The last thirty years of financial academia have done a good deal of work on constructing optimal portfolios based on things such as expected return, variance, and correlations. I am going to highlight a way to increase risk-adjusted returns by using systematic long/short strategies on an individual ticker. We are going to use the SPDR S&P 500 ETF (NYSEARCA:SPY) here for its preferable liquidity (and thus cheaper borrowing cost), its familiarity, and its implied diversification (because it is an ETF and not, say, an individual name that would have a large error term from earnings announcements, etc.).

## Choosing an Algorithm

We are going to be constructing very simple algorithms that only use one input in pricing the security. For those unfamiliar (or if algorithm is a scary word), all we are going to be doing is coming up with a systematic (no inputs from the trader/investor) "rule" for trading the security. The algorithm we are going to use is very simple: -1* (Yesterday's Price). So, for example, if SPY increased by 2% yesterday, we will short the S&P at a rate of 2%. If SPY increased the next day by 3%, we will short at a rate of 3%. The amount shorted will be adjusted for volatility later, but all that is important is that on the second day, we are shorting 50% more (3/2) than on the first day.

Exhibit 1 shows the daily returns of this strategy. Note that we have not introduced any sort of scale (and as such, a 0.05 return does not necessarily imply a 5% return).

*(All data from Yahoo Finance)*

*Exhibit 1: Daily Returns of Short-Term Reversion Strategy*

This reversion strategy carries with it some beneficial properties, namely that from 1994 to 2014, it returns on average .00255 units per year. This indicates a profitable strategy. The cumulative daily returns of the strategy are shown in Exhibit 2.

## Adjust for Volatility

To compare "apples to apples" our algorithm, we adjust the standard deviation of the daily returns by multiplying by a constant factor (think of this as leverage or cash held in the portfolio) such that the annual volatility of the algorithm is similar to that of the S&P. For the purposes of this article, I used the realized historical volatility of the S&P 500, 19.6%. If you are curious, the factor was 46.655 for the short-term algorithm.

This was done by taking the sample standard deviation over the past twenty years of the S&P 500 and annualizing it (because the observed standard deviation is a daily number) - yielding 19.642%. The annual standard deviation of the reversion strategy (before volatility adjusting) is 0.042% (this number is essentially meaningless because if I multiply all of my signals by a constant I can change it drastically). Thus, 19.642% / 0.042% = 46.65.

As a quick note, this implies that we are comparing our returns on a risk-adjusted basis, which is a very valuable analysis technique. The comparative Sharpe ratios of these return streams will be proportional to the returns that we calculate.

*Exhibit 2: Cumulative Returns of Short-Term Reversion Strategy, 1994-2014*

As you can see, this strategy performed incredibly well in 2008/2009, and if you look carefully, this strategy performs well when SPY is most volatile, and it performs not as well (although still positive on average) when SPY is tranquil. Exhibit 3 shows the returns of SPY versus the Short Term reversion algorithm.

*Exhibit 3: Comparative Returns of Short-Term Reversion Strategy vs. SPY, 1994-2014*

Fascinatingly, the Short-Term Reversion algorithm has cumulatively outperformed the S&P over the past 20 years, helped greatly by its performance in 2007-2009 and is uncorrelated to the underlying, SPY!

For sake of comparison, however, let's observe the year 2013, when volatility was relatively low and SPY posted a large return. As we can see, the strategy performed well, but not nearly as well as SPY.

*Exhibit 4: Comparative Returns of Short Term Reversion Strategy vs. SPY, 2013*

## Adding Momentum

If we see that a reversion strategy works well, let's observe if there is benefit in holding a momentum component to a portfolio. We have observed already that the S&P has been mean-reverting in the short-term (if we reversed the mean-reversion signals, we would simply get the inverse returns - largely negative), let's test if momentum is effective in the long-term. The signal we are going to use now is the cumulative return of the past 12 months of SPY. So if SPY was up 10% in the last year, we will purchase 10 units and if it was up 15% in the last year, we will purchase 15 units. Note that here we are trend-following (there is no -1 factor in the algorithm). This strategy's daily returns can be seen in Exhibit 5. We can observe that the volatility of the events of 2008/2009 caused a much larger effect for the momentum strategy than on the reversion strategy. This strategy returns an average of .008 units per year (these units cannot yet be compared to the reversion strategy returns until they are also volatility normalized).

*Exhibit 5: Daily Returns of Long Term Momentum Strategy*

## Putting It All Together

Let's observe the cumulative returns of SPY, the reversion strategy, and the momentum strategy.

*Exhibit 6: Comparative Returns of SPY, Reversion Strategy, and Momentum Strategy*

The momentum strategy has the most lackluster returns of the group, but because it is likely to be uncorrelated to our reversion strategy, perhaps we can combine the reversion & momentum strategy to produce returns that will lower the risk of our portfolio. On that note, here are the 20-year historical correlations of the daily returns of these strategies. Interestingly, short-term reversion is more correlated with SPY than long-term momentum!

SPY | Momentum | Reversion | |

SPY | -- | -0.2161406 | 0.214846 |

Momentum | -- | -- | -0.24769 |

Reversion | -- | -- | -- |

*Exhibit 7: Correlation Matrix of SPY, Reversion Strategy, and Momentum Strategy*

The negative historical correlation between reversion and momentum provide a good incentive to combine the reversion and momentum strategies to try and serve as a hedge against SPY.

## Combining Algorithms

We are going to combine the two algorithms, by normalizing the two signals and adding them together. With our volatility normalized, the new signal will be (-1*yesterday's percentage price change) + (past year's percentage price change). This strategy has a short-term reversion component and a long-term momentum component, and its returns compared to the S&P are much better!

*Exhibit 8: Comparative Returns of 50/50 Reversion & Momentum Portfolio versus SPY*

To put this in perspective, an investor who invested $10,000 in SPY on 1/1/1994 would have about $55,105 and an investor who invested $10,000 in Reversion/Momentum would have about $61,526, or about a 12% outperformance over the past 20 years. Of course, this is not taking into account trading costs (which could be rather high) or the fact that the performance of the strategy in 2008 and 2009 accounts for a large portion of the returns.

## Trading Costs

To best compare passive and active strategies, however, it is important to consider the effects of trading costs. If we assume a 1/10 of a basis point per trade ($1 in trading costs for every $100,000 under assets), an investor will lose 507.3 basis points over the 20-year period, or a lost alpha of 5%. Depending on the structure of the brokerage account an investor uses, however, the 50/50 momentum-reversion strategy may be cheaper than the reversion strategy because it would trade less. Furthermore, adding an element of passive long exposure will reduce trading costs.

## Conclusion

There is clearly evidence that an investor could benefit from introducing uncorrelated return streams to passively holding SPY.

**Additional disclosure:** I am short SPY as a hedge to the long positions in my portfolio.

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