7.11 CAPM Implementation
7.11.1 Factor Portfolios
For our comparison, we have chosen to explore 6 Fama-French factors and the Betting Against Beta portfolio:
• Fama-French Momentum Factor
• Fama-French SMB (Small Minus Big) Factor - Size
• Fama-French HML (High Minus Low) Factor - Value
• Fama-French RMW (Robust Minus Weak)
• Fama-French CMA (Conservative Minus Aggressive)
• Fama-French Market Minus Risk-free, value-weighted
• Betting-Against-Beta
The momentum factor is constructed from six value-weight portfolios formed on size and prior returns.
Specifically, from the average return on two high prior return portfolios minus the average of two low prior return portfolios, both on the the prior 2-12 months, meaning the last month is excluded:
Momentum = 1/2(Small High+Big High)−1/2(Small Low+Big Low) (65) The daily prior 2-12 return breakpoints are the 30th and 70th NYSE percentiles.
The 5 Fama-French factors are constructed using 6 value-weight portfolios formed on size and book-to-market, 6 value-weight portfolios formed on size and operating profitability, and 6 value-weight portfolios formed on size and investment. Specifically, SMB63 is the average return on nine small stock portfolios minus the average return on nine big stock portfolios:
SMB = 1/3(SMBB/M + SMBOP + SMBIN V) (66)
HML is the average return on two value portfolios minus the average return on two growth portfolios:
HML = 1/2(Small V alue+Big V alue)−1/2(Small Growth+Big Growth) (67) RMW is the average return on two robust operating profitability portfolios minus the average return on two weak operating profitability portfolios:
RMW = 1/2(Small Robust+Big Robust)−1/2(Small W eak+Big W eak) (68) CMA is the average return on two conservative investment portfolios minus the average return on two aggressive investment portfolios:
CMA = 1/2(Small Conservative+Big Conservative)−1/2(Small Aggressive+Big Aggressive) (69) Additionally, the Fama-French Market Minus Risk-free (Rm−Rf) is the excess return on the mar-ket and the value-weighted return of US stocks listed on NYSE, AMEX and NASDAQ in the CRSP
63See Appendix for the individual portfolios
database.
Lastly, the market-neutral Betting Against Beta (BAB) factors are long leveraged low-beta assets and short high-beta assets64. To construct the BAB factor, all securities in a country are ranked in ascending order on the basis of their estimated beta and the ranked securities are assigned to one of two portfolios: low-beta and high-beta. Then in each portfolio, securities are weighted by the ranked betas so that lower-beta securities have larger weights in the low-beta portfolio and the same for higher-beta securities.
Letz be then×1 vector of beta rankszi = rank(βit) at portfolio formation and letz= 10nz/nbe the average rank. n is the number of securities and 1n is an×1 vector of ones. The portfolio weights of the low-beta and high-beta portfolios are given bywL=k(z−z)− and wH =k(z−z)+, wherek is a normalizing constantk−1 = 10n|z−z|/2 andx+and x− indicate the positive and negative elements of a vector x. By construction, 10nwH = 1 and 10nwL= 1.
Both portfolios are rescaled to have a beta of one at portfolio formation. The BAB is then the self-financing zero-beta portfolio that is long in the low-beta portfolio and short in the high-beta portfolio:
rBABt+1 = 1 βtL
rLt+1−rf
− 1 βtH
rHt+1−rf
(70)
wherert+1L =r0t+1wL, rHt+1 =rt+10 wH, βtL=βt0wL, and βtH =βt0wH
64Betting Against Beta (2014)
8 Results
This section will be a run-through of all results and discussions of these. First, the performance of the each individual strategy will be evaluated in order to see how they match up against each other.
Second, we will look at how the ensemble learning method with the three strategies performed in order to see if there are significant advantages in creating an ensemble learning strategy of three fundamentally different methods. Lastly, we will introduce our strategies in a CAPM framework to see if the generate alpha against several factor portfolios.
8.1 Momentum Strategy
It is also important to stop up and consider what the realistic outcomes of the strategy could be.
Having the momentum strategy in mind, we do not expect incredible gains of more than 10% a year or even positive returns every year. Besides very few strategies doing this, the momentum strategy in particular, is a long-term, long-only, volatility-eliminating and stable investment strategy. The goal is to show strong performance in bull markets and to lose less than the benchmark index in bear markets. If this is possible, the strategy is very attractive over time.
The first result that comes to mind is how the strategy performed against the S&P 500 index. The figure below shows their cumulative returns over the period ranging from 07-01-2014 to 31-12-2019.
Figure 44: Cumulative returns and vol adjustment Naive vs HRP - Momentum
As we can easily see, the strategy is definitely able to generate higher return than the S&P 500 over the period for both asset allocation methods. If it was only making money and not beating the in-dex, time would have been wasted and there would be no point in implementing this strategy. It is clear to see how well it is performing during bull markets and an even bigger performance effect stemming from simply not being invested in stocks when the markets are bearish, which is
observ-able from the horizontal return periods, late 2018 in particular. However, this positive effect also has a negative downside. As we can see, the two strategies take a long time to start trading again after a period of not trading. Because of this, the strategy ”misses” potential gains in start 2019.
We also specifically notice the incredibly gains made in late 2016 after Trump was voted President of the United States. In addition, and as mentioned earlier, we can see that the positive feedback nature of momentum strategies is clear and that the strategy is prone to short momentum crashes.
The strategy is not able to prevent this, as it takes a mid-term and a long-term MA parameter into account. However, HRP allocation is able to soften this effect by providing a better variance structure.
HRP also beats the naive allocation method in cumulative returns. The HRP return profile looks very similar to 1/N, but it seems to be better at reducing losses, specifically at the dips in 2015. The higher return in HRP seem to come mainly from trading in 2019, as the cumulative returns are close to the same level for both allocations at the beginning of 2019. We can see a big difference on the bottom plots, which show the cumulative returns volatility matched to the benchmark. The Naive allocation end up very close to the benchmark. However, the HRP allocation is clearly above, indicating lower volatility.
We specifically expect the momentum strategy to have low volatility and low max. drawdown, as it is constructed to stay out of bearish markets. The figures below show the rolling volatility and the top 5 drawdown periods.
Figure 45: Rolling vol and top 5 drawdown periods Naive vs HRP - Momentum
We can see in the top figures above that the average Naive volatility is slightly closer to the 0.15 line, as it is an illustration of the annual volatility given in table 2. Both allocations have very similar rolling volatility profiles. However, HRP seem to be more compressed in some periods. Specifically, the period from the beginning to 2015 seem much flatter. Table 11 in the appendix show details for the top 5 drawdown periods shown above. We can see that the longest drawdown duration for 1/N is the worst drawdown period. This is not the case for HRP, where the second worst period is much longer than the worst period. It looks like the two worst periods have switched between HRP and 1/N, which we can also see above. The 2018-2019 period is worst for HRP and the 2015-2016 period is worst for naive allocation. These differences is probably due to the different nature that the two allocations methods have. Both have a worst drawdown period lower than S&P 500, which had a
drawdown period of -19.8% from 21-09-2018 to 24-12-2018.65
A quick overview of basic return metrics of how well it actually performed can be seen below.
Table 2: Momentum performance
Strategy Momentum
Asset Allocation Naive Diversification (1/N) Hierarchical Risk Parity
Cumulative Returns 90.7% 102.8%
Annual Returns 11.0% 12.1%
Annual Volatility 14.6% 14.2%
Sharpe Ratio 0.79 0.87
Sortino Ratio 1.10 1.24
Max. Drawdown -17.7% -17.5%
Daily VaR95% -1.8% -1.7%
Daily Turnover 9.6% 12.3%
α against S&P 500 0.05 0.07
β against S&P 500 0.66 0.59
Start Date 07-01-2014
End Date 31-12-2019
Total Months 74
It is worth noticing that HRP performs better at every metric in the table above, highlighted with bold numbers, except for Daily Turnover.
1. Returns: cumulative- and annual returns are higher for HRP.
2. Variance: annual volatility, max. drawdown and daily VaR are all lower for HRP.
3. Performance Criteria: as a result of greater returns and lower variance, Sharpe- and Sortino ratios are higher for HRP.
4. Cost: daily turnover is, however, higher for HRP.
5. CAPM: we observe that α is higher and β is lower for HRP against S&P 500
HRP asset allocation is, by construct, more comprehensive than naive 1/N and affected by more factors, including variance. So, it is no surprise that rebalancing asset sizes is a heavier load for HRP.
HRP allows high jumps between an asset’s weight. It could, potentially, drop from 90% allocation to 10% the next trading day. From the bottom figures in Figure 44 and from the volatility in the table, we see, as expected, that HRP is not as volatile as 1/N. We mentioned in section 3.3 that if we combine high-vol and low-vol stocks in a portfolio and allocate equally, the volatile stocks set the direction for the portfolio and the performance of less volatile stocks won’t matter enough. This effect is reduced with HRP allocation. We notice that both Sharpe- and Sortino ratio is higher for HRP, indicating a more attractive portfolio than the naive diversification. The daily Vale-at-Risk is
65Capital Spectator
constructed with a 95% tail ratio, so HRP and 1/N will not see daily losses greater than -1.7% and -1.8%, respectively, in 95% of the cases. Theα and β in the table is calculated against the S&P 500 benchmark. The HRP allocation is better on both parameters as it performs better in both return and systematic risk against the benchmark.