Introduction to Portfolio Construction Using Asset Classes

Now that we understand the importance of asset allocation, and how to estimate the target real rate of return you’ll have to earn to reach your accumulation goals, let's take a closer look at the different approaches that can be used to construct a portfolio of asset classes.

Historically, the most commonly used approach to asset allocation and portfolio construction is a methodology called "mean variance optimization" (MVO), which is an application of linear programming.

MVO starts with three assumptions about each asset class: the average rate of annual return it is expected to provide in the future, the expected standard deviation of its returns (i.e., the extent to which actual returns are distributed around the expected average), and the correlation of its returns with those on other asset classes (i.e., the extent to which asset class returns linearly move together). MVO then calculates an optimal asset allocation solution, that either minimizes expected risk (defined as portfolio standard deviation) for a target level of expected annual return, or that maximizes return for a target level of risk.

Unfortunately, the theoretically "optimum" portfolios produced by MVO have too often produced disappointing real world results. There are two broad reasons for this: problems with the methodology itself, and problems with the inputs it uses.

Criticisms of the MVO methodology include the following:

- Standard deviation does not align with most investors' intuitive understanding of risk, which is better described as the distribution of returns below the expected average, or the probability of failing to achieve some threshold portfolio return (and hence the goals and dreams that depend on that return).

- Correlation only captures linear (but not non-linear) relationships between asset class returns, and does a poor job of capturing the relationships between returns in the tails of a distribution (where contagion and herding across asset classes under extreme conditions can cause many correlations to sharply increase).

- MVO assumes that asset returns are normally distributed - that is, when plotted on a graph they have the familiar symmetrical bell-curve shape. In fact, most financial asset returns are not normal, are more tilted in a positive or negative direction (technically, they "skewed"), and have fatter tails than the normal distribution (i.e., they have a greater proportion of extreme returns, or, technically, greater "kurtosis").

- While MVO is a one period technique, most investors' goals extend across long time horizons. When the returns on an asset class vary (i.e., when standard deviation is greater than zero), the long term geometric average return will be less than the arithmetic average return used in most MVO analyses (this difference will be roughly equal to the arithmetic return less two times the standard deviation squared). Moreover, across multiple periods, asset class weights will not always equal the weights assumed by the MVO analysis, as rebalancing only happens at intervals.

- Returns, standard deviations and correlations aren't stable over time; for example, they have been quite different in the past under different economic conditions. The averages that are typically used in MVO analyses can assume away serious problems that can (and have) occur under adverse regimes (e.g., ones involving high uncertainty or high inflation).

- Too often, MVO analyses use historical data as the basis for calculating return, risk and correlation inputs. What is often not acknowledged is that these historical samples are only estimates of the true values of the returns, standard deviations and correlations produced by the underlying return generating process. For example, statistical software that calculates the historical average or mean return of a time series of asset returns also typically calculates the standard error of this estimate. The standard error is equal to the sample standard deviation divided by the square root of the number of data points used in the estimate. Assuming that the data come from a normal distribution (that is, one in the shape of the bell curve), there is a 67% chance that the true mean will lie within plus or minus one standard error of the sample mean, and a 95% chance that it will lie within two standard errors. Given the relatively short historical data series available for most asset classes, these standard errors are typically quite large relative to the sample mean. All else being equal, this should introduce a substantial amount of uncertainty into the MVO calculation - yet this uncertainty is seldom acknowledged or disclosed.

- A similar problem occurs in the case of standard deviation. The use of the sample standard deviation in an MVO analysis assumes that the underlying series of financial returns are independent from each other (technically, that they have zero autocorrelation). Yet this is seldom the case. As a result, assets whose returns have positive autocorrelation have underestimated standard deviations, while assets with negative autocorrelations have overestimates standard deviations.

There are a number of different approaches available that improve on both MVO's methodology, and the traditional approach of using unadjusted historical data for asset allocation inputs.

On the methodology side, if one believes that a complex adaptive system makes it impossible to forecast asset class returns, risks, and correlations, one can simply default to the so-called "1/N" or equally weighted approach. More specifically, this approach first identifies broadly defined asset classes that provide exposure to different underlying sources of risk and return. It then allocates equal amounts to index products (mutual funds or exchange traded funds) that provide low cost, passive exposure to these asset classes.

In contrast to equal weighting, if an investor or advisor believes that different asset classes perform better or worse under different economic scenarios, they could determine their allocation weights based on the expected probability of different scenarios developing.

Alternatively, if they assume that the relative ranking of asset class risks is more predictable than the ranking of asset class returns, then equal risk weighting would be an option (though this would also depend on the degree of uncertainty regarding underlying assumptions about future standard deviations and correlations).

The most sophisticated approaches to long-term asset allocation problems (such as the methodology we used in the past to construct our model portfolios) start with a candidate asset allocation, and then simulate a large number of possible portfolio returns that could result under combinations of different regimes (e.g., periods of high uncertainty, high inflation, and relatively normal times). These approaches employ evolutionary algorithms to identify additional candidate asset allocations, and test them in the same way.

Evolutionary algorithms and other so-called heuristic search techniques are used because these asset allocation models typically contain not only a number of portfolio constraints (e.g., the amount invested in commodities and timber cannot, in total, exceed 20% of the portfolio), but also a number of different goals (e.g., at least 95% probability of accumulating a certain amount by a certain date, with a maximum shortfall of no more than 20% under the target goal).

This complicated model structure makes it computationally impossible to either identify a single "optimum" solution, or to exhaustively search the space that contains every possible solution (technically, they are NP-hard problems). The only way to approach such problems is via evolutionary algorithms that search for solutions that have a high probability of achieving the specified goals under a wide range of possible future scenarios. While these solutions can be said to be robust (i.e., they have a reasonable chance of achieving a given set of goals in the face of irreducible uncertainty), one cannot say they are optimal in the MVO sense of being sure there are no better solutions available.

Similarly, there are a number of methodologies available for improving the quality of the asset class inputs used by different asset allocation methodologies. For example, standard deviation estimates can be adjusted to reflect the impact of autocorrelation. It is also possible to use distributions other than the normal (Gaussian or "Bell Curve") distribution to describe asset class returns.

Another approach is to reduce the size of the forecasting problem by making returns, standard deviations and correlations a function of a smaller number of factors (e.g., economic variables or statistically derived "principal components"). However, this still leaves one with the challenge of forecasting the future values of these factors, future returns for being exposed to them, and uncertainty about whether relationships between the factors and the asset class variables will remain stable in the future (which is unlikely in a continuously evolving complex adaptive system).

Yet another approach, known as the Black-Litterman model, assumes that markets are in equilibrium, and uses market capitalization weights to infer the market's assumptions about future asset class returns, standard deviations and correlations (BL then combines this prior assumption with an investor’s own views to generate the final set of assumptions). The major criticism of Black-Litterman is that it is invalid if markets are not in equilibrium. And we are definitely in the camp that believes that while markets are attracted to equilibrium, they are rarely in this state.

An alternative approach attempts to limit parameter estimation errors (e.g., future asset class returns, risks, and correlations) by adjusting (or "shrinking") these estimates derived from historical data or a forecasting model towards a central average, with the more extreme estimates (which are assumed to be most subject to error) being shrunk by the largest amounts (note that mathematically, the common asset allocation practice of setting multiple constraints on maximum asset class weights has a similar impact).

A final approach to improving asset allocation parameter estimates is model averaging, which combines estimates derived from multiple approaches, including historical data, shrinkage estimators, and factor or other econometric models. While there are many different methods for combining estimates that have been made using different methodologies, it has been shown that taking either the median estimate or a simple average often substantially improves forecast accuracy.

These asset allocation/portfolio construction methodologies and new developments in this area are something we wrote about frequently in The Index Investor (as you can see from our back issues). We believe that it is critical that investors understand the strengths and weaknesses of the methodology that underlies any asset allocation recommendation that they receive.

Having discussed the theory behind portfolio construction, let's move on to a real example: how our model portfolios performed between December 2002 and December 2017.

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