Optimal Diversification: Is It Really Worthwhile?

Optimal Diversification: Is It Really Worthwhile? By Ping Cheng, Ph.D. Assistant Professor of Finance Department of Finance and Economics Salisbury State University Salisbury, MD 21801 (410) 543-6327 (410) 546-6208 Fax pxcheng@ssu.edu and Youguo Liang, Ph.D. Principal Prudential Real Estate Investors 8 Campus Drive Parsippany, NJ 07054 (973) 683-1765 (973) 683-1788 Fax youguo.liang@prudential.com

Optimal Diversification: Is It Really Worthwhile? Executive Summary Recent research has demonstrated that the Markowitz efficient frontier is fuzzy and may consist of many statistically indistinguishable frontiers. Therefore, it opens the possibility that an efficient portfolio developed by mean-variance analysis may not be any more efficient than a naively diversified portfolio. Using an efficiency test developed by Gibbons, Ross, and Shanken in 1989, we find evidence to support that an efficient portfolio is statistically more efficient than a corresponding naively diversified portfolio when the portfolio formation period is the same as the period used for testing the efficiency difference. However, no evidence is found to support that an efficient portfolio is statistically more efficient than a corresponding naively diversified portfolio when the portfolio formation period differs from the test period. Thus, in practical terms, efficient portfolios may not be superior to naively diversified portfolios in a statistical sense. Introduction Long before the articulation and development of Modern Portfolio Theory (MPT) in the 1950s and 1960s, investors were well aware of the benefits of diversification. The age-old thinking of not putting all your eggs in one basket has been helping both institutional and individual investors avoid disastrous outcomes. As a matter of fact, Lefthouse (1997) has documented that the staff in a British firm, the Investment Registry, understood the role of covariance in portfolio risk as early as 1904. Armed with MPT, real estate academics and professionals have examined almost every possible way to diversify within real estate in the last two decades. Pension funds have been advised to seek diversification of managers and property types. The concepts of regional diversification and economic diversification have been refined. Today, optimal allocations or ranges of allocations within real estate are estimated, implemented and monitored as part of the overall portfolio strategy. Researchers have generally demonstrated that mean-variance analysis can result in portfolios that are more efficient than naive diversification (Hartzell, Hekman and Miles, 1986; Grissom, Kuhle and Walther, 1987; Malizia and Simons, 1991; Mueller, 1993). These studies typically calculate the efficient frontiers for various diversification schemes and compare them against a naively diversified portfolio to show marginal improvement in mean-variance efficiency. Few studies, however, have questioned whether the improvement is significant in a statistical sense. That is, are portfolios formed with MPT significantly more efficient than those formed with other approaches such as a naive diversification? Although improvement in portfolio efficiency, no matter how insignificant, is always beneficial, one must question whether such benefit outweighs the cost of portfolio rebalancing. Moreover, even if the benefit is significant on an ex post basis, will an efficient portfolio constructed from historical data be efficient in the implementation period? These are the questions that must be answered in order to understand the true benefit of applying MPT to real estate portfolio management. 1

Pagliari, Webb, and Del Casino (1995) find that an efficient portfolio in an ex post sense may not remain efficient for the near future. Further, recent studies have demonstrated that the so-called efficient frontier is not singularly delineated (Liang, Myer and Webb, 1996; Gold, 1995 and 1996; Ziobrowski, Cheng and Ziobrowski, 1997). Instead, the efficient frontier is a collection of many statistically indistinguishable frontiers. As a result, the efficient frontier calculated from MPT is fuzzy, and the optimal allocation is even fuzzier. The fuzziness of the results may be caused by: (1) the model s high sensitivity to key inputs; and (2) the information inefficiency in the real estate industry, which puts MPT at risk of being an error maximizer. A recent study by Rubens, Louton and Yobaccio (1998) tests the marginal benefits of adding domestic and international real estate to mixed-asset portfolios. Using a statistical method developed by Gibbons, Ross and Shanken (1989), Rubens et al. find that gains in portfolio efficiency from including real estate are not statistically significant. To add another perspective to the issue, our study redirects the focus on within-real estate diversification, and attempts to test whether MPT offers significant benefits to real estate investors. Within-real estate diversification has traditionally been studied along two dimensions, namely, geographic grouping and property types. Geographic/economic based diversification is well documented in the literature. Since early 1980s, researchers have used various techniques and databases to study the effect of applying MPT to real estate portfolios diversified at regional, MSA, and submarket levels (Malizia and Simons, 1991; Mueller and Ziering, 1992; Mueller, 1993; Ziering and Hess, 1995; Williams, 1996; Wolverton, Cheng and Hardin, 1998; and Cheng and Black, 1998). A number of studies have proposed geographic classifications and indicated that portfolio efficiency can be improved using mean-variance analysis. The general consensus of the literature is that (1) naive diversification across broad geographic areas is less than optimal; (2) geographical grouping based on economic fundamentals allows more efficient diversification; (3) metropolitan areas appear to be more appropriate than regional level for portfolio diversification; and (4) market segments exist in the submarket level and it is possible to achieve diversification within a region, a state, or a city. While research in applying MPT to real estate diversification is likely to continue, whether MPT offers real benefits for portfolio diversification is becoming an increasingly important issue. Unless we are convinced that MPT really works, continuing efforts on refining various diversification strategies may not add value. Therefore, this study re-examines some of the more popular geographic and property type diversification schemes to see whether portfolios formed with MPT are significantly superior to naively diversified portfolios. Methodology According to Sharpe (1964), a portfolio s risk-return characteristic can be measured by its Sharpe ratio, defined as excess return per unit of risk: k p k RF S = (1) σ p Where k p is the portfolio return; k RF is the risk-free rate; and σ p is the standard deviation of k p. Higher Sharpe ratio is associated with higher portfolio efficiency. Two portfolios with different mean-variance efficiencies must have different Sharpe ratios. Gibbons, Ross and Shanken (1989) 2

developed a simple method for testing the statistical difference between the Sharp ratios of two portfolios. Given a set of investment opportunities, any two portfolios, p i and p j (i, j = 1, 2, 3, ) can be constructed to exhibit Sharp ratios of S i and S j, respectively. If p i is an efficient portfolio while p j s efficiency is unknown, then it must be true that Si S j. To test the statistical significance of their difference, the proper null hypothesis can be stated as: H 0 : S i = S j (2) This hypothesis can be tested using the following test statistic: 2 2 1+ S i W = 1 (3) 2 1 + S j Note that W is a non-negative number because Si S j. Also, under the null hypothesis W is equal to zero, which implies that the two portfolios have similar mean-variance efficiencies. Large W will lead to rejection of the null hypothesis and conclude that the mean-variance efficiencies of the two portfolios are significantly different. Since W follows an uncommon Wishart distribution, a transformation is necessary to convert the test statistic into an F distribution: T ( T n 1) F = W (3) n( T 2) Where T is the number of observations in a time series, and n is the number of investment opportunities. It is shown in Gibbons et al. (1989) that under the null hypothesis, the transformed W-statistic, F, is centrally distributed as F (T, T-n-1). It should be noted that the power of the test is critically affected by the degree of freedom of the F-test. In other words, the ratio T/n must meet a certain threshold for the test to be sensitive. As a rule of thumb, Gibbons et al. suggest that the ratio be at least 3. This technique can be used to test whether the Sharpe ratio of a mean-variance efficient portfolio is significantly greater than that of a naively constructed portfolio a typical one is an equalweighted or value-weighted portfolio. This methodology is to be applied to three geographic diversification and two property type diversification schemes: 1. The four-region scheme (East, Midwest, South, and West) classified by the National Council of Real Estate Investment Fiduciaries (NCREIF). 2. The NCREIF eight sub-region scheme. The sub-regions are: Northeast, Mideast, Southeast, Southwest, East North Central, West North Central, Mountain, and Pacific. 3. The 15 largest U.S. metropolitan statistical areas (MSAs) by 1997 total population. The fifteen limit is set by two factors: (1) many MSAs do not have returns compiled by NCREIF until the 1980s; and (2) in order to preserve the power of the W-statistic, the number of MSAs has to be limited in relationship to the number of quarterly observations available. 4. The four NCREIF property-type scheme (office, retail, apartment, and industrial). 5. The seven NCREIF property-type scheme (CBD office, suburban office, mall retail, non-mall retail, apartment, warehouse, and non-warehouse industrial). Because the apartment sector is more homogeneous than office, retail, and industrials, it is not subdivided. 3

For each of these diversification schemes, an equally-weighted and a value-weighted portfolio are formed as the base (or naively diversified) portfolios for the sample period. Mean-variance efficient portfolios are constructed by solving the classical quadratic programming problem with the constraint of no short sale. Since there is an infinite number of optimized portfolios on an efficient frontier, it is impossible to test every single efficient portfolio against the base portfolios. Therefore, three portfolios are singled out for testing: the efficient portfolio with the maximum Sharpe ratio, the efficient portfolio that has the same expected return as the equallyweighted portfolio, and the efficient portfolio that has the same expected return as the valueweighted portfolio. The null hypothesis is that the two portfolios Sharpe ratios (one efficient portfolio and one naive portfolio) are not significantly different. Rejecting the null hypothesis will lead us to conclude that optimal diversification, by geographic location or by property type, significantly adds value. We test the mean-variance efficiency of each of these schemes on an ex post as well as an ex ante basis. For the ex post test, we use the entire sample period for formulating the portfolios and testing the efficiency differences. For the ex ante test, we divide the sample into two sub-periods: the first half for formulating the portfolios and the second half for testing the efficiency differences. It is necessary to note that in order to maintain the power of the test, both the formulating period and testing period (or the holdout period) must contain enough observations. For the four regions and eight sub-regions categories, the time series is long enough when divided equally into two sub-periods. However, for the other categories, the holdout period has to be overlapped with the formulating period. For example, there are only 66 observations in the entire time series for the 15 MSA diversification scheme; the formulating period includes the first 50 observations and the holdout or testing period includes the last 50 observations. Therefore, the holdout period is only partially different from the formulation period. However, this should not be a concern if the ex ante test fails to reject the null hypothesis of significant difference in portfolio efficiency, since it indicates that only partially different samples can add enough uncertainty to make an ex post efficient portfolio insignificantly different from a naive portfolio in the holdout period. Data The NCREIF Property Index and its regional, property-type, and MSA level subindices are used in the tests. Indices of the four regions have 82 quarterly observations covering the third quarter 1978 through the fourth quarter 1998. The number of observations becomes fewer for other subindices. For the eight sub-region indices, the data spans from the third quarter 1979 through the fourth quarter 1998. For the 15 MSAs, complete data series is only available for the period from the third quarter 1982 through the fourth quarter 1998. For the four property types and seven sub-property types, the data covers the fourth quarter 1984 through the fourth quarter 1998. These indices also contain aggregate property values, which are the basis for the construction of a value-weighted portfolio. Since most of the property values are appraisal-based, the return indices tend to understate the volatility of the real estate market (Geltner, 1993). This smoothing effect can cause serious biases when real estate performance is compared with other assets in a mixed-asset portfolio. However, with respect to the within-real estate diversification, which is 4

the focus of this study, the appraisal bias becomes a systematic error because it has a similar impact on all the properties included in the indices. Therefore, we do not adjust the data series for smoothing in this study. Results Exhibit 1 shows the ex post results for the geographical diversification schemes. Ex post means the time period used for constructing the portfolios is the same as the one used to test the efficiency difference. The upper panel summarizes the risk and return characteristics of the five portfolios. As previously stated, for each geographic category, we construct two passive or naive portfolios: an equal weighted (EqWt) and a value-weighted (ValWt). Three efficient portfolios are singled out for the efficiency tests. Portfolio MPT1 has the same expected return as the equally-weighted portfolio and portfolio MPT2 has the same expected return as the valueweighted portfolio. Portfolio MPT-MaxS is the efficient portfolio with the maximum Sharpe ratio. The risk-free rates are the average quarterly total returns on 30-day Treasury bills for the corresponding period. (See Exhibit 1 on page 9.) The lower panel summarizes the significance tests. At the four-region level, none of the tests is significant, indicating that the three MPT portfolios provide no efficiency improvement over the naive portfolios. In other words, the efficient portfolios are statistically indifferent from the corresponding naive portfolios in terms of efficiency. The results are different for the eight NCREIF sub-regions. Three out of the four F-statistics show at least a significance level of 5 percent. For the 15 Large MSAs category, all F-statistics are significant or marginally significant. These results confirm findings of earlier studies that suggest, except for the most broad geographic areas, ex post portfolios formed with MPT exhibit mean-variance efficiency over naive portfolios. The ex ante results, however, depict a very different picture. Ex ante means the sample period used to formulate the efficient portfolios is different from the period used to test the efficiency. Exhibit 2 shows the ex ante results for the geographic diversification schemes. The formulation period predates, or at least partially predates, the testing or holdout period. The ex post efficient portfolios do not always have higher Sharpe ratios than the naive portfolios in the holdout period. When they do, the efficiency tests indicate a p-value of nearly one, suggesting that these portfolios are not statistically more efficient than the naive portfolios in the holdout period. (See Exhibit 2 on page 10.) Similar results are observed with diversification by property type. Exhibit 3 displays the ex post results. The four property type category shows little evidence of significant efficiency improvement of the MPT portfolios over the naive ones. Except for the comparison between portfolio MPT-MaxS and portfolio ValWt being significant at the 10 percent level, all other three tests are insignificant. The effect of optimization is more apparent for the seven property type category, where portfolio MPT-MaxS exhibits significant efficiency improvement over both EqWt and ValWt. However, the other two MPT portfolios fail to beat their naive counterparts with statistical significance. (See Exhibit 3 on page 11.) 5

If the ex post results are not impressive, the ex ante results are even less so. Exhibit 4 shows that all ex ante tests are insignificant with very large p-values. It is worth noting, however, that although the risk reductions are not statistically significant, their magnitudes are noticeable. (See Exhibit 4 on page 12.) Conclusions Using a simple statistical test, this study examines the effectiveness of Modern Portfolio Theory when applying to within real estate diversification. Three geographic and two property type diversification schemes are tested for mean-variance efficiency improvement of MPT portfolios over naively diversified portfolios. There is evidence to support that the MPT portfolios are statistically more efficient than naive portfolios when the portfolio formulation period is identical to the holdout or testing period. Of course, this is not possible in the real world. When the formulation period differs from the testing period, there is no evidence to support that the MPT portfolios are statistically more efficient than naively diversified portfolios. These conclusions are rather suggestive than conclusive. One must keep in mind that no statistically significant benefit being found does not mean that there is no benefit at all with using MPT. For investors who welcome every bit of risk reduction regardless of how slight the chance is, MPT may still be perceived as useful. Future research could use the same test to revisit some of the previous studies that proposed various diversification strategies, and obtain additional evidence as to the ex ante performance of all those strategies. 6

References Cheng, P. and R.T. Black, Geographic Diversification and Economic Fundamentals in Apartment Markets: A Demand Perspective, Journal of Real Estate Portfolio Management, 1998, 4:2, 93-106. Geltner, D., Temporal Aggregation in Real Estate Return Indices, AREUEA Journal, 1993, 21:2, 141-166. Gibbons, M.R., S.A. Ross and J. Shanken, A Test of the Efficiency of a Given Portfolio, Econometrica, 1989, 57:5, 1121-1152. Gold, R.B., Why the Efficient Frontier for Real Estate is Fuzzy, Journal of Real Estate Portfolio Management, 1995, 1:1, 59-66. 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. Grissom, T.V., J. L. Kuhle, C. H. Walther, Diversification Works in Real Estate, Too, Journal of Portfolio Management, 1987, 13:2, 66-71. Hartzell, D., J. Hekman, and M. Miles, Diversification Categories in Investment Real Estate, AREUEA Journal, 1986, 14:2, 230-254. Lefthouse, S., International Diversification, Journal of Portfolio Management, 1997, 24:1, 53-56. Liang, Y., F.C.N. Myer, and J.R. Webb, The Bootstrap Efficient Frontier for Mixed-Asset Portfolios, Real Estate Economics, 1996, 24:2, 247-256. Malizia, E.E. and R.A. Simons, Comparing Regional Classification for Real Estate Portfolio Diversification, Journal of Real Estate Research, 1991, 6:1, 53-78. Miles, M. and T. McCue, Diversification in the Real Estate Portfolio, Journal of Financial Research, 1984, 7:1, 57-68. Mueller, G.R., Refining Economic Diversification Strategies for Real Estate Portfolios, Journal of Real Estate Research, 1993, 8:1, 55-68. Mueller, G.R., B.A. Ziering, Real Estate Portfolio Diversification Using Economic Diversification, Journal of Real Estate Research, 1992, 7:4, 375-386. Pagliari, J.L. Jr., 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. 7

Rubens, J. H., D. A. Louton and E. J. Yobaccio, Measuring the Significance of Diversification Gains, Journal of Real Estate Research, 1998, 16:1, 73-86. Sharpe, W. F., Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance,1964, 19:3, 425-442 Williams, J.E., Real Estate Portfolio Diversification and Performance of the Twenty Largest MSAs, Journal of Real Estate Portfolio Management, 1996, 2:1, 19-30. Wolverton, M.L., P. Cheng, W. G. Hardin III, Real Estate Portfolio Risk Reduction Through Intracity Diversification, Journal of Real Estate Portfolio Management, 1998, 4:1, 35-41. Ziobrowski, A.J., P. Cheng, and B.J. Ziobrowski, Using a Bootstrap to Measure Optimum Mixed-Asset Portfolio Composition: A Comment, Real Estate Economics, 1997, 25:4, 695-704. Ziering, B. and R. Hess, A Further Note on Economic versus Geographic Diversification, Real Estate Finance, 1995, 12:3, 53-60. 8

Exhibit 1 Testing the Efficiency of Select Portfolios on the Efficient Frontier: Ex Post Results in the Case of Geographic Diversification Four NCREIF Regions (3q 78-4q 98) Eight NCREIF Sub-Regions (3q 79-4q 98) Fifteen Large MSAs (3q 82-4q 98) Portfolios Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio EqWt 2.251 1.790 0.277 2.108 1.726 0.338 1.711 1.941 0.093 ValWt 2.270 1.860 0.277 2.177 1.859 0.351 1.721 1.950 0.098 MPT1 2.251 1.546 0.321 2.108 1.071 0.545 1.711 0.258 0.701 MPT2 2.270 1.506 0.342 2.177 1.039 0.629 1.721 0.274 0.698 MPT-MaxS 2.235 1.325 0.362 2.338 1.110 0.733 1.873 0.459 0.748 Risk-free rate 1.755 1.524 1.530 Efficiency Test N F-test p-value N F-test p-value N F-test p-value MPT1 vs. EqWt 82 0.480 0.750 78 1.450 0.192 66 1.647 0.095 MPT2 vs. ValWt 82 0.737 0.569 78 2.140 0.043 66 1.625 0.101 MPT-MaxS vs. EqWt 82 0.996 0.415 78 3.364 0.003 66 1.877 0.049 MPT-MaxS vs. ValWt 82 1.002 0.412 78 3.264 0.003 66 1.872 0.050 Notes: 1. EqWt = the equal-weighted portfolio; ValWt = the value-weighted portfolio; MPT1 is a mean-variance efficient portfolio with the same expected return as EqWt; MPT2 is a mean-variance efficient portfolio with the same expected return as ValWt; MPT-MaxS is the mean-variance efficient portfolio with the highest Sharpe ratio; risk-free rate = average quarterly total returns of 30-day T- bills for the corresponding period. 2. NCREIF = National Council of Real Estate Investment Fiduciaries. The four NCREIF regions are East, Midwest, South, and West. The eight NCREIF sub-regions are: Northeast, Mideast, Southeast, Southwest, East North Central, West North Central, Mountain, and Pacific. The fifteen MSAs are the top fifteen MSAs by population. 3. N = number of quarterly observations used for constructing the efficient test statistic, F. 4. Ex post means that the time period used for efficiency test is the same as the one used for constructing the efficient portfolios. 9

Exhibit 2 Testing the Efficiency of Select Portfolios on the Efficient Frontier: Ex Ante Results in the Case of Geographic Diversification Four NCREIF Regions Eight NCREIF Sub-Regions Fifteen Large MSAs Portfolios Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio EqWt 1.356 1.824 0.020 1.245 1.840 0.016 1.347 2.035 0.011 ValWt 1.346 1.890 0.014 1.224 1.977 0.004 1.346 2.018 0.011 MPT1 1.373 1.861 0.028 0.835 1.041-0.366 1.748 1.599 0.265 MPT2 1.339 1.872 0.010 0.937 1.277-0.218 1.748 1.599 0.265 MPT-MaxS 1.367 1.675 0.028 0.852 1.050-0.347 1.872 1.619 0.338 Risk-free rate 1.320 1.216 1.325 Efficiency Test N F-test p-value N F-test p-value N F-test p-value MPT1 vs. EqWt 41 0.004 1.000 39 NM NM 50 0.165 1.000 MPT2 vs. ValWt 41 NM NM 39 NM NM 50 0.165 1.000 MPT-MaxS vs. EqWt 41 0.004 1.000 39 NM NM 50 0.269 0.996 MPT-MaxS vs. ValWt 41 0.006 1.000 39 NM NM 50 0.269 0.996 Notes: 1. EqWt = the equal-weighted portfolio; ValWt = the value-weighted portfolio; MPT1 is a mean-variance efficient portfolio with the same expected return as EqWt; MPT2 is a mean-variance efficient portfolio with the same expected return as ValWt; MPT-MaxS is the mean-variance efficient portfolio with the highest Sharpe ratio; risk-free rate = average quarterly total returns of 30-day T- bills for the corresponding period. 2. NCREIF = National Council of Real Estate Investment Fiduciaries. The four NCREIF regions are East, Midwest, South, and West. The eight NCREIF sub-regions are: Northeast, Mideast, Southeast, Southwest, East North Central, West North Central, Mountain, and Pacific. The fifteen MSAs are the top fifteen MSAs by population. 3. N = number of quarterly observations used for constructing the efficiency test statistic, F. 4. NM = not meaningful due to a negative Sharpe ratio or the W-statistic being negative. 5. Ex ante means that the time period used for efficiency test differs from the one used for constructing the efficient portfolios. In the case of four NCREIF regions and eight NCREIF sub-regions, the first half of the sample are used to construct the efficient portfolios and the second half are used to construct the efficiency test statistic, F. In the case of fifteen large MSAs, the first 50 observations are used to formulate the efficient portfolios and the recent 50 observations are used for constructing the F-statistic (the two periods overlap because there are only 66 observations in total). 10

Exhibit 3 Testing the Efficiency of Select Portfolios on the Efficient Frontier: Ex Post Results in the Case of Property Type Diversification Four Property Types (4q'84-4q'98) Seven Property Types (4q'84-4q'98) Portfolios Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio EqWt 1.738 1.608 0.215 1.528 1.591 0.085 ValWt 1.610 1.727 0.126 1.542 1.651 0.090 MPT1 1.738 1.207 0.286 1.528 0.501 0.270 MPT2 1.610 1.136 0.191 1.542 0.506 0.294 MPT-MaxS 1.828 1.037 0.420 1.693 0.545 0.550 Risk-free rate 1.393 1.393 Efficiency Test N F-test p-value N F-test p-value MPT1 vs. EqWt 57 0.459 0.765 57 0.475 0.848 MPT2 vs. ValWt 57 0.276 0.892 57 0.565 0.781 MPT-MaxS vs. EqWt 57 1.677 0.169 57 2.130 0.058 MPT-MaxS vs. ValWt 57 2.128 0.090 57 2.122 0.059 Notes: 1. EqWt = the equal-weighted portfolio; ValWt = the value-weighted portfolio; MPT1 is a mean-variance efficient portfolio with the same expected return as EqWt; MPT2 is a meanvariance efficient portfolio with the same expected return as ValWt; MPT-MaxS is the meanvariance efficient portfolio with the highest Sharpe rati; risk-free rate = average quarterly total returns of 30-day T-bills for the corresponding period. 2. NCREIF = National Council of Real Estate Investment Fiduciaries. The four property types are apartment, industrial, office, and retail. The seven property types are: apartment, warehouse, non-warehouse industrial, CBD office, suburban office, mall retail, and non-mall retail. 3. N = number of quarterly observations used for constructing the efficient test statistic, F. 4. Ex post means that the time period used for efficiency test is the same as the one used for constructing the efficient portfolios. 11

Exhibit 4 Testing the Efficiency of Select Portfolios on the Efficient Frontier: Ex Ante Results in the Case of Property Type Diversification Four Property Types Seven Property Types Portfolios Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio EqWt 1.474 1.780 0.099 1.313 1.792 0.009 ValWt 1.311 1.661 0.008 1.280 1.831-0.009 MPT1 1.438 1.284 0.109 1.397 1.141 0.087 MPT2 1.401 1.275 0.081 1.340 1.039 0.041 MPT-MaxS 1.432 1.113 0.121 1.413 0.978 0.119 Risk-free rate 1.298 1.298 Efficiency Test N F-test p-value N F-test p-value MPT1 vs. EqWt 40 0.018 0.999 40 0.036 1.000 MPT2 vs. ValWt 40 0.060 0.993 40 NM NM MPT-MaxS vs. EqWt 40 0.043 0.996 40 0.067 0.999 MPT-MaxS vs. ValWt 40 0.134 0.969 40 NM NM Notes: 1. EqWt = the equal-weighted portfolio; ValWt = the value-weighted portfolio; MPT1 is a mean-variance efficient portfolio with the same expected return as EqWt; MPT2 is a meanvariance efficient portfolio with the same expected return as ValWt; MPT-MaxS is the meanvariance efficient portfolio with the highest Sharpe ratio; risk-free rate = average quarterly total returns of 30-day T-bills for the corresponding period. 2. NCREIF = National Council of Real Estate Investment Fiduciaries. The four property types are apartment, industrial, office, and retail. The seven property types are: apartment, warehouse, non-warehouse industrial, CBD office, suburban office, mall retail, and non-mall retail. 3. N = number of quarterly observations used for constructing the efficient test statistic, F. 4. NM = not meaningful due to a negative Sharpe ratio. 5. Ex ante means that the time period used for efficiency test differs from the one used for constructing the efficient portfolios. The first 40 observations are used to formulate the efficient portfolios and the recent 40 observations are used for constructing the F-statistic (the two periods overlap because there are only 57 observations in total). 12