Updated Asset Class Assumptions and Evaluations

In order to update our model portfolios asset allocations, we need to update our assumptions about future asset class risk and returns. As the previous article has made clear, there is an irreducible level of uncertainty that accompanies this process. In fact, the only thing we can say with confidence is that our estimates will most likely turn out to be wrong. It is for this reason that we use an equally weighted portfolio as our ultimate performance benchmark, since it assumes that neither future returns nor risks can be forecast with any accuracy beyond luck. This raises an obvious question: why do we believe this is not the case?

The most important reason is that basic differences in the return generating processes for different asset classes (e.g., bonds, commercial property, and equity) suggest that there will be stable differences in the dispersion (i.e., riskiness) of their returns. This means that the ranking of asset classes according to their standard deviations should remain relatively stable over time. This economic hypothesis is supported by the statistical fact that you can improve the accuracy of an estimate of standard deviation by increasing the frequency with which data from a given period (e.g., 1989 to 2004) are sampled (e.g., by using monthly instead of annual data). To be sure, this isn't always true, as volatility (standard deviation) varies (or “clusters”) over time. But it changes much less than the ranking of asset classes by their relative returns over different periods.

Given everything written in the previous article, we are much less confident about our -- or anybody else's -- ability to accurately forecast future asset class returns. At best, we can limit the size of the inevitable estimation errors we will make. To do this we combine two unavoidably flawed approaches to the estimation problem: historical data and the outputs from a forecasting model.

The use of historical data contains a number of pitfalls. The first is uncertainty about the extent to which the sample of data you are using represents the “true” distribution of results the may be produced by the return generating process. This is a particular concern with respect to so-called “extreme events”, or periods in which large gains or losses are experienced. Does your sample contain all the extreme events a return process might produce? It is for this reason that the arcane subject of “extreme value theory” is so popular with hedge fund mangers who trade in highly leveraged derivative instruments.

One way to deal with this problem is to convert your sample into a distribution of returns that can be described using just a few variables - e.g., the mean (average) and standard deviation (a measure of dispersion around the mean) for a normal distribution, or “bell curve.” However, this raises another issue: what is the right distribution to use? The normal distribution has some real attractions, because it simplifies a number of calculations. Unfortunately, a look at the data shows that the distributions of returns for many financial assets aren't quite normal. Typically, they are “off center” (technically, they are “skewed”) and they have “fatter tails” (technically, they have excess “kurtosis”) than a normal distribution. In practice, this leads to arguments about (a) what other distribution to use (e.g., a lognormal or Student's T), and (b) whether it matters. The latter question is addressed by Cremers, Krtizman and Page in their paper “Optimal Hedge Fund Allocations: Do Higher Moments Matter?” They find that the question turns on the shape of what is known as an investor's “utility” function, which is a measure of their sensitivity toward investment gains and losses. They find that for the most common models of investor utility (technically, power utility functions), using the normal distribution in asset allocation will produce an acceptable result.

Yet another issue is whether the returns generating process underlying the historical data you use has remained constant (or “stationary”) over time. If it has not, then estimates derived from historical data that includes the previous process will be poor predictors of future returns. Unfortunately, statisticians continue to argue about the best way to test for these so-called “structural breaks” or “non-stationarities.” Some analyses find them, and others don't, leaving investors with more uncertainty. Our instinct is that insofar as the economy is a complex adaptive system, the return generating process for many asset classes is likely to have some structural breaks, raising questions about the wisdom of relying solely on historical data to project future returns.

One technique that has been invented to deal with the problem of estimation errors when using historical data is called “shrinkage.” Its basic intuition is that the accuracy of an estimate will be improved if outlying data are “shrunk” towards a common reference point. One such point is known as the “grand mean”, which in our case would be the average return on all the asset classes included in our analysis. However, this raises two other issues. The first is how much to shrink each asset class's average return. Different authors have produced many different equations that attempt to improve on the everyday “let's split the difference” heuristic (see, for example, “Bayes-Stein Estimation for Portfolio Analysis” by Philippe Jorion, and “Optimal Estimation of the Risk Premium for the Long Run” by Jacquier, Kane and Marcus). Other authors have argued that the simple approach works quite well in many situations.

The second issue is the fact that even the “grand mean” - the average of the average expected return for each asset class - is still based on the original sample data. This has led to a search for other “shrinkage targets” that would add new information, and in so doing hopefully raise the accuracy of the resulting estimate. In finance, one approach to this is to use the output from a forward-looking return forecasting model as the shrinkage target.

However, this introduces another source of uncertainty: model error. As we have seen, in a complex adaptive system that gives rise to non-linear results, is difficult if not impossible to construct an accurate model of the return generating process for most asset classes. And even if we could, changes in that process (or copying by other investors) would inevitably invalidate our model at some point in the future. And how can one be certain that the model one decides to use is the most accurate one available? The simple answer is that you can never be sure of this. So what is an investor to do?

Our solution to this problem is to use an equilibrium model to forecast future returns. We know that most of the time, financial markets will not be in equilibrium. However, we also believe that markets are at least attracted to equilibrium, even if they rarely attain it. Specifically, we ask the question, what real rate of return would an investor require, in equilibrium (where the returns supplied equaled the returns demanded), to hold this asset class? To answer it, we take a so-called “building block” approach, that begins with the current yield on real return bonds (our proxy for the risk free rate), and adds various return premia to them based on the relative riskiness of different asset classes. These premia are shown in the table below:

For all these asset classes, our estimates of future risk (standard deviation) were based on the combination of the historical 1989-2004 results, plus a set of results for domestic equities and bonds covering 1900 to 2004 that is found in the Global Investment Returns Yearbook by Dimson, Marsh and Staunton. These were rounded to avoid the appearance of excessive precision on our part.

This leaves us with the issue of how to combine our historically based return estimates with estimates derived from our forecasting model. A recent paper “Forecast Combinations” by Allan Timmerman (an acknowledged expert in the field) concludes that simple methods often work best. Another paper, “Structural Breaks and the Performance of Forecast Combinations” by Timmerman and Marco Aiolfi presents evidence that forecast combinations are more accurate than individual forecasts because they better incorporate the affect of structural breaks. We are also persuaded by the inherent logic of the “KISS” (keep it simple, stupid) principle. All of this leads us to the use of a simple approach (50/50 weighting) to combine our historical and model based return estimates

The following tables show our historical and model based estimates of future real returns on different asset classes. The historical table shows returns from 1989 to 2004. This period covers a relatively wide range of financial market events (e.g., the 1998 debt market problems, and the internet bubble). However, we also note that the underlying economic conditions were relatively benign during this period, with inflation generally declining, and real growth fairly steady. As a result, estimates derived from the 1989 to 2004 data probably have some limitations with respect to their coverage of the entire return generating process for most asset classes (especially the extreme events that may be possible).

Four additional qualifications are also in order. First, the data for commercial property reflects traded property securities, and not property that is directly owned. Hence, our estimates will differ from those produced by companies that measure property returns (usually using appraisal based methods, that understate risk). Second, the data for timber is based on a U.S. index. In the past, the returns on this index have diverged from those on other national indexes. Unfortunately, we have no easy basis for combining the returns on these different indexes. However, we also note that in recent years, as investment in timberland has become more popular among institutions, these differences seem to be narrowing. Third, we used the Goldman Sachs Commodities Index for that asset class, as it has the longest available data series. Finally, for equity market volatility we used the VIX index, which measures the implied volatility on S&P 500 options. This has a longer data series than similar indexes (e.g., the VSTOXX) that measure volatility in other equity markets.

Last but not least, in the following tables we present three pieces of data for each asset class. First, its average arithmetic annual return. Second, the standard deviation of those returns. We then adjust the average annual return to reflect relative risk (technically, we subtract one half the variance, which is the standard deviation squared) to derive an estimate of the compound annual (or geometric average) return that would be realized by an investor who held that asset class (an no other) over a long period of time.

| Global Asset Class Returns | This Month's Letters to the Editor: Fed Chairman Bernanke; Is My FA Generating Alpha?; and New iShares in the UK | Updated Asset Class Assumptions and Evaluations | Equity Market Valuation Update | This Month's Issue: Key Points | Uncertainty |