Asset Allocation Review: Dealing with Uncertainty

We began our asset allocation analysis back in May, with a review of historical and expected future real returns across various asset classes and currencies. Last month, we used our future risk and return estimates as inputs into a simulation optimization process, to produce portfolios with the highest probability of achieving compound target real returns of 3%, 5%, and 7% over a twenty year period.

This month, we will begin by using the historical data for each asset class as inputs into the same simulation optimization process. We will then discuss the limitations of our approach, and various proposals for how they might be overcome. We will conclude this month with our final recommended twenty year target return portfolios. Next month, we will conclude our asset allocation analysis with a discussion of how our recommendations change when the investment horizon is shortened to ten years.

Portfolios Based on Historical Data

The inputs for this month's target return portfolios were estimated from 32 years of monthly real returns data, covering the period from 1971 to 2002. They are summarized in the following table:

We used these historical inputs to generate the model target return portfolios using our simulation optimization process. As we did last month (when we generated target return portfolios using our future asset class rate of return forecasts), we set limits on the maximum amount that could be allocated to certain asset classes, including 40% to foreign bonds, and 20% each to commercial property, commodities, and emerging markets equity. The model portfolios are as follows:

A couple of items in this table might need a bit of further explanation. The "Probability of Achieving Target Return" refers to the target compound annual rate of return at the end of the twenty year holding period. The "Expected Average Annual Return" is the expected rate of return for the portfolio in any single year, rather than over the full twenty years. Generally speaking, the average annual (also known as arithmetic) return will be higher than the average compound (also known as geometric) return, because the latter reflects the full impact of poor annual returns on the achievement of your long term financial goal. A simple example can help make this clear. Consider an investment which, over three years, has annual returns of 20%, (30%), and 20%. If you invested 100 at the beginning of the period, at the end of three years you would only have 100.8 in your account, for a compound return of only 0.27% over the three year period. Your average annual return, however, would be much higher: 3.33%! So while the Expected Average Annual Return and Standard Deviation give you an indication of the performance a model portfolio is likely to deliver in any given year, what really counts is the portfolio's probability of achieving your long term compound annual return target.

Another interesting point about these portfolios is the relatively heavy weights they give to bonds, property, and commodities, and the relatively low weights (at least compared to what you often see in the media) they give to equities. The logic behind these allocations is the same as in our return example above. When you are trying to achieve a long term compound annual rate of return goal, avoiding big losses is just as important as achieving big gains. Our multiyear simulation optimization model takes this into account, and generates higher weightings to relatively less risky non-equity asset classes (to minimize the chances of suffering a big loss) than other asset allocation models (e.g., single period mean/variance optimization models).

Long time readers will also notice two other differences from previous year's asset allocation reviews. First, we have included three, rather than four target return portfolios. In the past, we have developed model portfolios for target nominal returns of 6%, 8%, 10% and 12%. Assuming a long term rate of inflation of 3%, these are equivalent to target real return portfolios of 3%, 5%, 7%, and 9%. However, if you believe that our forecasts for future real rates of return on different asset classes are on target (see last month's as well as the May and June issues for these forecasts and the logic behind them), it is going to be extremely difficult to achieve a compound annual rate of return of 9% in the years ahead. When we used our simulation optimization model to generate portfolios with the highest probability of achieving this goal, we usually ended up with results very similar to our 7% target return portfolios (though with a lower probability of achieving the target rate of return). Given this, it seemed prudent not to include the 9% target real rate of return portfolios in this year's analysis.

The second major difference is the absence of any review of our "benchmark beating" portfolios this year. As you know, these are model portfolios whose objective is to either deliver higher returns than a domestic benchmark (e.g., 80% equity/20% bonds; 60% equity/40% bonds, or 20% equity/80% bonds) while taking on the same amount of risk, or to deliver the same returns while taking on less risk. Basically, this comes down to a matter of philosophy. In a nutshell, we believe that the purpose of good investment management is to ensure that over the long term, the value of your assets matches or exceeds the value of your liabilities (e.g., the amount of capital you need to achieve your retirement income goal). Our target return portfolios are based on this belief. However, our benchmark beating portfolios are not -- they focus on relative annual returns, rather than achieving one's long term goals. When we started The Index Investor back in 1997, we felt we had to include the latter, because we sensed that relative annual performance was important to a substantial percentage of our potential readers. However, following the substantial bear market we experienced in the last two years, we now believe that this percentage has grown smaller, as more and more people have realized that it is long term performance (and not bragging rights at the pub about last quarter's performance) that really counts. So while we will continue to publish our "benchmark beating" portfolios (there is some value in tradition, after all), we do not plan to update them as frequently as we have in the past.

Dealing with Uncertainty

As we have discussed in previous articles, the main problem with using historical sample data in an asset allocation analysis is estimation uncertainty. You simply can't be sure that the inputs derived from your sample reflect the "true" value for the population as a whole. And even if they do, you can't be sure that the process which generated the historical return data will remain unchanged ("stationary" in stats-speak) in the future. This problem is most acute for average returns; by increasing the frequency of the data collection within a given year (e.g., from quarterly to monthly), the estimation uncertainty associated with the standard deviation and correlation of returns can be substantially reduced. With average returns, however, the only way to reduce estimation uncertainty is to use a longer historical sample (assuming, as noted above, that the underlying returns generating process doesn't change). Generally speaking, the impact of estimation errors (that is, differences between your sample estimate and the true values of a variable) is a function of two factors: the length of your sample period and of your forecasting horizon. As we have seen, using a longer sample period improves the accuracy of your estimate. In contrast, the longer your forecasting horizon, the greater the potential impact of an estimation error, due to the compounding effect. When the forecast horizon is longer than the sample period, expected returns will tend to be biased upwards (note, this is not the case in our analysis, where the forecast horizon is 20 years, but the sample period is 32 years). More broadly, as a rule of thumb, the greater the ratio of forecasting horizon to sample period, the more an investor should suspect assets with high expected returns, due to the potential impact of estimation errors.

Four different approaches have been proposed to reduce the potential impact of estimation errors.

First, you can impose constraints, and limit the maximum weight that can be given to certain asset classes. Statistically, the most logical asset classes to constrain are those whose returns and/or risk are high relative to the average for all the asset classes being used. These are the asset classes that are most likely to be subject to estimation error (e.g., see "Risk Reduction in Large Portfolios" by Jagannathan and Ma, which can usually be found by searching on either google.com or www.ssrn.com ). We have used this approach in our analysis.

Second, if you suspect that the returns generating process may not be stable over time, you can weight your sample data so that more recent data points count more heavily (e.g., see "Time Weighted Portfolio Optimization" by Lee and Stevenson). While we recognize that the means, standard deviations, and correlations in our sample change over time (e.g., many of the latter move closer to 1.0 when markets are falling, then fall back when markets are rising), we are not convinced that this in fact represents a changing in the underlying returns generating process (indeed, there is evidence on both sides of this question). In our analysis, we have assumed that it is a stable process with ups and downs that are likely to average out over our twenty year holding period.

A third approach to minimizing the impact of estimation uncertainty is to explicitly recognize it in your optimization process. This is known as resampling or bootstrapping. To apply it, one starts with the historical distribution of returns for a given asset class (that is, with its mean and standard deviation), repeatedly draws a new sample of returns from it (i.e., one "resamples" it), and then calculates the mean and standard deviation of the new sample. By repeating this process many times, you develop a probability distribution for the mean and standard deviation statistics themselves -- the shape of these probability distributions reflecting the degree of uncertainty about the "true" values of these statistics. Practically, explicitly recognizing estimation uncertainty in this manner makes it clear that portfolios with very different asset weights may be, in the statistical sense, equivalents (and in so doing confirm the point that when it comes to asset allocation, some judgement is required!). On the other hand, an important criticism of this approach is that it is still only based on your sample distribution -- actual reduction of estimation uncertainty (as opposed to making its impact explicit) requires the introduction of additional information (see "Portfolio Choice and Estimation Risk" by Herold and Maruer). Our simulation optimization process uses the bootstrapping approach, as it resamples a distribution (based on either historical or forecast data) to develop a probability distribution of the likely results from a given asset allocation.

The fourth approach to minimizing the impact of estimation uncertainty combines the inputs derived from a historical sample with inputs derived from some prior view of the returns generating process (e.g., combine .67 times your sample asset class weights with .33 times the weights in your prior view). This is also known as the "shrinkage" approach to managing estimation error (e.g., see Jorion, "Bayes-Stein Estimation for Portfolio Analysis") . However, using this approach introduces another source of uncertainty, about the correctness of the model that you use to form your prior view (e.g., see our article on factor models in this issue, as well as "Stock Return Predictability and Model Uncertainty" by Dov Amarov, "Comparing Asset Pricing Models" by Pastor and Stambaugh, and/or "A Shrinkage Approach to Model Uncertainty and Asset Allocation" by Zhenyu Wang).

Different authors have suggested different models that could be used to form your prior view of expected asset class returns, risks and correlations.

If you do not believe that your historical sample data provide any useful information about future returns or risks, you should use an equally weighted portfolio (EWP) as your prior view. If you believe that your historical sample provides more information about risk than it does about returns, you should use what is known as the Minimum Variance Portfolio (MVP) as your prior view. The MVP is the combination of asset weights that minimizes the standard deviation of your portfolio (taking correlations into account). In other words, it is the least risky portfolio you can form given the asset classes you have decided to include in your analysis. The following table shows the minimum variance portfolios for our six different currencies (based on historical real standard deviations and correlations from our 1971-2002 sample):

Finally, if you believe that your sample provides useful information about both risk and return, you should use the global market portfolio as your prior. Theoretically, (assuming markets are efficient and in equilibrium) this is the portfolio which maximizes expected return per unit of expected risk. Very closely related to this is another common approach to asset allocation, known as the Black-Litterman model (as described in their paper "Global Portfolio Optimization"), which shrinks the optimal portfolio that results from an investor's future return forecast toward the global market portfolio. Black and Litterman's underlying assumption is that markets are generally in equilibrium, with the expected return on each asset class balancing the available supply and demand of securities. To the extent that a forecast of future returns implies an optimal portfolio that has different weights from the market portfolio, it represents a departure from this equilibrium condition. To adjust for the possibility that this is a result of estimation error rather than actual disequilibrium, Black and Litterman shrink the optimal portfolio back toward the global market portfolio.

However, the global market portfolio itself is not without problems. First, there is the question of what asset classes belong in it. While debt and equity surely belong (though people argue about whether and how to include privately placed issues), a number of writers have argued against including commercial real estate and commodities. While the former is undoubtedly a large asset class, most of it is illiquid, and its returns data tend to be compromised by the fact that valuations are based on subjective opinions rather than objective market prices. The reason commodities aren't included is to avoid double counting -- for example, a substantial portion of the value of the world's physical oil is already implicitly included in the value of the bonds and shares issued by the world's oil companies.

A second issue with the global market portfolio is that it is not at all easy to obtain timely information about the size and composition of the global bond market. Unlike equities, which trade on exchanges, bonds all trade privately ("over the counter"), which makes data about them much more difficult to collect. For example, in 2001 the world's three leading references on the global bond market (produced by Merrill Lynch, the Bank for International Settlements, and the International Monetary Fund) produced three different estimates for its market value (in billions of U.S. dollars): $32,972, $35,327, and $41,792. The good news, however, was that they were all more or less in agreement with each other as to the share of outstanding bonds that had been issued in different currencies.

The asset class weights in the global market portfolio are constantly changing over time, as investors change their required rates of return on different asset classes. Unfortunately, as data about the market values of different asset classes only becomes available with a significant time lag (and even then is still subject to significant uncertainty), you can never, as a practical matter, be completely sure of the current weights in the global market portfolio.

As a point of reference, we have calculated the following global market portfolio weights for different currencies and assets based on 2002 data, which is the most recent available:

Most people reading this table will have the same reaction: I never realized the bond market was bigger than the equity market! Undoubtedly, this results from the latter getting much more publicity (and having more readily available data) than the former. However, when you think about it, it makes sense. Most corporations employ more debt than equity on their balance sheets. Then add to this the debt issued by governments (which don't issue equity), and by various asset backed security vehicles (e.g., mortgage backed bonds) which also have very little equity supporting them. When you consider all these factors, it is no surprise that the bond market is bigger. But that much bigger? Keep in mind that at the end of last year, equity market values were well below their 2000 peaks, while very low interest rates had caused bond market valuations to be very high. On a long term basis, a 60/40 split seems more realistic than the 70/30 split at the end of 2002.

Using the 2002 data, we have calculated what the global market portfolio looked like from the perspective of our six different currency regions:

Obviously, the most striking thing about this table is the heavy weight given to foreign bonds in the global market portfolios. However, we also need to keep in mind that most investors have an aversion not only to risk, but also to regret. Studies have shown that most people are willing to move away from the optimal return/risk portfolio (which, theoretically, is the global market portfolio) if such a move reduces the chances of them underperforming a popular benchmark or peer group (which is usually based on domestic market results), or suffering substantial negative returns during certain periods of time (e.g., see Chow, "Portfolio Selection Based on Risk, Return, and Relative Performance" or "How Much Foreign Stock?" by Herold and Maurer). Given this, as a practical matter we do not believe that many investors are comfortable holding allocation to foreign bonds suggested by the global market portfolio. Hence, in our simulation optimization model, we have limited the maximum allocation to foreign bonds to 40% of our model portfolios.

So where does this leave us?

To derive our final model portfolio allocations, we chose to employ a shrinkage approach by combining the portfolios we created using data from our historical sample (which is subject to estimation uncertainty) with the portfolios based on our forecast of future returns (which is subject to model uncertainty). Given the length of our sample period, and the rather dodgy track record of market forecasters, we decided to give the portfolio based on the sample data a weighting of .67, and the portfolio derived from our forecasts a weighting of .33. These weights are purely subjective, and you should feel free to change them when forming your own portfolio. You may also find it interesting to replace either of our portfolios with the global market portfolio, again using relative weights of your choosing. Frankly, there is no settled theory on the "right" way to make this decision; at best, looking at the results of a variety of approaches can help to inform one's judgement.

The resulting model target real return portfolios are shown in the following tables. Please note that the asset class weights in the first column are based on historical sample data, the weights in the second column are based on our estimates of future asset class returns, and the weights in the last column reflect 2/3 of the historical and 1/3 of the future weights.

Next month, we plan to conclude our asset allocation review with a look at how shortening the investment horizon from twenty to ten years changes our model portfolio weights. We also plan to compare our model portfolios with those actually employed by a number of successful institutional investors. We also know that many of our readers are both keenly interested and very knowledgeable about the challenges involved in asset allocation. So we close with a final request: if there are other asset allocation topics you'd like us to include in next month's issue, please don't hesitate to get in touch!

| Model Portfolio Update | The Confusing World of Factor Models | Product and Strategy Notes | Global Asset Class Returns | Asset Allocation Review: Dealing with Uncertainty | Equity Market Valuation Update |