The Maximum Drawdown as a Risk Measure: the Role of Real Estate in the Optimal Portfolio Revisited

Transcription

1 The Maximum Drawdown as a Risk Measure: the Role of Real Estate in the Optimal Portfolio Revisited Foort Hamelink * and Martin Hoesli ** This draft: June 24, 2003 Abstract We investigate the role of real estate in a mixed-asset portfolio when the maximum drawdown (hereafter MaxDD), rather than the standard deviation, is used as the measure of risk. We argue that the MaxDD concept is one of the most natural measures of risk, and that such a framework can help reconcile the optimal allocations to real estate and the effective allocations by institutional investors. The empirical analysis is conducted from the perspective of Swiss pension funds who are faced with legal constraints on the weights that can be allocated to the various asset categories and pertains to the period We show that most portfolios optimized in Return/MaxDD space, rather than in Return/Standard Deviation space, yield a much lower MaxDD, while only a slightly higher standard deviation (for the same level of return). The reduction in MaxDD is highest for portfolios situated half-way on the efficient frontier, typically close to those held by pension funds. Also, the reported weights for real estate are much more in line with the actual weights to real estate by institutional investors. * Lombard Odier Darier Hentsch, Vrije Universiteit and FAME, Foort@Hamelink.com ** University of Geneva (HEC and FAME) and University of Aberdeen (Business School), martin.hoesli@hec.unige.ch Please address correspondence to: Martin Hoesli, University of Geneva, HEC, 40 boulevard du Pont-d Arve, CH-1211 Geneva 4, Switzerland. 1

2 The Maximum Drawdown as a Risk Measure: the Role of Real Estate in the Optimal Portfolio Revisited I. Introduction The positive role of real estate has been well documented in the literature (for a review, see Hoesli and MacGregor, 2000, chapter 10). It is common knowledge for instance that real estate returns are lowly (positively or negatively) correlated to those of stocks and bonds (Ziobrowski and Ziobrowski, 1997). Several studies have suggested that the optimal weight that should be allocated to real estate in mixed-asset portfolios is in the 15-30% range (Ennis and Burik, 1991; Ziobrowski and Ziobrowski, 1997). Institutional investors allocate, however, a substantially lower weight to real estate than that reported in the literature (Chun and Shilling, 1998; Geltner and Miller, 2001). Several studies have attempted to provide explanations for this discrepancy. It has been argued for instance that the weight which should be allocated to real estate is much more in line with the actual institutional weight when an asset-liability framework is used rather than an asset only framework (Chun et al., 2000; Craft, 2001), or when real estate market imperfections are considered (Kallberg et al., 1996). Some authors also stress that weights are reported with error, and hence that caution should be exercised when implementing a diversification strategy as the confidence bounds around the efficient frontier allocations are very broad (Liang et al., 1996). Yet an alternative approach to addressing the issue of errors in parameters is to use Bayes-Stein estimators (Jorion, 1985 and 1986; Stevenson, 2001). As most of the errors in parameters concern the expected means, the idea is to shrink the sample averages toward a common mean. By so doing, the calculated portfolio allocations have been shown to be more stable than when classical mean-variance analysis is used. We follow another route in trying to reconcile the optimal and effective allocations to real estate in mixed-asset portfolios. We argue that the standard deviation may not be the appropriate measure of risk, but rather that the maximum drawdown (MaxDD) should be used. The MaxDD is defined as the loss suffered when an asset is bought at a local maximum, and sold at the next local minimum. MaxDD was first introduced by Grossman 2

3 and Zhou (1993), but the framework was limited to an univariate series. Our study goes much further as it extends the MaxDD framework to the multi-asset setting. This path of research is quite similar in spirit to that used by Sing and Ong (2000). These authors use downside risk as their measure of risk and show that the allocation to real estate does not change substantially. Downside risk, however, is defined in terms of standard deviation of the returns below a given threshold and does not consider the serial correlations found in the return series, as does MaxDD. The empirical analysis is undertaken using Swiss data for the period and taking into account the recommended legal ceilings for Swiss pension funds on investment in each of the asset classes. Ceilings have been set for pension funds to ensure that they do not take too much risk. This is achieved by setting lower ceilings for riskier asset classes than for less risky assets (see section III). These ceilings are only guidelines in the case of large pension funds, however, as they can depart from these ceilings if they show that they comply with the principles of diversification and liquidity. We consider Swiss and foreign bonds and stocks, Swiss and foreign securitized real estate, and Swiss and foreign direct real estate. Our results show that the gains in terms of maximum drawdown are quite substantial, while the losses in terms of standard deviation are usually limited. As far as the optimal allocation to real estate is concerned, our results are much more in line with actual holdings by institutions. Our results thus provide a contribution to the debate on the reconciliation of optimal and actual allocations to real estate by institutional investors. The paper is structured as follows. We review the literature in section II, while in section III we present the data and method that we use. Our results are discussed in section IV. Finally, section V contains some concluding remarks. II. Review of the Literature The positive role of real estate in diversifying mixed-asset portfolios has been well documented in the literature. Real estate returns have been shown to be lowly correlated with those of stocks and bonds. Several authors have suggested an optimal weight for real estate in U.S. mixed-asset portfolios in the 15-30% range (Fogler, 1984; Ennis and Burik, 1991; Ziobrowski and Ziobrowski, 1997). All of these studies have been conducted in a Modern 3

4 Portfolio Theory (MPT) framework. Further evidence of the usefulness of real estate as a portfolio diversifier is provided by Chaudhry et al. (1999). Using co-integration techniques, they find that stocks have an inverse long-run relationship with real estate. They also find that the overall impact of stocks on the real estate market is much less than its impact on bonds and T-bills 1. Institutional investors worldwide hold on average a lower percentage in real estate than the weight suggested in the literature. This is somewhat puzzling and reconciling these weights has been the focus of several studies. It can be argued for instance that the low weight allocated to real estate by institutional investors is due to market imperfections, such as illiquidity and lumpiness. Kallberg et al. (1996) examine the role of real estate investments in a portfolio context incorporating the real estate imperfections of indivisible assets and no short sales. Their study suggests that a 9% allocation to real estate is optimal, a result which is much more in line with the actual holdings of institutions than the weights which have been reported in earlier studies. However, their analysis also indicates an optimal allocation to bonds of less than 2%. Thus, as stated by Geltner and Miller (2001), the overall mixed-asset portfolio suggested by their model is at least as different from the empirical pension fund allocation as is that implied by classical portfolio theory. Another possible explanation is that institutions invest in the long run, while such optimal weights are derived using monthly, quarterly or annual time increments. It seems reasonable to assume that higher correlations between real estate and financial assets, and hence less diversification benefits, exist in the long run. Unfortunately, no long enough time series of real estate returns exist which would enable the testing of such an hypothesis. Another possible explanation which has been suggested for the discrepancy between theoretical and empirical allocations to real estate is that institutions use an asset-liability optimization framework, rather than an asset only framework, when making investment decisions. In other words, institutions aim at optimizing their net wealth portfolio, subtracting the present value of future liabilities from the present value of assets. In such a context, institutions would place stronger emphasis on assets that are negatively correlated to the present value of liabilities, e.g. on bonds, and hence less emphasis on real estate. Chun et al. (2000) report that 1 Fraser et al. (2002) use Granger causality tests and cointegration techniques to demonstrate that there is no long-run relationship between real estate returns and those of either bonds or stocks. 4

5 the optimal weight which should be allocated to real estate is approximately 10% rather than 20-30% (see also Craft, 2001). If pension obligations, however, are sensitive to inflation, then real estate s inflation-hedging effectiveness may still call for a large allocation to real estate, even in an asset-liability framework (see also Geltner and Miller, 2001, p. 544). Some authors have suggested that confidence bounds should be constructed around the efficient frontiers to assess the statistical significance of results. Liang et al. (1996) use bootstrap simulation and show that the confidence intervals are large enough to render efficient frontiers effectively useless 2. They argue that the efficient frontier is fuzzy and the weight vectors even fuzzier. The real estate weights observed in institutional portfolios are thus not statistically different from those reported in theoretical studies. Bootstrapping, however, relies on very loose assumptions regarding the distribution of asset returns; large confidence bounds thus would be expected. Moreover, the confidence intervals for the weights of the other asset classes are wide as well. Thus, these results highlight the fact that caution should be exercised when constructing a mixed-asset portfolio, rather than suggest that optimal mixed-asset portfolio diversification is useless. An interesting line of research is that of the impact of parameter uncertainty on optimal portfolio selection. It has been shown that the practical application of portfolio analysis is hampered by estimation error, especially in expected returns. Variances and covariances are also unknown, but are more stable over time (Jorion, 1985, and Chopra and Ziemba, 1993 for common stocks; Stevenson, 2001 for international real estate stocks). Given these results, it would seem appropriate to increase the reliability of the point estimates for the optimal asset allocations. One possibility for reducing estimation error is to constrain the allocations, thereby forcing greater spread across the assets (Frost and Savarino, 1988). The choice of constraints is arbitrary, however. Another possibility is to use the Bayes-Stein shrinkage approach (Jorion, 1985). When such an approach is used, the means are shrunk towards a common value (the common mean, i.e. the mean across all assets considered). Jorion (1986) uses this approach to investigate the benefits of international stock portfolio diversification. He finds that the out-of-sample performance of the optimal portfolio is substantially increased. Also, he argues that the benefits from diversification are more likely to accrue from a reduction in risk, rather than gains in average returns. Stevenson (2001), using data 2 See also Ziobrowski et al. (1997), Rubens et al. (1998), Ziobrowski et al. (1999), and Cheng and Liang (2000). 5

6 pertaining to international real estate securities, finds an increased stability in calculated portfolio allocations in comparison to the classical mean-variance tangency approach, and significant improvements in out-of-sample performance. Yet a further possible explanation could be that the theoretical weights that have been reported in the literature are flawed as real estate returns are in several instances not normally distributed, and hence that Markowitz portfolio optimization may be inadequate. It has been suggested that in such cases a downside risk framework should be used, i.e. that risk should be measured as the deviation below a pre-specified target rate of return. Sing and Ong (2000) show that the weights allocated to real estate do no change in any substantial manner when downside risk is considered, while Cheng and Wolverton (2001) argue that comparing the results obtained in such a framework with those obtained in a classic modern portfolio theory framework is not as straightforward as it may appear. The latter authors argue that the results could only be compared fairly if a common risk measure were available, which is not yet the case. The non-normality of returns is one reason why Markowitz portfolio optimization may be inadequate. The other potential source of problem is the serial dependence of the returns. If returns were independently distributed over time, then the probability of observing negative returns month after month as has been the case in the three years following March 2000 is almost zero. The concept of MaxDD is an answer to these two potential pitfalls of the traditional Markowitz framework. Grossman and Zhou (1993) present the theoretical framework of MaxDD, but in the context of one single asset only. Particularly appealing is the fact that this approach does not rely on any a priori distribution or time-dependence of the returns. III. Data and Method 1. Data All series are total return series and the returns are computed at the annual frequency. The world stock and bond indices, as well as the GPR index of securitized real estate are taken from Datastream. The Swiss stock and bond indices are from Pictet. The Swiss direct real estate index is the IAZI/CIFI index of apartment buildings in Switzerland. The index is 6

7 constructed using the hedonic method and does not suffer from the smoothing associated with appraisal-based indices. Institutions in Switzerland typically invest in apartment buildings (approximately 85% on average), so that using an index of residential real estate is appropriate. To proxy for foreign real estate returns, we use data for two very large real estate markets for which a long history of return series is available (i.e. those of the U.S. and the U.K.) The U.S. index is the NCREIF Property Index (NPI), whereas the Invesment Property Databank (IPD) index is used for the U.K. It is well known that such indices suffer from smoothing, so we perform all analyses with desmoothed returns. We use the Geltner (1993) approach to desmoothing in that we assume that the standard deviation of real estate lies halfway between that of stocks and that of bonds 3. Annual returns for the various asset classes are reported in Table 1, while summary statistics are contained in Table 2. Swiss direct real estate does not exhibit very favorable risk and return characteristics as compared to bonds: it has a slightly lower return and a higher risk. This is due to the strong impact of the real estate crash of the late 1980s-early 1990s. In contrast, the very bullish real estate markets of the 1970s are not included in the sample period. The usual conclusion that stocks are the riskiest asset class is not that pronounced when the MaxDD is used as the measure of risk. As a matter of fact, U.S. direct real estate has the largest MaxDD over the period under investigation. Figure 1 contains comparisons of the performance of Swiss and foreign investments for the various categories of assets. Foreign assets in general exhibit greater variability than Swiss assets; this is of course in part due to currency movements. Swiss pension funds are subject to recommended ceilings on the various asset classes. These are to unsure that the portfolio allocations do not lead to unconsidered risks being taken. Obviously, the riskier the asset category, the lower the ceiling. The total allocation to stocks for instance cannot exceed 50%, whereas an allocation of 100% to Swiss bonds is accepted. The limit on real estate holdings is 55% (50% in Swiss real estate and 5% in foreign real estate). The limit on real assets (real estate and stocks) is set at 70%. These guidelines appear in Table 3. These ceilings are mandatory for smaller institutions, whereas they only constitute 3 There are several alternative ways of desmoothing appraisal-based return indices, and as shown by Corgel and deroos (1999), the choice of the method being used impacts upon the optimal allocation to real estate in a mixed-asset portfolio. In our case, the smoothing issue does not affect our results very much as the legal ceiling on foreign real estate investments for Swiss pension funds is 5%, so that the allocation to this type of real estate is in any case marginal. For this reason, no sensitivity analyses to the desmoothing assumption are performed. 7

8 guidelines for larger pension funds. The latter can depart from these ceilings if they can show that their investment policy is in accordance with the principles of diversification and liquidity. Given that the ceilings have been set primarily using such principles, large deviations from these are unlikely. Portfolio managers hence are faced in most cases with an optimization issue under these constraints. At the end of the paper, we discuss the implications of these constraints in terms of the efficiency of the optimal portfolios. < INSERT TABLE 1 HERE > < INSERT TABLE 2 HERE > < INSERT TABLE 3 HERE > < INSERT FIGURE 1 HERE > 2. Method a) Maximum Drawdown: Definition and Intuition The MaxDD is formally defined as the loss suffered when the asset is bought at a local maximum, and sold at the next local minimum. Figure 2 gives an illustration of the MaxDD on the series of World bonds. This series is a good example because the value of the MaxDD is significant. It clearly shows that world bond investments in the mid-eighties were not profitable for a Swiss investor: yields went up, and the U.S. Dollar went down. < INSERT FIGURE 2 HERE > Figure 2 shows cumulated annual log returns since After five years of positive returns, the total return index reached a local maximum of 133% (cumulated log returns) at the end of During the next three years, the index dropped from 133% to 43% (cumulated log returns), which represents a loss of 58% for an investment made at the end of 1984 and kept for three years. The MaxDD criterion considers the 58% loss as the appropriate measure of risk, regardless of the time length over which this loss occurred (three years in this example), and regardless also of the time needed to recover the loss (10 years in this example). 8

9 We believe that the MaxDD has several important advantages over alternative measures of risk. Semi-variance, one of the most popular risk measures, considers standard deviation only over negative outcomes, typically those that constitute the risk of a portfolio. While this measure may seem appealing, it does not take into account the serial correlation of returns. A drop in equities of 25% is perceived differently by most investors when it is the third consecutive year that it occurs, than when it happens for the first time. Whether this is rational or not is subject to debate, but it seems obvious that had World stocks generated a 7.1% return every year since 1979 (which is the average return over ), none of the pension funds would be experiencing funding problems today. The source of their current problems is that World stocks lost 51% as of then end of 2002, compared to year-end Although it might appear counter-intuitive, the fact that time is not taken into account in the MaxDD criterion is probably very representative of the risk perception by most investors. How relevant is, for instance, the fact that World stocks fell by 51% in the past three years, rather than in two or four years? Time duration matters much less than the magnitude of the fall in prices. Value-at-risk (VAR) is another example of an appealing alternative measure of risk, as it considers both a probability and an absolute level of loss. But contrary to what is the case with the MaxDD, VAR considers a pre-defined time frame over which the loss may occur, and therefore lacks to fully incorporate the time-dependence of financial series. The MaxDD criterion might appear simplistic for several reasons. First, the MaxDD for a given series applies to one particular period in the past. In the example above, when we add more observations to the series of World Bond returns, as long as there is not a new drawdown, larger in magnitude than the previous one, the MaxDD of the series will remain unchanged. This is not true for the standard deviation, for instance, where adding more observations to the sample will modify the historical figure. The MaxDD criterion is therefore very stable over time and always reminds of the worst historical situation. Keeping in mind past crises might not be a bad idea in the current environment: the coming years may very well be years of increasing interest rates, and a declining dollar, making unhedged fixed income investments particular risky. How risky? The answer might be the risk of a loss in the order of magnitude of 58%, the historical drawdown. b) Frontiers under the Maximum Drawdown criterion 9

10 It is straightforward to extend Markowitz s Mean-Variance or Mean-Standard Deviation (hereafter M-SD) framework to a Mean-Maximum Drawdown (hereafter M-MaxDD) framework. A portfolio is said to be efficient in M-MaxDD space when no other portfolio yields the same level of return with a lower level of MaxDD, or when no other portfolio yields a higher level of return, for the same level of MaxDD. In the next section, we will compare optimal portfolios obtained in a M-SD framework with those obtained in a M-MaxDD framework. The difficulty lies in comparing the two sets of results, as in a way we compare apples with oranges (Cheng and Wolverton, 2001). A portfolio which is optimal in M-SD space is, by definition, dominating portfolios obtained through optimization under any alternative measure of risk. In other words, a portfolio on the efficient frontier in M-SD space with a given level of return has, by definition, a higher MaxDD than an efficient portfolio in M-MaxDD space with the same level of return. Comparisons have therefore to be done on a trade-off basis, the gain in terms of MaxDD relative to the loss in terms of SD. IV. Results Four different sets of results were generated, based on four different samples of returns. The first set considers unhedged returns in Swiss Francs, where direct U.K. and U.S. real estate series have been desmoothed. The second set uses the raw (smoothed) returns, rather than the unsmoothed returns, for direct U.K. and U.S. real estate. The two other data sets differentiate by the fact that they consider hedged, rather than unhedged returns. We mainly discuss the results obtained with the first set of data (unhedged, desmoothed) because this is the most relevant framework for most institutional investors, especially in Switzerland where the cost of hedging foreign currency exposure is high due to the historically low level of Swiss interest rates. The hedging is accomplished through forward currency contracts with an expiration of one year. We assume that the manager hedges the full initial investment at the beginning of the period (of one year). It should be noted that this is not really a full hedge, as the hedge only applies to the initial amount invested and not to the returns realized on the investment. 10

11 Typically, when positive returns are generated, the hedge will prove insufficient, and when negative returns are generated, the position will be over-hedged. We also discuss the results obtained with the other datasets where necessary. In the first part, we discuss the trade-off between the two measures of risk (SD and MaxDD) when one or the other framework is used to optimize portfolios. In the second part, we investigate the compositions of the optimal portfolios. 1. Trade-off between SD and MaxDD Figure 3, Panel A represents the traditional efficient frontier in M-SD space, taking into account the constraints to which the typical Swiss pension fund manager is subject to. Starting from the bottom, the three frontiers are: (1) that containing stocks and bonds only, (2) that with stocks, bonds and Swiss direct and indirect real estate, and finally (3) that with all assets described in the data section. The direct U.K. and U.S. real estate series are desmoothed. Without surprise, adding Swiss real estate to the basic portfolio of stocks and bonds allows a reduction in risk for the same level of expected return (see also Hoesli and Hamelink, 1996 and 1997). Adding also international real estate yields an even larger risk reduction, even though the constraints are very tight (a maximum allocation of 5% to overseas real estate only is permitted) and therefore the benefits of adding real estate are less significant as when no restrictions apply. It is also because of these constraints that all frontiers do not originate from the same low risk, low return point, and do not end at the same high risk, high return point. While Panel A is the traditional picture, Panel B in Figure 3 shows similar frontiers in M- MaxDD space. A striking result is the limited reduction in MaxDD for low risk portfolios and the very high level of reduction in MaxDD for high-risk portfolios. For instance, the MaxDD of a portfolio with an expected return of 6.5% is reduced by 50% when real estate (both Swiss and foreign) is added to the portfolio. For low risk portfolios, it seems that adding real estate to the portfolio does not significantly reduce the MaxDD. < INSERT FIGURE 3 HERE > 11

12 The conclusions reported above are not substantially different when direct U.K. and U.S. real estate returns are not desmoothed. This is largely because the maximum allocation to foreign real estate is in any case limited to 5%. The frontiers in M-MaxDD space that include real estate are only slightly shifted to the left, but are extremely close to one another for low levels of risk. The frontiers in M-SD space are, not surprisingly, more affected when smoothed direct real estate returns are used. Looking at the samples of hedged returns, the global picture remains largely unchanged, although all frontiers are closer together and more to the left hand-side of the graphs, due to lower overall variances. The above frontiers cannot directly be compared with one another, as one is in M-SD space, the other in M-MaxDD space. It is therefore informative to compute for each portfolio on the M-MaxDD frontier the corresponding level of SD, and to compute for each portfolio on the M-SD frontier the corresponding level of MaxDD. Figure 4, Panel A, represents in M- MaxDD space the two frontiers. < INSERT FIGURE 4 HERE > Not surprisingly, the portfolio obtained through M-MaxDD optimization always has a lower MaxDD than the corresponding portfolio obtained through M-SD optimization. It could not be otherwise. For very high levels of risk, both frontiers are very close to one another. For a portfolio with an expected return of 6.5% (a medium level of risk), for instance, the difference is significant: the MaxDD is reduced from 9.5% to 6%, with the same level of return (6.5%). For lower levels of risk, the difference is less pronounced. For instance, optimizing in M- MaxDD space reduces the MaxDD of a 5.5% return portfolio only from 2.9% to 1.8%. Similar conclusions can be drawn from the other samples (raw returns and/or hedged returns). The next comparison is in M-SD space, where we compare the portfolios obtained through M- MV and M-MaxDD optimization in terms of means and standard deviations. The results are presented in Figure 4, Panel B and show that for higher levels of risk the loss in terms of higher SD is small. For instance, for the 6.5% return portfolio for which the gain in terms of MaxDD was substantial, the SD increases only from 8.0% to 8.8%. For lower levels of risk and return, the difference is more pronounced. The 5.5% return portfolio sees its SD increase from 4.1% to 5.7%. For very low levels of risk, it is interesting to note that there is almost no increase in SD when the optimization is done in M-MaxDD space. 12

13 As we said before, the comparison between the two frontiers is not straightforward. All depends on what is considered the most appropriate measure of risk. If it is SD, there is no harm if the MaxDD of some portfolios is high, as MaxDD is not the measure of risk. If MaxDD is the chosen measure, then we should not worry about some portfolios having a high SD, as SD is not measuring risk. Although we believe that MaxDD is in many cases much more representative than SD, SD is in most cases the framework of reference for investors. In order to better assess the increase in terms of MaxDD versus the loss in terms of SD when portfolios are optimized in M-MaxDD space, we summarize in Table 4 the figures for five portfolios on the frontiers. All portfolios, except the very first one, which is the minimum variance portfolio and the minimum MaxDD portfolio, respectively, have the same expected returns. As all frontiers were formed by optimizing 50 portfolios, we refer to the 5 portfolios in the table as Portfolios 1, 10, 20, 30, 40, and 50. For each portfolio, we compute the difference in SD obtained for each level of return with the two optimization methods. We also compute the difference in MaxDD in an similar way. < INSERT TABLE 4 HERE > An overall picture is given in Figure 5, which shows for all 50 portfolios the level of loss in terms of SD versus the level of gain in terms of MaxDD. Although for many portfolios the loss in SD is relatively high (around 1.5%) relative to the gain in MaxDD (around 1%), a few portfolios yield very significant drops in MaxDD, for only slight SD increases. This is the case for many more portfolios when the smoothed direct real estate returns are used instead of the desmoothed returns. For none of the portfolios is the increase in SD above 1.2%, while for many portfolios the decrease in MaxDD is above 3%. The same conclusions are reached when the two hedged samples are used. < INSERT FIGURE 5 HERE > 2. Composition of optimal portfolios In this section, we analyze the composition of portfolios that are optimal under the M-SD criterion and under the M-MaxDD criterion. In order to make relevant comparisons, we 13

14 aggregate the various assets into four categories: (1) bonds, (2) stocks, (3) direct and indirect Swiss real estate, and (4) direct and indirect foreign real estate. Bonds comprise both Swiss and foreign fixed income securities, stocks encompass both Swiss and foreign stocks, Swiss real estate contains both direct and securitized real estate, while foreign real estate comprises U.K. direct, U.S. direct and World securitized real estate. Figure 6, Panel A reports the weights for the traditional case where portfolios are optimized in the M-SD space. < INSERT FIGURE 6 HERE > The relative weights of each of these asset categories are represented for the 50 portfolios on the frontiers. The horizontal axis is the standard deviation of each of these portfolios. The impact of the investment constraints that apply to Swiss institutional investors is clearly visible. Foreign real estate is limited to 5% of the total portfolio, and this is the optimal allocation to this asset class whatever the level of risk. The optimal allocation to Swiss real estate is around 35% and very constant over all portfolios. This result is both in line with previous research (Hoesli and Hamelink, 1996 and 1997), and in sharp contradiction to the actual allocation to real estate by the typical pension fund. On average, Swiss pension funds hold only 15% of their assets in Swiss real estate. In Panel B, we show the composition of optimal portfolios under the MaxDD criterion. The allocation to fixed income securities is much larger. The allocation to real estate is much more variable than is the case when the traditional M-SD criterion is used. The 10 least risky portfolios (out of 50) have allocations to real estate (both Swiss and foreign) between 40% and 15%. In the second quintile, this allocation drops to zero and remains very low. In the next two quintiles, the allocation to real estate increases, but remains below 15%. It is only in the last quintile that the percentage of total assets invested in real estate increases again above 40%. For low levels of risk, typically those portfolios of interest to institutional investors, the optimal holding in real estate under M-MaxDD is therefore much lower than what is traditionally obtained under M-SD. This is an interesting finding, especially in light of the actual real estate holdings by Swiss pension funds. The 15% that the latter hold in real estate is exactly around the optimal level suggested by the M-MaxDD criterion for portfolios with lower levels of risk. Furthermore, comparing the M-SD and the M-MaxDD optimal weights, 14

15 it appears that the excess allocation to real estate in the M-SD case has been replaced by an allocation to bonds, rather than stocks. A very similar picture emerges with the raw (smoothed) data. The two hedged samples, although less close to reality than the unhedged samples, both show slightly higher allocations to real estate, which is not surprising given the significantly reduced level of volatility when returns are hedged. Finally, it is of interest to examine the allocation to real estate if there were no constraints to investment in the various asset classes. These are reported in Figure 7. At the bottom end of the frontier, the allocations are very similar. The allocation to foreign real estate, however, grows rapidly when it is not restricted by the 5% limit. In fact, this allocation to foreign real estate is driven by World real estate securities which exhibit favorable return and risk parameters. As from the middle of the efficient frontier, the allocation to real estate is in excess of 40%. At the upper end of the frontier, the allocation is exclusively in real estate, with foreign investments representing a large fraction of that allocation. V. Conclusion In this paper we maintain that standard deviation is not necessarily the unique measure that should be used to ascertain the riskiness of assets when constructing efficient portfolios. An alternative measure of risk is the maximum drawdown which measures the loss from a local maximum to the next local minimum. This measure is intuitively appealing in that institutional investors for instance should be concerned with the maximum loss on the various asset classes. The traditional mean-variance framework can easily be adapted to a meanmaximum drawdown framework, and the optimal weight which should be allocated to the various asset classes can be determined. The aim of this paper is to ascertain the role of real estate in diversifying mixed-asset portfolios when a maximum drawdown framework is considered. The empirical analysis is conducted from the perspective of Swiss pension funds. We consider the various asset classes that are considered by these investors, i.e. Swiss and foreign stocks and bonds, Swiss and foreign securitized real estate, and Swiss and foreign direct real estate investments. The overseas direct investments are proxied by means of the U.S. and 15

16 U.K. direct real estate indices. The constraints on investments in the various asset classes that apply to Swiss pension funds are taken into account in the analysis. Even though these constraints are not compulsory for all investors, they should be fairly representative of the investment policy by Swiss pension funds. Our results suggest that most portfolios optimized in Return-MaxDD space yield a much lower MaxDD than when a traditional mean-variance framework is used, and that such portfolios only exhibit a slightly higher standard deviation. Hence, using such a framework appears to enhance the MaxDD quite a bit, while the losses in terms of standard deviation are only limited. An interesting contribution of the paper lies in attempting to reconcile the optimal allocation to real estate and the actual allocation by institutional investors. It is well known that while the suggested allocation to real estate is in the 20-30% range, the actual allocation is in most countries less than 10%. In Switzerland, the actual proportion of assets invested in real estate is a bit higher at 15%. The optimal allocations to real estate using a MaxDD framework are much more in line with the actual allocations. This is an interesting finding: from that perspective, the real estate holdings by institutions appear not to be that inefficient. This is not to suggest that institutional investors necessarily use a MaxDD framework when allocating funds across asset classes, but only that their investment policy is not that inefficient when considered from the perspective of the maximum drawdown. 16

17 References Chaudhry, M. K., Myer, F. C. N. and Webb, J. R. (1999) Stationarity and cointegration in systems with real estate and financial assets, Journal of Real Estate Finance and Economics, 18:3, Cheng, P. and Liang, Y. (2000) Optimal diversification: is it really worthwhile?, Journal of Real Estate Portfolio Management, 6:1, Cheng, P. and Wolverton, M. L. (2001) MPT and the downside risk framework: a comment on two recent studies, Journal of Real Estate Portfolio Management, 7:2, Chopra, V. K. and Ziemba, W. T. (1993) The effect of errors in means, variances and covariances on optimal portfolio choice, Journal of Portfolio Management, 19:2, Chun, G. H., Ciochetti, B. A. and Shilling, J. D. (2000) Pension-plan real estate investment in an asset-liability framework, Real Estate Economics, 28:3, Chun, G. H. and Shilling, J. D. (1998) Real estate asset allocations and international real estate markets, Journal of the Asian Real Estate Society, 1:1, Corgel, J. B. and deroos, J. A. (1999) Recovery of real estate returns for portfolio allocation, Journal of Real Estate Finance and Economics, 18:3, Craft, T. M. (2001) The role of private and public real estate in pension plan portfolio allocation choices, Journal of Real Estate Portfolio Management, 7:1, Ennis, R. M. and Burik, P. (1991) Pension fund real estate investment under a simple equilibrium pricing model, Financial Analysts Journal, 47:3, Fogler, H. R. (1984) 20% in real estate: can theory justify it?, Journal of Portfolio Management, 10:2, Fraser, W. D., Leishman, C. and Tarbert, H. (2002) The long-run diversification attributes of commercial property, Journal of Property Investment and Finance, 20:4, Frost, P. A. and Savarino, J. E. (1988) For better performance: constrain portfolio weights, Journal of Portfolio Management, 15:1, Geltner, D. (1993) Estimating market values from appraised values without assuming an efficient market, Journal of Real Estate Research, 8:3, Geltner, D. and Miller, N. G. (2001) Commercial Real Estate Analysis and Investments, Upper Saddle River (NJ): Prentice Hall. Grossman, S. J., and Zhou, Z. (1993) Optimal investment strategies for controlling drawdowns, Mathematical Finance, 3, Hoesli, M. and Hamelink, F. (1996) Diversification of Swiss portfolios with real estate: results based on a hedonic index, Journal of Property Valuation and Investment, 14:5, Hoesli, M. and Hamelink, F. (1997) An examination of the role of Geneva and Zurich housing in Swiss institutional portfolios, Journal of Property Valuation and Investment, 15:4, Hoesli, M. and MacGregor, B. D. (2000) Property Investment Principles and Practice of Portfolio Management, Harlow: Longman. Jorion, P. (1985) International portfolio diversification with estimation risk, Journal of Business, 58:3,

18 Jorion, P. (1986) Bayes-Stein estimation for portfolio analysis, Journal of Financial and Quantitative Analysis, 21:3, Kallberg, J. G., Liu, C. H. and Greig, D. W. (1996) The role of real estate in the portfolio allocation process, Real Estate Economics, 24:3, Liang, Y., Myer, F. C. N. and Webb, J. R. (1996) The bootstrap efficient frontier for mixedasset portfolios, Real Estate Economics, 24:2, Rubens, J. H., Louton, D. A. and Yobaccio, E. J. (1998) Measuring the significance of diversification gains, Journal of Real Estate Research, 16:1, Sing, T. F. and Ong, S. E. (2000) Asset allocation in a downside risk framework, Journal of Real Estate Portfolio Management, 6:3, Stevenson, S. (2001) Bayes-Stein estimators and international real estate asset allocation, Journal of Real Estate Research, 21:1/2, Ziobrowski, A. J., Caines, R. W. and Ziobrowski, B. J. (1999) Mixed-asset portfolio composition with long-term holding periods and uncertainty, Journal of Real Estate Portfolio Management, 5:2, Ziobrowski, A. J., Cheng, P. and Ziobrowski, B. J. (1997) Using a bootstrap to measure optimum mixed-asset portfolio composition: a comment, Real Estate Economics, 25:4, Ziobrowski, B. J. and Ziobrowski, A. J. (1997) Higher real estate risk and mixed-asset portfolio performance, Journal of Real Estate Portfolio Management, 3:2,

19 Table 1: Returns on the various asset classes, (in Swiss Francs, unhedged). Note: returns on U.S. and U.K. direct real estate have been desmoothed. Swiss indirect real estate World indirect real estate Swiss direct real estate Swiss bonds (domestic debtors) Swiss bonds (foreign debtors) U.K. direct U.S. direct real estate real estate Swiss stocks World stocks World bonds % 42.8% 2.4% 37.0% 23.8% 6.1% 33.1% 2.3% 2.3% 33.1% % 10.1% 14.7% -8.1% 13.8% -11.9% -0.9% 1.9% 1.9% 2.6% % 46.9% 9.1% -2.6% -1.9% 13.3% 24.5% 12.0% 12.0% 47.8% % 36.4% 14.4% 5.0% 35.0% 27.3% 32.1% 3.4% 3.4% 23.7% % 24.2% 15.3% 4.1% 37.9% 4.5% 27.9% 3.4% 3.4% 48.8% % 24.3% 4.3% 6.9% -17.9% 61.4% -0.6% 6.4% 5.9% -18.9% % 26.8% 17.6% -9.3% -22.1% 9.7% -0.5% 6.1% 6.0% -26.5% % -9.1% 4.6% 34.6% -15.8% -27.5% -22.6% 5.0% 4.9% -30.1% % 57.7% 7.0% 49.9% 35.7% 23.6% 47.1% 4.9% 4.3% 47.3% % 8.5% 7.3% -2.0% 5.0% 22.6% 25.4% -4.6% -4.0% 12.7% % -34.7% -0.8% -23.1% -28.7% -19.3% -36.1% 1.8% 1.2% -23.4% % 15.4% -2.5% 2.5% -25.3% 17.7% 20.3% 8.3% 8.2% 29.7% % -4.0% -2.6% -13.0% 8.1% 17.6% 4.8% 12.0% 13.5% 22.5% % 56.5% -3.2% 31.8% 19.6% 50.8% 20.2% 13.0% 12.3% 15.0% % -19.0% -0.5% -0.3% 6.9% -7.6% -14.1% -0.6% -0.5% -21.4% % -2.7% 1.9% -13.8% -2.2% 23.1% 3.5% 12.3% 11.6% -7.3% % 40.4% -7.0% 46.5% 38.4% 18.3% 34.0% 5.4% 5.9% 41.5% % -7.3% 0.5% 26.3% 35.5% 55.2% 31.2% 5.7% 5.2% 20.2% % -14.8% 6.2% 3.5% 16.1% 15.4% 12.2% 5.7% 4.2% 2.0% % 18.8% 8.3% 31.5% 12.8% 11.7% 47.2% -0.5% 0.4% 28.2% % 11.7% 13.0% 1.4% 16.3% 11.9% -9.7% 3.4% 3.7% 5.7% % 2.4% -4.7% 4.3% -5.3% -22.0% -13.2% 3.8% 3.9% 7.1% % -10.9% 6.9% 2.6% -12.4% -26.0% -37.7% 10.2% 8.6% -19.2% Table 2: Summary statistics, (in Swiss Francs, unhedged). The table shows the main statistics on the various asset classes. All returns are annual, unhedged and in Swiss francs. Returns on U.S. and U.K. direct real estate have been desmoothed. Mean SD Mean/SD Cumul MaxDD Swiss indirect real estate 6.4% 12.7% % World indirect real estate 11.2% 25.6% % Swiss direct real estate 4.7% 6.9% % U.K. direct real estate 7.6% 19.8% % U.S. direct real estate 5.4% 23.0% % Swiss stocks 9.5% 24.7% % World stocks 7.1% 27.5% % Swiss bonds (domestic debtors) 5.2% 4.4% % Swiss bonds (foreign debtors) 5.1% 4.2% % World bonds 7.5% 27.3% %

20 Table 3: Constraints applying to Swiss pension funds. Lower Bound Constraints Upper Bound 0% real estate (direct & indirect, Swiss & Foreign) + stocks (Swiss & Foreign) 70% 0% Swiss & Foreign stocks 50% 0% Swiss bonds (foreign debtors) + Foreign bonds 30% 0% Foreign stocks & bonds 30% 0% Swiss direct & indirect real estate 50% 0% Swiss stocks 30% 0% Swiss bonds (foreign debtors) 30% 0% Foreign bonds 20% 0% Foreign stocks 25% 0% Foreign direct & indirect real estate 5% Table 4: Comparison of portfolios optimized in M-SD, and in M-MaxDD space. Portfolios 1,, 50 refer to 50 portfolios on the efficient frontiers. All portfolios, except the first one, are constructed to have the same mean. The columns Standard Deviation report the SD for the portfolios obtained under M-SD and under M-MaxDD optimization, as well as the difference. By construction, M-SD optimized portfolios dominate M-MaxDD portfolios under the SD criterion. The columns Maximum Drawdown report the MaxDD for the portfolios obtained under M-SD and under M-MaxDD optimization, as well as the difference. By construction, M-MaxDD optimized portfolios dominate M-SD portfolios under the MaxDD criterion. Panel A: Unhedged raw returns, in Swiss Francs. Standard Deviation M-SD M-MaxDD optimization optimization Difference M-SD optimization Maximum Drawdown M-MaxDD optimization Difference Mean Portfolio 1 (M-SD optimization) 5.15% Portfolio 1 (M-MaxDD optimization) 5.15% 2.78% 3.19% 0.41% -0.45% -0.35% 0.10% Portfolio % 3.58% 4.13% 0.55% -1.88% -1.68% 0.20% Portfolio % 5.34% 6.44% 1.10% -4.21% -3.13% 1.08% Portfolio % 7.48% 8.36% 0.88% -9.77% -6.38% 3.39% Portfolio % 10.26% 10.85% 0.59% % % 0.51% Portfolio % 14.87% 14.87% 0.00% % % 0.00% Panel B: Unhedged desmoothed returns, in Swiss Francs. Standard Deviation M-SD M-MaxDD optimization optimization Difference M-SD optimization Maximum Drawdown M-MaxDD optimization Difference Mean Portfolio 1 (M-SD optimization) 5.11% Portfolio 1 (M-MaxDD optimization) 4.98% 2.99% 3.12% 0.13% -1.16% -0.57% 0.59% Portfolio % 3.83% 4.76% 0.94% -2.74% -2.06% 0.68% Portfolio % 5.49% 7.10% 1.62% -4.75% -3.55% 1.21% Portfolio % 7.56% 8.71% 1.15% -8.57% -6.54% 2.03% Portfolio % 10.39% 10.85% 0.46% % % 0.67% Portfolio % 15.20% 15.20% 0.00% % % 0.00% 20

21 Figure 1: Cumulated returns (in log) on the various asset classes (in Swiss francs, unhedged). Panel A: Direct Real Estate (raw returns and desmoothed). Raw Returns Swiss direct real estate U.K. direct real estate U.S. direct real estate Desmoothed Returns Swiss direct real estate U.K. direct real estate U.S. direct real estate Indirect Real Estate Swiss indirect real estate World indirect real estate Panel B: Other asset classes. Stocks 3 Swiss stocks World stocks Bonds Swiss bonds (domestic debtors) Swiss bonds (foreign debtors) World bonds

22 Figure 2: Illustration of the Maximum Drawdown concept. The series pertains to World bonds in Swiss Francs (unhedged). The large loss of 58% over three years (between December 1984 and December 1987) is due to a combination of higher yields around the World and to a declining U.S. Dollar World Bonds local maximum maximum draw down local minimum after previous maximum

23 Figure 3: Efficient frontier taking into account the investment constraints that apply to Swiss pension funds. Note: both graphs are based on desmoothed returns. 7.5% PANEL A: Optimization in M-SD space. 7.0% All assets Financial assets and real estate Financial assets only 6.5% 6.0% 5.5% 5.0% Standard Deviation 4.5% 0% 2% 4% 6% 8% 10% 12% 14% 16% PANEL B: Return / MaxDD space. 7.5% 7.0% 6.5% 6.0% 5.5% All assets 5.0% Financial assets and real estate Financial assets only Maximum Draw Down 4.5% 0% 5% 10% 15% 20% 25% 23

24 Figure 4: Return and Risk (SD or MaxDD) characteristics of portfolios optimized in M-SD space and in M-MaxDD space. When optimization is done in M-MaxDD space, optimal portfolios have inevitably higher SD than their counterparts optimized in M-SD space (for a give level of mean return). For portfolios with a mean return around 6.5% the gain from MaxDD optimization is substantial, while SD increases only slightly. For lower levels of return, the increase in SD is more important. When raw returns (instead of desmoothed returns) are used, the increase in SD is small for all portfolios. PANEL A: Portfolios in Mean-MaxDD space, desmoothed returns. 7.5% 7.0% MV.Optimization DD.Optimization 6.5% 6.0% 5.5% 5.0% Maximum Drawdown 4.5% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% PANEL B: Portfolios in Mean-SD space, desmoothed returns. 7.5% 7.0% MV.Optimization DD.Optimization 6.5% 6.0% 5.5% 5.0% Standard Deviation 4.5% 2% 4% 6% 8% 10% 12% 14% 16% PANEL C: Portfolios in Mean-SD space, raw returns. 7.5% 7.0% MV.Optimization DD.Optimization 6.5% 6.0% 5.5% 5.0% Standard Deviation 4.5% 2% 4% 6% 8% 10% 12% 14% 16% 24