Global Market Strategy
J.P. Morgan Securities Ltd.
London
August 7, 2007
INVESTMENT STRATEGIES: NO. 35
Markowitz in tactical asset allocation
• 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 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% insample 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 work2. 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. RibeiroAC
(44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Jan Loeys
(44-20) 7325-5473
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
www.morganmarkets.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Table 1. Cross-market Momentum: out of sample; Jan 06 - May 07
Statistic
Average Return
Excess Return over EW
Standard Deviation
Sharpe Ratio
Max (monthly)
Min (monthly)
Value
15.29%
3.71%
6.72%
1.43
4.95%
-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.
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 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 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. 7791, 1952.
2
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 BlackLitterman 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.
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
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.
Table 2. Basic Statistics – Asset Classes – 1994-2007
%, Bloomberg codes in capitals
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
Statistic
Annualized Returns
MSCI North America – NDDUNA
10.5
MSCI Europe – NDDUE15
11.3
MSCI Asia Pacific – NDDUP
3.0
MSCI EM – NDUEEGF
5.3
Real Estate – GPRJPPLU
13.6
EMBI – JPEMCOMP
11.4
Commodities – DJAIGTR
9.6
Global GBI – JHDCGBIG
6.4
Cash – JPCAUS3M
4.4
Standard Deviation
16.5
16.8
18.8
17.8
10.9
14.2
13.4
2.8
0.2
Source: JPMorgan.
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 govern3. 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
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
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
Box 2. Risk Budgeting in Asset Allocation
250-day
Returns
MSCI North America
MSCI Europe
MSCI Asia Pacific
MSCI EM
Real Estate
EMBI
Commodities
Global GBI
0.17
0.83
0.06
0.21
0.13
0.93
0.03
0.38
0.31
2.47
-0.23
-2.13
0.27
1.60
0.06
0.54
0.28
2.04
0.11
0.77
0.22
1.56
0.01
0.10
0.27
3.14
-0.03
-0.18
0.14
0.64
0.09
0.72
0.38
2.61
0.32
1.67
0.12
0.72
0.04
0.29
0.27
1.91
0.09
0.62
0.16
0.63
-0.08
-0.42
Volatility
MSCI North America
MSCI Europe
MSCI Asia Pacific
MSCI EM
Real Estate
EMBI
Commodities
Global GBI
0.71
8.84
0.58
6.56
0.38
4.18
0.45
3.44
0.48
2.04
0.39
3.83
0.74
6.84
0.38
2.00
0.75
9.30
0.58
5.37
0.34
3.90
0.49
3.75
0.60
2.76
0.51
4.49
0.81
9.97
0.53
3.16
0.73
7.63
0.60
4.67
0.45
5.06
0.41
3.14
0.58
2.27
0.53
4.35
0.86
10.44
0.54
2.89
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 level6. This
vol cap is intentionally very close to the volatility of the
equally-weighted benchmark.
Source: JPMorgan
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
4
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.
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
Table 4. Dynamic Markowitz – Basic Statistics for different lookbacks
Table 5. Comparison Strategies
Statistics
125-day
Average Excess Return (over cash)
Total Return – Geometric Average
Standard Deviation
Sharpe Ratio
Max (monthly)
Min (monthly)
185-day
10.7%
15.6%
7.8%
1.37
6.0%
-9.4%
8.7%
13.4%
7.9%
1.10
6.3%
-7.3%
250-day
7.7%
12.5%
7.7%
1.00
6.8%
-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
Statistics
Average Excess Return
Total Return – Geometric Average
Standard Deviation
Sharpe Ratio
Max (monthly)
Min (monthly)
EquallyWeighted
Static
Markowitz
Risk
Budgeting
5.7%
10.3%
7.3%
0.77
6.10%
-13.9%
7.2%
11.9%
7.0%
1.03
6.70%
-13.8%
5.8%
10.5%
5.2%
1.12
6.4%
-6.5%
Source: JPMorgan
Table 6. Comparison to Momentum Strategies
700
600
Statistics
Strategy
500
MSCI NA
400
300
200
GBI
100
Equally -Weighted
Average Excess Return
Standard Deviation
Sharpe Ratio
Alpha (DM over Mom)
T-stats
Beta (DM on Mom)
Relative
Risk-adjusted Rel.
Momentum
Momentum
10.3%
9.2%
1.13
4.1%
3.64
0.73
9.8%
8.3%
1.19
2.5%
3.01
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).
0
1994
1996
1998
2000
2002
2004
2006
Source: JPMorgan
Chart 2. Average Portfolio
Cash
14%
MSCI North America
10%
MSCI Europe
10%
Global GBI
11%
MSCI Asia Pacific
6%
MSCI EM
10%
Commodities
11%
EMBI
14%
Real Estate
14%
Source: JPMorgan
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
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 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
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
Chart 3. Comparing with Component Strategies
Sharpe ratio
Table 7a. Information Ratios for DM against comparison strategies
1.6
1.4
RiskRisk
Relative Adjusted
Static Budgeting Momentum
Markowitz
1.2
1.0
0.8
0.4
185-day
250-day
Equally-weighted Portfolio
Average Portfolio
Static Markowitz
Risk Budgeting
0.94
0.95
0.58
0.94
0.57
0.59
0.27
0.58
0.40
0.41
0.10
0.40
Table 7b. Alpha relative to different comparison strategies
0.2
Equallyweighted
Mean
variance
Momentum
Dynamic
Markowitz
Source: JPMorgan
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.
6
125-day
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.
0.6
0
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)
6.4
4.7
6.5
4.8
5.7
3.3
3.0
4.4
2.8
4.5
2.9
3.2
1.4
0.8
3.8
2.0
3.8
2.1
2.8
0.6
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
Comparison Strategies
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)
125-day
185-day
250-day
4.6
3.3
4.6
3.3
3.6
2.0
2.1
3.3
3.2
2.0
3.2
2.1
2.2
0.9
0.6
2.0
2.7
1.4
2.8
1.4
1.9
0.4
0.2
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.
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
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 tstatistics indicating that those alphas are statistically
significant.
Table 8. Long-short Strategies
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.
Statistics
Flexible LS
5.3%
5.4%
0.97
4.5%
2.99
0.12
4.8%
5.6%
0.85
5.2%
3.26
-0.07
Average Excess Return
Standard Deviation
Sharpe Ratio
Alpha – EW
T-stats
Beta – EW
Source: JPMorgan
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
(days)
(months)
Volatility Constraint
60
3
8%
Vol-cap LS
Portfolio
Constraints
25%
Average Exc.
Return (cash)
5.7%
Standard
Deviation
8.0%
Sharpe Ratio
0.71
Alpha
(over EW)
Beta
(EW)
0.7%
t-stat
0.52
125
3
8%
25%
10.7%
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%
10.7%
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
0.87
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%
10.7%
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
Source: JPMorgan
7
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
Table 10. Effect of excluding asset classes
Chart 5. DM information ratio vs past asset class dispersion
IR
1.8
1.6
1.4
y = 41.054x - 0.1071
1.2
R 2 = 0.1235
1.0
0.8
0.6
0.4
0.2
0.0
Past Dispersion
-0.2
-0.4
Excluding Asset
Average Exc. Return
(cash)
Information
Ratio (EW)
8.9%
9.9%
11.7%
9.4%
9.1%
10.3%
10.3%
10.5%
10.7%
0.56
0.75
1.11
0.69
0.61
0.87
0.85
0.97
0.93
MSCI North America
MSCI Europe
MSCI Asia Pacific
MSCI EM
Real Estate
EMBI
Commodities
Global GBI
No exclusions
0.000
Source: JPMorgan
Chart 4. Effect of excluding asset classes on Sharpe ratios
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
No exclusions
Global GBI
Commodities
EMBI
Real Estate
MSCI EM
MSCI Asia Pacific
MSCI Europe
0.00
MSCI North America
0.005
0.010
0.015
0.020
0.025
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.
Chart 6. DM information ratio vs past EW volatility
IR
1.8
1.6
1.4
y = -9.4171x + 1.0844
1.2
R 2 = 0.1742
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
Past Volatility - Equally-weighted Portfolio
-0.4
0.00
0.02
0.04
0.06
0.08
0.10
0.12
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.
Source: JPMorgan
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.
8
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
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Chart 7. DM information ratio vs past EW return
IR
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
Chart 8. Rolling 250-day dispersion in performance
2.5%
y = 0.8393x + 0.3364
y = -2E-06x + 0.105
R2 = 0.8728
2.0%
R 2 = 0.0614
1.5%
1.0%
0.5%
Past Return - Equally-weighted Portfolio
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
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 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.
Year
0.0%
1995
1998
2001
2004
2006
Source: JPMorgan. 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
Sharpe Ratio
13.8%
0.95
15.2%
1.18
15.5%
1.23
Alpha (over EW)
t-stats
Standard Deviation
Max (monthly)
Min (monthly)
0.88%
1.24
5.9%
7.3%
-12.1%
2.00%
2.71
5.7%
7.4%
-8.2%
2.36%
3.18
5.8%
7.5%
-8.2%
Source: JPMorgan
Longer periods and intra-asset class
We 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, 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 equities8. Here we test
whether we can add value by using this optimization
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.
9
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
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.
Volatility timing/risk budgeting in asset
allocation
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 signal10. 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.
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 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.
10
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 itself13.
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.
J.P. Morgan Securities Ltd.
Ruy Ribeiro (44-20) 7777-1390
ruy.m.ribeiro@jpmorgan.com
Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
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Global Market Strategy
Markowitz in tactical asset allocation
August 7, 2007
Investment Strategies Series
This series aims to offer new approaches and methods on investing and trading profitably in financial markets.
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
2002
6. 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
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
11. A Fair Value Model for US Bonds, Credit and Equities, Nikolaos Panigirtzoglou and Jan Loeys, Jan 2005
29. Relative Value in Tranches I, Dirk Muench, November
2006
12. JPMorgan Emerging Market Carry-to-Risk Model,
Osman Wahid, February 2005
30. Relative Value in Tranches II, Dirk Muench, November
2006
13. Valuing cross-market yield spreads, Nikolaos
Panigirtzoglou, January 2006
31. Exploiting carry with cross-market and curve bond
trades, Nikolaos Panigirtzoglou, January 2007
14. Exploiting cross-market momentum, Ruy Ribeiro and
Jan Loeys, February 2006
32. Momentum in Money Markets, Gianluca Salford, May
2007
15. A cross-market bond carry strategy, Nikolaos
Panigirtzoglou, March 2006
33. Rotating between G-10 and Emerging Markets Carry,
John Normand, July 2007
16. Bonds, Bubbles and Black Holes, George Cooper,
March 2006
34. A simple rule to trade the curve, Nikolaos Panigirtzoglou,
August 2007
12