Model Portfolios Based on Expected Future Returns

Over the past two months, we have reviewed different asset classes, including their historical and estimated future real (inflation adjusted) returns and standard deviations. This month, to stimulate our thinking on the critical subject of asset allocation, we are using the estimated future returns as inputs, and using them to develop model target return portfolios. Specifically, we will use them to develop model portfolios whose objective is to achieve compound annual real returns of either 3%, 5%, or 7% over a twenty year period.

Are these our new recommended model portfolios? No, they are not. We try not to adjust our model portfolios too often, lest we simply chase performance, and in effect become inadvertent market timers. Rather, we see this month's analytical effort as one of a number of different inputs into a decision making process that will conclude with our "official" model portfolio update in November.

We should also note, yet again, the limitations of our approach. As we noted last December, "many investors don’t fully appreciate that the historical inputs used in asset allocation models are themselves only statistical estimates of the "true" values for these variables…Because of the possibility of estimation error, many portfolios with different asset allocations are statistically indistinguishable from one another in terms of their expected risk and return." On top of this, we have introduced another layer of uncertainty through our estimation of the future average annual real returns for different asset classes.

Moreover, while most asset allocation models (including the ones we use) assume that the returns on different asset classes are normally distributed (i.e., when plotted on a graph they form a "bell curve"), strictly speaking this is usually not true. In fact, the return distributions for many asset classes have "fatter tails" than would be the case in a normal distribution. Statistically, this means extreme events are more likely to happen than would be the case if the returns were normally distributed. How much more likely? Fortunately, a 19th century Russian mathematician named Pafnuty Chebyshev worked this out. In the case of a normal distribution, the range defined as the mean (average) plus or minus two deviations is supposed to cover about 95 percent of possible outcomes, while the three standard deviations are supposed to cover 99 percent. Chebyshev showed that if the distribution isn’t normal, you would need about four standard deviations to cover 95 percent of the possible outcomes, and about seven standard deviations to capture 99 percent. Unfortunately, the assumption of normality is practically necessary to make many asset allocation models computationally feasible. However, most investors don't realize that as a result of this assumption, a models may provide a false sense of confidence about the maximum percent of value that their portfolios could lose over any period of time. Practically, this means that there is more risk inherent in high standard deviation asset classes (like equities) than most people realize, and that a more conservative asset allocations are probably more effective in the long term (a point we’ve taken to heart in the construction of our target return model portfolios).

A final issue that affects asset allocation models is the fact that the underlying economic processes that generate the return distributions they use as inputs are not themselves stable (or, as they say in statistics, they aren't "stationary"). The evidence in support of this observation is quite strong: for example, standard deviations (also known as volatility) are not stable across time; rather, they tend to cluster in "regimes" of high and low values for this variable. The same is true for the correlations of returns between asset classes: there is a lot of data that says that correlations tend to increase during bad times, and then move apart during good times.

When you take all these issues into consideration, it becomes clear that, at best, the asset allocations in our model portfolios are rough estimates rather than exact conclusions about the "best" way to achieve a given target return over a given period of time.

We should also say a few words about the modeling approach we used to develop these model portfolios. We start with initial portfolio weights for each asset class (i.e., with an initial asset allocation). We also start with our estimates for the average annual real annual returns for each asset class, the standard deviation of historical real returns between 1971 and 2002 , and the correlation of returns between different asset classes. The latter are also based on the71-02 data, except where that data is missing (e.g., in the case of returns on commercial property). Where that is the case we substitute the correlation from the U.S. market. We begin our calculations by randomly sampling from the distributions of returns for each asset class to generate twenty different annual returns for each asset class. Each year, we use the asset class weights (which we rebalance to their target weights each year) and asset class returns to calculate the annual return for the portfolio. We then use the twenty annual portfolio returns to calculate the compound annual return for the portfolio. This simulation (that is, the random generation of returns and calculation of the portfolio return) is repeated 1,000 times for the initial asset allocation. This gives us a distribution of returns for the given asset allocation. We then use this distribution of returns to calculate the probability of achieving the target return. For example, assume our target return is 5%. If the average of our distribution of returns is also 5%, then the probability of achieving our target rate of return would be 50%. If the average of our distribution of returns is greater than 5%, then the probability of achieving our target return would be more than 50%.

We then go back and generate a new asset allocation, repeat the above simulation process, and compare the resulting probabilities of achieving our target rate of return. Unfortunately, we cannot, due to computational constraints, consider all possible combinations of asset weights. To limit our search, we use non-linear optimization software (technically, it uses a combination of scatter and tabu search, with a neural network accelerator). In those cases where two or more different asset allocations both produce the highest probability of achieving the target return, we further compare them on the basis of their respective standard deviations (preferring the allocation which produces less volatile portfolio returns over the twenty year period). Strictly speaking, because we do not (and cannot) evaluate every possible combination of weights, we cannot be sure that this approach will produce "the single best" asset allocation (that is, the one with the highest probability of achieving our target return, at the lowest standard deviation, given the input assumptions we've used). What we can say with confidence is that the solutions this procedure identifies are almost certainly among the best that exist. Given the possible estimation errors attached to the input assumptions we use, and the complexity of the calculations involved, this is practically the best one can hope to achieve.

With respect to the relative importance of those estimation errors, an important point was made by Chopra and Ziemba in their 1993 Journal of Portfolio Management article entitled "The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice." These authors' key finding was that the estimation error of the average return is ten times as important as the estimation error of the[standard deviation], which in turn is twice as important as the estimation error of the [correlation]. There are a number of steps one can take to limit the potential impact of these estimation errors. The one we have chosen to use is to limit the maximum weight that can be given to some asset classes in our model portfolios. These include limits on foreign bonds (40% maximum), property (20%), commodities (20%), foreign equity (40%), and emerging markets equity (20%). We believe that this leaves sufficient room for effective portfolio diversification, while limiting the potential impact of estimation errors.

With all these considerations in mind, let's take a look at the model portfolios that result. The following table shows the weights for different asset classes in portfolios whose objective is to achieve a compound annual real rate of return of 3%, 5%, or 7% over twenty years. Assuming future inflation of 3% per year, these portfolios correspond to our current 6%, 8% and 10% (nominal) target return portfolios. We have shown the 9% target real rate of return portfolio, as it is essentially the same as the 7% portfolio (though with a lower probability of achieving its goal). Along with the asset weights, the following table also shows the expected average annual rate of return on each portfolio, the standard deviation of expected returns, and the probability of achieving its target compound annual rate of return over twenty years.

Of the three model portfolios in this table, it is the 7% target return one that we find most interesting. Our input assumptions assume that the highest average annual real returns will be realized on commodities (8.1%), emerging markets equities (7.5%), and foreign bonds (7.0%). These are also the only asset classes whose expected annual returns exceed the portfolio's target return of 7%. It is therefore no surprise that in all three cases, we see allocations that are equal to the maximum limits we have set for each asset class. While relaxing these limits would enable higher allocations to these asset classes, and a higher probability of achieving the 7% target rate of return, we hesitate to do so because we strongly suspect that the higher the level of expected return, the higher the likelihood that estimation error may be involved. This reinforces a point we have already made: these portfolios are suggestive, but certainly not definitive answers when it comes to our asset allocation decision. Over the next couple of months we will be presenting additional analyses to you, to ensure that our final model portfolio recommendations are based on a number of different perspectives.

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