Journal of Real Estate Portfolio Management: Real Estate in the Real World: Dealing with Non-Normali

Continued from page 1.

The primary goal of this research is to produce an empirical allocation model that will be used on a regular basis by a large real estate pension fund advisor. As such, the process began by identifying the role of the model within the decision-making process of the advisor and by defining a "minimal set" of desirable properties the model must have to achieve this goal and be consistent with the current state of knowledge. The overarching consideration is that this model is but one part of a broader, more expansive strategic investment decision-making process used by the advisor. The model is not meant to be a "stand alone" application to drive specific investment decisions. Indeed, we have very strong reservations against the use of quantitative mean-variance allocation models, even those incorporating the modifications we present, in a mechanical fashion. Simply put, the available real estate data is not sufficiently precise or robust to support highly targeted allocation decisions-an observation supported by many researchers (e.g., Graff, Young, and Schoenberg, 1995; and Gold, 1996). On a more specific basis, this model is intended for use within the real estate sector only, and at this point, is restricted to providing allocation forecasts across the national property markets. In addition, this model is intended for use as a quantitative benchmark-that is, as an empirical means of developing a set of optimal property allocations that are consistent with the investor's forecasts of real estate fundamentals. We, like most other researchers, use the NCREIF property indices as our primary data source. Restricting the empirical scope of the model helps mitigate some (though not all) of the empirical difficulties other research has noted stemming from the construction and composition of the NCREIF Index. We should also note that we are still in the formative stages of research on this effort and that much work remains; thus, our current research should be considered very much a work-in-progress.

As described above, the mean-variance model is used extensively for asset allocation by a wide range of investors and across a broad spectrum of investment categories. Commercial real estate has been no exception, as researchers have explored the usefulness of this model for providing diversification both within the real estate asset class, and in the mixed-asset portfolio (e.g., Mueller and Mueller, 2003). More generally, financial researchers have, over the past three decades, systematically examined and tested virtually every facet of the mean-variance model, including each of the primary assumptions of the underlying asset pricing model, and the ex-ante and ex-post implications of the model, especially the optimality of the predicted allocations. In our research, we made no attempt to be thoroughly exhaustive in the review of this body of research; instead, we distilled the relevant current research in real estate allocation into three basic properties to which we think all allocation models should conform:

(1) Conditionally Predictive. The underlying model of returns or asset prices should be explicitly forward-looking and incorporate the effects of lagged returns as well as current and future exogenous market factors. In particular, the effects on these exogenous factors of the out-of-sample predictive power of the model should be testable statistically. Since private commercial real estate returns-at least as measured by the NCREIF Index-have been shown to be conditionally forecastable, then such predictions should be used in place of the more naïve "historical estimates," such as past means and volatilities, that are often used as inputs into mean-variance models. This also helps ensure that the model's forecast of returns and of higher moments, especially conditional volatilities, are consistent with the investor's views of future market conditions (e.g., rents, occupancies, net absorption, new construction, etc.). More sophisticated models also permit the calculation (usually via Monte Carlo methods) of true out-ofsample uncertainty, thus providing a set of meanvariance inputs more consistent with MPT and better reflective of the underlying risk of the market sector.2

(2) Non-normal Returns. The underlying asset modeling and forecasting process must explicitly account for the observed non-normality of real estate returns. Deviations from normality have significant implications for optimal portfolio construction, since the negative skewness and leptokurtosis ("fat-tail" behavior) of returns can have a dramatic impact on the distribution of future portfolio performance. In particular, the type of non-normality usually seen in asset returns implies that there is greater likelihood of larger-thanaverage returns compared to those predicted by the normal distribution (leptokurtosis), and that the distribution will be skewed to the left, implying more downside risk than suggested by the normal. Ideally, the modeling process should also account for serial persistence in volatilities, say through some sort of ARCH process, should such behavior be shown to be statistically significant.3

(3) Reflect Investor Risk Tolerance and Constraints. It seems a rather obvious point that a useful optimization model should adequately reflect the risk tolerances and constraints (either strategic or statutory) of the investor. In this regard, the standard Markowitz mean-variance model often fails. Risk is most typically measured as the standard deviation of returns. In such a model, a risk-averse investor is as sensitive to returns about their expectation as returns below their expectation. This symmetry of return preferences, however, is often at odds with the observed attitudes of investors toward risk. In many instances, investors are far more concerned with the downside of the return distribution (i.e., the likelihood that returns fall below some threshold value). By contrast, positive return surprises are viewed favorably. This introduces an asymmetry into the optimal allocation problem that needs to be considered.

In the interest of disclosure, the other important real-world complications, such as illiquidity, tax considerations and statutory investment restrictions, are not dealt with in the model. This can be a serious impediment to the use of any quantitative asset allocation models, especially as precise guides to investment strategy.

Modeling Non-normal Conditional Return

Investment researchers have long recognized that many asset returns and prices exhibit systematic and persistent deviations from normality. In particular, asset returns-especially high-frequency returns-are skewed, leptokurtic and conditionally heteroscedastic (asset volatilities change over time and over the business cycle). Research on real estate returns confirms these findings (e.g., Myer and Webb, 1997; and Brown, 2000). Such behavior has profound implications for asset pricing models (and hence optimal portfolio construction). Accordingly, a voluminous body of financial and econometric research has been dedicated to exploring ways of dealing with these deviations from normality.

The Noncentral Student-T Distribution

One common way of accounting for deviations from normality is to adopt a more flexible statistical distribution of returns consistent with the aforementioned characteristics. This is the approach adopted in this paper. Specifically, a model of real estate returns is developed using the noncentral Student-t distribution. This distribution is a generalization of the well-known symmetric Student-t, which has been very successfully used in modeling a wide range of data with outliers (e.g., Lange, Little and Taylor, 1989). In economics, Geweke (1993) has shown that the symmetric Student-t has been shown to have clear advantages for modeling macroeconomic time series. More recently the Student-t has been adopted by research in finance to model asset returns and credit derivatives (e.g., Mashal and Zeevi, 2002). The symmetric Student-t is leptokurtic (e.g., has "fatter tails" than the normal distribution). As a result, it can better handle volatile time series with heavy tails. In many cases, however, economic and financial data is skewed as well as fat-tailed. For instance, stock return data often contains spells of "crashes," making the data left-skewed. A more general distribution is required to suitably model this skewness (while also accounting for leptokurtosis). Here the noncentral Student-t is adopted for this purpose.

The noncentral Student-t generalizes the symmetric Student-t by introducing a noncentrality parameter that controls the degree of skewness. To illustrate the effect of this generalization, consider first how the symmetric Student-t compares to the normal distribution. The tails of the symmetric Student-t are controlled by a single degrees-of-freedom parameter, v. Exhibit 1 contrasts a family of Student-t densities with different v values with the standard normal. Smaller values of v correspond to more leptokurtic distributions (note: as v [arrow right] ∞), the symmetric Student-t converges to the normal). In all cases, however, the Student-t distribution, like the normal, is not skewed.

The noncentral Student-t allows for skewness using a single parameter, commonly denoted δ. Positive values of δ skew the density to the right, while negative values skew to the left. Exhibit 2 depicts a family of noncentral Student-t (denoted NCT), where the degrees-of-freedom parameter is fixed at ν = 1 and the noncentrality parameter δ varies. From comparative purposes, we also plot the symmetric Student-t with ν = 1. The skewness induced by varying δ is apparent, as is the preservation of leptokurtosis.

Surprisingly, though the noncentral Student-t clearly supports the skewed and heavy-tailed behavior required for asset pricing, very few empirical studies have made use of this distribution. Exceptions are papers by Lahiri and Teigland (1987) and Dasgupta and Lahiri (1992), each of which used the noncentral Student-t in analyzing economic survey data. Part of this difficulty is that the noncentral Student-t is computationally complex, and presents some formidable estimation challenges (see Appendix B for further details). In a recent paper, however, Tsionas (2002) develops a computationally efficient Bayesian estimator to support inference in linear regression models with noncentral Student-t disturbances. We adopt this estimator for modeling real estate returns.

Returns Data

The private equity returns are calculated by using the NCREIF property indices. The sample of quarterly data spans the interval 1978:Q1 through 2003:Q3, a total of n = 103 observations. Returns were calculated as the one-period log difference in the published Index values for each of the four major property types: offices, retail, industrial and apartments. Descriptive statistics are shown below in Exhibit 3, while Exhibit 4 shows return history over the sample period. The contemporaneous correlations are shown Exhibit 5. Returns for the office, retail and industrial sectors show evidence of negative skewness, while the apartment sector appears positively skewed. Office and apartment returns show evidence of leptokurtosis,4 while the retail and industrial sector returns are thinner-tailed than the normal (corroborating the findings of Myer and Webb. 1991). The Jarque-Bera test strongly rejects the null of i.i.d. normality at less than the 1% level. As an additional diagnostic, the nonparametric kernel density estimates of returns for each sector were computed and contrasted with a normal distribution having the same sample mean and variance as each sector (Exhibits 6-9). From this it seems clear that the normal is not likely to be an adequate model of return distributions.

In Bayesian statistics, unknown parameters are estimated by integrating the posterior density function, which, via Bayes' Theorem, is the product of the prior density functions of the unknown parameters selected by the researcher and the likelihood function corresponding to the data model (e.g., Equation (5) in Appendix B). Specifically, it can be shown that a Gibbs Estimator (or Gibbs Sampling) can used for the noncentral Student-t linear model. Briefly, Gibbs Sampling is an iterative technique where a complex probability density function can be approximated by sampling from the (usually) simpler conditional density functions comprising the full pdf (see Gelfand and Steel, 1990). Typically, most Bayesian posterior density functions can be integrated analytically. In addition, for even modest sized problems, the dimensionality of the posterior pdf precludes the use of direct numerical integration. With Gibbs sampling, the posterior density is first expressed as a sequence of conditional density functions. These conditionals (which of course depend on both the prior distributions and the likelihood function) are for many problems much simpler to sample. Under certain conditions, it can be shown that random draws from these conditional density functions will converge to the draws taken directly from the complete posterior pdf as the number of draws increases.5 We outline the basic Gibbs sampling procedure for the noncentral Student-t linear model in Appendix B. For more complete details the reader is referred to Tsionas (2002).

Sample Results

The noncentral Student-t Gibbs Sampler is computationally demanding. As a result, specification searches are conducted first using conventional ordinary least squares (OLS), and then the noncentral Student-t estimator is applied to candidate models. A sample equation for office returns is given in Exhibit 10 (note: this is a representative set of results, not the final specification).

Bayesian estimates are generated using 1,500 Gibbs trials. To avoid "start-up" problems, we discard the first 500 iterations. Exhibit 10 presents OLS estimates along with the posterior medians and semi-interquartile ranges of the noncentral Student-t parameters β, σ^sup 2^, ν and δ. For comparative purposes, estimates for δ = 0 are also reported (i.e., the conventional symmetric Student-t).

The conditional normality of the β vector suggests that the student-t point estimates should not differ substantially from OLS estimates. This is generally confirmed by the empirical results. The degrees-of-freedom parameter ν exceeds 2.0 for both the symmetric and noncentral Student-t, reflecting the leptokurtosis seen in historical office returns. The skewness parameter, however, though negative, is close to zero, and its dispersion is quite wide, as seen by the semi-interquartile range estimate of 0.189. Indeed, under the posterior density of δ, the p(δ

Generating Forecasts

Bayesian forecasting is often more challenging than forecasting from non-Bayesian models. For a non-Bayesian model, one simply "solves" the estimated equation, taking account of any recurrence structure if a multi-step forecast is required. For Bayesian models, the forecasting process involves estimating the posterior predictive distribution of y. This takes into account both the stochastic variation of the dependent variable and the uncertainty of the model parameters. Using a Gibbs Sampler, this involves augmenting the historical X data with the forecast (exogenous) X and then sampling from the conditional density of the dependent variable, collecting the Gibbs samples at these points. An important advantage of this approach is that by estimating the posterior predictive distribution, one can also immediately calculate any function of the predictive, including the mean, median, variance and higher moments. In our case, this means estimating both the posterior predictive density of out-of-sample returns and the volatility of returns-the two key inputs of asset allocation. Using the Gibbs Sampler, the one step-ahead posterior predictive returns were generated for each of the four asset types. Representative results for the office sector are shown in Exhibit 11.

Optimal Conditional Portfolios

The Markowitz MPT model is, without a doubt, the most widely used method of quantitative asset allocation. Researchers have long-recognized, however, that the traditional Markowitz approach has two serious drawback. First, mean-variance optimization tends to perform poorly when asset returns are skewed. More precisely, the assumption that asset returns follow a symmetric bell-shaped distribution (such as the normal) implies that the efficient frontier will systematically include a lower proportion of negatively skewed assets for a given level of returns than is optimal. Second, the implied symmetry of the covariance-based measure of risk ignores investor risk-aversion. A more complete discussion of these issues can be found in Sing and Ong (2000).

Indeed, in 1959 Markowitz himself recognized the "asymmetrical efficiencies" in mean-variance analysis and suggested an alternative measure, based on the idea of semi-variance (i.e., the variability of returns below some target threshold value). This semi-variance approach has considerable intuitive appeal. The usual mean-variance approach treats return deviations from the mean (expected return) in a symmetric fashion-that is, unexpectedly high returns are considered as sub-optimal as unexpectedly low returns. This tends to fly in the face of both common sense and experience. For instance, consider the case of a pension investor. This class of investor generally has a statutory obligation to allocate investments in such as way as to achieve a prespecified threshold return to minimize the likelihood that future pension obligations cannot be met. This suggests that optimal portfolios should be constructed with an eye more towards avoiding returns below the threshold level than the conventional mean-variance model. Interestingly, despite the appeal of semi-variance methods, Markowitz ultimately discarded it in favor of the mean-variance approach, citing the relative computational simplicity of the latter.

The optimal solution to Equation (5), subject to the standard set of portfolio weight constraints and a target rate of return τ, yields a new downside-risk efficient frontier.

To help illustrate some of the differences in frontiers, the historical returns are used to compute the standard covariance matrix, and the CLPM covariance matrix, as defined in Equation (3). Then a set of expected returns is selected, which for expositional ease, is set to {apartment, office, retail, industrial} = {0.015, 0.025, 0.03, 0.05}. Then n = 2 is selected for the downside risk estimator (i.e., minimizing the semi-variance) and τ = 0 and τ = 0.01 are set as the downside return thresholds. The resulting frontiers are shown in Exhibit 12.

It should be noted that the MPT efficient frontier is not directly comparable with its downside-risk counterpart, since the underlying risk preferences of investors are fundamentally different. However, some of the effects on the optimal portfolio construction can be illustrated. Under the downside-risk model, the investor's aversion to negative returns (τ = 0) has the effect of pushing the frontier to the left (i.e., the investor will tolerate less risk for a given level of return). For instance, for expected returns = 3%, the difference in quarterly risk is 96 basis points. Increasing the value of τ has the effect of pushing the downside frontier closer to the mean-variance frontier as the investor becomes more risk-tolerant (i.e., will accept higher levels of risk for a given return).

Next, out-of sample frontiers are developed based on the noncentral Student-t model. To develop volatility inputs for the mean-variance and downside risk models, one-step ahead forecasts are generated for each of the four property sectors, using the approach outlined above. Then the most recent twenty-eight quarters of return history for each sector are taken, and the corresponding one-period return forecast for each of the 1,000 Gibbs trials are appended.6 Thus, each of the Gibbs trials generates a new set of return series for each of the four property sectors. The standard covariance matrix and the CLPM covariance matrix are calculated for each set. Next, the mean of each of these sequences of covariance matrices, across all 1000 trials is calculated. Finally, using the same expected return vector as in the example above, the efficient frontiers for the mean-variance and downside-risk models are calculated. For the downside model the return threshold is set to τ = 0.025. The results are shown in Exhibit 13.

As was the case in the historical example, the forecast downside-risk frontier lies to the left of the standard mean-variance frontier. We also varied τ somewhat and found that values τ > 3 induce the downside risk frontier to intersect and finally move to the right of the mean-variance frontier. Again, these results should be viewed strictly as illustrative. A more complete analysis involves multi-period return forecasting and a more extensive review of the effects of downside risk aversion on optimal portfolio construction. These results, along with a more comprehensive return forecasting model, will be published in a forthcoming paper.

Conclusion

In response to the precipitous crash in the commercial real estate markets in the mid-1980s, the Markowitz MPT model has been adopted as a risk mitigation and diversification tool by institutional real estate investors. Despite its widespread use, researchers have recognized that the conventional mean-variance model suffers from a number of important limitations when applied to commercial real estate. This paper presents a summary of some work-in-progress developing a with-in real estate property sector allocation model that better reflects the underlying statistical properties of real estate returns and allows for a more flexible (and arguably) realistic measure of risk. We adopt a newly developed Bayesian approach that permits us to model and conditionally forecast property returns that are both skewed and leptokurtic. We use these estimates to drive a downside-risk allocation model that addresses some of the critical limitations of the conventional MPT framework. Results to date show promise that a tractable and more useful allocation model will emerge.

Endnotes

1. The assumption of normality can be relaxed to some extent. The theory can accommodate conditionally normal returns, as well as ARCH-effects.

2. Strictly speaking, MPT defines optimal allocation in terms of investors' expectation of future returns and future risk-these are ex-ante concepts, quite distinct from the ex-post estimates that are frequently used.

3. We test for ARCH(p) effects and cannot reject the null hypothesis that no ARCH effects are present. The current model can be modified to handle ARCH/GARCH volatility structures.

4. We report excess kurtosis. The normal distribution has excess kurtosis equal to 3.

5. Gibbs Sampling is a special case of Markov Chain Monte Carlo (MCMC) analysis. MCMC techniques are built upon the idea that, provided certain regularity conditions are met, we can simulate from a complex density p by generating a Markov Chain of draws from a presumably simpler density whose stationary distribution is π, up to a constant of proportionality.

6. A preferred method would be to model the error terms as draws form a multivariate density. This approach, however, is significantly more complex than the approach presented here. To the best of the authors' knowledge, a multivariate noncentral Student t distribution has not been developed in the literature. It may, however, be possible to approximate this approach using statistical copula functions, which allow for the use of complex joint densities using known marginal densities.

References

Bawa, V. S. and E. B. Lindenberg, Capital Market Equilibrium in a Mean-Lower Partial Moment Framework, Journal of Financial Economics, 1977, 5, 189-200.

Brown, R. J., Tier II Real Estate, Real Estate Research Institute, 2000.

Dasgupta, S. and K. Lahiri, A Comparative Study of Alternative Methods of Quantifying Qualitative Survey Responses Using NAPM Data, Journal of Business and Economics Statistics, 1992, 10, 391-400.

Gelfand, A. E. and M. F. J. Steel, Sampling Based Approaches to Calculating Marginal Densities, Journal of the American Statistical Association, 1990, 85, 398-409.

Geltner, D., B. D. MacGregor and G. M. Schwann, Appraisal Smoothing and Price Discovery in Real Estate Markets, Urban Studies, 2003, 40, 1047-64.

Geweke, J., Bayesian Treatment of the Independent Student-t Linear Model, Journal of Applied Econometrics, 1993, 8, S19-S40.

Gold, R. B., The Use of MPT for Real Estate Portfolios in an Uncertain World, Journal of Real Estate Portfolio Management, 1996, 2:2, 95-106.

Graff, Richard A., Michael S. Young and R. J. Schoenberg, Mean-Variance Efficient Portfolios: Fool's Gold in the Investment Strategy Treasure Chest, RREEF working paper, 1995.

Graff, R. A. and M. S. Young, Real Estate Return Correlations: Real-World Limitations on Relationships Inferred from NCREIF Data, Journal of Real Estate Finance and Economics, 1996, 13, 121-42.

Johnson, N. L., S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, second Edition, New York, NY: Wiley, 1995.

Lahiri, K. and C. Teigland, On the Normality of Probability Distributions of Inflation and GNP Forecasts, International Journal of Forecasting, 1987, 3, 269-79.

Lange, K. L., R. J. A. Little and J. M. G. Taylor, Robust Statistical Modeling Using the ? Distribution, Journal of the American Statistical Association, 1989, 84, 881-96.

Markowitz, H., Portfolio Selection, New York, NY: John Wiley & Sons, 1959.

Mashal, R. and A. Zeevi, Beyond Correlation: Extreme Co-Movements between Financial Assets, Columbia University Working Paper, 2002.

Mueller, A. G. and G. R. Mueller, Public and Private Real Estate in a Mixed-Asset Portfolio, Journal of Real Estate Portfolio Management, 2003, 9:3, 193-203.

Myer, F. C. N. and J. R. Webb, Are Commercial Real Estate Returns Normally Distributed?, NCREIF White Paper, 1991.

Pagliari, J. L., J. R. Webb and J. J. Del Casino, Applying MPT to Institutional Real Estate Portfolios: The Good, the Bad, and the Uncertain, Journal of Real Estate Portfolio Management, 1995, 1:1, 67-88.

Sing, T. F. and S. E. Ong, Asset Allocation in a Downside Risk Framework, Journal of Real Estate Portfolio Management, 2000, 6:3, 213-23.

Tsionas, E. G., Bayesian Inference in the Noncentral Student-t Model, Journal of Computational and Graphical Statistics, 2002, 11:1, 208-221.

Viezer, T. W, Evaluating 'Within Real Estate' Diversification Strategies, Journal of Real Estate Portfolio Management, 2000, 6:1, 75-95.

Weisstein, E. W, Noncentral Student's t-Distribution, 2004, From Mathworld, a Wolfram Web Resource, http:// mathworld.wolfram.com/NoncentralStudentst-Distribution. html.

Worzala, E. M. and V. L. Bajtelsmit, Real Estate Asset Allocation and the Decisionmaking Framework Used by Pension Fund Managers, Journal of Real Estate Portfolio Management, 1997, 3:1, 47-56.

by Mark S. Coleman*

Asieh Mansour**

* Chatham Research Alliance, Arlington, MA 02476 or mark.coleman@chathamresearch.com.

** RREEF, San Francisco, CA 94111-5853 or amansour@rreef.com.