markowitz in tactical asset allocation

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Markowitz in tactical asset allocation

Global Market Strategy

J.P. Morgan Securities Ltd.

London August 7, 2007

Ruy M. RibeiroAC

(44-20) 7777-1390

[email protected]

Jan Loeys(44-20) 7325-5473

jan .loeys@jp morgan.com

www.morganmarkets.com

I N V E S T M E N T S T R A T E

G I E S : N O .

3 5

• 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 and0.77 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 momentum1. 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% in-

sample alpha.

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.

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

The certifying analyst is indicated by an AC. See page 11 for analyst

certification and important legal and regulatory disclosures.

We thank Vadim di Pietro for valuable comments and discussion.


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This Dynamic Markowitz strategy, as we call it, performed

well over our 1994-2007 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 equallyweighted portfolios, as well as those based solely on long-

term 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 riskPersistence 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 2006. 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. 77-

91, 1952.

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%

Source: JPMorgan. 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 2006.

Box 1. Drawbacks in standard portfolio optimiza-

tion and possible solutions

Many portfolio allocation applications assume stationarityin 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 compo-

nents 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.


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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, 2007.

Our aim is to replicate the current potential choices of a

typical pension fund with a focus on easily tradable assets:

1) 1994-2007, 9 asset classes, daily4;

2) 1971-2007, 6 asset classes, daily5.

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” meanthat 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.

Table 2. Basic Statistics – Asset Classes – 1994-2007%, Bloomberg codes in capitals

Statistic Annualized Returns Standard Deviation

MSCI North America – NDDUNA 10.5 16.5

MSCI Europe – NDDUE15 11.3 16.8

MSCI Asia Pacific – NDDUP 3.0 18.8

MSCI EM – NDUEEGF 5.3 17.8

Real Estate – GPRJPPLU 13.6 10.9

EMBI – JPEMCOMP 11.4 14.2

Commodities – DJAIGTR 9.6 13.4

Global GBI – JHDCGBIG 6.4 2.8

Cash – JPCAUS3M 4.4 0.2

Source: JPMorgan.

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 refer-

ence 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 serialcorrelation. 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 resultsIn 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


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returns in these asset classes. To this end, we run regres-

sions 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 calculat-

ing an efficient frontier of these 6-month rolling returns and

risks, and picking a point along it at the 8% risk level6. 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 ashort summary. In particular, since our measures of expected

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.

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 risk-

free 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.

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

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 0.17 0.28 0.38

0.83 2.04 2.61

MSCI Europe 0.06 0.11 0.32

0.21 0.77 1.67

MSCI Asia Pacific 0.13 0.22 0.12

0.93 1.56 0.72

MSCI EM 0.03 0.01 0.04

0.38 0.10 0.29

Real Estate 0.31 0.27 0.27

2.47 3.14 1.91

EMBI -0.23 -0.03 0.09

-2.13 -0.18 0.62

Commodities 0.27 0.14 0.16

1.60 0.64 0.63

Global GBI 0.06 0.09 -0.08

0.54 0.72 -0.42

Volatility

MSCI North America 0.71 0.75 0.73

8.84 9.30 7.63

MSCI Europe 0.58 0.58 0.60

6.56 5.37 4.67

MSCI Asia Pacific 0.38 0.34 0.45

4.18 3.90 5.06

MSCI EM 0.45 0.49 0.41

3.44 3.75 3.14

Real Estate 0.48 0.60 0.58

2.04 2.76 2.27

EMBI 0.39 0.51 0.53

3.83 4.49 4.35

Commodities 0.74 0.81 0.86

6.84 9.97 10.44

Global GBI 0.38 0.53 0.54

2.00 3.16 2.89

Source: JPMorgan


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return and risk are noisy, due to estimation error and possi-

ble 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

invested in at least 3 asset classes and there is no short

selling7. The constraints are modified in the robustness

section of this paper.

The Dynamic Markowitz strategy outperforms each of theindividual asset classes, both in total returns and on a risk-

adjusted 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 volatil-

ity. Chart 3 presents the strategy’s average monthly alloca-

tions, showing the strategy uses all selected assets over

time.

Comparing with component strategiesOur 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

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 0.77 1.03 1.12

Max (monthly) 6.10% 6.70% 6.4%

Min (monthly) -13.9% -13.8% -6.5%

Source: JPMorgan

Table 6. Comparison to Momentum Strategies

Relative Risk-ad justed Rel .

Statistics Momentum Momentum

Average Excess Return 10.3% 9.8%Standard Deviation 9.2% 8.3%

Sharpe Ratio 1.13 1.19

Alpha (DM over Mom) 4.1% 2.5%

T-stats 3.64 3.01

Beta (DM on Mom) 0.73 0.88

Source: JPMorgan. 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).

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%



Max (monthly) 6.0% 6.3% 6.8%

Min (monthly) -9.4% -7.3% -7.4%

Source: JPMorgan. 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

Source: JPMorgan

0

100

200

300

400

500

600

700

1994 1996 1998 2000 2002 2004 2006

Strategy

MSCI NA

GBI

Equally-Weighted

Source: JPMorgan

Chart 2. Average Portfolio

MSCI North America10%

MSCI Europe

10%

MSCI Asia Pacific

6%

MSCI EM

10%

Real Estate14%

EMBI

14%

Commodities

11%

Global GBI

11%

Cash14%

MSCI North America10%

MSCI Europe

10%

MSCI Asia Pacific

6%

MSCI EM

10%

Real Estate14%

EMBI

14%

Commodities

11%

Global GBI

11%

Cash14%

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.


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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 haveselected. 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 0.94 0.57 0.40

Average Portfolio 0.95 0.59 0.41

Static Markowitz 0.58 0.27 0.10

Risk Budgeting 0.94 0.58 0.40

Source: JPMorgan. 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


Equally-weighted Portfolio (regression) 6.4 4.4 3.8

Equally-weighted Portfolio (exc. return) 4.7 2.8 2.0

Average Portfolio (regression) 6.5 4.5 3.8 Average Portfolio (exc. return) 4.8 2.9 2.1

Static Markowitz (regression) 5.7 3.2 2.8

Static Markowitz (exc. return) 3.3 1.4 0.6

Risk Budgeting (regression) 3.0 0.8 0.3

Risk Budgeting (exc. return) 4.7 2.8 2.0

Source: JPMorgan. 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


Equally-weighted Portfolio (regression) 4.6 3.2 2.7

Equally-weighted Portfolio (exc. return) 3.3 2.0 1.4

Average Portfolio (regression) 4.6 3.2 2.8

Average Portfolio (exc. return) 3.3 2.1 1.4

Static Markowitz (regression) 3.6 2.2 1.9

Static Markowitz (exc. return) 2.0 0.9 0.4

Risk Budgeting (regression) 2.1 0.6 0.2

Risk Budgeting (exc. return) 3.3 2.0 1.4

Source: JPMorgan. 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 1994-2007.

Source: JPMorgan

Chart 3. Comparing with Component StrategiesSharpe ratio

StaticMarkowitz

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Equally-

weighted

RelativeMomentum

Dynamic

Markowitz

RiskBudgeting

Risk-

Adjusted

Mean

variance

Momentum

StaticMarkowitz

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Equally-

weighted

RelativeMomentum

Dynamic

Markowitz

RiskBudgeting

Risk-

Adjusted

Mean

variance

Momentum


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Each of the panels in Table 7 reports additional metrics of

outperformance with respect to the non-momentum strate-

gies. Similar results hold with respect to momentum strate-

gies, 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 versionsThe 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 excess-

return 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 non-

directional 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 portfolioconstraints above. In this case, we reintroduce the standard

Sharpe ratio maximization.

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 infact equal to their information ratios.

Robustness analysisWe 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 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 0.97 0.85

Alpha – EW 4.5% 5.2%

T-stats 2.99 3.26

Beta – EW 0.12 -0.07

Source: JPMorgan

Source: JPMorgan

Table 9. Robustness to change in parametersAlpha Beta

over

60 3 8% 25% 5.7% 8.0% 0.71 0.7% 0.52 0.87

125 3 8% 25% 1 0.7% 7.8% 1.37 5.7% 4.13 0.83

185 3 8% 25% 8.7% 7.9% 1.10 3.7% 2.68 0.84

250 3 8% 25% 7.7% 7.7% 1.00 2.9% 2.14 0.82

125 1 8% 25% 11.2% 7.5% 1.49 6.6% 4.78 0.78

125 2 8% 25% 10.3% 7.5% 1.37 5.7% 4.15 0.79

125 3 8% 25% 1 0.7% 7.8% 1.37 5.7% 4.13 0.83

125 6 8% 25% 9.9% 8.1% 1.22 4.6% 3.35 0.89

125 3 6% 25% 9.4% 6.4% 1.47 5.3% 4.64 0.67

125 3 7% 25% 10.2% 7.2% 1.42 5.6% 4.34 0.76

125 3 8% 25% 1 0.7% 7.8% 1.37 5.7% 4.13 0.83

125 3 9% 25% 11.3% 8.4% 1.35 5.9% 4.00 0.89

125 3 8% 20% 9.9% 7.4% 1.33 5.0% 4.01 0.81

125 3 8% 25% 10.7% 7.8% 1.37 5.7% 4.13 0.83

125 3 8% 33% 11.0% 8.1% 1.35 6.0% 3.91 0.83

125 3 8% 50% 11.5% 8.4% 1.37 6.6% 3.87 0.80

Sharpe Ratio t-statVolatility Constraint

Standard

Deviation

Observation Period

(days)

Rebalancing Frequency

(months)

Portfolio

Constraints

Average Exc.

Return (cash)


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decrease the number of months used to compute recent

risk and recent performance and the rebalancing fre-

quency. 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 theresults 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.

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

Source: JPMorgan

The return statistics remain positive and attractive even

when we depart significantly from the benchmark param-

eters. 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 shortlookback 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

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.401.60

M S C I N o r t h A m

e r i c a

M S C I E u

r o p e

M S C I A s i a P a

c i f i c

M S C I E M

R e a l E s t a t e

E

M B I

C o m m o d

i t i e s

G l o b a l

G B I

N o e x c l u s

i o n s

Chart 4. Effect of excluding asset classes on Sharpe ratios

Source: JPMorgan

Chart 5. DM information ratio vs past asset class dispersionIR

Source: JPMorgan. 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.

y = 41.054x - 0.1071R2 = 0.1235

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.21.4

1.6

1.8

0.000 0.005 0.010 0.015 0.020 0.025

Past Dispersion

Chart 6. DM information ratio vs past EW volatilityIR


equally-weighted portfolio computed using 250 business days.

y = -9.4171x + 1.0844

R2 = 0.1742

-0.4

-0.2

0.00.2

0.4

0.6

0.8

1.0

1.21.4

1.6

1.8

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Past Volatility - Equally-weighted Portfolio


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Longer periods and intra-asset classWe also tested the strategy, with a longer sample, starting

in 1974. 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, perform-

ance 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 equities8. Here we test

whether we can add value by using this optimization

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 equally-

weighted 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, there-

fore, 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.

Table 11. Basic Statistics – DM with Longer Sample for differentlookback periods

Statistic 125-day 185-day 250-day

Average Return 13.8% 15.2% 15.5%


Alpha (over EW) 0.88% 2.00% 2.36%

t-stats 1.24 2.71 3.18


Max (monthly) 7.3% 7.4% 7.5%

Min (monthly) -12.1% -8.2% -8.2%

Source: JPMorgan

Chart 7. DM information ratio vs past EW returnIR

Chart 8. Rolling 250-day dispersion in performance

Source: JPMorgan. 250-day dispersion is the 250-day average of the cross-sectional standard deviation in

asset classes returns based on daily information.

y = -2E-06x + 0.105

R2 = 0.8728

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

1995 1998 2001 2004 2006

Year


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.

y = 0.8393x + 0.3364

R2 = 0.0614

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.21.4

1.6

1.8

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4

Past Return - Equally-weighted Portfolio

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 429–

451. For international equities momentum using individual stocks, see for

example Rouwenhorst, K.G., International Momentum Strategies. Journal

of Finance, Vol. 53, No. 1, pp. 267-284, February 1998.


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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 momentum9, 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 perform-

ance vis-à-vis a simple momentum signal10. However, DM

and the relative momentum-based strategy both outperforma 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 time-series 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.

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 usingunhedged 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 2001

12.Johannes, M., Polson, N., and J. Stroud, Sequential Optimal Portfolio

Performance: Market and Volatility Timing, Working Paper, Columbia

Business School, 2002.

13.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.

Volatility timing/risk budgeting in assetallocationThere 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 inthese strategies is attributable to return persistence and its

correlation to changes in risk and not risk itself 13.

ConclusionThis 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 takeadvantage 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.


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[email protected]

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the research analyst denoted by an “AC” on the cover or within the document individually certifies, with respect to each security or issuer that the research analyst covers

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Ruy Ribeiro (44-20) 7777-1390

[email protected]

17. JPMorgan FX Hedging Framework , Rebecca Patterson

and Nandita Singh, March 2006

18. Index Linked Gilts Uncovered , Jorge Garayo and Francis

Diamond, March 2006

19. Trading Credit Curves I , Jonny Goulden, March 2006

20. Trading Credit Curves II , Jonny Goulden, March 2006

21. Yield Rotator , Nikolaos Panigirtzoglou, May 2006

22. Relative Value on Curve vs Butterfly Trades, Stefano Di

Domizio, June 2006

23. Hedging Inflation with Real Assets, John Normand, July

2006

24. Trading Credit Volatility, Saul Doctor and Alex Sbityokov,

August 2006

25. Momentum in Commodities, Ruy Ribeiro, Jan Loeys and

John Normand, September 2006

26. Equity Style Rotation, Ruy Ribeiro, November 2006

27. Euro Fixed Income Momentum Strategy, Gianluca Salford,

November 2006

28. Variance Swaps, Peter Allen, November 2006

29. Relative Value in Tranches I , Dirk Muench, November

2006

30. Relative Value in Tranches II , Dirk Muench, November

2006

31. Exploiting carry with cross-market and curve bond

trades, Nikolaos Panigirtzoglou, January 2007

32. Momentum in Money Markets, Gianluca Salford, May

2007

33. Rotating between G-10 and Emerging Markets Carry,

John Normand, July 2007

34. A simple rule to trade the curve, Nikolaos Panigirtzoglou,

August 2007

1. Rock-Bottom Spreads, Peter Rappoport, Oct 2001

2. Understanding and Trading Swap Spreads, Laurent

Fransolet, Marius Langeland, Pavan Wadhwa, Gagan

Singh, Dec 2001

3. New LCPI trading rules: Introducing FX CACI , Larry

Kantor, Mustafa Caglayan, Dec 2001

4. FX Positioning with JPMorgan’s Exchange Rate

Model , Drausio Giacomelli, Canlin Li, Jan 2002

5. Profiting from Market Signals, John Normand, Mar

20026. A Framework for Long-term Currency Valuation,

Larry Kantor and Drausio Giacomelli , Apr 2002

7. Using Equities to Trade FX: Introducing LCVI, Larry

Kantor and Mustafa Caglayan, Oct 2002

8. Alternative LCVI Trading Strategies, Mustafa

Caglayan, Jan 2003

9. Which Trade, John Normand, Jan 2004

10. JPMorgan’s FX & Commodity Barometer , John

Normand, Mustafa Caglayan, Daniel Ko, Nikolaos

Panigirtzoglou and Lei Shen, Sep 2004

11. A Fair Value Model for US Bonds, Credit and Equi-

ties, Nikolaos Panigirtzoglou and Jan Loeys, Jan 2005

12. JPMorgan Emerging Market Carry-to-Risk Model ,

Osman Wahid, February 2005

13. Valuing cross-market yield spreads, Nikolaos

Panigirtzoglou, January 2006

14. Exploiting cross-market momentum, Ruy Ribeiro and

Jan Loeys, February 2006

15. A cross-market bond carry strategy, Nikolaos

Panigirtzoglou, March 2006

16. Bonds, Bubbles and Black Holes, George Cooper,

March 2006

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