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
[email protected]
and
Youguo Liang, Ph.D.
Principal
Prudential Real Estate Investors
8 Campus Drive
Parsippany, NJ 07054
(973) 683-1765
(973) 683-1788 Fax
[email protected]
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:
S=
k p − k RF
(1)
σp
Where kp is the portfolio return; kRF is the risk-free rate; and σp is the standard deviation of kp.
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, pi and pj (i, j = 1, 2, 3, …)
can be constructed to exhibit Sharp ratios of Si and Sj, respectively. If pi is an efficient portfolio
while pj’s efficiency is unknown, then it must be true that S i ≥ S j . To test the statistical
significance of their difference, the proper null hypothesis can be stated as:
H0 : S i = Sj
(2)
This hypothesis can be tested using the following test statistic:
2
 1+ S 2 
i 
−1
W =
(3)
 1+ S 2 
j 

Note that W is a non-negative number because S i ≥ 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
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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
Portfolios
EqWt
ValWt
MPT1
MPT2
MPT-MaxS
Risk-free rate
Efficiency Test
MPT1 vs. EqWt
MPT2 vs. ValWt
MPT-MaxS vs. EqWt
MPT-MaxS vs. ValWt
Four NCREIF Regions
(3q’78-4q’98)
Mean Std Dev Sharpe Ratio
2.251 1.790
0.277
2.270 1.860
0.277
2.251 1.546
0.321
2.270 1.506
0.342
2.235 1.325
0.362
1.755
N
F-test
p-value
82
0.480
0.750
82
0.737
0.569
82
0.996
0.415
82
1.002
0.412
Eight NCREIF Sub-Regions
(3q’79-4q’98)
Mean Std Dev Sharpe Ratio
2.108 1.726
0.338
2.177 1.859
0.351
2.108 1.071
0.545
2.177 1.039
0.629
2.338 1.110
0.733
1.524
N
F-test
p-value
78
1.450
0.192
78
2.140
0.043
78
3.364
0.003
78
3.264
0.003
Fifteen Large MSAs
(3q’82-4q’98)
Mean Std Dev Sharpe Ratio
1.711 1.941
0.093
1.721 1.950
0.098
1.711 0.258
0.701
1.721 0.274
0.698
1.873 0.459
0.748
1.530
N
F-test
p-value
66
1.647
0.095
66
1.625
0.101
66
1.877
0.049
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 Tbills 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
Portfolios
Four NCREIF Regions
Eight NCREIF Sub-Regions
Fifteen Large MSAs
Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio
EqWt
ValWt
MPT1
MPT2
MPT-MaxS
Risk-free rate
Efficiency Test
1.356
1.346
1.373
1.339
1.367
1.320
N
1.824
1.890
1.861
1.872
1.675
0.020
0.014
0.028
0.010
0.028
F-test
41
41
41
41
0.004
NM
0.004
0.006
MPT1 vs. EqWt
MPT2 vs. ValWt
MPT-MaxS vs. EqWt
MPT-MaxS vs. ValWt
1.840
1.977
1.041
1.277
1.050
0.016
0.004
-0.366
-0.218
-0.347
p-value
1.245
1.224
0.835
0.937
0.852
1.216
N
2.035
2.018
1.599
1.599
1.619
0.011
0.011
0.265
0.265
0.338
p-value
1.347
1.346
1.748
1.748
1.872
1.325
N
F-test
F-test
p-value
1.000
NM
1.000
1.000
39
39
39
39
NM
NM
NM
NM
NM
NM
NM
NM
50
50
50
50
0.165
0.165
0.269
0.269
1.000
1.000
0.996
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 Tbills 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
Seven Property Types
(4q'84 - 4q'98)
(4q'84 - 4q'98)
Portfolios
Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio
EqWt
ValWt
MPT1
MPT2
MPT-MaxS
Risk-free rate
1.738
1.610
1.738
1.610
1.828
1.393
1.608
1.727
1.207
1.136
1.037
0.215
0.126
0.286
0.191
0.420
1.528
1.542
1.528
1.542
1.693
1.393
1.591
1.651
0.501
0.506
0.545
0.085
0.090
0.270
0.294
0.550
N
57
57
57
57
F-test
0.459
0.276
1.677
2.128
p-value
0.765
0.892
0.169
0.090
N
57
57
57
57
F-test
0.475
0.565
2.130
2.122
p-value
0.848
0.781
0.058
0.059
Efficiency Test
MPT1 vs. EqWt
MPT2 vs. ValWt
MPT-MaxS vs. EqWt
MPT-MaxS vs. ValWt
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
Portfolios
Four Property Types
Seven Property Types
Mean Std Dev Sharpe Ratio Mean Std Dev Sharpe Ratio
EqWt
ValWt
MPT1
MPT2
MPT-MaxS
Risk-free rate
Efficiency Test
1.474
1.311
1.438
1.401
1.432
1.298
N
1.780
1.661
1.284
1.275
1.113
0.099
0.008
0.109
0.081
0.121
F-test
40
40
40
40
0.018
0.060
0.043
0.134
MPT1 vs. EqWt
MPT2 vs. ValWt
MPT-MaxS vs. EqWt
MPT-MaxS vs. ValWt
1.792
1.831
1.141
1.039
0.978
0.009
-0.009
0.087
0.041
0.119
p-value
1.313
1.280
1.397
1.340
1.413
1.298
N
F-test
p-value
0.999
0.993
0.996
0.969
40
40
40
40
0.036
NM
0.067
NM
1.000
NM
0.999
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).
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