Homework #6
This section is not graded, and you do not need to submit your answers. But you are expected to consider these issues and be ready to discuss them.
1.
2.
3.
What is novel about the AQR Momentum product under construction compared to the various momentum investment products already offered?
Name three reasons the momentum investment product will not exactly track the momentum index, (ie. why the strategy will have tracking error.)
When constructing the momentum portfolio, AQR ranks stocks on their returns from month t − 12 through t − 2. Why don’t they include the t − 1 return in this ranking?
Investigating Momentum
In this section, we empirically investigate some concerns regarding AQR’s new momentum product. On Canvas, find the data file, “momentum data.xlsx”.
• The first tab contains the momentum factor as an excess return: r ̃mom.
• The second tab contains returns on portfolios corresponding to scored momentum deciles.
- – rmom(1) denotes the portfolio of stocks in the lowest momentum decile, the “losers” with the lowest past returns.
- – rmom(10) denotes the portfolio of stocks in the highest momentum decile.
• The third tab gives portfolios sorted by momentum and size.
– rmomSU denotes the portfolio of small stocks in the top 3 deciles of momentum scores.
– rmomBD denotes the portfolio of big-stocks in the bottom 3 deciles of momentum scores.
mom:FF
Note that the Fama-French momentum return, r ̃ mom:FF 1 momBU momSU
, given in the first tab, is constructed by FF as,
r ̃ = 2 (r + r
1 momBD momSD ) − 2 (r + r
) (1)
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1. Is momentum still profitable?
The excess returns of lucrative trading strategies often disappear once the strategy is well-known. The first widely-cited paper on momentum was published in 1993. Have momentum returns or risk changed since then? The AQR case takes place at the end of 2008. Have momentum returns changed in 2009-2021?
Investigate by filling out the summary statistics below for the full-sample and three sub-samples.
mom:FF
(a) Using the data provided, fill in Table 1 with the appropriate stats for r ̃ . Table 1: Momentum performance over time.
Subsample
1927-2021 1927-1993 1994-2008 2009-2021
mean vol
Sharpe
skewness
mv
corr. to r ̃ corr. to r ̃
2.
- (b) Has momentum changed much over time, as seen through these subsample statistics?
- (c) Does this data support AQR’s argument that momentum is an important piece of the ideal portfolio? What if mean returns to momentum are in actuality near zero due to transaction costs—would there still be evidence here that momentum is valuable?
The case is about whether a long-only implementation of momentum is valuable. Construct your own long-only implementation:1,
momU:FF 1 momBU momSU f r ̃ =2(r +r )−r
Note that this is following the FF approach of treating big and small stocks separately. This would be very similar to a scaled version of,
momU 1 mom(8) mom(9) mom(10) f r ̃ =3r +r +r −r
Fill out Table 2 for the data in the period 1994-2021.
Is long-only momentum as attractive as long-short momentum with respect to mean, volatil-
ity, and Sharpe Ratio?
1
momU
For the question below, use the FF-style, r ̃ . t
(a) (b)
momU:FF
Note that we must subtract the risk-free rate to get the excess return of this portfolio, r ̃ .
2
1994-2021 Long-and-short, (r ̃ )
mean vol
Sharpe
mv
Table 2: Long-only momentum performance.
skew corr. to r ̃ corr. to r ̃
Long-only (r ̃ )
mom:FF momU:FF
(c) Is long-only momentum as diversifying as long-short momentum with respect to market and value premia?
3. Is momentum just data mining, or is it a robust strategy?
Assess how sensitive the threshold for the “winners” and “losers” is in the results. Specifically,
(d) Show a plot of the cumulative product of 1 + r ̃ and 1 + r ̃ over the 1994-2021 subsample.2
we compare three constructions:
• long the top 1 decile and short the bottom 1 deciles
momD3 1 mom(8) mom(9) mom(10) r ̃ =3r +r +r
1 10 1 3
= rmom(k) − rmom(k)
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r ̃ =r −r • long the top 3 deciles and short the bottom 3 deciles
k=8 k=1
• long the top 5 deciles and short the bottom 5 deciles
momD1 mom(10)
mom(1)
1 mom(1)
−3r +r +r
1 10 1 5
momD5 mom(k) mom(k)
r ̃ =5 r −5 r k=6 k=1
- (a) Compare all three constructions, (in the full-sample period,) by filling out the stats in the table below for the period 1994-2021.
- (b) Do the tradeoffs between the 1-decile, 3-decile, and 5-decile constructions line up with the theoretical tradeoffs we discussed in the lecture?
- (c) Should AQR’s retail product consider using a 1-decile or 5-decile construction?
momD3
2This is not quite a cumulative return given that these are excess returns, which is why we must add 1 before taking the cumulative product.
(d) Does r ̃ have similar stats to the Fama-French construction in (1). Recall that con- struction is also a 3-decile, long-short construction, but it is segmented for small and large stocks. Compare the middle row of Table 3 with the top row of Table 2.
mom:FF momU:FF
mom(2)
mom(3)
3
1994-2021
mean vol
Sharpe
skewness
mv
Table 3: Robustness of momentum construction.
corr. to r ̃
corr. to r ̃
momD1
r ̃ momD3
r ̃ momD5
r ̃
4. Does implementing momentum require trading lots of small stocks—thus causing even larger trading costs?
For regulatory and liquidity reasons, AQR is particularly interested in using larger stocks for their momentum baskets. (Though they will launch one product that focuses on medium-sized stocks.)
Use the data provided on both small-stock “winners”, rmomSU, and small-stock “losers”, rmomSD, to construct a small-stock momentum portfolio,
rmomS = rmomSU − rmomSD ttt
Similarly, use the data provided to construct a big-stock momentum portfolio, rmomB = rmomBU − rmomBD
ttt
(a) Fill out Table 4 over the sample 1994-2021.
Table 4: Momentum of small and big stocks.
1994-2021 All stocks, r ̃
Small stocks rmomS t
Large stocks rmomB t
mean vol
Sharpe
skewness
corr. to r ̃[m]
mom:FF
(b) Is the attractiveness of the momentum strategy mostly driven by the small stocks? That is, does a momentum strategy in large stocks still deliver excess returns at comparable risk?
5. In conclusion, what is your assessment of the AQR retail product? Is it capturing the important features of the Fama-French construction of momentum? Would you suggest any modifications?
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3 Extensions
This section is not graded, and you do not need to submit your answers. We may discuss some of these extensions.
1. In Section 2 we analyzed whether Momentum changes substantially when we modify the con- struction of the factor. Let’s examine that question for the Value factor.
Re-do Tables 2, 3, and 4 but for the decile and size portfolios of the Value factor. Get this data from Ken French’s website.3
Based on these statistics,
(a) Is the long-only version of Value substantially different? (b) Is the 1 or 5 decile version of Value substantially different?
(c) Does the 3 decile version of Value look like the Fama-French version? (d) Do the Big Value and Small Value factors look similar?
3 https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html Specifically, see the following data sets:
• Size-sorted portfolios for value. (For instance, “big stocks value”, rvB): “6 Portfolios Formed on Size and Book- to-Market (2 x 3)”
• Decile portfolios for value, rv(k): “Portfolios Formed on Book-to-Market”
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