1. Hasib Ahmed Phuvadon Wuthisatian Atsuyuki Naka
Behavior of Momentum in the Foreign Exchange Market: Evidence from Portfolio Approach
Hasib Ahmed
Phuvadon Wuthisatian
Atsuyuki Naka

2. Introduction Momentum Strategy Momentum in other asset classes:
the trend of the market that is winner stocks will continue to rise while loser stocks tend to keep falling (Jegadeesh and Titman, 1993).
Momentum in other asset classes:
Equity, currency, bond, international assets.
Foreign Exchange (FX) market
Most highly traded market in the world
Average trading of $5.1 trillion per day (BIS, 2016).

3. Existence of Momentum
Jegadeesh and Titman (1993) observe the trading strategy according to the past performance, buying well-performed stocks and selling poorly-performed stocks, they find that this strategy generates positive returns from holding period of 3 to 12 months approximately 12% per annum.
Moskowitz, Ooi, and Pedersen (2012), which they test for 58 instruments and find that a strong significance of stock return predictability based on the past performance for all the instruments. They also document that the excess returns of these instruments reverse over longer horizon suggesting that momentum strategies disappear after certain period of time.
Menkhoff, Sarno, Schmeling, and Schrimpf (2012b) use cross section of currencies and report spread in excess return of up to 10% per annum using winner minus losers (WML) strategy for foreign currencies.
Okunev and White (2003) use the cross-sectional analysis of eight major currencies. Using month to month performance, they find momentum strategy creates approximately 6% per annum.

4. Motivation
Currency momentum is particularly interesting, because it provides high returns unrelated to carry trade returns.
It does not make much sense theoretically that such huge abnormal return for momentum strategy in the currency market exists and arbitrageurs are not exploiting it enough to make the strategy (closer to) obsolete.
One explanation could be the possibility of problem of large crashes in the FX momentum.
So, our objective for this study is to look for the problem of large crashes in the FX momentum.
If crashes contributes to the returns of the strategy, then we should be able to observe the source of returns whether from winner or loser portfolio.

5. Momentum Crashes
Daniel and Moskowitz (2016) provide an empirical evidence of momentum crashes and implement the dynamic momentum strategy based on forecast of momentum’s mean and variance to improve the momentum strategies. Their result suggest that the dynamic momentum strategy can help double the alpha and Shape ratio. Their finding also extends to other asset classes such as foreign exchange (FX) market.
Note that their study is based on 10 developed currencies – potential selection bias.
We follow their approach to investigate whether the crashes contribute to the abnormal returns in currency momentum by extending up to 66 currencies.

6. Risk-Managed Momentum
Barroso and Santa-Clara (2015) explain that risk-managed momentum should provide higher mean return, less volatility, less skewness, and higher Sharpe ratio than the plain momentum strategy.
We believe that using risk-managed momentum of winner-minus-loser (WML) portfolio would benefit investors to hedge themselves against the financial crises.

7. Contributions
(i) unlike stock market, out sample does not behave as it is predicted by Daniel and Moskowitz (2016) result,
(ii) the loser portfolio, however, acts the same way as it does in stock market, and
(iii) the source of returns from WML is mainly from loser portfolio rather winner portfolio.

8. Data Source: Barclays Bank International, Reuters (Datastream).
Years: December, 1984 through December, 2015.
The currencies are required to have at least five years of spot and forward rate data.
Final sample: 66 currencies.

9. Methodology Monthly excess returns for holding foreign currency k as,
𝑟 𝑥 𝑡+1 𝑘 ≡ 𝑖 𝑡 𝑘 − 𝑖 𝑡 −∆ 𝑠 𝑡+1 𝑘 ≈ 𝑓 𝑡 𝑘 − 𝑠 𝑡+1 𝑘
Given that covered interest parity (CIP) holds even at very short horizons (Akram, Rime, and Sarno (2008)), 𝑖 𝑡 𝑘 − 𝑖 𝑡 equals 𝑓 𝑡 𝑘 − 𝑠 𝑡 𝑘 .
The last date observations are used as observation for the month.
The end of month net return
long position: 𝑟 𝑥 𝑡+1 𝑙 = 𝑓 𝑡 𝑏 − 𝑠 𝑡+1 𝑎
short position: 𝑟 𝑥 𝑡+1 𝑠 = −𝑓 𝑡 𝑎 − 𝑠 𝑡+1 𝑏
If a currency stays in the portfolio at the end of month:
long position: 𝑟 𝑥 𝑡+1 𝑙 = 𝑓 𝑡 𝑏 − 𝑠 𝑡+1
short position: 𝑟 𝑥 𝑡+1 𝑠 = −𝑓 𝑡 𝑎 − 𝑠 𝑡+1

10. Portfolio Construction
We then construct the net excess returns into portfolios based on the lagged returns under previous formation period, 𝑓=1,3,6,9,12 months, and these portfolios are held in holding period, ℎ=1,3,6,9,12 months.
The winner minus loser (WML) portfolios are formed by taking long position in the winner portfolio and short position in the loser portfolio.
The portfolios, in our analysis, are dividing into five deciles ranging from lowest 20% of excess return up to top 20% excess return.

11. Sources of Momentum Return
We follow Daniel and Moskowitz (2016) to estimate the time-varying betas of winner and loser portfolios using 126-day rolling market model regression with daily data. The estimation of betas is calculated based on the ten daily lags of market return as follows:
𝑟 𝑖,𝑡 𝑒 = 𝛽 0 𝑟 𝑚,𝑡 𝑒 + 𝛽 1 𝑟 𝑚,𝑡−1 𝑒 +…+ 𝛽 10 𝑟 𝑚,𝑡−10 𝑒 + 𝜀 𝑖,𝑡
where 𝑟 𝑚 𝑒 is the daily market return.
Then, we sum the estimation coefficients 𝛽 𝛽 1 +…+ 𝛽 10 to determine the betas for winner and loser portfolios.

12. 𝑟 𝑖,𝑡 = (𝛼 0 + 𝛼 𝛽 𝐼 𝛽,𝑡−1 ) +(𝛽 0 + 𝛽 𝛽 𝐼 𝛽,𝑡−1 ) 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡
Option-like behavior
We use four-time series regressions on a set of independent variables.
Excess monthly market return index 𝑅 𝑚,𝑡 𝑒 in month t
A bear market indicator is characterized by 𝐼 𝛽,𝑡−1 . We assign the value equal to one if the past 12-month return is negative and zero otherwise.
Up-market indicator is defined as 𝐼 𝑢, 𝑡 . The value is equal to one if the excess return of currency portfolio is greater than risk- free in month t.
Up-Down market indicator is defined as 𝐼 𝛽, 𝑡−1 𝐼 𝑢,𝑡 . The variable is assigned to extent to capture the trend of up-down market.
𝑟 𝑖,𝑡 = 𝛼 0 + 𝛽 0 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡
𝑟 𝑖,𝑡 = (𝛼 0 + 𝛼 𝛽 𝐼 𝛽,𝑡−1 ) +(𝛽 0 + 𝛽 𝛽 𝐼 𝛽,𝑡−1 ) 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡
𝑟 𝑖,𝑡 = (𝛼 0 + 𝛼 𝛽 𝐼 𝛽,𝑡−1 ) +(𝛽 0 + 𝐼 𝛽,𝑡−1 ( 𝛽 𝛽 + 𝐼 𝛽, 𝑡−1 𝐼 𝑢,𝑡 ) 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡

13. Momentum Optimality
We analyze further for optimality of momentum portfolio. We run a regression by assigning the variable to indicate either bear or bull market. The regression is described as:
𝑟 𝑖,𝑡 = (𝛼 0 + 𝛼 𝛽 𝐼 𝛽,𝑡−1 ) +(𝛽 0 + 𝐼 𝛽,𝑡−1 ( 𝛽 𝛽 + 𝐼 𝑢, 𝑡 𝛽 𝛽,𝑢 ) 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡
𝑟 𝑖,𝑡 = (𝛼 0 + 𝛼 𝐿 𝐼 𝐿,𝑡−1 ) +(𝛽 0 + 𝐼 𝐿,𝑡−1 ( 𝛽 𝐿 + 𝐼 𝑢, 𝑡 𝛽 𝐿,𝑢 ) 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡

14. Risk-managed momentum
I estimate the realized variance in month 𝑡 as
𝑅 𝑉 𝑖,𝑡 = 𝑗=0 20 𝑟𝑥 𝑖, 𝑑 𝑡 −𝑗 2 ,
An AR(1) regression of realized variances will produce the realized volatility of momentum and market.
I’ll also get the persistence measure of risk ( 𝑎 1 coefficient).
Using the monthly return of momentum as 𝑟 𝑥 𝑡+1 𝑘 and daily return as 𝑟 𝑥 𝑑+1 𝑘 , the variance forecast will be
𝜎 𝑊𝑀𝐿, 𝑡+1 2 =21 𝑗=0 125 𝑟 𝑥 𝑊𝑀𝐿, 𝑑 𝑡 −𝑗 2 /126
As WML is a zero-investment and self-financed strategy, it can be scaled without constraints.
So, the returns are scaled as
𝑟 𝑥 𝑊𝑀 𝐿 ∗ ,𝑡+1 = 𝜎 𝑡𝑎𝑟𝑔𝑒𝑡 𝜎 𝑡+1 𝑟 𝑥 𝑊𝑀 𝐿 ,𝑡+1
where 𝑟 𝑥 𝑊𝑀 𝐿 is the unscaled momentum and 𝑟 𝑥 𝑊𝑀 𝐿 ∗ is scaled momentum. 𝜎 𝑡𝑎𝑟𝑔𝑒𝑡 is a constant target level of volatility.
Then, I compare the performances of 𝑊𝑀 𝐿 ∗ and 𝑊𝑀𝐿.

19. Table 6: Market Timing Regression
Table 6: Market Timing Regression. The table represents the estimated coefficients from regressions of monthly excess return of WML, winner, and loser portfolio. The regression models are: model (1) - equation (3): 𝑟 𝑖,𝑡 = 𝛼 0 + 𝛽 0 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡 unconditional market model, model 2 - equation (4): 𝑟 𝑖,𝑡 = (𝛼 0 + 𝛼 𝐵 𝐼 𝐵,𝑡−1 ) +(𝛽 0 + 𝛽 𝐵 𝐼 𝐵,𝑡−1 ) 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡 conditional bear market indicator, and model (3) and (4) - equation (5): 𝑟 𝑖,𝑡 = (𝛼 0 + 𝛼 𝐵 𝐼 𝐵,𝑡−1 ) +(𝛽 0 + 𝐼 𝐵,𝑡−1 ( 𝛽 𝐵 + 𝐼 𝑈, 𝑡−1 𝛽 𝑈,𝑡 ) 𝑅 𝑚,𝑡 𝑒 + 𝜀 𝑡 the up-market indicator. α0 represents the intercept of the regression model, αB is the bear market indicator, β0 is the market excess return, βB is the up-market indicator, and βU, B is the up-down market indicator. The t-test is reported as Newey-West t-statistic test. The t-tests are reported in parentheses. *, ** shows the significance of 5%, and 10%.
Coefficient
Variable
WML
Winner
Loser 1 2 3 4 α0
Constant (Alpha)
0.0034
0.0016
0.0031
0.0024
0.0003
(2.22)**
(-0.43)
(-0.44)
(1.01)
(1.58)
(1.60)
(2.79)**
(0.30)
(-1.89)*
(-1.91)*
(-0.76) αB
Alpha in Bear Market
0.0014
0.0091
0.0061
(0.50)
(2.54)**
(-0.62)
(1.54)
(1.23)
(2.43)** β0
Market Excess Return
0.0147
0.5868
0.5530
0.8303
1.1214
1.1100
(0.21)
(6.82)**
(6.92)**
(6.55)**
(21.67)**
(23.56)**
(23.87)**
(23.88)**
(-17.97)**
(-9.02)**
(-9.08)**
(-9.51)** ΒB
Bear Market indicator
(-9.89)**
(-4.85)**
(-6.53)**
(-4.62)**
(-5.84)**
(-6.81)**
(-3.30)**
(-4.78)**
Β U, B
Up-Down Market
(-3.41)**
(-2.31)**
(-3.22)**
(-2.89)**
(-2.33)**
(-1.03)
R-square
0.0001
0.2159
0.2407
0.2269
0.5675
0.6542
0.6640
0.6618
0.4741
0.5361
0.5431
0.5355

24. Conclusion
We analyze the momentum strategy based on winner minus loser portfolio.
We also explore the possible sources of the returns of momentum strategy using Daniel and Moskowitz (2016) approach.
Our result, however, contradicts to what they find.
We find that the momentum strategy is mainly from the loser portfolio, which indicates the return during the financial stress attributed to the return in the momentum strategy.
We also test for the risk-managed momentum; however, our result does not indicate much improvement of implementing such strategy.