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Jan 13, 2012 Comments Welcome

Tail Risk in Momentum Strategy Returns

Kent Daniel, Ravi Jagannathan and Soohun Kim

Abstract

Price momentum strategies have historically generated high positive returns with little

systematic risk. However, momentum strategies incur periodic but infrequent largelosses. During 13 of the 1002 months in our 1927:07-2010:12 sample, losses to a US-equity momentum strategy exceeded 20 percent per month. We characterize theserisk-spikes with a two state hidden Markov model, with turbulent and calm states.These states are persistent, and forecastable using ex-ante instruments. Each of the 13months with losses exceeding 20 percent per month occurs during a turbulent month,meaning a month when the predicted probability of the hidden state being turbu-lent exceeds 0.5. During turbulent months, momentum returns are more volatilite,and average -0.65 %/month. A dynamic momentum strategy that avoids turbulentmonths has a Sharpe Ratio 0.32 double that of the static momentum strategy. Thepredictability of momemtum tail-risk renders momentum an even stronger anomaly.

Columbia University, Graduate School of Business (Daniel) and Northwestern University, Kellogg School

(Jagannathan and Kim). This paper originally appeared under the title Risky Cycles in Momentum Re-

turns We thank Raul Chhabbra, Francis Diebold, Robert Engel, Robert Korajczyk, Jonathan Parker,

Prasanna Tantri, the participants at the Fourth Annual Conference of the Society for Financial Economet-

rics and the participants of seminars at Northwestern University, the Indian School of Business, the Securities

and Exchange Board of India, and the Shanghai Advance Institute for Finance, Shanghai Jiao Tong Univer-

sity for helpful comments on the earlier version of the paper. We alone are responsible for any errors and

omissions.


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1 Introduction

Relative strength strategies, also known as price momentum strategies, have been and con-

tinue to be popular among traders. Quantitative strategies used by active money managersgenerally employ at least some form of momentum,1 and even non-quantiative mutual fund

managers appear to utilize momentum in their trading decisions.2

Price momentum in stocks can be described as the tendency of stocks with the highest

(lowest) past return to outperform (underperform) the broader market. Price momentum

strategies exploit this pattern by taking a long position in past winners and an equal short

position in past losers. While such strategies produce high abnormal returns, they also

generate infrequent, but large losses.

Over the 1002 month sample period we examine in this paper July 1927 through

December 2010 the particular momentum strategy we examine generated monthly returns

in excess of the return on the risk free asset with a mean of 1.18%, a standard deviation of

8%, and with little systematic risk as measured by a standard CAPM or a Fama and French

(1993) three factor model. However, the strategy returns have an empirical distribution

that is both strongly leptokurtic and highly left skewed. There are thirteen months with

losses exceeding 20%. In its worst month the strategy experiences a loss of 79%. Were

momentum returns generated from an i.i.d. normal distribution with mean and variance

equal to their sample counterparts, the probability of realizing a loss of more than 20% in

thirteen or more months in a sample of 1002 months would be 0.03%, and the probability of

incurring a single loss of 79% or worse would be 3 1024.The skewed and leptokurtic distribution of momentum strategy returns suggests that

momentum returns may be drawn from a mixture of normal distributions. To capture this,

we develop a variation of the two state hidden Markov regime switching model (HMM) of

Hamilton (1989), where the market is calm in one state and turbulent in the other.

However, our switching model captures not only changes in the volatility of the underlying

return generating processes, but also other features of the joint distribution of momentum

1Swaminathan (2010) estimates that about one-sixth of the assets under management by active portfoliomanagers in the U.S. large cap space is managed using quantitative strategies.

2Jegadeesh and Titman (1993) note that . . . a majority of the mututal funds examined by Grinblatt andTitman (1989; 1993) show a tendency to buy stocks that have increased in price over the previous quarter.

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strategy returns and market returns, many of which are documented in the literature and

some of which are new.3 In particular, momentum returns have the features of a written call

on the stock index, and the nature of this option-like feature changes with the underlying

state.

Based on this model, we infer the state of the economy by observing changes in joint dis-

tribution of momentum returns with past and contemporaneous stock index returns. Con-

sistent with results in the literature, we find that momentum returns have the features of

a written call on the stock index. The hidden states are persistent, and can be estimated

ex-ante using the switching model.

Our estimation reveals that distribution from which returns are drawn is more volatile

when the market is in the unobserved turbulent state. We find that all the thirteen months

with losses exceeding 20% occur when the predicted probability of the market being in

the turbulent state exceeds 50%. The probability of observing a loss of 20% or more in

thirteen or more months during the 200 months in the sample when we classify the market

as turbulent increases to 91%, and the large momentum strategy losses become less black

swan-like. Linear regression models of momentum returns and GARCH models of time

varying volatility are not as successful as the HMM in identifying time periods when large

losses are more likely to occur. Our findings are consistent with Ang and Timmerman

(2011) who argue in favor of using regime switching models to capture abrupt changes in

the statistical properties of financial market variables.

Use of the ex ante probability of the turbulent state that we extract from the switching

model allows us to construct a dynamic momentum strategy that outperforms the standard,

static strategy. We find that the monthly Sharpe Ratio of momentum returns in months

with a predicted probability of being in the turbulent state exceeding 0.5 is -0.05; and the

monthly Sharpe Ratio for the other months is 0.32. The corresponding numbers for the

excess returns on the stock index portfolio are 0.09 and 0.13. This high Sharpe-ratio in the

calm state makes the momentum a still stronger anomaly.

The rest of the paper is organized as follows. We provide a brief review of related

literature in Section 2. We describe the data and develop the econometric specifications of

the hidden Markov model for characterizing momentum returns in Section 3. We discuss the

3Section 2 provides a review of this literature

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empirical findings in Section 4 and conclude in Section 5.

2 Related Literature

Levy (1967) is one of the early academic articles to document the profitability of stock price

momentum strategies. However, Jensen (1967) raised several issues with the methodology

employed by Levy (1967). Perhaps as a consequence, momentum received little attention

in the academic literature until Jegadeesh and Titman (1993), who developed with a clever

method for constructing portfolios based on relative price momentum in stocks that was

rigorous and replicable. A number of studies have subsequently confirmed the Jegadeesh

and Titman (1993) findings using data from markets in a number of countries (Rouwen-

horst, 1998), and in a number of asset classes (Asness, Moskowitz, and Pedersen, 2008). As

Fama and French (2008) observe, the abnormal returns associated with momentum are

pervasive.

Korajczyk and Sadka (2004) show that momentum profits cannot be explained away by

transactions costs and so remain an anomaly. Chabot, Ghysels, and Jagannathan (2009)

find that momentum strategies earned anomalous returns even during the Victorian era

with very similar statistical properties, except for the January reversal, presumably because

capital gains were not taxed in that period. Interestingly, they also find that momentum

returns exhibited negative episodes once every 1.4 years with an average duration of 3.8

months per episode.

The additional literature on momentum strategies is vast, and can be grouped into

three categories: (a) documentation of the momentum phenomenon across countries and

asset classes (b) characterization of the statistical properties of momentum returns, both in

the time-series and cross-section and (c) theoretical explanations for the momentum phe-

nomenon. We make a contribution to (b). In what follows we provide review of only a few

relevant articles that characterize the statistical properties of momentum returns, and refer

the interested reader to the comprehensive survey of the momentum literature by Jegadeesh

and Titman (2005).

Our work is directly related to several other papers in the literature. First, a number of

authors, going back to Jegadeesh and Titman (1993), have noted that momentum strategies

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can experenience severe losses over extended periods. In particular, Griffin, Ji, and Martin

(2003) document that there are often periods of several months when momentum returns

are negative.

In addition, a large number of papers have explored the time-varying nature of the risk of

momentum strategies. Kothari and Shanken (1992) note that, for portfolios formed on the

basis of past returns, the betas will be a function of past market returns. Using this intuition,

Grundy and Martin (2001) show that beta of momentum returns become negative when

the stock market has performed poorly in the past. Rouwenhorst (1998) documents that

momentum returns are nonlinearly related to contemporaneous market returns i.e., that

their up market beta is less than their down market beta.4 Daniel and Moskowitz (2011) find

that this non-linearity is present only following market losses. Building on Cooper, Gutierrez,

and Hameed (2004), who show that the expected returns associated with momentum profits

depend on the state of the stock market, Daniel and Moskowitz (2011) also show even after

controlling for the dynamic risk of momentum strategies, average momentum returns are

lower following market losses, and when market volatility is high. They find evidence of

these features not only in US equities, but also internationally, and in momentum strategies

applied to commodity, currency, and country equity markets. Finally, they show that the

factors combine to result in large momentum strategy losses in periods when the market

recovers sharply following steep losses.5

4To our knowledge, Chan (1988) and DeBondt and Thaler (1987) first document that the market beta ofa long-short winner-minus-loser portfolio is non-linearly to the market return, though DeBondt and Thaler(1987) do their analysis on the returns of longer-term winners and losers as opposed to the shorter-termwinners and losers we examine here. Rouwenhorst (1998) demonstrates the same non-linearity is present forlong-short momentum portfolios in non-US markets. Finally Boguth, Carlson, Fisher, and Simutin (2010),building on the results of Jagannathan and Korajczyk (1986), note that the interpretation of the measures ofabnormal performance in Chan (1988), Grundy and Martin (2001) and Rouwenhorst (1998) is problematic.

5Ambasta and Ben Dor (2010) observe a similar pattern for Barclays Alternative Replicator return thatmimics the return on a broad hedge fund index. They find that when the hedge fund index recoverssharply from severe losses the replicator substantially underperforms the index. Also, Elavia and Kim(2011) emphasize the need for modeling changing risk for understanding the recent weak performance of

quantitative equity investment managers who had over two decades of good performance.

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3 Data and Econometric Specifications

3.1 Data

Price momentum strategies using stocks can be constructed in a variety of different ways.

For this study we utilize a momentum strategy based on the momentum decile portfolio

returns available in Ken Frenchs Data Library.6 These portfolios are formed by first ranking

stocks based on past 11 months of returns and grouping them into ten decile portfolios. The

returns on each of the ten decile portfolios are measured in the second month after the return

measurement period.7 Most of our analysis will concentrate on the zero-investment portfolio

which, at the end of each month, invests $1 in the top decile portfolio of past winners and,

and takes a short position of $1 in the bottom decile portfolio of past losers. The return tothis strategy is the difference between the top and bottom decile returns.

Data for the CRSP value-weighted market return, the risk-free rate, and the Fama and

French (1993) size and value portfolios SMB and HML are also taken from Ken Frenchs

Data Library.

3.2 Data Summary Statistics

Table 1 presents estimators of the moments of the monthly momentum strategy returns,and of the CRSP value-weighted portfolio return, net of the risk-free rate. The monthly

momentum strategy returns averages 1.18% over this 1002 month period, with a monthly

Sharpe Ratio of 0.15. In contrast, the realized Sharpe ratio of the market is only 0.11. The

alpha of momentum strategy is 1.75%/month with respect to the Fama and French (1993)

three factor model, and when the three Fama and French factors are combined with the

momentum portfolio the Sharpe Ratio almost doubles to 0.30.

While the momentum strategy has a higher in sample Sharpe Ratio than the stock index

portfolio, Table 1 shows that it exhibits strong excessive kurtosis and negative skewness. The

excess kurtosis is also evident from the infrequent but very large losses to the momentum

6The library is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.htmlThe specific momentum decile portfolios we use are those designated by 10 Portfolios Prior 12 2.

7Skipping the month after the return measurement period is done both so as to be consistent with themomentum literature, and is done so as to minimize market microstructure effects and to avoid the short-horizon reversal effects documented in Jegadeesh (1990) and Lehmann (1990).

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Table 1: Estimates of First Four Moments of Momentum and Market ReturnsThis table presents estimates of the first four moments of the distribution of monthly returnsfor MOM the zero-investment momentum portfolio used in the study and for the CRSPvalue weighted (market) portfolio net of the risk-free rate. The sample period is 1927:07-2010:12.

Mean Std.Dev Skewness Kurtosis

MOM 1.18% 8.00% -2.45 21.02MktRf 0.62% 5.46% 0.16 10.44

strategy during the 1002 months in the sample period. Given the low unconditional volatility

of the momentum strategy, if returns were normal the probability of observing a month with

a loss exceeding 42% in a sample of 1002 months would be one in 29,000, and the probability

of seeing five or more months with losses exceeding 42% would be almost zero. Yet the

lowest five monthly returns in the sample are: -79%, -60%, -46%, -44%, and -42%, a rare

black swan like occurrence from the perspective of someone who believes that the time series

of monthly momentum returns are generated from an i.i.d. normal distribution.

In Section 3.7 we describe a hidden Markov model, based on the framework of Hamilton

(1989), which we use to capture the behavior of momentum returns. Specifically, we model

momentum returns with a mixture of normal distributions where the parameters of the

normal distributions depend on past and contemporaneous history in possibly nonlinear ways

to accommodate the negative skewness of momentum strategy returns. Even though the

distribution of returns conditional on the hidden state and the current market is normal,

this model is able to capture the unconditional skewness and kurtosis of the momentum

returns as documented above.

In what follows we first describe our rationale for modeling the joint distribution of

momentum strategy returns and current and past market returns the way we do.

3.3 Dependence of Beta on Past Market Return

Grundy and Martin (2001), using a result which to our knowledge was first pointed by out

by Kothari and Shanken (1992), argue that the momentum portfolio beta will be a function

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of the market return over the return measurement period.8

The intuition for the Kothari and Shankens result is as follows: If the market return

is negative over the return measurement period then, from a Bayesian perspective, a firm

which earned a negative return i.e. which moved down with the market is likely to have

a higher beta than a positive return firm. Conversely, when the market return was strongly

postive over the return measurement period, the past winners will tend to be high beta

stocks, and the past losers low beta stocks.

Because the momentum portfolio is long past winners and short past losers, the intuition

of Kothari and Shanken (1992) and Grundy and Martin (2001) suggests that the momentum

portfolio beta should be negative in bear markets when the market return was negative

over the return measurement period and positive in bull markets.

If we define the IBt as a indicator that the market-state is Bearish, meaning that the

return on the market over the return measurement period from t 12 through t 2 monthsis negative, i.e.:

IBt 1 if R

M(t12, t2) > 00 otherwise

(1)

this suggests the following specification for the joint distribution of momentum and stock

market returns

Rmomt = + 0 + BIBt RMt + t.Estimation of this specification over our full sample yields:

0 B

estimate 0.01 0.16 1.29t-stat 5.66 2.93 17.16

Consistent with the findings in the literature noted above, B is economically and statistically

negative, but the abnormal return, controlling for the dynamic market risk, is still strong.

8See pp. 195-198 of Kothari and Shanken (1992).

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3.4 Dependence of Beta on Current Market Return

As noted in Section 2, a number of authors have noted that the contemporaneous return

of past-return sorted portfolios are nonlinearly to contemporaneous market return.9 In the

language of Henriksson and Merton (1981), the up-beta of momentum portfolios is lower

than their down-beta. This option-like feature of portfolios is captured with the following

specification:

Rmomt = +

0 + U IUt

RMt + t

Where our up-market indicator, IUt , is an indicator variable which is one when the market

return is above zero in month t. Estimation of this specification over our full sample yields:

0 U

estimate 0.03 0.07 0.89t-stat 9.96 0.95 7.53

Notice that up-beta estimate our momentum portfolio is lower than the down-beta estimate

by 0.89, and is again strongly statistically significant, consistent with the literature noted

above. Also, here it is important to note that the estimated is no longer a valid mea-

sure of abnormal performance, since the contemporaneous up-market indicator uses ex-post

information.10

3.5 Beta and Market Recoveries

Daniel and Moskowitz (2011) find that momentum portfolios incur large losses when the

market recovers sharply following large losses. For example, the worst five months with the

largest losses on the momentum portfolio we discussed have the pattern in Table 2, consistent

with the observations in Daniel and Moskowitz (2011).

In order to capture this effect, we follow Daniel and Moskowitz (2011) and define the

recovery indicator function that takes the value of 1 whenever both the bear-market indicator

is 1 and the contemporaneous market return is postive (i.e., when IBt IUt = 1), where IU,t isand indicator variable which is 1 when the contemporaneous market return is positive, and

9See, in particular, footnote 4.10See Jagannathan and Korajczyk (1986) and, more recently, Boguth, Carlson, Fisher, and Simutin (2010).

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Table 2: Patterns of Historical Momentum Crashes

Month Rmomt (%) RMt (%)

24n=1 R

Mtn(%)

1932/8 -78.96 37.60 -68.35

1932/7 -60.11 33.72 -75.452009/4 -45.89 11.05 -44.111939/9 -43.94 15.96 -21.501933/4 -42.33 38.37 -59.46

is zero otherwise:

IUt

1 if RMt > 0

0 otherwise(2)

This motivates the following specification for the momentum return generating process.

Rmomt = +

0

+B IBt+R IBt IUt > 0

RMt + t (3)

Estimation of this specification yields:

0

B

R

estimate 0.02 0.14 0.91 0.66t-stat 7.65 2.61 9.33 5.94

The results can be summarized as follows:

0 > 0 : 0 is the average beta in bull (i.e., non-bear). The point estimate is positiveand statistically significant, though small.

B < 0 : The momentum beta is strongly negative when the measurement periodmarket return is negative that is in a bear-market.

R < 0 : The momentum beta is even more negative when the contemporaneous marketreturn is positive, in a bear market-state.

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3.6 Final Specification

The specification of (3) by Daniel and Moskowitz (2011) emphasizes the role of past market

return on the option-like feature of momentum return. They show that the option-like feature

of momentum return depends on the past market return being negative by examining the

specification of the form given in equation (3). Since our objective is to infer the hidden

state by examining the joint behavior of momentum returns and stock index returns, we

consider the following slightly more general specification for the stochastic process governing

the evolution of the stock index and momentum returns that nests (3) as a special case.

Rit = +

L ILt

+ LU

IL

t IU

t

+ B IBt

+ BU IBt IUt

RMt + t (4)

where i denotes the Winner, the Loser or the Momentum (i.e., Winner - Loser) portfolios.

Here, we have added one final indicator: a BulL market indicator which is one when the

measurement-period market return is greater than zero, so that

ILt

(1

IBt

) (5)

In the regression equation of (4), L/B represent the beta when the market return during

the measurement period is positive/negative. When momentum returns have written call

option like feature the sign of LU and BU will be negative, and BU will be more negative

indicating the accentuated option-like feature following declining markets. The dependence

of momentum return beta on past and contemporaneous stock index return is summarized

below.Market State

BulL Bear

RMtUp L + LU B + BU

Down L B

(6)

Table 3 present the results of estimating equation (4). As before, L > B. Also, as in

Daniel and Moskowitz (2011), BU is negative. However, notice that LU is also negative

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Table 3: Momentum components in Bull and Bear Markets

Portfolio

winner loser mom

0.01 -0.01 0.02(8.97) (-5.47) (8.27)

L 1.45 1.10 0.38(33.47) (16.52) (4.28)

LU -0.27 0.26 -0.53(-3.84) (2.51) (-3.66)

B 0.89 1.59 -0.70(22.10) (26.47) (-8.37)

BU -0.28 0.53 -0.81(-4.93) (6.22) (-6.85)

though less negative than BU. Noting that LU is similar to BU for the Winner return, we

can observe that the stronger option like feature of momentum returns following past down

market is mainly due to the behavior of the loser component of momentum returns.11

We therefore use equation (4) in developing our Hidden Markov Model specification.

3.7 A Hidden Markov Model of Momentum Strategy Returns

We now define a Hidden Markov Model (HMM) of momentum strategy returns based on the

specification given in equation (4). We use St to denote the unobserved random underlying

state of the economy, and st the realized state of the economy. We assume that there are

two possible states, one is Calm (C) and the other Turbulent (T).

Our specification for the momentum portfolio return generating process is based on equa-

tion (4), but we allow the coefficients to vary by state:

Rmomt = (St) +

L(St) I

L

t

+ LU(St) ILt I

Ut

+ B(St) IBt

+ BU(St) IBt I

Ut

RMt + (St)

momt . (7)

11The difference in statistical significance between our results and those of Daniel and Moskowitz (2011)is because of the lower residual variance that is achieved with this specification.

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Here momt is i.i.d. N(0, 1), so (St) denotes the residual volatility, which is a function ofthe state. As before, the ex-ante bulL and Bear market indicators ILt and I

Bt , and the ex-post

Up market indicator IUt are as defined in equations (1), (2), and (5), respectively.

In addition we define the return generating process for the market as

RMt = (St) + M (St) Mt , (8)

3.8 Maximum Likelihood Estimation

We estimate the parameters of the hidden Markov model using the maximum likelihood

method described in Hamilton (1989).

We begin by defining the vector of observable variables of interest as of time t Yt, as:

Yt =

ILt , ILt I

Ut , I

Bt , I

Bt I

Ut , R

momt , R

Mt , 1

,

We let yt denote the realized value of Yt.

We let St denote the random state at time t (which, in our setting, is either calm or

turbulent, and st denote the particular realized value of the state at date t.

We let Pr (St = st|St1 = st1) denote the probability of moving from state st1 at timet 1 to state st at time t.12 Also, Ft1 denotes the information set at time t 1, i.e.,{yt1, , y1}.

Note that the unconditional probabilities of the two states are determined by the transi-

tion probabilities. In our esimation, we set Pr (S0 = s0) to the unconditional probabilities,

based on the two parameters governing the transition probabilities.

Suppose we know the value of Pr (St1 = st1|Ft1). Then, Pr (St = st|Ft1) is given by:

Pr (St = st

|Ft1) = st1

Pr (St = st, St1 = st1

|Ft1)

=st1

Pr (St = st|St1 = st1) P r (St1 = st1|Ft1) , (9)

which can be computed since we know the elements of the transition probability matrix,

12Here, we use Pr(x) to denote the probability of the event x when x is discrete, and the probabilitydensity ofx when x is continuous.

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Pr (St = st|St1 = st1).Next, we can compute the joint conditional distribution of the hidden state at time t, st

and observable variables at time t, yt as follows.

Pr (yt, St = st|Ft1) (10)= Pr (yt|St = st,Ft1) P r (St = st|Ft1)

The second term on the RHS is given by (9) and the first term is the state dependent

likelihood ofyt, which under the distributional assumption from (7) is given by

Pr (yt

|St = st,

Ft1) =

1

mom (st)2exp

1

2mom (st)

2(mom (St)

momt )

2

(11)

1mom (st)

2

exp 1

2M (st)2

M (St)

Mt

2(12)

where

mom (St) momt = R

momt (St)

St, I

Lt , I

LUt , I

Bt , I

BUt

RMt

M (St) Mt = R

Mt (St) .

Given Pr (yt, St = st|Ft1) from (10), we can integrate out St for St = {turbulent, calm}and compute Pr (yt|Ft1) the conditional likelihood of yt as:

Pr (yt|Ft1)=st+1

Pr (yt, St = st|Ft1) .

Now consider the log likelihood function of the sample given by:

L =Tt=1

log(Pr(yt|Ft1)) , (13)

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which may be maximized numerically to form estimates of the parameter set

=

(C) , L (C) , LU (C) , B (C) , BU (C) , mom (C)

(T) , L

(T) , LU

(T) , B

(T) , BU

(T) , mom (T) (C) , M (C) , (T) , M (T)

Pr (St = C|St = C) , Pr (St = T|St = T)

.

We showed how to compute Pr (yt|Ft1) given Pr (St1 = st1|Ft1) . To compute Pr (yt+1|Ft)we need to know Pr (St = st|Ft) . This is accomplished by Bayesian updating procedure asfollows.

Pr (St = st|Ft) = Pr (St = st|yt,Ft1)=

Pr (yt, St = st|Ft1)Pr (yt|Ft1) (14)

where the last equality of (14) is due to Bayes rule. Using (14) to compute Pr (St = st|Ft)and the algorithm described above, we can compute Pr (yt+1|Ft). This way we can generatethe likelihood Pr (yt+1|Ft) for t = {0, 1, , T1} and compute L =

Tt=1 log(Pr(yt|Ft1))

given specific values for the unknown parameters that we need to estimate.

We estimate the values of the models of the hidden Markov model parameters by maxi-

mizing the log likelihood given by (13), i.e.,

ML = arg max

L () (15)

A consistent estimator of the asymptotic covariance matrix of the estimated parameters,

() , is given by: () = H (ML)1 ,where H () is the Hessian ofL ().

13

13We estimate H (ML) numerically using the Matlabs function fminunc .

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4 Empirical Results

Table 4 summarizes the maximum likelihood estimates of the model parameters. The param-

eter estimates are consistent with the observations in Grundy and Martin (2001). In bothhidden states, we can see that L > B, 14 i.e., the beta of momentum returns is smaller

when the average return on the market index during the portfolio formation period is neg-

ative. Also, both of LU and BU, are negative consistent with the written call option-like

features of momentum returns discussed in section 3.6. While both the calm and turbulent

states are persistent, the probability of remaining in the calm state starting from the calm

state is higher.

Table 4: Maximum Likelihood Estimates of HMM ParametersThis table presents the parameters of the HMM model, estimated using the momentumstrategy return and the market return over the full sample. As indicated below, the MLestimates ofmom and M are each multiplied by 100. The Log-Likelihood for this estimationis 3.14 103. The AIC is 6.23 103, and the BIC is 6.15 103.

STATE

Calm Turbulent

Parameter MLE t-stat MLE t-stat 1.96 7.11 2.91 2.77L 0.51 4.43 0.26 1.39

LU 0.45 2.55 0.63 1.85B 0.33 2.05 0.73 4.56BU 0.34 1.58 1.02 4.28mom (102) 4.10 23.16 10.22 19.44 0.99 6.85 0.50 0.84M (102) 3.64 31.54 8.91 18.68Pr (St= st|St1= st) 0.97 83.15 0.91 26.68

Table 5 provides the maximum likelihood estimates of the differences in parameter values

in the calm and the turbulent states and the corresponding t-values. BU

the parameterthat captures the effect of market recovery following sharp losses on the sensitivity of mo-

mentum returns to market returns is significantly different across the two states. B is

marginally different across hidden states, indicating that it is easier to differentiate hidden

states following past-down market than past-up market. The variance of the market return

14t-stat on the null that L = B is 3.85 for calm state and 3.58 for turbulent state.

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as well as the residual variance of the momentum return in the two hidden states are sig-

nificantly different. The returns are about twice as volatile in the turbulent states when

compared to the calm states. This difference in parameters helps us to infer the hidden

state.

Table 5: A Test of Equality of the Parameters in the Two Hidden States

MLE t-statC T 0.94 0.85LC LT 0.25 1.10LUC LUT 0.18 0.44BC BT 0.40 1.73BUC BUT 0.68 2.08C

mom T

mom 0.06 11.39C T 1.48 2.40CM TM 0.53 11.15

Table 6 shows that how many of large positive and negative returns occur when the

predicted probability of the turbulent state is p or more, where p takes any of the values from

10% to 50% in steps of 10%. When we estimate the model using all the sample observations,

all the 13 months with losses of more than 20% occur in months with Pr (St = T

|Ft1) > 0.5.

There are 11 months of when momentum return exceed 20%. and 8 of those 11 months occur

during months with Pr (St = T|Ft1) > 50%. Each entry represents the number of detectedlarge gains(loses)/the number of total large gains (losses) in the sample.

We also examine the ability of the hidden Markov specification to predict the state of

the economy in real time, following the suggestion of Amisano and Geweke (2011). For that

purpose we use an expanding window to estimate the model parameters and that gives us

real time predicted probability of the next month being in the turbulent state for 400 months.

In particular, for each month t = T 399, , T, we estimate the model parameters usingmaximum likelihood using data for the months {1, , t 1}. The first period of our out ofsample test is September 1977 and the last is December 2010.

From table 7 we can see that there were 5 months with losses exceeding 20% and all

of them occur during months with predicted probability of being in the turbulent state in

excess of 50%. In contrast, there were 4 months with gains exceeding 20% but 3 of them

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Table 6: Ability of the HMM to Capture Extreme Gains and Losses

Pr (St= T

|Ft1) # Gains captured/Total #Gains #

is more than 10% 12.5% 15% 17.5% 20% of Months50% 37/69 25/42 17/27 10/14 8/11 20640% 43/69 28/42 18/27 11/14 9/11 24130% 46/69 30/42 20/27 11/14 9/11 27120% 47/69 30/42 20/27 11/14 9/11 30910% 55/69 34/42 23/27 14/14 11/11 381

Pr (St= T|Ft1) # Losses captured/Total #Losses #is more than 10% 12.5% 15% 17.5% 20% of Months

50% 32/53 27/35 24/31 19/21 13/13 206

40% 34/53 27/35 24/31 19/21 13/13 24130% 37/53 29/35 26/31 19/21 13/13 27120% 41/53 33/35 30/31 21/21 13/13 30910% 42/53 33/35 30/31 21/21 13/13 381

occur during months with predicted probability of being in the turbulent state exceeding

50%. The asymmetry is probably due to the fact that in the momentum return generating

process we use, steep stock market losses, being more likely in turbulent markets, will result

in assigning a higher probability to the underlying state being turbulent. Steep losses willalso result in the loser stocks will be highly levered, and accentuated option-like features.

Consequently, when the market recovers sharply the momentum strategy which has a short

position in the losers will also lose sharply. However, because of the accentuated option

like features, when the market falls further, the loser stocks in the portfolio will not gain

as much, leading to the asymmetric pattern we observe, i.e., the prevalence of steep losses

during predicted turbulent markets relative to gains. We find that this pattern is observed

primarily when the economy is in the turbulent state.

The association between large losses and the predicted probability of being in the turbu-

lent state is illustrated in Figure 1 using the entire sample and in Figure 2 using an expanding

window in real time.

The vertical axis is the realized return in a month and the horizontal axis is the predicted

probability of the month being in a turbulent state. As can be seen,the scatter plot fans out

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Table 7: Out of Sample Ability of the HMM to Capture Extreme Moves

Pr (St= T

|Ft1) # Gains captured/Total #Gains #

is more than 10% 12.5% 15% 17.5% 20% of Months50% 12/29 10/18 7/12 4/5 3/4 7440% 13/29 11/18 7/12 4/5 3/4 7930% 16/29 12/18 7/12 4/5 3/4 9020% 19/29 14/18 9/12 4/5 3/4 10610% 21/29 14/18 9/12 4/5 3/4 130

Pr (St= T|Ft1) # Losses captured/Total #Losses #is more than 10% 12.5% 15% 17.5% 20% of Months

50% 14/26 11/16 8/12 6/7 5/5 7440% 14/26 11/16 8/12 6/7 5/5 7930% 15/26 12/16 9/12 6/7 5/5 9020% 15/26 12/16 9/12 6/7 5/5 10610% 15/26 12/16 9/12 6/7 5/5 130

with lot more dispersion when we move from the left to the right, providing a visual measure

of the ability of the hidden Markov model to predict turbulent months.

The behavior of momentum and stock market returns during months with predicted

probability of being in the turbulent state in excess of 50% are summarized in the table 8.

Table 8: Summary Statistics of Returns in Turbulent and Calm Months

Fcst. # of MOM Returns MKT-Rf ReturnsState months Mean (%) SD (%) SR Mean (%) SD (%) SRCalm 796 1.66 5.19 0.32 0.57 4.37 0.13

Turbulent 206 -0.65 14.27 -0.05 0.80 8.55 0.09All 1002 1.18 8.00 0.15 0.62 5.49 0.11

Of the 1002 months, 206 months are turbulent and 796 months are calm. Momentum

returns are on average negative though not statistically significantly so during turbulent

months, and almost three times as volatile as calm months. In contrast momentum returns

average 1.66% during calm months with an associated Sharpe Ratio of 0.32, twice the Sharpe

Ratio for all months for the stock market portfolio. To the extent that turbulent months are

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.8

0.6

0.4

0.2

0

0.2

0.4

0.6

Predicted Probability of Turbulent State

RealizedMomentumR

eturn

Figure 1: Realized Momentum versus Predicted Probability (in sample)

Table 9: Summary Statistics of Returns in Turbulent and Calm Months (Out ofSample)

Fcst. # of MOM Returns MKT-Rf ReturnsState months Mean (%) SD (%) SR Mean (%) SD (%) SRCalm 326 1.71 5.48 0.31 0.58 4.31 0.14

Turbulent 74 -0.94 13.42 -0.07 0.49 5.83 0.08All 400 1.22 7.64 0.16 0.57 4.62 0.12

predictable and can be avoided the momentum phenomenon becomes more of a puzzle.

Interestingly, the average return on the stock market portfolio is the same during turbu-

lent as well calm months even though the market is almost twice as volatile during turbulent

months, which is consistent with the evidence presented in Breen, Glosten, and Jagannathan

(1989). We also report the result with out of sample estimation with Table 9.

When we classify months with predicted probability of the unobserved state being in

the turbulent state exceeding 50% as being turbulent and the other months as calm, as we

noted earlier, all large losses (exceeding 20%) occur during turbulent months. While both

calm and turbulent months exhibit persistence, the former is less so. Calm months have an

average duration of 16 months and turbulent months have an average duration of 4 months.

As can be seen from figure 3 turbulent months occur when the economy is in a recession as

well as when the economy is in an expansion.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

Predicted Probability of Turbulent State

RealizedMomentumR

eturn

Figure 2: Realized Momentum versus Predicted Probability (out of sample)

As can be seen from Table 10 and 11, during the 1002 months in our sample, the mo-

mentum portfolio formation period stock index returns were negative only for 303 months,

i.e., 30% of the months. However, in the 206 months classfied as belonging to the turbu-

lent class (i.e., months with probability of the hidden state being turbulent exceeding 0.50),

formation period stock returns were negative for 124 months, i.e., 60% of the months. This

is consistent with the view that declining markets are more likely when the hidden state is

turbulent.

Table 10: Past market return and the likelihood of hidden state being turbulent in sample

Pr (St = T|Ft1)Formation Period Return 50% 50%

Positive 82 617Negative 124 179

Past 2 year Return

50%

50%Positive 91 666Negative 114 119

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1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 20200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr(S

t=T|F

t

1)

NBER Recession

Turbulent Periods

Figure 3: Turbulent Months and NBER Recessions

Table 11: Past market return and the likelihood of the hidden state being turbu-lent out of sample

Pr (St = T|Ft1)Formation Period Return 50% 50%

Positive 26 260Negative 48 66

Past 2 year Return 50% 50%Positive 37 280Negative 37 46

4.1 Alternative Specifications

4.1.1 Linear Model of Expected Momentum Strategy Return

In this section we examine whether we can model the expected momentum strategy return

as a linear function of observed variables that describe the state of the market. To answer

this question we consider the linear model considered by Daniel and Moskowitz (2011) given

below:

Rmomt = 0 + 1

I12n=2

RMtn


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Table 12: Linear Model for Predicting Momentum Strategy Return

0 1 R2adj

Estimate 0.02 -105.95 4.23t-stat 6.83 -6.64

As can be seen, the estimated value of 1, the effect of past daily volatility on the future

mean momentum return, is highly significant with a t-statistic of6.64. However, this linearmodel is not as effective as the HMM when it comes to predicting months in which large

losses are likely to occur. This is not surprising, as the objective of this linear model is not

to forecast volatility as we do here, but rather to forecast mean return.

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.020.8

0.6

0.4

0.2

0

0.2

0.4

0.6

RealizedMomentumR

eturn

Fitted Momentum Return

Figure 4: Realized and Fitted Momentum Returns: Linear Model

Figure 4 gives a plot of the realized momentum strategy returns against their predicted

counterpart of the analogue of the linear model in (16) that we estimate. As can be seen,

the HMM may add some ability to forecast large losses relative to the linear model.

4.1.2 GARCH Model of Momentum Strategy Return Volatility

In this section, we compare the results of Markov regime-switching model with those of

GARCH15 model for the joint movement of momentum returns and market returns. We

15See Bollerslev (1986)

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assume that momentum returns are related to the average stock market return during the

portfolio formation period as well as the contemporaneous stock market return in nonlinear

ways in equation (7) of the hidden Markov model specification. Also, we let the volatility

of market returns has an effect on the variance of momentum returns in a GARCH(1,1)

manner. We estimate the following specification:

Rmomt = +

LILt

+LUILUt

+BIBt

+BUIBUt

R

Mt + mom,t (17)

RMt = + M,t (18)

where mom,t

M,t

N

0

0

, t 0

0 t

and t and t are specified as follows in a Bivariate GARCH:

2t = 0 + p12t1 + q1

2mom,t1 + p2

2t1 + q2

2M,t1 (19)

2

t = 0 + p32

t1 + q32

M,t1 (20)

With the variance specification of (19) and (20), we try to capture the spillover of the

variance of market to the variance of momentum. p2 and q2 are parameters to measure

the effect of the past variance of market on the current variance of momentum return in a

GARCH manner. We estimate the model specified from (17) to (20) through ML method

using Matlab.

Table 13 gives the parameters in the variance equation of GARCH(1,1). The estimated

values of L is similar in magnitude to what we obtained using OLS. However, unlike for

OLS, the estimated value of B or BU is smaller than the regression results. This is to

be expected, since the nonlinear recovery response occurs mostly during volatile market

conditions, past-down market is correlated with turbulent hidden states, and those periods

are down weighted when estimating the model parameters with GARCH effects through ML

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Table 13: Bivariate Model Parameter Estimates

Mean-ReturnParams MLE t-stat

1.70 5.84L 0.38 3.90

LU -0.33 -1.98B -0.57 -6.22

BU -0.43 -3.10 0.75 5.72

GARCH of MOMParams MLE t-stat

0 3.70 2.48p1 0.27 2.81q1 0.35 5.18

p2 0.26 2.22q2 0.22 3.34

GARCH of MKTParams MLE t-statal

0 0.61 2.91p3 0.86 45.93q3 0.12 6.09

procedure.

Given the return specification of (17), it is not trivial to calculate the conditional vari-ance of momentum return. However, with the assumption that the market return is normally

distributed and independent of the residual terms of the momentum return, we can calcu-

late the conditional variance of momentum return using the property of truncated normal

distribution for the changing beta conditional on the current market return in the equation

of (17).

0 5 10 15 20 25 30 35 4080

60

40

20

0

20

40

Inferred Standard Deviation of Momentum Return

RealizedMomentumR

eturn

Figure 5: Realized Momentum vs GARCH forecast of Volatility

Figure 5 shows the plot of realized momentum returns against the predicted standard

deviations of the returns. Figure 5 is similar to figures 1 and 2 except that the predicted

standard deviation of momentum return replaces the predicted probability of being in the

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Table 14: HMM-GARCH Comparison

HMM GARCHThreshold # of Sharpe Ratio # of Sharpe Ratio

Loss Months Calm Turbulent Months Calm Turbulent-10% 990 0.27 0.15 881 0.40 0.14

-12.50% 517 0.35 0.09 634 0.39 0.10-15% 396 0.36 0.05 374 0.37 0.03

-17.50% 287 0.36 -0.01 354 0.35 0.04-20% 148 0.31 -0.08 229 0.33 -0.02

turbulent state.

In order to evaluate the performance of the hidden Markov model, we classified a month

as being turbulent when the predicted probability for the turbulent state exceeded a giventhreshold, say 50%. In a similar fashion, for the GARCH model, we classify a month as being

turbulent when the predicted standard deviation of momentum return during the month is

higher than a given threshold standard deviation level.

Table 14 compares the minimum number of months that will be classified as being tur-

bulent so as to include all months with losses exceeding a given loss level under hidden

Markov and GARCH models. As can be seen, the hidden Markov model is better in identi-

fying months that are likely to have large losses in the following sense. The hidden Markov

model will classify 145 of the 1002 months as being turbulent if the turbulent months were

to contain all the months with losses exceeding 20% In contrast, the GARCH specification

will classify 229 of the 1002 months as being turbulent for capturing all months with losses

exceeding 20%. The comparison with the realized Sharpe Ratios is also interesting. With

20% threshold, the Sharpe Ratios of turbulent month returns are -0.08 and -0.02 for the

hidden Markov model and GARCH model. The Sharpe Ratios for the corresponding calm

month returns are 0.31 and 0.33. Even though the GARCH model classifies much more

months as turbulent, in terms of Sharpe Ratio separation, GARCH is about as efficient as

the hidden Markov model. The advantage of the hidden Markov model is in its ability to

predict months with higher probability of large losses.

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4.1.3 Alternative Hidden Markov Model Specifications

this section has been revised substantially to take into account the new specifi-

cation

Recall the hidden Markov model for the joint movement of momentum and market return

given in equations (7) and (8). We examine four different versions of the mean equation by

imposing alternative sets of restrictions:

Model 1: L = B and LU = BU = 0

Model 2: LU = BU = 0

Model 3: L = B

Model 4: LU = 0

That is, Model 1 is a standard one factor market model. Model 2 takes allows the beta to

depends on the past market return. Model 3 is designed to capture the option-like features of

momentum returns, and Model 4 allows for differential up- and down-market betas following

down markets.

Table 15 shows the estimated parameter values for the four restricted versions of the

hidden Markov model. Interestingly, the volatility of the market return and the momentum

strategy residual returns in the two states are about the same in all four models, and the

turbulent state is much less persistent.For all the four models, as before, we classify a month as being turbulent when the

predicted probability of the unobserved underlying state being turbulent exceeds 50%. Table

16 compares the fraction of large gains and losses exceeding a given threshold level that

occur during turbulent months in the four restricted models and the unrestricted model we

considered earlier. Note that during turbulent months the sensitivity of momentum strategy

returns to the stock market return rises sharply in magnitude. Therefore it should come as no

surprise that model 1 which is the standard market model for momentum return generating

process but with state dependent parameters performs almost as well as the full model.

Model 4 that corresponds to the specification of Daniel and Moskowitz (2011) about the

same as the full unrestricted model, suggesting that the increased turbulent state sensitivity

of momentum return to stock index return when the market recovers following a decline

helps most in learning about the hidden state from the data.

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Tab

le15:

Max

imum

Likel

ihoo

dEst

ima

teso

fAlterna

tive

HMM

Spec

ifica

tions

Mo

del

1

Mo

del

2

M

ode

l3

Mo

del

4

Ca

lm

Turbu

lent

Ca

lm

Turbu

len

t

Ca

lm

Turbu

len

t

Ca

lm

Turbu

len

t

1.4

7

-0.9

2

1.4

9

-0.6

0

2.0

1

4.0

1

1.4

6

0.14

(8.1

3)

(-0

.96)

(8.5

1)

(-0

.68)

(6.9

3)

(3.4

5)

(7.7

0)

(1.62)

L

0.0

9

-0.9

3

0.2

6

-0.0

5

0.3

9

-0.2

6

0.2

5

-0.

00

(1.5

7)

(-9

.66)

(4.5

3)

(-0

.32)

(3.3

3)

(-1

.74)

(-4

.29)

(-0.

02)

LU

-0.46

-1.2

7

(-2

.66)

(-5

.16)

B

0.0

9

-0.9

3

-0.5

6

-1.3

1

0.3

9

-0.2

6

-0.4

5

-0.

85

(1.5

7)

(-9

.66)

(-6

.35)

(-12

.23)

(3.3

3)

(-1

.74)

(-2

.91)

(-5.

59)

BU

-0.46

-1.2

7

-0.1

3

-0.

80

(-2

.66)

(-5

.16)

(-0

.63)

(-3.

57)

mom

4.4

5

12

.23

4.1

7

11

.00

4.4

0

11

.14

4.1

2

10.4

0

(27

.11)

(17

.26)

(24

.41)

(17

.08)

(26

.58

)

(18

.04)

(24

.35)

(18

.54)

0.9

8

-0.7

4

0.9

6

-0.6

1

1.0

1

-0.0

1

0.9

8

-0.

01

(6.9

3)

(-1

.09)

(6.7

2)

(-0

.92)

(7.1

5)

(-1

.10)

(6.8

4)

(-0.

85)

M

3.7

0

9.4

9

3.6

8

9.3

2

3.6

6

9.1

0

3.6

5

8.99

(33

.11)

(17

.09)

(32

.38)

(15

.91)

(32

.80

)

(18

.42)

(32

.77)

(17

.91)

Pr

(St=

st|St1=

st

)

0.9

7

0.8

8

0.9

6

0.8

6

0.9

7

0.9

1

0.9

7

0.90

(81

.44)

(18

.78)

(80

.85)

(18

.76)

(96

.51

)

(27

.26)

0.9

0

(24

.64)

Log-L

ike

lihoo

d(103)

3.0

6

3.1

2

3.0

8

3.1

3

AIC

(103)

-6.1

0

-6.2

2

-6.13

-6.2

3

BIC

(10

3)

-6.0

5

-6.1

5

-6.06

-6.1

5

SharpeR

atio

0.3

0

-0.0

6

0.2

8

-0.0

3

0.3

0

-0.0

2

0.3

2

-0.

05

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Table 16: Fraction of Large Gains(Losses) that Occur in Turbulent Months

Gains greater than or equal to: # ofModel

10%

12.5%

15%

17.5%

20% Months

1 32/69 22/42 16/27 10/14 8/11 1732 34/69 24/42 17/27 10/14 8/11 1733 40/69 28/42 18/27 10/14 8/11 1974 35/69 25/42 17/27 10/14 8/11 202

Full 40/69 25/42 18/27 10/14 8/11 206

Losses worse than or equal to: # ofModel 10% 12.5% 15% 17.5% 20% Months

1 29/53 26/35 23/31 18/21 13/13 1732 28/53 24/35 21/31 16/21 12/13 1733 31/53 27/35 24/31 19/21 13/13 1974 32/53 27/35 24/31 19/21 13/13 202

Full 33/53 27/35 25/31 20/21 13/13 206

The scatter plots of the momentum strategy return against the predicted probability of

the unobserved state being turbulent for the four restricted models are given in Figure 6.

The patterns are similar across all four models, which is consistent with the results in Table

16.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.5

0

0.5

Predicted Probability

RealizedMomentum

Model 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

0.5

0

0.5


RealizedMomentum

Model 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

0.5

0

0.5


RealizedMom

entum

Model 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.5

0

0.5


RealizedMomentum

Model 4

Figure 6: Alternative HMM specifications

5 Conclusion

Relative strength strategies, also known as momentum strategies are widely used by active

quantitative portfolio managers and individual investors. These strategies generate large

positive returns on average with little systematic risk as measured using standard asset

pricing models and remain an anomaly.

In this paper we studied the returns on one such momentum strategy. During the period

covering July 1927 - December 2010 the returns on that widely studied strategy using U.S.

stocks generated an average monthly return in excess of 1.18% and an alpha of 1.75%/month

with respect to the three Fama and French (1993) factor model. Momentum strategy returns,

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when combined with the Fama and French three economy wide pervasive factor returns, gives

rise to a portfolio with a Sharpe Ratio of almost 0.30 per month.

However momentum strategies also incur infrequent but rather large losses. There were

five months with losses exceeding 42%/month in the sample of 1002 months. The probability

of such an event occurring if momentum strategy returns were independently and normally

distributed would be close to zero. We show that such periodic tail risk in momentum

returns can be captured by a two state hidden Markov model, where one state is turbulent

and the other is calm. We find that it is possible to predict which of the two hidden state

the economy is in with some degree of confidence All the 13 months with losses exceeding

20%/month occur during turbulent months, i.e., months when the predicted probability of

the hidden state being turbulent exceeds 0.5. Momentum returns averaged -0.65%/month

during turbulent months, with a Sharpe Ratio of -0.05. When such turbulent states are

avoided, the monthly Sharpe Ratio of momentum strategy returns increases to 0.32 and

price momentum poses still more of a challenge to standard asset pricing models.

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