INVESTMENT STRATEGIES: NO. 35. Markowitz in tactical asset allocation. Starting with cross-market momentum

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1 J.P. Morgan Securities Ltd. London Classical mean variance portfolio optimization, conceived by Harry Markowitz in 1952, is used frequently for long-term strategic asset allocation, but not for tactical asset allocation. We show, however, that adding Markowitz to a momentum-based tactical asset allocation significantly enhances returns. This Dynamic Markowitz strategy produced a Sharpe ratio of 1.37 since 1994, compared to 1.13 for a basic cross-market momentum strategy and 0.77 for an equally weighted portfolio. JPMorgan has developed a new family of dynamic asset allocation indices based on Dynamic Markowitz. Starting with cross-market momentum Last year, we launched a tactical asset allocation strategy based on relative return momentum 1. Given a choice of 10 asset classes, the strategy invests equally in the five that performed best over the past 6 months and ignores the rest. The strategy has performed well out of sample, earning 15.3% pa since Jan 2006, or 3.7% over the equally weighted portfolio of various equity, bond, credit, EM, real estate, hedge funds, and commodity asset classes (see Table 1, next page). This excess return is in line with the strategy s 4.5% insample alpha. INVESTMENT STRATEGIES: NO. 35 Within sample, the cross-market momentum strategy was robust to many alternative specifications and periods. One potential enhancement that we did not examine and that is receiving strong interest from investors is applying mean variance portfolio optimization based on Harry Markowitz path-breaking work 2. Markowitz optimization involves calculating an efficient frontier of all possible portfolios that provide the highest expected return for each level of portfolio risk. In practice, investors do not make much use of mean variance optimization as they find the results too sensitive to inputs (See Box 1). When they use it, it is for long-term strategic asset allocation but never for shorter-term asset allocation around a given benchmark. We find here, to our satisfaction as economists, that when combined with our momentum-based tactical asset allocation strategy, Markowitz optimization does have significant added value. We thank Vadim di Pietro for valuable comments and discussion. The certifying analyst is indicated by an AC. See page 11 for analyst certification and important legal and regulatory disclosures. Ruy M. Ribeiro AC (44-20) Jan Loeys (44-20) jan.loeys@jpmorgan.com Contents Starting with cross-market momentum 1 Persistence in risk and return 2 Data, methodology and results 3 Comparing with component strategies 5 Long-short versions 7 Robustness Analysis 7 When does this strategy work well? 9 Longer periods and intra-asset class 9 Volatility timing/risk budgeting in asset allocation10 Conclusion 10

2 Table 1. Cross-market Momentum: out of sample; Jan 06 - May 07 Statistic Value Average Return 15.29% Excess Return over EW 3.71% Standard Deviation 6.72% Sharpe Ratio 1.43 Max (monthly) 4.95% Min (monthly) -2.57%. In this table, we perform an out-of-sample analysis of the cross-market relative momentum strategy proposed in Ribeiro and Loeys, Exploiting Cross-Market Momentum, Investment Strategies No. 14, Feb This Dynamic Markowitz strategy, as we call it, performed well over our sample period. Using returns and risks data for only the previous six months, the strategy delivered an annualized return of 15.6% with annualized volatility of 7.9%, significantly outperforming equally weighted portfolios, as well as those based solely on longterm Markowitz, or exclusively on return momentum. The performance of the Dynamic Markowitz strategy comes from three sources: 1. Momentum in asset class returns as best performing asset classes in the recent past are also more likely to outperform in the near future. 2. Persistence (clustering) in asset class volatility and correlation. 3. Stability in total risk exposure as we can combine asset classes to maintain a reasonably constant total volatility, thus introducing a volatility timing feature in the strategy. This paper is organized as follows. First, we rehearse the reasons why returns and risk should be persistent. Second, we describe the data, the testing methodology and our choices of alternative strategies that serve as points of comparison. Third, we test the Dynamic Markowitz strategy using a diversified set of asset classes. Fourth, we consider two long-short versions of the strategy. Fifth, we analyze the performance over time and in relation to market conditions. Sixth, similar strategies are tested using longer data or international portfolios of bonds or equities only. Persistence in returns and risk Persistence in performance and risk is an empirical question. Loosely speaking, a variable that exhibits persistence is one 1. See Ribeiro and Loeys, Exploiting Cross-Market Momentum, Investment Strategies No. 14, Feb Please visit morganmarkets.com for an updated version (revised Table 1, Chart 4 and Chart 5). 2. Harry Markowitz, Portfolio Selection, Journal of Finance, Volume 7, pp , Box 1. Drawbacks in standard portfolio optimization and possible solutions Many portfolio allocation applications assume stationarity in asset returns. The structure of both expected returns and covariances are estimated with long data. This implicitly assumes an i.i.d. process for returns. There is, however, strong evidence of predictability in stock returns when conditioned on price ratios and other predictable components in other asset classes. There are many other drawbacks as well. This allocation methodology has a backward-looking bias as it indirectly computes the optimal static portfolio for a particular past sample. Another negative feature of mean variance optimization is that allocations are very sensitive to parameters estimates, making the likely estimation error of expected returns and covariances a serious issue. Another unfortunate feature of this approach is that, at certain horizons, we find that expected and realized returns are negatively related, as decreases/increases in prices may just reflect an increase/decrease in long-term expected returns (but not for the horizons we are considering in this paper). There are two sets of solutions to these problems. The first type of solution addresses the problems directly. This can be achieved, for example, with the use of shrinkage methods for the estimation of the covariance matrices (via factor models, principal components, etc.) and the use of equilibrium returns for the base scenario of expected returns above which views can be added (i.e. the Black- Litterman model). The second type of solution just hides the dirt under the carpet by using, for instance, portfolio constraints. This approach implicitly acknowledges that inputs are poorly estimated and tries to minimize the impact of possible parameter misspecification. For simplicity and transparency, in this paper we use the dirt under the carpet approach. It should be stressed, however, that we are using very noisy measures of expected returns. For example, even if momentum is indeed a feature of the markets considered here, nothing tells us how the magnitude, sign, or relative rank of past returns relates to the magnitude of expected future returns. For the case of the covariance matrix, using daily returns helps reduce estimation error. However, as with expected returns, the estimated covariance matrix is but a noisy forecast of true future risk. 2

3 that tends to remain above (below) its long-run mean when it is currently above (below) its long-run mean. For the case of returns, the term momentum is often used to describe this phenomenon, whereas the term clustering is used in reference to volatility. The stronger the persistence, the longer it takes for the variable to return to its long-run mean. If both returns and risk are persistent, we will benefit from using more timely information in an allocation model. In fact, empirically, there is strong support for persistence in both returns and volatility. For example, in a cross-market momentum paper (Exploiting Cross-Market Momentum), we tested for the persistence in returns (i.e., momentum), and found supporting evidence. The evidence for the case of volatility is even stronger. The standard modelling of volatility in asset returns relies on autoregressive processes. These models account for mean-reversion as volatility is expected to revert to a mean level, but are also consistent with volatility clustering, as periods of higher/lower than average volatility are likely to be followed by high/low volatility. As usual, the why? is harder to answer. For the case of returns, behavioural finance arguments appear to provide the most convincing explanations. Two basic cognitive biases are commonly suggested as reasons for momentum: underreaction and overreaction. With underreaction, prices are slow to react to news, and returns exhibit positive serial correlation. In the case of overreaction, investors use past price movements to infer future price movements, and push prices in the same direction as previous price moves, again resulting in positive autocorrelation. Despite the empirical evidence, momentum in stock returns remains somewhat controversial as it implicitly implies market inefficiency. The idea that risk is persistent is less controversial, as is not inconsistent with market efficiency. Moreover, it is widely agreed that shocks to volatility have persistent effects. One possibility is that the magnitude of the economic shocks change in a persistent way. Additionally, the investor s uncertainty on information provided by the economic news flows may also vary over time depending on the underlying state of the economy. Data, methodology, and results In our previous paper, we included all components that are usually considered asset classes by market participants, but here we take a more conservative stance. We drop the less liquid asset classes that may also have shorter time series Table 2. Basic Statistics Asset Classes %, Bloomberg codes in capitals Statistic Annualized Returns Standard Deviation MSCI North America NDDUNA MSCI Europe NDDUE MSCI Asia Pacific NDDUP MSCI EM NDUEEGF Real Estate GPRJPPLU EMBI JPEMCOMP Commodities DJAIGTR Global GBI JHDCGBIG Cash JPCAUS3M and/or may suffer from data quality issues, such as hedge funds and high-yield. Results for alternative portfolios including these asset classes are available upon request. Our analysis takes the point of view of a USD based investor. All returns are daily and all statistics are annualized based on daily information. We use two different data sets to account for limitations in availability of reliable series for certain asset classes. In both cases, the time series end in June 29, Our aim is to replicate the current potential choices of a typical pension fund with a focus on easily tradable assets: 1) , 9 asset classes, daily 4 ; 2) , 6 asset classes, daily 5. Table 2 shows the basic statistics for the 9 asset classes, considered in the shorter sample. We initially focus on this sample in order to compare to the results in our previous paper. They are USD cash (3-month Libor), global govern- 3. We apply a similar concept by using both realized volatility and correlation in daily returns. Recent research has shown that realized risk based on high frequency data can be superior to using parametric models of the ARCH family. We compute standard deviations using rolling windows of past months (for example, 125 daily returns). We also used an iterative GARCH-approach that adds new data as we advance in time. In the GARCH case, standard deviations will tend to revert to a long-term mean that is constantly updated. 4. Table 2 includes the bloomberg codes for all the total return indices used. Whenever they were not available for the full sample we used the closest proxy. Details available upon request. 5. In this case, the strategy allocates into US equities (Fama-French market factor series), World ex US equities (Datastream), Bonds (GBI US, and before that, a return series constructed using Constant Maturity 10-year bond yield), Commodities (DJAIG, and before that, GSCI), Real Estate (NAREIT) and Cash (JPMorgan Cash Index, and before that Fama-French Cash Return). Other variations are available upon request. NAREIT is monthly in the beginning of the sample. 3

4 Table 3. Regression coefficients and t-stats (in italics) 3-month returns/volatility on past x-business day returns/volatility Asset Classes 125-day 185-day 250-day Returns MSCI North America MSCI Europe MSCI Asia Pacific MSCI EM Real Estate EMBI Commodities Global GBI Volatility MSCI North America MSCI Europe MSCI Asia Pacific MSCI EM Real Estate EMBI Commodities Global GBI ment bonds hedged in USD (JPMorgan s GBI), emerging market bonds (EMBI), four equity markets (North America, Europe, Asia Pacific, and emerging markets, all in USD), real estate, and commodities. The Dynamic Markowitz strategy is based on two principles: momentum and mean variance optimization. The first exploits the empirically documented predictability of risk and return over medium-term horizons, executed in our base case by using returns and risks over the past 6 months (125 business days). Accordingly, our first task is to test the empirical support for momentum (persistence) in risks and Box 2. Risk Budgeting in Asset Allocation Here we discuss one of the multiple interpretations of risk budgeting, an allocation strategy defined in terms of risk exposure per asset or asset class. We will not discuss risk budgeting in its general context, only its application to asset allocation. Risk budgeting is certainly an important organizational and managerial tool, especially when investment decisions are not fully centralized. It also provides a simple way to define target exposures of alpha strategies expected to have low correlation. For example, we use this approach in our GMOS publication. In fact, later we will show that a simple risk budgeting approach outperforms the optimal static mean variance efficient portfolio. However, we also show that risk budgeting underperforms full-blown dynamic asset allocation because the former ignores important changes in both expected returns and correlations. returns in these asset classes. To this end, we run regressions of quarterly returns on past 60-, 125-, 185-, 250-day returns and repeat the process for volatility. In both cases, we use an report coefficients and t-statistics based on Newey-West adjusted standard errors to account for the overlapping nature of the regressions. Table 3 shows that most of the indices exhibit positive (and mostly statistically significant) autocorrelations in both returns and volatility. The second feature of our strategy is executed by calculating an efficient frontier of these 6-month rolling returns and risks, and picking a point along it at the 8% risk level 6. This vol cap is intentionally very close to the volatility of the equally-weighted benchmark. We will not discuss here the practical issues in implementing traditional portfolio optimization and reserve Box 1 for a short summary. In particular, since our measures of expected 6. Or less if the maximum risk portfolio has less than 8% risk. In this optimization, we are ignoring three small but relevant results in portfolio optimization. First, we wrongly assume that the volatility of the risk-free rate is indeed risk (time variation in rates is not risk as the cashflow of the riskfree investment is known in advance regardless of the rate level). Second, we ignore the two-fund separation property. Third, we use past return information for the risk-free asset, even though current yield is sufficient to determine future performance. The reason is that these simplifying assumptions have minor effects, but make the problem simpler to readers less familiar with the results of traditional portfolio optimization. 4

5 Table 4. Dynamic Markowitz Basic Statistics for different lookbacks Statistics 125-day 185-day 250-day Average Excess Return (over cash) 10.7% 8.7% 7.7% Total Return Geometric Average 15.6% 13.4% 12.5% Standard Deviation 7.8% 7.9% 7.7% Sharpe Ratio Max (monthly) 6.0% 6.3% 6.8% Min (monthly) -9.4% -7.3% -7.4%. This table present basic statistics of the DM strategy when parameters are based on three distinct lookback periods. Chart 1. Cumulative Performance of DM Strategy Chart 2. Average Portfolio Global GBI 11% Commodities 11% Cash 14% EMBI 14% Strategy Equally -Weighted MSCI North America 10% Real Estate 14% MSCI Europe 10% MSCI NA GBI MSCI Asia Pacific 6% MSCI EM 10% return and risk are noisy, due to estimation error and possible misspecification, we introduce portfolio constraints. Specifically, we compute a constrained solution that sets weights for risky asset classes between 0% and 25%. We allow up to 50% investment in the proxy for cash in order to guarantee that the volatility constraint will always be satisfied. Therefore, at any point in time the portfolio is Table 5. Comparison Strategies Equally- Static Risk Statistics Weighted Markowitz Budgeting Average Excess Return 5.7% 7.2% 5.8% Total Return Geometric Average 10.3% 11.9% 10.5% Standard Deviation 7.3% 7.0% 5.2% Sharpe Ratio Max (monthly) 6.10% 6.70% 6.4% Min (monthly) -13.9% -13.8% -6.5% Table 6. Comparison to Momentum Strategies Relative Risk-adjusted Rel. Statistics Momentum Momentum Average Excess Return 10.3% 9.8% Standard Deviation 9.2% 8.3% Sharpe Ratio Alpha (DM over Mom) 4.1% 2.5% T-stats Beta (DM on Mom) DM stands for Dynamic Markowitz strategy and Alpha shows the alpha of the DM strategy over each of these simple momentum rules resulting from the regression (Jensen s Alpha). invested in at least 3 asset classes and there is no short selling 7. The constraints are modified in the robustness section of this paper. The Dynamic Markowitz strategy outperforms each of the individual asset classes, both in total returns and on a riskadjusted basis, as illustrated in Chart 1. Table 4 shows that performance is stronger for the 125-business day lookback period, but this result is particular to the more recent data used here, as we will examine in more depth later on. The strategy s delivered volatility is close to but below our target volatility in all cases, reflecting the fact that past volatility does indeed provide a reasonable forecast of future volatility. Chart 3 presents the strategy s average monthly allocations, showing the strategy uses all selected assets over time. Comparing with component strategies Our Dynamic Markowitz (DM) strategy provides strong results, but how good are they compared to other strategies and where do the returns really come from? To answer these 7. In order to make the solution faster and more reliable, we use discrete steps in the weights. For example, a 5% step implies that the possible weights are 0%, 5%, 10%, 15%, 20% and 25%. Results with continuous weights are marginally different. 5

6 Chart 3. Comparing with Component Strategies Sharpe ratio Equallyweighted Risk- Risk Relative Adjusted Static Markowitz Budgeting Momentum Mean variance Momentum Dynamic Markowitz questions, we build a number of comparison portfolios that contain some, but not all of the elements of the DM strategy. As a reference point, we also report the returns to a simple equally-weighted portfolio (EW). Next come two portfolios that contain elements of mean variance optimization. The first is what we call Static Markowitz. It is a fixed allocation calculated from an efficient frontier that uses the full period risks (sample covariance matrix) and delivered returns of the component asset classes. By definition, this strategy represents the most efficient constant-weight portfolio that an investor could have selected. This strategy is not feasible in the sense that investors could not have known ex ante the returns and risk parameters. A related comparison portfolio is one based on Risk Budgeting where the investors maintains constant risk allocations to each asset class, using rolling recent volatilities, but not returns (see Box 2). The fixed-risk allocations are based on the portfolio weights from the Static Markowitz strategy in conjunction with the average full sample volatility of each asset class. Once again, this is a proxy for the most efficient risk-weighted portfolio, but it is not feasible ex ante. Next, we present two comparison portfolios based on momentum in recent asset class returns (Relative Momentum) or in Sharpe ratios (Risk-Adjusted Relative Momentum). First, we apply the simple relative momentum approach where we go long only 4 best performers out of the 9 asset classes using equal weights. Second, we apply a modified version of the relative momentum approach, where we rank asset classes by the past normalized excess return, i.e. recent Sharpe ratios. Table 7a. Information Ratios for DM against comparison strategies Comparison Strategies 125-day 185-day 250-day Equally-weighted Portfolio Average Portfolio Static Markowitz Risk Budgeting This table shows information ratios for the Dynamic Markowitz strategy against different comparison strategies. Average portfolio is the portfolio that uses constant portfolio weights equal to the allocation shown in Chart 2. These calculations do not account for differences in volatility/beta as we do in 7b and 7c. Table 7b. Alpha relative to different comparison strategies Comparison Strategies 125-day 185-day 250-day Equally-weighted Portfolio (regression) Equally-weighted Portfolio (exc. return) Average Portfolio (regression) Average Portfolio (exc. return) Static Markowitz (regression) Static Markowitz (exc. return) Risk Budgeting (regression) Risk Budgeting (exc. return) This table shows two alternative measures of alpha against different comparison strategies. In one case, we simply compute the excess return over the respective comparison. In the other case, we use the intercept of an OLS regression on the respective comparison strategy (Jensen s Alpha). The regression accounts for the difference in risk of the strategies that is clear in the different level of volatility. Average portfolio is the portfolio that uses constant portfolio weights equal to the allocation shown in Chart 2. Table 7c. Alpha t-stats Comparison Strategies 125-day 185-day 250-day Equally-weighted Portfolio (regression) Equally-weighted Portfolio (exc. return) Average Portfolio (regression) Average Portfolio (exc. return) Static Markowitz (regression) Static Markowitz (exc. return) Risk Budgeting (regression) Risk Budgeting (exc. return) This table shows the t-statistics of the alphas reported in table 7b. Average portfolio is the portfolio that uses constant portfolio weights equal to the allocation shown in Chart 2. These comparisons allow us to investigate the sources of the added value in our Dynamic Markowitz strategy. As depicted in Chart 3, both mean variance optimization and momentum make significant contribution to the improvement in Sharpe ratios, with momentum being more important. Additionally, all these strategies have a higher Sharpe ratio than the equally-weighted basket. Tables 5 and 6 on the previous page summarize the performance statistics of these five portfolios over the period

7 Each of the panels in Table 7 reports additional metrics of outperformance with respect to the non-momentum strategies. Similar results hold with respect to momentum strategies, but excluded for the sake of conciseness. Panel A reports that the information ratios of DM with respect to the alternative strategies are all positive. Panel B shows that DM excess returns over the non-momentum strategies are all positive, and even more so when the differences in volatility/ beta are accounted for. Panel C reports corresponding t- statistics indicating that those alphas are statistically significant. Long-short versions The benchmark strategy is long-only because it is easier to implement for most investors. In this section, we analyse the implications for alpha of long-short strategies. We consider two variations. First, we compute a constrained non-directional excessreturn portfolio, that we call Vol-cap long-short. This strategy selects weights of between -25% and 25% such that the sum of the weights on the risky assets is zero and all capital is invested in cash. This portfolio is said to be nondirectional because the net exposure to risky assets is zero. We set a volatility cap of 6%. Second, we compute a more flexible version, we call Flexible long-short, where we remove the vol limit. The leverage of this portfolio is free to move over time, only limited indirectly by the portfolio constraints above. In this case, we reintroduce the standard Sharpe ratio maximization. Table 8. Long-short Strategies Statistics Vol-cap LS Flexible LS Average Excess Return 5.3% 4.8% Standard Deviation 5.4% 5.6% Sharpe Ratio Alpha EW 4.5% 5.2% T-stats Beta EW Table 8 shows that the long-short strategies have a smaller Sharpe ratio than the long-only version, but the reason for this is the lack of exposure to the overall market. One could add beta to the strategy to increase the Sharpe ratio, but this hides the true alpha. Since the long-short strategies are constructed to be non-directional, their Sharpe ratios are in fact equal to their information ratios. Robustness analysis We will show that the optimal rule above is only one of the many rules that work, reassuring us that the idea makes sense and that the above results do not arise from pure luck. There is no theory to tell us exactly when we should rebalance the portfolio or how many months should be used to compute past risk and return. Hence, as a robustness test, we report the performance of the DM strategy for slightly altered rules. The following robustness checks are considered: 1. changing observation period, rebalancing frequency and portfolio and volatility constraints: We increase and Table 9. Robustness to change in parameters Observation Period Rebalancing Frequency Portfolio Average Exc. Standard Alpha Beta (days) (months) Volatility Constraint Constraints Return (cash) Deviation Sharpe Ratio (over EW) t-stat (EW) % 25% 5.7% 8.0% % % 25% 10.7% 7.8% % % 25% 8.7% 7.9% % % 25% 7.7% 7.7% % % 25% 11.2% 7.5% % % 25% 10.3% 7.5% % % 25% 10.7% 7.8% % % 25% 9.9% 8.1% % % 25% 9.4% 6.4% % % 25% 10.2% 7.2% % % 25% 10.7% 7.8% % % 25% 11.3% 8.4% % % 20% 9.9% 7.4% % % 25% 10.7% 7.8% % % 33% 11.0% 8.1% % % 50% 11.5% 8.4% %

8 Table 10. Effect of excluding asset classes Average Exc. Return Information Excluding Asset (cash) Ratio (EW) MSCI North America 8.9% 0.56 MSCI Europe 9.9% 0.75 MSCI Asia Pacific 11.7% 1.11 MSCI EM 9.4% 0.69 Real Estate 9.1% 0.61 EMBI 10.3% 0.87 Commodities 10.3% 0.85 Global GBI 10.5% 0.97 No exclusions 10.7% 0.93 Chart 5. DM information ratio vs past asset class dispersion IR y = x R 2 = Past Dispersion Chart 4. Effect of excluding asset classes on Sharpe ratios MSCI North America MSCI Europe MSCI Asia Pacific MSCI EM Real Estate EMBI Commodities Global GBI No exclusions DM Information ratio is the information ratio of the DM strategy with respect to the equally-weighted portfolio computed using 250 business days and dispersion is the 250-day average of the cross-sectional standard deviation in asset classes returns based on daily information. Chart 6. DM information ratio vs past EW volatility IR y = x R 2 = Past Volatility - Equally-weighted Portfolio DM Information ratio is the information ratio of the DM strategy with respect to the equally-weighted portfolio computed using 250 business days. decrease the number of months used to compute recent risk and recent performance and the rebalancing frequency. We consider the effect of relaxing or tightening the portfolio or volatility constraints. 2. excluding asset classes: We consider the effect of excluding some of the asset classes, showing that the results do not rely on any one particular asset class. Changing observation, rebalancing, constraints... Table 9 reports the effect of changing the assumed values of four basic parameters: ranking period, rebalancing period, volatility constraint, and portfolio constraints. We consider a reasonable range of parameters that are compatible with both momentum and the fluctuations in risk and we also avoid extreme allocations. The return statistics remain positive and attractive even when we depart significantly from the benchmark parameters. The alpha against the equally-weighted portfolio is statistically and economically significant in almost all reported cases and remains significant even outside the reported ranges. The Sharpe ratio is above 1.0 in almost all cases, with the exception of the rules with very short lookback periods. Our base case scenario (shaded) is not even the optimal strategy in this sample, as other variations have higher alphas, though sufficiently close. For example, both more frequent rebalancing and lower volatility constraints appear to be beneficial. Excluding asset classes A strategy is robust when it is not highly sensitive to the asset classes that are used. Our analysis shows that not one 8

9 Chart 7. DM information ratio vs past EW return IR y = x R 2 = Past Return - Equally-weighted Portfolio DM Information ratio is the information ratio of the DM strategy with respect to the equally-weighted portfolio computed using 250 business days and past EW return is the return of the equally-weighted portfolio in the past 250 business days. of the asset classes are essential for the performance of this strategy. We compute the returns of the strategy when only 7 out of 8 risky asset classes are included (risk-free is always used) and consider all 8 possible combinations. Table 10 shows the average excess return over cash and information ratios for all possible exclusions, while Chart 4 provides a visual test of the stability of Sharpe ratio stability. When does this strategy work well? The strategy outperforms its equally-weighted and more active benchmarks almost always and its alpha has weak correlation with market returns in most cases. Here we look at the relation between DM s alpha (above equally-weighted portfolio) and the return and volatility of the equallyweighted portfolio, as well as with the dispersion in returns of the underlying asset classes (see Charts 5-7). This allows us to understand which market conditions are most conducive to positive performance and not directionality, as we compare to the past realization of these variables. There is a more clear correlation to dispersion (positive) and volatility (negative). Chart 8 shows a steady decrease in dispersion and, therefore, an increase in correlation among asset classes over this period. Even though information ratios may remain positive, excessive correlation is overall the least favourable scenario for a momentum strategy. The negative relation to volatility may be partially mechanical as the volatility cap may become too restrictive when overall volatility is very high. Chart 8. Rolling 250-day dispersion in performance 2.5% 2.0% 1.5% 1.0% 0.5% 0.0%. 250-day dispersion is the 250-day average of the cross-sectional standard deviation in asset classes returns based on daily information. Table 11. Basic Statistics DM with Longer Sample for different lookback periods Statistic 125-day 185-day 250-day Average Return 13.8% 15.2% 15.5% Sharpe Ratio Alpha (over EW) 0.88% 2.00% 2.36% t-stats Standard Deviation 5.9% 5.7% 5.8% Max (monthly) 7.3% 7.4% 7.5% Min (monthly) -12.1% -8.2% -8.2% y = -2E-06x R 2 = Longer periods and intra-asset class We also tested the strategy, with a longer sample, starting in In this test, we use allocations of between 0% and 33.3% as we only have 6 asset classes. Even though we are forced to use fewer assets because of lack of data, performance remains interesting with an alpha above 2% a year for longer lookbacks. Table 11 reports the main statistics for this new strategy. Using longer lookback periods delivered higher returns. In general, the DM strategy performs only slightly worse with this smaller set of asset classes. The contribution of momentum in international allocation within an asset class has been previously analyzed in the academic literature, particularly for equities 8. Here we test whether we can add value by using this optimization Year 8. For international equities using indices, see for example Bhojraj, S. and Swaminathan, B, Macromomentum: Returns Predictability in International Equity Indices, The Journal of Business, volume 79 (2006), pages For international equities momentum using individual stocks, see for example Rouwenhorst, K.G., International Momentum Strategies. Journal of Finance, Vol. 53, No. 1, pp , February

10 procedure. We do this for international country equity indices (Datastream indices) and also short-term bond indices (1-year or 2-year constant maturity JPMorgan Country Indices, for example). We choose these bond indices as previous work shows that short-term bonds are more prone to momentum 9, as they are more sensitive to short-term changes in economic expectations. In both cases, we use the full available sample of daily total returns for Australia, Canada, Denmark, Germany, Japan, Sweden, the UK and the US. Once again, we set the volatility cap close to the volatility of the corresponding equally-weighted basket. In general, for the intra asset class cases we find that adding recent volatility/risk information does not improve performance vis-à-vis a simple momentum signal 10. However, DM and the relative momentum-based strategy both outperform a passive benchmark, with Sharpe ratios of slightly above 1 in the case of equities, versus 0.65 for the corresponding equally-weighted equity portfolio. Similar results hold when restricting attention to bonds, with DM showing a slight edge over the simple relative momentum strategy when using 185- or 250-day lookback periods. One of the reasons both strategies perform similarly is that movements in volatility within an asset class are more highly correlated than those between asset classes. A principal component analysis of asset volatilities shows that the first principal component explains only 54% percent of the timeseries variation in variances in the cross-market case, versus 81% for the stocks-only case (where performance in DM and relative momentum are more similar). Furthermore, there is also less dispersion in the levels of risk within an asset class, and, therefore, volatility information is somewhat less useful. Volatility timing/risk budgeting in asset allocation There have been many practitioner publications analyzing the benefits of risk budgeting in asset allocation. In our opinion, some of these studies exaggerate the contrast between asset and risk allocation, implicitly making the wrong assumption that asset allocation has to be a static problem. Several academic papers have also analyzed these issues and the benefits of using timely volatility information. For instance, Fleming et al (2001) 11 analyze a portfolio allocation problem with equities, bonds, gold and cash using constant expected returns but a time-varying covariance matrix in a rule that rebalances daily. Johannes et al (2002) 12 perform a similar analysis with the S&P 500 index and cash and attempts to model the persistence in expected returns. A word of caution, however, is warranted. Part of the return in these strategies is attributable to return persistence and its correlation to changes in risk and not risk itself 13. Conclusion This analysis shows that it makes sense to exploit return and risk momentum/persistence in standard mean variance asset allocation, when considering a diversified set of asset classes. Nonetheless, the same caveats that we raised in Exploiting Cross-Market Momentum apply here. For example, persistence in risk and momentum in returns may disappear or change over time, as investors take advantage of these empirical regularities. And if they do, the direction of a possible transformation is not clear. Moreover, an asset allocation decision should not be based solely on persistence/momentum arguments. Value considerations, for example, are also vital to optimal asset allocation. One should view the dynamic rules presented in this paper as an overlay strategy to an existing portfolio, creating a separate and important source of alpha. 9. See Salford, G., Momentum in Money Markets, Investment Strategies No. 32, JPMorgan for an analysis of momentum in individual bond markets. 10.These strategies have a strong currency component as we are using unhedged returns. Results with hedged returns are also available. 11.Fleming, J., Kirby, C., and B. Ostdiek, The Economic Value of Volatility Timing, The Journal of Finance, Vol LVI, No. 1, Feb Johannes, M., Polson, N., and J. Stroud, Sequential Optimal Portfolio Performance: Market and Volatility Timing, Working Paper, Columbia Business School, Related to the empirical risk-return trade-off (negative) which contradicts the theoretical risk-return trade-off (positive). It is beyond the scope of the paper to discuss problems with the interpretation of the empirical trade-off, but it is our view that better use of filtering and conditional information can explain this apparent puzzle. Recent academic literature supports this view. 10

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