizon. If stocks, bonds, and the GNMA portfolio all lost money, the strategy would still provide a return equal to that realized on Treasury bills, less 3.8 percent.

Now suppose that it is essential to the investor to maintain a five-year return of no less than that of Treasury bills less 2 percent, while retaining as much of the upside potential of stocks, bonds, and GNMAs as possible. The costs of a best-return strategy can be chosen as

(See Case #4 in Table 26.4.) This choice of costs would assure a minimum performance of Treasury bill return less 2 percent, while keeping the possibility open to participate in the performance of the three riskier assets if any one of them turns out to do well.

If mortgages were less important to the investor than stocks and bonds, perhaps the following cost assignment may be a preferred choice:

(See Case #5 in Table 26.4.) This cost allocation would attribute lower costs to stocks and bonds than the previous case, and higher costs to the less important mortgage portfolio.

Table 26.4 lists a number of possible alternatives for the cost allocation. Note that these are just a few possibilities out of an infinite range of feasible cost allocations, with no particular meaning to the order in which the cases are listed.

Strategy Implementation

The strategy is executed by a dynamic allocation of investment funds among the several assets. The amounts allocated to the individual assets are maintained to be proportional to the partial derivatives, with respect to the asset values, of the valuation function (the same function that is also used initially to determine the costs of the strategy). The required allocation changes continuously as a function of the asset performance to date, and the remaining time to the horizon.

An example of the strategy is provided in Table 26.5. The strategy is simulated over a one-year investment horizon from January 1, 1981, to December 31, 1981, using the four assets described. The risk parameters are those listed in Table 26.4, as measured over a prior period from January 1979 to December 1980. The costs, allocated uniformly, are 8.8 percent for each asset. The simulations assume monthly rebalancing, with transaction costs of 0.25 percent round-trip (since the rebalancing can be executed by trading futures).

The initial allocation was 36.1 percent, 8.7 percent, 28.8 percent, and 26.4 percent among stocks, bonds, mortgages, and cash, respectively. One month later, based on the market moves over the month, the allocation was changed to 23.3 percent, 6.8 percent, 34.5 percent, and 35.5 percent, respectively, for a turnover of 13.6 percent. The rebalancing is continued each month until the horizon date.

Table 26.5 lists, for each rebalancing period, the last month performance and the performance since inception of the four assets, as well as the scheduled performance of the plan (the performance, calculated from the valuation formula, that is expected from the strategy given the performance of the individual assets), and the actual performance of the plan before and after transaction costs.

The summary of the strategy performance is provided in Table 26.6. Over the one-year horizon, the annual continuously compounded returns for the four assets were −5.1 percent for stocks, 7.0 percent for bonds, 0.2 percent for mortgages, and 14.9 percent for cash. The scheduled return was 6.2 percent, equal to the best of the four asset returns (Treasury bills in this case) less 8.8 percent. The actual performance of the plan was 5.8 percent before and 5.5 percent after transaction costs, very close to the schedule. The difference between the actual and promised performance is due to monthly (rather than continuous) rebalancing and to the actual risk parameters over the investment horizon differing from the assumed values (which were estimated over a previous period).