William N. Goetzmann & Susan M. Wachter

Our analysis suggests that the use of raw historical data for the construction of international portfolios is likely to be dangerously misleading, due to uncertainty regarding inputs to the mean-variance optimzation. The aggregation procedures we use reduce this uncertainty and provide guidelines for prudent cross-border diversification.

All of these trends are true for real estate as well. A decade ago it was virtually impossible to obtain performance information about international real estate investments. Today, such data are available for established as well as emerging real estate markets. In the 1980's and 1990's international real estate investors actively pursued a global development strategy, and investment portfolio managers sought high returns and diversification in cross-border real estate deals. In a fashion similar to the world's stock markets, the highest returns to real estate investment over the recent period have been in emerging real estate markets, most notably the Far East. While international investment in real estate has yielded high returns for some investors, it engenders significant risk. For example, investors who sought the safety of international real estate markets in the 1990's experienced shocking declines in both U.K. and Japan and lackluster performance in most continental European markets as well. In theory, international diversification lowers investor risk. In practice, this diversification has some serious pitfalls.

In this paper, we focus on the problem of international real estate diversification by examining the cross-sectional behavior of the international office market. We find clear evidence of a global market "crash" in 1992, which was preceded in most cases by declining property values since the end of the 1980's. We explore the role of international real estate investments in the portfolio. We address some of the pitfalls of international diversification through the use of a statistical method, "cluster analysis" to identify meaningful groups of office markets that moved together over the last eight years. The clustering method identifies three major "families" of commercial property markets, and shows them to be relatively robust groups. These groups are then used in a mean-variance optimization framework to estimate the potential reduction in risk, if any, that a U.S. investor might have achieved through international diversification.

Our analysis suggests that the use of raw historical data for the construction of international portfolios is likely to be dangerously misleading, due to uncertainty regarding inputs to the mean-variance optimization. The aggregation procedures we use reduce this uncertainty, and provide guidelines for prudent cross-border diversification. One approach to the data problem taken by Liu and Mei (1994) is to gather higher quality data. They use returns to investable property trusts in a number of other countries to examine the benefits of diversification. Where proprty trusts exist and data are available, this is obviously an attractive alternative to the use of appraisal data. However many cross-border real estate investment decisions must be made on the basis of differnt kinds of data.

**The ICPA Data**

**
**For ten years, International Commercial Property Associates has maintained a
database of commercial rents and yields for a number of countries around the world.
The rent data is gathered by Hillier-Parker in the U.K. and affiliates in Southeast Asia
and Australia and Landauer in the U.S. Other real estate research firms report
information to ICPA for rents in Europe, Canada and Scandinavia. These data, despite
certain limitations, provide a fascinating glimpse of the world real estate crash over the
late 1980's and late 1990's.

In our analysis of the International Commercial Property Associates [ICPA] data,
we focus on world office markets. These are defined by ICPA as first class, or class
A office spaces in excess of 5,000 square feet in prime locations in principal cities.
These are total returns, i.e. income and appreciation, however the appreciation is *not*
based upon appraisals, but upon changes in capitalized rents. The rents are asking
rents. This may make it difficult to exactly identify crashes in the office market, since
effective rents typically lead asking rents in declining markets. In addition, it is difficult
to determine from the ICPA datasources the extent to which the rents could be
considered "net." Without cleare evidence on this, we make the assumption that they
are not. Thus, while effective rents are desirable, they are not easily obtained. Yields
are implicitly a function of appraised property values and net operating income in each
market. Thus, the estimated total return T_{i,t} for country i at time t is:

Summary statistics about these return series are instructive. The equally-weighted global average returns to investment in office markets was 13.5% in dollar-denominated terms over the period. This return compares favorably with the total return to investment in the S&P 500 over the 1986 through 1993 period (14.1%), however the risk of global office market investment was considerably greater. Whether calculated in dollar terms, or in terms of local currency, the volatility of most office markets around the world far exceeded that of the U.S. The average standard deviation was higher than 25% per year. The volatility of dollar-denominated returns in certain markets was extraordinary over the period. Hong Kong, Malaysia, Portugal, Spain, Switzerland and Taiwan all had standard deviations exceeding 35% per year. The volatility is largely due to changes in the capitalized rents. This method of estimating values implicitly assumes that new asking rents are good proxies for the expected future rents. Sometimes this is a reasonable assumption, and sometimes it is not. Property managers are often able to diversify their exposure to shocks in rents by diversification across lease maturities. The simplicity of the capitalized rent series does not account for this. Despite the limitations of the dataset, it provides a useful basis for comparison of the performance of office markets around the world. While estimates risk and return are approximate, they clearly suggest that overseas markets have been more volatile than those in the U.S.

**The Global Real Estate Crash**

Figure 1 shows the cumulative wealth indices for all of the countries. Note that most markets were flat or down over the second half of the sample period. In particular, we find that the 1992 downturn in rents and property values in the United States was a world-wide phenomenon, experienced by twenty-one of the twenty-four markets studied. In addition, the drop in office building values was as severe as it was broad. In several countries, property values decreased by more than thirty percent. Figure 2 shows the wealth indices converted to U.S. dollar returns, and the basic trends are unaffected. The message of this cross-sectional analysis of the ICPA data is clear. The recent U.S. crash was a global crash. There were few safe havens in the early 1990's for investors in global office markets. Figures 3 and 4 plot the average annual return for each index in local currency and dollar terms. From 1986 through 1990, a negative return for any country was rare. 1991 and 1993 were mixed years, but in 1992, the year of the global real estate crash, only three of the twenty-four countries had positive returns.

**Why Did it Happen?**

From the perspective of the United States market, it is tempting to explore domestic reasons for the real estate downturn. Potential causes include the 1986 tax law change, shifting bank and insurance regulation and evolving U.S. pension policy. Fergus and Goodman (1994) among others suggest that the magnitude of the U.S. crisis was due to the rapid constriction in bank credit. None of these is sufficient to explain such a widespread global downturn, however. While the U.S. is one of the world's most important economies, direct U.S. real estate investment across the globe is unlikely to have caused the crash. Current research suggests a number of contributing international factors. Renaud (1994), for instance, attributes the crash to a massive surge of Japanese international investment capital in the 1980's. He traces a cascade of international property investment that may have led to price disequilibrium in office markets. In addition, Renaud (1994) and Hendershott and Kane (1992) have linked the 1980's building frenzy to easy credit and lax regulation in the U.S., Japan and elsewhere.

While Japanese capital increases and global institutional changes certainly provided the context for the global crash, there is some evidence for an underlying economic basis for the crisis. Figure 5 show the percentage change in gross domestic product [GDP] for each of the countries in the database over the period. Note that a global contraction in production began in 1988 and 1989. Virtually all GDP growth rates in the sample were lower in 1991 and 1992 than in the 1987-1990 period. While insufficient time-series data exist to test for causality between the two variables, it is doubtful that the real estate crash precipitated the global contraction in production, but the reverse is certainly plausible. In fact, it is likely that the demand for office real estate values respond to fluctuations in demand from the production sector. The story told by these figures is a simple one. Global office markets move up and down with global business cycles. Such movements may have occurred regardless of credit expansions or constrictions, bank regulatory changes, or the Japanese financial crisis. None-the-less, each of these could have been exacerbating factors, since none is properly exogenous.

It is appealing to look for fluctuations in demand for office space as a possible explanation for the office market slump, if only for purposes of parsimony. The coincidence of the global recession and the real estate crash alone is insufficient to explain why the decrease in demand for office space was unanticipated. Figure 5 suggests that trends in GDP growth might have been forecastable by developers, bankers and mortgage lenders, however this conjecture is 20/20 hindsight. A test of the rationality of developers and lenders during the 1980's awaits future analysis that takes into account the forecasts of GDP at that time, as well as anticipated future office space.

Regardless of the root causes for the global crash, it is valuable to know the degree to which exposure to a common set of economic variables drives returns in the world's office markets. One empirical approach to this question is to extract latent variables from historical data. To this end, we calculated the principal components from the matrix of total returns. Since we have more countries than time periods, the maximum number of orthogonal principal components is limited to eight. The variance explained by these factors (in order of importance) is: .44, .21, .13, .09, .06, .04, .015, 0, 0. In other words, 44% of the variation in global office market returns is captured by a single factor. Whether this factor is global inflation, global production or some unidentified source of risk is an open empirical question. An examination of the weights comprising the first principal component suggest that it is close to being and equal-weighted index formed from the individual series. Thus there appears to be at least one, strong, common factor that drove international office markets in the recent past. While diversification might help to reduce some of the extraordinary non-systematic risk implied by the high volatility of many of the markets, there is undoubtedly a high lower bound on the benefits of spreading risks internationally. In the next section, we address some major issues confronting the creation of a global real estate portfolio, and provide a method for reducing errors associated with portfolio creation.

**International Diversification: Applying Mean-Variance Analysis**

Asset allocation using mean-variance analysis has become increasingly popular
among institutional investors over the past decade. In fact, it has been a major impetus
towards global investment because it typically indicates that international diversification
reduces risk. In theory, mean-variance analysis identifies a set of portfolios that
maximize the investor's expected return for each level of risk. This set of portfolios is
called the *efficient frontier* since it represents all of the undominated portfolios in risk and
return space. In practice, the identification of the efficient frontier is hampered by
problems of estimation. An exact identification of the efficient frontier is only possible
when the future means, standard deviations and correlations of each asset in the
choice set are known with certainty. Typically, these inputs are estimated from
historical time-series data. Thus, errors are large when only a short historical period is
available for analysis. The effect of these errors on the investment decision can be
dramatic. Assets with high means tend to dominate the portfolios on the efficient
frontier. This is the case even when the high means are due to estimation error. For
example, when the number of assets included in the choice set grows large, the chance
that one will have an unusually high mean *due entirely to chance* also increases. Figure
6 is an efficient frontier based upon historical means and standard deviations for
twenty-five countries, taken from annual data, 1987 through 1993. It is created using
The Ibbotson Associates EnCorr Optimizer, which allows positivity constraints on asset
holdings. This constraint accounts for the flattened shape of the frontier. Due to the
great number of assets, and low correlations among several of them, it appears that an
investor could have achieved an extraordinarily low level of risk over the period. The
minimum variance portfolio has only 6% annual standard deviation. Oddly enough, one
country is actually on the frontier, suggesting that, for one level of risk, there is no need
to diversify at all -- Portugal is the only investment required! Indeed, for most levels of
investor risk aversion, Portugal dominates the frontier. The minimum variance portfolio
is no more reasonable. It is comprised of 54% Norwegian office space, 20% Hong
Kong, 14 % Malaysia, 10% Singapore and the rest in Swiss properties. In other words,
the model as specified suggests that even conservative investors should place most
of their money in a small, emerging real estate markets, based solely upon eight years
of unusually high performance. The model implicitly assumes that the Portuguese
market will continue to grow at its historical rate in the future. When inputs are
uncertain, it typically sends most investors off chasing past winners, rather than
identifying markets with reasonable prospects for future growth. Not only does a naive
application of mean-variance analysis steer investors towards past winners rather than
future winners, in some cases it leads them away from prudent diversification across
several markets.

**Reducing Mean-Variance Input Error**

The misleading investment advice given by a naive application of mean-variance
optimization in the preceding section is due primarily to the poor quality of statistical
inputs to the model. One approach to reducing estimation error is to optimize over a
few sub-indices for which means, standard deviations and correlations can be well
estimated. Elton and Gruber (1971) suggest that a useful first step towards mean-variance
optimization is to aggregate individual assets into asset classes. If assets
within classes can be expected to move together, and have similar risk and return
characteristics, this will greatly improve the results of the mean-variance analysis. For
instance, consider a simple case in which there are three asset classes. These classes
can be treated as separate populations, and the individual asset return histories can
be regarded as samples from the population. Each asset within a class shares the
same mean, standard deviation and correlation with respect to the other two classes.
Suppose one wished to estimate statistics about the asset class, rather than the
individual assets comprising it. In this case, the standard error of the mean estimate
will decrease in the square root of the number of included assets. In practice, since
assets are *not* drawn from identical populations, there is a tradeoff between errors due
to assignment to the wrong group versus errors due to estimation uncertainty.

Because the time-series' data from IPCA is so short, and the appreciation is estimated via capitalization of rents, rather than appraisals or transactions, the estimation error in the means is likely to be dramatic. For this reason, we selected a relatively high level of aggregation i.e. three "families." To estimate these families, we apply a commonly-used clustering algorithm, k-means, to the time-series of returns. The procedure is described in more detail in the next section.

**K-Means Cluster Analysis**

A useful first step to forming a diversified portfolio is to identify a few groups of assets that behave differently across groups and similarly within groups. There are a number of different approaches to grouping assets. Following Elton and Gruber (1971) and our own recent work, Abraham, Goetzmann and Wachter (1994) and Goetzmann and Wachter (1994) on residential housing indices and commercial rent indices, we apply the k-means clustering algorithm to country returns in order to identify meaningful clusters of countries which moved together in the past, and which are likely to move together in the future.

The procedure, developed by Hartigan (1975), assumes a fixed number of groups within the sample of observations and chooses an allocation of observations to groups to minimize within-group sums of squared deviations from the mean. For the global office markets, we assume that there are J = 1...k distinct groups of office markets in the world, where a group is defined as more than one of the j = 1...N office markets whose time-series of returns resemble each others.' The clustering procedure allocates market j to group J in order to minimize the within-group sum of squared deviations from the group average. That is, we assume that the returns for any member of group J can be expressed as:

And the algorithm seeks assignments of countries to groups in order to minimize the sum of squared errors:

To find a global minimum requires calculation of the SSE for every possible combination of countries to groups -- a huge problem. Local optima may be found by beginning with a set of means in T space, corresponding to a set of k existing observations, and then seeking switches that reduce the SSE.

One problem is that the k-means procedure requires pre-specification of the number of groups. While it is natural to think of a North American group, a European group and an Asian Pacific group, perhaps the global markets are more naturally represented by a simple division into two markets, or a more complex division into five or ten independent clusters. To address this issue empirically, we use a bootstrapping procedure .

The bootstrap can be used to estimate confidence bands about the clusters, and
also to estimate how many clusters appear in the data. Let R_{i,t} represent the percent
return for country i at time t, where t ranges from 1986 through 1993, and i includes
twenty-four countries. R_{.,t} is the vector of t returns for all countries in the sample at time
t. For the bootstrap, we create a pseudo-history of returns by randomizing with
replacement over the dimension of time, to generate t*, a vector of length eight, for
which dates between 1986 through 1993 have been drawn with replacement with equal
probability. We then create a pseudo-history R* by selecting the cross-section of returns
that correspond to the bootstrapped dates: R_{.,t*} corresponding to each new date in the
pseudo-history. That is:

and t* is a random element of {1986...1993}

In order to determine whether the frequency of association is high or low, and in order to test whether it is significantly different from the null hypothesis of no association, we need benchmark frequencies. These are calculated by bootstrapping the data under the null hypothesis. To reproduce the distribution of the pairwise associations under the null hypothesis that there are no relationships among the countries and there are no differences in means, we created a pseudo-history by drawing with replacement from the entire sample, mixing across countries as well as time. That is, we do not preserve differences across rows or columns.

where t* is a random element of {1...T}

i* is a random element of {1...N}

**How Many Groups?**

To estimate the number of clusters in the data, we calculate the mean squared error difference between the cells in the association frequency table, and the median association frequency under the null. The number of clusters for which this value is maximized provides the greatest level of rejection of the null. We found that the mean squared deviation from the null hypothesis for each pre-specified cluster number grows quickly after two groups, and then drops after seven (see Figure 7). Three groups is near the maximum. Four through seven groups do almost as good a job at rejection of the null as does three, however. Figure 8 shows the geographical relationships among the clusters. Cluster one contains Belgium, Canada, Denmark, France, Germany, Holland, Ireland, Italy, Malaysia, Portugal, Singapore and the U.S.. Cluster two contains Australia, Finland, Spain, Sweden, Switzerland and the U.K. Cluster three contains Hong Kong and Taiwan. Figure 9 shows the clustering when seven groups are specified. With greater freedom to split countries into more groups, all four of the southeast Asian markets separate into groups. European countries separate from the U.S. and Canada and the group containing U.K. and Australia remains nearly unchanged.

**Tests of Robustness**

How robust are these clusters? Could they simply be a result of chance similarities among a set of independent random variables? In Table 2, we show the result of robustness studies of the clusters. The table provides a frequency of association table for all countries when three groups are specified, as well as the 5%, 50% and 95% frequency quantiles from the bootstrap under the null. Without these intervals, one could not tell whether an association frequency of .4 was suggestive of association or of dissociation. Given three clusters, an association frequency between 37% and 48% is suggestive, but not convincing evidence of association. The table suggests that ambiguous relationships between countries are relatively infrequent. Canada has the most association frequencies falling between the 5% and 95% thresholds, with seven. For instance, we cannot reject the hypothesis that Canada clusters with U.K. as well as Sweden, Belgium, France, Holland, Denmark, Malaysia and Norway. On the other hand, Australia, Sweden, Hong Kong and Taiwan are all relatively unambiguous in their relationships with other counties. In certain cases, these ambiguities are telling. For instance, even though they typically cluster in different groups, we cannot reject the hypothesis that Singapore and Australia belong in the same group, or that Singapore and Hong Kong belong in the same group. In other words, we cannot reject the hypothesis that the southeast Asian cluster extends beyond Hong Kong and Taiwan to include Singapore, and perhaps even Australia.

Figure 10 plots the association frequencies for four selected countries, U.S., U.K., Hong Kong and Taiwan. A bar over .48 high indicates a significant association between cities, and a bar less than .28 high indicates a significant dissociation. Notice that U.S. is strongly associated with the greatest number of other countries including much of Europe, while the U.K. has its own distinct group, including Australia. Oddly enough, although k-means groups Singapore with the U.S. using the available data, the association frequencies indicate that this is a low probability event. Singapore does not typically group with the U.S., the U.K. or with China. Its only constant partner is Portugal.

In sum, the cluster analysis indicates a few features of the global office market
that may be useful for investment decisions. First, the U.S. and the U.K. separate into
different clusters at both high and low levels of aggregation. Most countries on the
continent separate from the U.K., and from the U.S. when seven groups are specified.
While we cannot reject the hypothesis that the southeast Asian countries (excluding
Australia) group together, it appears that their differences become important as more
groups are specified. These clusters provide some general guidelines for international
real estate diversification . First, it is clear that North America is *not* a separate group.
At high levels of aggregation, it clusters with both European and Asian markets. In fact,
it appears to be the dominant country in the largest cluster when three groups are
specified. This is not surprising considering the role of the U.S. in the global economy.
Even at lower levels of aggregation it still groups with Scandinavian and European
countries. Given the relative comfort with which U.S. real estate investors have with
investing in other English-speaking countries, it is encouraging to find that the U.S.
clusters *apart* from U.K. and Australia. The bad news is that neither country served as
a safe haven during the recent crash. Indeed the performance of portfolios based
upon the three major clusters indicates that cluster 2, containing UK peaked in 1988.

**Optimization Over Groups**

Our asset classes conform to equal-weighted portfolios across each country in the k-means cluster. The primary effect of using this reduced set of data is that correlations across the three portfolios are relatively high: from 39% to 62%. Since k-means separates observations in the space of returns, it chooses groups such that they "move differently" from each other. Thus we might expect low correlations between clusters. But once asset classes are created and estimation error is reduced, international correlations are relatively high. This is not surprising, given the pervasiveness of the 1992 crash. However, it suggests that the benefits to international real estate diversification may be overstated.

Figure 11 shows the indices for the three sub-indices and Figure 12 shows the efficient frontier derived from the three sub-indices. The composition of the minimum variance portfolio is now 12% in Portfolio one, 24 % in Portfolio two and 64 % in Portfolio three. Its volatility (20%) is much higher than before, since spuriously low correlations are absent, and the investor is constrained from holding an all-US portfolio.. As investor risk tolerance increases, the proportion invested in Portfolio two increases. The Scandinavian countries drop out, while U.S. and European holdings decline. Properties in the Asian countries replace them. The recommendations derived from optimizing over sub-indices tell a different story from the previous analysis that included all countries.

**
**Ex Post**Selection Bias, and Regression Towards the Mean**

Although cluster analysis may reduce input uncertainty by aggregating like
assets into classes, it cannot completely eliminate biases inherent in using historical
data for mean-variance inputs. Since the procedure clusters in the space of historical
returns, it will tend to group assets with high past returns together, even when these
past returns are due to chance. While we may see *ex post* that the Portuguese market
was a big winner, and the Asian markets formed a cluster that provided valuable
diversification, it may have been difficult, if not impossible to discover this *ex ante*.
These markets tend to group together, precisely because they are outliers. Thus, as a
group, they are likely to regress toward the mean in future years. This is
equally true for the big losers. Portfolio two is nearly unrepresented among efficient
portfolios, due to its low return. Thus, the optimizer makes a useful recommendation
that an investor avoid what appears to be a dominated asset. In other words, the
optimizer tells us to stay away from U.K., Australian, Spanish and Scandinavian office
markets. This is not surprising, given the magnitude of the 1992 crash in some of
these countries, but it ignores the possibility that these markets have "bottomed out."
It is straightforward to explore the implications of such a forecast. If we wish to
assume that the Asian markets have peaked and the U.K. market has bottomed out,
we simply decrease the mean for group three and increase the mean for group two.
In other words, we "shrink" the mean estimates toward the average. This shrinkage
presumes some future regression towards the mean.

While such shrinkage may appear *ad hoc*, there is a broad statistical literature
offering guidelines for the magnitude of shrinkage desired. Most of it is based upon a
simple intuition. Suppose for a moment that one had no economic information at all to
distinguish among the different real estate markets. This state of ignorance
corresponds to a "diffuse prior" with regard to the risk, return and correlation structure
of the asset classes. In this circumstance, all means and standard deviations and off-diagonal elements of the correlation matrix would be set equal to each other. When
these diffuse prior inputs are used in the mean-variance optimizer, they result in a
minimum-variance portfolio that is equally weighted across all asset classes. This
diffuse prior portfolio is useful, since the level of precision with which the optimization
inputs are measured lays somewhere between a diffuse prior and the point estimates
based upon historical data. Given the short history of returns collected by ICPA, it
seems logical to strongly shrink the inputs towards each other, even for high levels of
investor risk tolerance. The shrinkage thus builds in the anticipated regression towards
the mean that is a likely result of clustering on historical returns.

**Conclusion**

The analysis of the ICPA data reveals some interesting features of the world real estate returns, and provides some guidance to global investors in the office market. The ICPA data clearly indicates that the U.S. real estate crash was part of a global trend. This is useful information for policy makers, but disappointing for investors seeking international diversification.

While the plethora of new return data about international markets is tempting to institutional investors who seek to improve portfolio risk-return profiles, we recommend using the data judiciously. Simply using historical statistical inputs steers investors towards a risky strategy of chasing past winners. Cluster analysis represents an important intermediate stage of aggregation. Our application of k-means results in a set of three groups that capture differences among European, Scandinavian, Iberian and Asian markets. Statistics from these sub-groups result in a mean-variance analysis that appear to be well diversified across global real estate markets for most portions of the efficient frontier. More conservative portfolios are tilted towards Continental Europe and the U.S., while more aggressive portfolios are tilted towards Asia and the Iberian countries. We recommend a further precautionary step for institutional investors, that is, shrinking portfolio weights towards the "diffuse prior" portfolio.

Mean-variance optimization is a useful tool for evaluating the effect of cross-sectional asset relationships upon portfolio risk. But when long-term equilibrium expected returns cannot be reliably estimated from the data, mean variance optimization can be misleading. This is precisely the circumstance confronting international real estate investors. There is just enough information available to run an optimizer and to identify an efficient frontier, however there are serious pitfalls inherent in equating differences among historical returns with differences in expected returns.

**Notes**

1. The data for analysis was generously provided by International Commercial Property Associates. We thank Roger Ibbotson, Patric Hendershott, Bob Edelstein, the participants in the AREUEA 1995 conference and the Berkeley Real Estate Workshop for helpful suggestions. All errors are the sole responsibility of the authors. We thank Ibbotson Associates for making the EnCorr software available for our use.

2. See, for example, Harvey (1994).

3. See, for example, Jorion (1989).

4.These include Deutsche Immobilien Partner in Germany, Huoneistomarkkinoiti in Finland, Jan Henning Hansen Eindomstaksering in Norway and the Regional Group of Companies in Canada.

5.Countries are: Australia, Belgium, Canada, Denmark, Finland, France, Germany, Holland, Hong Kong, Indonesia, Ireland, Italy, Japan, Malaysia, Norway, Portugal, Singapore, Spain, Sweden, Switzerland, Taiwan, Thailand, UK and USA. Japan, Indonesia and Thailand lack data for the early years, and thus are not included in the cluster analysis.

6. We provide summary statistics only for countries with data beginning in 1986.

7. See Capozza (1994), for a discussion of domestic explanations for the U.S. real estate crisis.

8. We found little difference in results depending upon whether factors were extracted from local currency returns or dollar-based returns.

9.Although the smalll-sample distribution of the variance explained by the first principal component is difficult to calculate analytically, we bootstrapped it by repeatedly scrambling the time dimension for each series, so that the years no longer lined up, and then extracted the principal components. For i.i.d. returns, this simulates a null hypothesis of no common factor structure. We rejected the null hypothesis for the first principal component at the .002 probability level, based upon 1000 bootstraps. The median bootstrap value was 32%, however, suggesting that random alignments of the series' could produce seemingly important principal components. None of the other principal components were unusual, however. This would seem to provide support for a single factor structure to global office market returns, however Brown (1989) shows that a multiple factor return generating process can give rise to a variance-covariance matrix with a single, large principal component such as the one identified here.

10.See Brown and Chen (1983) and Jorion (1986).

11.See Best and Grauer (1989) and Broadie (1993) for results on the effect of estimation error in the means.

12. Since there are more assets than time periods used to estimate the correlation matrix for the program, without a positivity constraint it would be possible to identify a portfolio with zero variance.

13. Panton, Lessia and Joy (1976) apply the algorithm to the world's equity markets.

14. The rejection function does *not* simply plot variance explained by grouping. If it did,
then the explained variance would be maximized by the number of groups equal to the
number of observations, irrespective of structure in the data.

15.These frequencies are 28%, 37 % and 48% respectively.

16. Note that the diffuse prior portfolio is still comprised of sub-indices, since each country is a portfolio of the office buildings within it. A proper specification would weight each office building equally, resulting in a "space-weighted" portfolio -- something akin to a capital-weighted security portfolio.

Local Currency Dollar DenominatedArith. Stand. Arith. Stand.

Mean Dev. Mean Dev.

Australia 9.35 31.33 7.63 21.05

Belgium 19.36 23.90 13.25 14.27

Canada 11.29 17.34 10.00 12.69

Denmark 0.82 15.62 -3.13 10.04

Finland 9.08 34.31 7.38 24.77

France 17.07 30.55 11.75 20.62

Germany 20.43 19.95 14.63 13.80

Holland 20.01 21.61 13.88 14.18

HongKong 35.25 38.89 35.13 39.19

Ireland 12.86 17.59 10.25 9.07

Italy 21.51 41.91 18.25 30.17

Malaysia 11.13 37.10 12.00 35.62

Norway 1.68 16.41 1.13 13.25

Singapore 27.70 44.52 36.75 23.24

Portugal 36.75 23.24 22.50 38.93

Spain 23.93 51.64 18.25 39.07

Sweden 07.70 44.11 05.75 33.28

Switzerl. 10.95 61.81 3.50 47.08

Taiwan 39.47 42.96 32.25 41.80

UK 07.96 33.31 5.75 24.84

USA 07.75 9.10 7.75 09.10

Averages 16.76 31.29 13.55 24.57

Each table entry represents the frequency with which two countries appeared in the same cluster, given 1,000 bootstraps. To evaluate the significance of association and dissociation between two countries, the frequency may be compared to the distribution of association frequencies expected under the null hypothesis of no associations. These are : 5% quantile = .28, 50% quantile = .37, 95% quantile = .48

AUS SPA SWE SWI UK BEL CAN FIN FRA GER HOL IRE ITA HK

AUS 1.00 0.36 0.45 0.39 0.55 0.07 0.45 0.59 0.13 0.08 0.09 0.19 0.12 0.27

SPA 0.36 1.00 0.78 0.57 0.47 0.15 0.13 0.37 0.33 0.08 0.10 0.06 0.35 0.04

SWE 0.45 0.78 1.00 0.76 0.69 0.08 0.22 0.54 0.24 0.04 0.01 0.06 0.26 0.02

SWI 0.39 0.57 0.76 1.00 0.75 0.03 0.14 0.42 0.12 0.01 0.00 0.01 0.14 0.01

UK 0.55 0.47 0.69 0.75 1.00 0.12 0.31 0.66 0.25 0.10 0.10 0.12 0.26 0.04

BEL 0.07 0.15 0.08 0.03 0.12 1.00 0.39 0.28 0.80 0.85 0.88 0.67 0.76 0.05

CAN 0.45 0.13 0.22 0.14 0.31 0.39 1.00 0.57 0.34 0.46 0.41 0.66 0.29 0.09

FIN 0.59 0.37 0.54 0.42 0.66 0.28 0.57 1.00 0.42 0.26 0.28 0.37 0.40 0.10

FRA 0.13 0.33 0.24 0.12 0.25 0.80 0.34 0.42 1.00 0.66 0.73 0.50 0.93 0.03

GER 0.08 0.08 0.04 0.01 0.10 0.85 0.46 0.26 0.66 1.00 0.84 0.78 0.65 0.07

HOL 0.09 0.10 0.01 0.00 0.10 0.88 0.41 0.28 0.73 0.84 1.00 0.71 0.71 0.06

IRE 0.19 0.06 0.06 0.01 0.12 0.67 0.66 0.37 0.50 0.78 0.71 1.00 0.45 0.10

ITA 0.12 0.35 0.26 0.14 0.26 0.76 0.29 0.40 0.93 0.65 0.71 0.45 1.00 0.03

HK 0.27 0.04 0.02 0.01 0.04 0.05 0.09 0.10 0.03 0.07 0.06 0.10 0.03 1.00

POR 0.24 0.26 0.05 0.01 0.08 0.56 0.22 0.17 0.48 0.50 0.57 0.31 0.49 0.28

SIN 0.31 0.19 0.03 0.01 0.06 0.30 0.22 0.16 0.29 0.22 0.34 0.16 0.28 0.31

TAI 0.25 0.06 0.02 0.01 0.04 0.19 0.11 0.11 0.11 0.19 0.20 0.21 0.09 0.84

DEN 0.21 0.02 0.06 0.18 0.24 0.26 0.46 0.15 0.12 0.36 0.28 0.49 0.09 0.04

MAL 0.06 0.03 0.00 0.00 0.01 0.54 0.33 0.13 0.37 0.68 0.61 0.58 0.38 0.12

NOR 0.22 0.02 0.07 0.19 0.25 0.25 0.46 0.16 0.12 0.36 0.27 0.48 0.09 0.04

US 0.21 0.04 0.06 0.02 0.07 0.34 0.55 0.22 0.18 0.45 0.39 0.66 0.15 0.10

(Continued)

Table II

POR SIN TAI DEN MAL NOR US

AUS 0.24 0.31 0.25 0.21 0.06 0.22 0.21

SPA 0.26 0.19 0.06 0.02 0.03 0.02 0.04

SWE 0.05 0.03 0.02 0.06 0.00 0.07 0.06

SWI 0.01 0.01 0.01 0.18 0.00 0.19 0.02

UK 0.08 0.06 0.04 0.24 0.01 0.25 0.07

BEL 0.56 0.30 0.19 0.26 0.54 0.25 0.34

CAN 0.22 0.22 0.11 0.46 0.33 0.46 0.55

FIN 0.17 0.16 0.11 0.15 0.13 0.16 0.22

FRA 0.48 0.29 0.11 0.12 0.37 0.12 0.18

GER 0.50 0.22 0.19 0.36 0.68 0.36 0.45

HOL 0.57 0.34 0.20 0.28 0.61 0.27 0.39

IRE 0.31 0.16 0.21 0.49 0.58 0.48 0.66

ITA 0.49 0.28 0.09 0.09 0.38 0.09 0.15

HK 0.28 0.31 0.84 0.04 0.12 0.04 0.10

POR 1.00 0.67 0.40 0.09 0.32 0.08 0.12

SIN 0.67 1.00 0.31 0.04 0.22 0.04 0.10

TAI 0.40 0.31 1.00 0.06 0.15 0.06 0.12

DEN 0.09 0.04 0.06 1.00 0.55 0.99 0.80

MAL 0.32 0.22 0.15 0.55 1.00 0.54 0.67

NOR 0.08 0.04 0.06 0.99 0.54 1.00 0.80

US 0.12 0.10 0.12 0.80 0.67 0.80 1.00

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