1. Partial Moment Momentum
Yang Gao a, Henry Leung a, and Stephen Satchell a,b
a University of Sydney Business School b Trinity College, University of Cambridge
Presented by:
Stephen Satchell;
11th Financial Risks Forum
27thMarch, 2018

2. Momentum trading strategy
Cross-sectional momentum strategies are employed by buying previous winners and selling previous losers. We follow a similar method to Jegadeesh and Titman (1993) J × K trading model.
J-month Formation period
Stocks are sorted based on their past performance.
K-month Holding period
Long winners (best-performing) while short losers (worst-performing).
1-month lag between formation and holding periods
Doing so can avoid short-term reversals.
Rebalancing
We revise the weights on 1/Kth of the securities in the entire portfolio in any given month t and carry over the rest from the previous month.
Zero-net position
Buy or short sell cash asset.

3. Why volatility matters
Momentum profits depend “critically” on the state of the market.
Chordia and Shivakumar (2002) and Cooper, Gutierrez, and Hameed (2004)
Momentum is profitable during economic expansionary periods, to be more specific, is profitable only following the periods of market gains.
Momentum profits benefit from persistent trends of the market which can be predicted by market volatility.
Barroso and Santa-Clara (2015) and Daniel and Moskowitz (2016)
Vol Scaling of the weights of momentum portfolios increases the Sharpe ratio of the plain momentum strategy.

4. But volatility itself is not good enough
Volatility is larger when markets fall than when they rise.
Future volatility is related more to past negative returns than past positive returns.
The economic consequences of volatility in falling or rising markets are not equal.
However, such volatility-scaled strategies, in both
a cross-sectional momentum setting [Barroso and Santa-Clara (2015) & Daniel and Moskowitz (2016)]
and a time-series momentum setting [Moskowitz, Ooi and Pederson (2012)]
are unable to distinguish between upside and downside risk.

5. Why might upside versus downside risk matter
Momentum crashes
Chordia and Shivakumar (2002); Cooper, Gutierrez, and Hameed (2004); Daniel and Moskowitz (2016).
Short selling constraint in adverse times
Ali and Trombley (2006); Gao and Leung (2017).
Investors risk preference
Daniel, Hirshleifer, and Subrahmanyam (1998); Hong and Stein (1999).
Thus, we propose two partial moments-based strategies (to be defined), which have the potential to better capture market trends and reduce losses during market turbulence using partial moment decompositions of squared market returns.

6. Main findings
We also find that momentum profits benefit from persistent trends of the market which can be predicted by market volatility.
We find that partial moments-based momentum trading strategies significantly outperform plain momentum strategy and volatility-scaled strategies [Barroso and Santa-Clara (2015) & Daniel and Moskowitz (2016)].
An explanation of this strong profitability is that momentum benefits from good versus bad risk. In the context of the conference good risk is emerging

7. Data Monthly and daily US equity data sourced from the CRSP.
Whole sample period: January 1927 to December 2016.
Our sample includes common stocks (CRSP share code 10 or 11) of all firms listed on NYSE, Amex and Nasdaq (CRSP exchange code 1, 2 or 3).
Awful Data but US journals like it
Proxy for the market portfolio and the risk-free rate
The value-weighted index of all listed firms in the CRSP.
1-month Treasury bill rate.

8. Partial moments
Following the literature: Andersen, Bollerslev, Diebold & Ebens (2001), Barndorff-Nielson (2002) and Baruník, Kočenda & Vácha (2016)
We define monthly realised variance 𝑅𝑉 as
𝑅𝑉 𝑡 = 𝑖=1 𝑛 𝑟 𝑖,𝑡 2
We define lower partial moment 𝑅𝑃𝑀 − and higher partial moment 𝑅𝑃𝑀 + as
𝑅𝑃𝑀 𝑡 − = 𝑖=1 𝑛 𝑟 𝑖,𝑡 2 𝐼 𝑟 𝑖,𝑡 <0
𝑅𝑃𝑀 𝑡 + = 𝑖=1 𝑛 𝑟 𝑖,𝑡 2 𝐼 𝑟 𝑖,𝑡 ≥0
There is an identity
𝑅𝑉 𝑡 = 𝑅𝑃𝑀 𝑡 − + 𝑅𝑃𝑀 𝑡 +

9. Relative Strength Momentum Forecastability
Momentum is forecastable even when returns are independently and identically distributed (𝑖𝑖𝑑).
Under the assumption that 𝑟 𝑡 is 𝑖𝑖𝑑 𝜇, 𝜎 2 , it follows by elementary calculation that
𝑉𝑎𝑟 𝑅 𝑡 =𝐸 𝑟 𝑡 2 𝑟 𝑡−1 2 − 𝐸 2 𝑟 𝑡 𝑟 𝑡−1
𝐶𝑜𝑣 𝑅 𝑡 , 𝑅 𝑡−1 =𝐸 𝑟 𝑡 𝑟 𝑡−1 2 𝑟 𝑡−2 −𝐸 𝑟 𝑡 𝑟 𝑡−1 𝐸 𝑟 𝑡−1 𝑟 𝑡−2
This leads to an autocorrelation coefficient as 𝜌= 𝜇 2 𝜎 2 𝜎 4 +2 𝜇 2 𝜎 2 . If we interpret the signal–to-noise ratio of the strategy as 𝑆𝑁= 𝜇 𝜎 , then 𝜌= 𝑆𝑁 𝑆𝑁 2 . Thus, if the strategy has a signal–to-noise ratio of 0.5, then returns will appear to have an autocorrelation coefficient of 0.17 although the underlying data are pure white noise.
Furthermore, as SN becomes large, we reach an upper bound for 𝜌 of ½.
Assuming that 𝐸( 𝑟 𝑡 )=0, we now turn to
𝑉𝑎𝑟 𝑅 𝑡 2 =𝐸 𝑟 𝑡 4 𝑟 𝑡−1 4 − 𝐸 2 𝑟 𝑡 2 𝑟 𝑡−1 2
𝐶𝑜𝑣 𝑅 𝑡 2 , 𝑅 𝑡−1 2 =𝐸 𝑟 𝑡 2 𝑟 𝑡−1 4 𝑟 𝑡−2 2 −𝐸 𝑟 𝑡 2 𝑟 𝑡−1 2 𝐸 𝑟 𝑡−2 2 𝑟 𝑡−1 2
Let 𝜇 𝑗 =𝐸( 𝑟 𝑡 𝑗 ). Then 𝜌= 1 𝜇 4 𝜇
This shows that when underlying returns are white noise, the lower is the forecastability of momentum volatility and the higher is the kurtosis. If we take this to be the kurtosis of semi-annual index returns, we might expect a number near 5 and thus, the (spurious) autocorrelation might be of a similar magnitude to before.
Thus, we do not use momentum volatility in guiding our strategies but instead use a value-weighted market index.

10. Estimation
We use daily data for our different volatility estimators and use last month as our forecast for next month.
Alternatively, we also use a VAR(1) model based on 𝑅𝑃𝑀 + and 𝑅𝑃𝑀 − .
Method 2 does not do as well as method 1.
Old result hard to beat a random walk out of sample one period ahead.

11. Adapted Sortino ratio (an improvement on Sharpe ratio)
A performance measure better capturing the downside risk
We define monthly realised variance 𝑅𝑉 as
𝐴𝑑𝑎𝑝𝑡𝑒𝑑 𝑆𝑜𝑟𝑡𝑖𝑛𝑜 𝑟𝑎𝑡𝑖𝑜= 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 2∗𝐷𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑆𝑒𝑚𝑖𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
where
𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 𝑡 = 𝑅 𝑡 − 𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑇𝑎𝑟𝑔𝑒𝑡 𝑅𝑒𝑡𝑢𝑟𝑛 𝑡
𝐷𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑆𝑒𝑚𝑖𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛= 𝑖=1 𝑁 (𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖 − 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 ) 2 𝑁 𝐼 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖 <0
Our adaptation differs from the standard Sortino (1994) ratio with target return equal to the riskless rate, as, in the event that mean returns are zero and 𝑅𝑃𝑀 − = 𝑅𝑃𝑀 + , we recover the Sharpe ratio.

12. PM-based momentum strategies
These have the potential to better capture market trends and avoid huge losses during market turbulence using partial moment decompositions of squared market returns.
1. Partial moment momentum strategy (PMM)
Switching positions of the winner and loser portfolios during the holding periods depending upon current estimates of partial moments ( 𝑅𝑃𝑀 − & 𝑅𝑃𝑀 + ).
2. Extended partial moment-decomposed momentum strategy (PMD)
Scaling the weights of winner and loser portfolios towards favourable/unfavourable volatility signals and holding an off-setting position in cash.

13. Partial moments and reference points
Medians of 𝑅𝑃𝑀 𝑡 + and 𝑅𝑃𝑀 𝑡 − as the reference points for upper and lower partial moments.
We divide market states into four conditions. Each quadrant represents one condition based on upper and lower partial moments and their reference points.
Momentum profits benefit from persistent trends of the market. Thus, we posit, for example,
Condition 1 represents an environment of high market volatility, which is not conducive to momentum trading profits.
Condition 3 reflects an environment when market trends tend to persist in the same direction. In this case, we expect high profitability as the outcome of momentum based strategies.
Low
Profitability
High
Profitability

14. Method 1, PMM construction
PMM: Switching positions of the winner and loser portfolios during the holding periods depending upon current estimates of market partial moments ( 𝑅𝑃𝑀 − & 𝑅𝑃𝑀 + ).
During the 1-month holding period from month t to t+1, we compare monthly upper and lower partial moments during the period [t-1, t] with their reference points and switch positions of winners, losers, and the cash assets.
For example, for PMM strategy 4 (PMM_S4), if condition 1 applies during the period [t-1, t] in which upper and lower partial moments are all higher than their reference points, then we close out our positions in both winners and losers. The PMM strategy return for month t+1 is 0.

15. PMM performance on a 11×1 basis. Four (sub)sample periods.
Reference points measured during the whole sample period.
PMM_S1 to PMM_S6: PMM strategies
WML strategy: 11×1 plain momentum strategy.
Sharpe ratio, adapted Sortino ratio and returns are annualised.
PMM:
Better risk-adjusted returns
do not work well when the market is calm

16. Method 2, PMD strategy
PMM strategies, by switching positions of winner and loser portfolios, do not work well when the market is calm.
Thus, we propose a different type of partial moments-based momentum strategy, named PMD strategies: an extension of the volatility-based momentum strategy in Barroso and Santa-Clara (2015) and Daniel and Moskowitz (2016) where they do not differentiate between upside or downside risk.
We extend by tilting our strategy long or short towards favourable/unfavourable volatility signals and holding an off-setting position in cash.

17. +( 𝜑 2 ( 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − ))− 𝜑 1 ( 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − )) 𝑟 𝑓,𝑡+1
Method 2, PMD strategy
The return of this strategy is denoted as
𝑟 𝑝,𝑡+1 = 𝜑 1 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − 𝑟 𝑤,𝑡+1 − 𝜑 2 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − 𝑟 𝑙,𝑡+1
+( 𝜑 2 ( 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − ))− 𝜑 1 ( 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − )) 𝑟 𝑓,𝑡+1
with the constraint
𝜑 1 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − + 𝜑 2 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − = 2 𝜎 𝑡𝑎𝑟 𝑅 𝑉 𝑡
where
𝜑 1 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − = 2 𝜎 𝑡𝑎𝑟 𝑅 𝑉 𝑡 𝑅𝑃𝑀 𝑡 + 𝑅𝑃𝑀 𝑡 + + 𝑅𝑃𝑀 𝑡 −
𝜑 2 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − = 2 𝜎 𝑡𝑎𝑟 𝑅 𝑉 𝑡 𝑅𝑃𝑀 𝑡 − 𝑅𝑃𝑀 𝑡 + + 𝑅𝑃𝑀 𝑡 −
Zero-net positions
If 𝑅𝑃𝑀 𝑡 + = 𝑅𝑃𝑀 𝑡 − , then we have a conventional long-short portfolio with scaling 𝜎 𝑡𝑎𝑟 𝑅 𝑉 𝑡 as in Barroso and Santa-Clara (2015), formula (5), page 115.

18. Method 2, PMD strategy (with leverage consideration)
In practice, leverage is an issue in long–short portfolios.
Many institutional hedge funds have strict restrictions on leverage, with 200% leverage being a typical upper bound. The previous popularity of funds provides evidence that leverage is not unconstrained in practice.
Leverage is defined as the sum of absolute value of long and short weights (ignoring cash positions) so that our abovementioned strategies have a leverage as in formula
𝜑 1 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − + 𝜑 2 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − = 2 𝜎 𝑡𝑎𝑟 𝑅 𝑉 𝑡
If it exceeds the upper bound, to rescale our weights to obey the 200% leverage condition, we need to change our scaling to
𝜑 1 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − =2 𝑅𝑃𝑀 𝑡 + 𝑅𝑃𝑀 𝑡 + + 𝑅𝑃𝑀 𝑡 −
𝜑 2 𝑅𝑃𝑀 𝑡 + , 𝑅𝑃𝑀 𝑡 − =2 𝑅𝑃𝑀 𝑡 − 𝑅𝑃𝑀 𝑡 + + 𝑅𝑃𝑀 𝑡 −
with corresponding positions in cash.

19. PMD performance on a 11×1 basis. Four (sub)sample periods.
PMD: unconstrained PMD strategy.
PMD_C: a 200% leverage-constrained PMD strategy.
WML strategy: 11×1 plain momentum strategy.
Sharpe ratio, adapted Sortino ratio and returns are annualised.

20. Comparison of performance
As we previously discussed, the higher the kurtosis, the lower the forecastability.
Higher and more significant returns (PMD)
Better risk-adjusted returns
Skewness is reduced and even turns positive
Normality test
Null: normally distributed

21. PMD strategies versus the Barroso and Santa-Clara (2015) volatility-scaled momentum strategy
See BSC (2015)
table 3, page 116.
on a 11×1 basis.
Four (sub)sample periods.
PMD/PMD_C: unconstrained/ 200% leverage-constrained PMD strategy.
BSC/BSC_C: unconstrained/ 200% leverage-constrained scaled momentum strategy by BSC (2015).
WML strategy: 11×1 plain momentum strategy.
Sharpe ratio, adapted Sortino ratio and returns are annualised.

22. Other robustness checks
All these tests reveal robust results.
PMM & PMD on a 6×6 basis
To show the effectiveness of our PM-based momentum strategies following Jegadeesh and Titman (1993).
Out-of-sample analysis
1927 to 1999 as in-sample.
Dynamic PMD
Fit in-sample partial moments into a VAR(1) model and construct out-of-sample PMD using forecasted partial moments.
For further reading kindly see Section 5 and Section 6 of the working paper. Available on SSRN (submission no ).

23. Practical Matters No transaction costs.
No consideration of tradability and liquidity. If it is in CRSP, we use it. Maybe ok for factor but not for strategy.
Methods used in Asness, Moskowitz, and Pedersen (2013) “P3-P1” momentum portfolios might be more practical (large and liquid stocks only; tercile sort rather than decile sort).

24. Conclusion
We also find that momentum profits benefit from persistent trends of the market which can be predicted by market volatility.
We find that partial moments-based momentum trading strategies  significantly outperform plain momentum strategy and volatility-scaled strategies
We suggest that this great profitability is due to the investment opportunities that arise from being able to distinguish good and bad risk and that partial moments are not traded out

25. Thank you! Yang Gao University of Sydney Business School