Momentum Average and TAA

Damian in the comments asked if I could use the average of four returns: the 1 month, 3 month, 6 month, and 12 month to rank the components of the momentum strategy from the previous post.

The results were as I expected. The momentum strategy isn't that good more than six months out and averaging the good Sharpe strategy with the bad Sharpe strategy really doesn't improve the returns. Intermediate trends are strong enough for ETFs that that should be the focus. I was able to improve on his idea by changing it to an average of 1 month, 3 month, and 4 month returns and the 1 month and 4 month are an improvement still above the momentum average. They still don't outperform the j=4, k=2 model in backtests, but they could outperform other models and there's no reason why it can't outperform in the future. The two bottom strategies at least relieve some of the burden on the 4 month.


I should also note that I am continually refining these strategies including the calculation of interest, margin, and the risk parity weights. I'll always post a comparison to a base model whenever it is different from anything posted in the past. The difference in returns from this model to the original reflects better accounting of interest returns. I also took out the previous inclusion of the style ETFs. Sector momentum has previously been far more important than style momentum and I expect that to continue in the future.

Another Post regarding recessions and TAA

I previously posted regarding the connection between returns of a TAA portfolio and when a recession occurs. This follow up with discuss the methodology and results from adjusting an investment strategy based on a quantitative estimate of the future probability of a recession.

The TAA models are good at reducing risk, but to increase return to something comparable to an equity index it seems silly to use the same amount of margin at all times. The post above gives evidence that lowering the amount of leverage in times when there is a prediction of a U.S. recession will increase the Sharpe ratio relative to a similarly levered model.

To incorporate this insight into the TAA model, as I used the same probit modeled I discussed in the first post I made regarding probit models. To use this model in an investing environment, I used a warm-up period of 200 months and then estimated the model each month to get a probability of a recession 12 months out. Since the NBER doesn't have dates for recessions in the past two years (I guestimated), I stopped the model 24 months prior to the end of February and used the coefficients as of that date with the data available to get the remaining predictions. Using this data I created two possible ways to scale in and out for leverage. I looked back four months for predictions in both cases. I did this for three reasons: the data constantly gets revised, I'm not sure always what is available on any given date and I want to be safe, and the lag time seems optimal (12 keeps you unlevered during the rebound, 1 is too sensitive, 6 works just as well, but I wanted to keep it shorter). So looking back four months, the first method will lever the portfolio when the estimated probability is greater than 50% and not lever the portfolio below that. The second method is similar except it provides three baskets levering a full amount, 50% of the full amount and none at all.

The results for the first and second method are comparable, but the second method slightly underperforms the first on a Sharpe ratio basis (the return is lower and the standard deviation is lower, but not by enough to offset the return). It's possible to fiddle with the parameters to improve the results, I'd rather just ignore it for the simplicity of the first model. The chart below gives the portfolio statistics for the first model with 0 leverage (and no probit data), 50% leverage, and 100% leverage. The first two columns are the base 0% and 50% and the probit columns represent the model with dynamic leverage ratios. Similar to before, I report the evenly weighted portfolios and the risk parity portfolios.

The results confirm my original intuition(!) and then the models even outperform what I had suspected would happen. Essentially the model reduces your leverage heading into a recession and then quickly puts it back on. Surprisingly, comparing these returns with the raw results from the last post on TAA and probits, shows an even better return for this method using the model than reducing leverage when a recession happened. I do use domestic equity volatility as a factor in the model which could help the investor get out when volatility increases (it is a small component compared to interest rates and money supply data though). Anyway, the results show an increase of the 50% leveraged portfolio's Sharpe ratio by almost .1 by reducing standard deviation substantially and keeping the returns constant. The 50% portfolio, previously with a horrible Sharpe ratio relative to the 0%, now is roughly comparable (with equal and risk parity weights). With 100% leverage, the risk parity probit model now has a Sharpe ratio equivalent to the equal weight 0% and 50% equal weight portfolios (but with more return and risk). However, the 100% leveraged probit models dramatically outperform their cousins without the probit model. Again, this is due to substantially reducing volatility by reducing leverage in periods leading up to recessions.

Given the success of the probit model in reducing volatility and keeping returns high, I now plan to investigate leverage as a function of the volatility of each asset class (individually). I'm guessing there would be similar results, but without some of the messier complications of using the probit model (not sure when data is released and it's relatively intensive computationally). Assuming I can find daily data from the site I got the monthly data, it shouldn't be that much of a problem and I can just use the percentage of large up and down days in a quarter as a proxy for volatility of each asset class.

A Preliminary Post regarding recessions and TAA

I was hoping to incorporate my recession model into decisions about margin into the TAA model previously blogged about (here and here), but things got away from me. The model is in Matlab and output recession predictions based on the information available at the time which was the big challenge so it really won’t be that much work beyond that. Hopefully, I can post that information tomorrow, but today I wanted to just give a brief update on the returns of the even Asset Allocation (20% in US stocks, Foreign Stocks, commodities, bonds, and REITs) vs. the returns of the Tactical version during times of recession.

I regressed the returns of the TAA strategy and the returns of the 50% levered TAA strategy against the returns on the comparable AA strategy (unlevered or levered) and a binary variable equal to 0 if the economy is not in a recession and 1 if the economy is in a recession. I’m saving the reader the time of reviewing the results, but the AA returns are highly significant (as expected) and the coefficient on the binary variable is not significant at the 5% level, but is at the 10% level for both levered and unlevered. I’m willing to concede that the effect on the adjusted R squared of the binary variable is very small, but the important thing is the sign of the coefficient and not necessarily how much of the variation is explained.

In both versions, the coefficient on the AA variable is roughly .61, but the intercept and the coefficient on the binary variable are roughly .003 and .004 in the unlevered and levered versions, respectively. These numbers are based on monthly returns, the annualized numbers are roughly 3.5% and 5%. For those without a statistical background, that means that in CAPM terms, the alpha is positive and larger for the levered version and, more importantly, both models outperform the passive strategies during recessions by a statistically and economically significant margin.

I tried to alternative strategies that could be considered a middle road. I developed two new returns variables based on the TAA data, the first was the unlevered TAA returns when there was no recession and levered otherwise and vice versa for the second. Looking at the regression results for these two gives evidence of being able to target which coefficient will be higher. Avoiding leverage in a recession increases the intercept to .004 and keeps the other coefficient flat, but avoiding leverage unless in a recession keeps the intercept, but the coefficient on the binary variable goes up to .004. Which is more important and why does it matter?

Ultimately looking at the return characteristics (below) yields the answer. As expected, the TAA model has the best Sharpe ratio; however, by avoiding margin during times of recession, you can increase the Sharpe ratio relative to a similarly leveraged portfolio. I haven’t been sure if I calculate the margin returns properly. Intuitively you would think that it would be double the returns minus the cost of debt, but with the TAA model you only margin the positions that you are in since the cost of margin is greater than the cost of cash. Furthermore, there are historical periods where the broker’s call rate gets so insanely large that it wouldn’t make sense to use margin, but this model, at present times, still uses it.

To conclude, by using a model that can predict recessions (which I conveniently have), you can improve the Sharpe ratio of leveraged portfolios. One possible note is that my model works to predict a recession within the next 12 months. It begins to show indications several months prior to a recession with plenty of time to take some off. Unfortunately this also means that there are several different ways to test it. Finally, I suspect that controlling the use of margin based on expected returns of the portfolio will also prove to enhance returns.

Component Tactical Asset Allocation: Part 3

This is the final of three posts on Tactical Asset Allocation. Part 1. Part 2.

The subject of this post is the theory of the market process and tactical asset allocation and why I believe that the former implies the latter will be a more successful strategy than buying and holding index funds.

The fundamental implication of the Efficient Markets hypothesis is that if information will be quickly incorporated in security prices. Mathematically this implies that stocks follow a random walk with jumps when new is absorbed by the market. Based on this assumption, the ideal investment strategy under the Capital Asset Pricing Model is to hold the market portfolio for the long-term (I know I’m ignoring Treynor-Black).

Beyond the typical objections to EMH, there are two main insights from the market process school that leads me to reject the EMH and I believe they are the most important observations of the school.

First, there are significant government interventions that make the market remarkably inefficient. The most important is the business cycle and monetary problems that in the 20^th century has almost always been caused by central banking. The standard Mises-Hayek theory of the business cycle is that by pushing the interest rates lower than the natural rate of interest, a central bank encourages investment beyond what would happen in the free market. Eventually the investment works its way into a boom for the producers of consumers’ goods and prices begin to increase. In order to stop the boom (to prevent runaway inflation), the central bank must raise rates. This action ends the boom and the process to correct the malinvestments made in the boom period. Since capital intensive industries are more exposed to interest rates, they typically feel more pain than consumers’ goods industries.

Moreover, governments intervene in other ways that are frequently not understood well enough by investors or they don’t always understand the implications. Economics is not a terribly difficult discipline and figuring out the implications of dumb government policies isn’t that difficult. If the government subsidizes ethanol production, farmers will use less land for the production of the typical agricultural products and those prices will have to rise. If there is a war in the Middle East, it is likely that oil (unless they have refineries in the country in question) and defense stocks will typically increase in value. Investors react in the short-term, but over the course of 6 months to a year or longer, these plays are still profitable. I’m not sure whether it is the uncertainty of these situations or that investors systemically do not think the government is as bad as it is (possible given the state of business school economics courses, also it is very difficult to quantitatively test), but investors do not react strongly enough.

Second, the market process school emphasizes the role of the entrepreneur in moving the market to equilibrium prices. EMH doesn’t care about why prices behave in certain ways, it merely attempts to model them. However, what is seen as a random walk are actually deliberate actions taken by entrepreneurs engaging in speculation and arbitrage. Entrepreneurs who have greater foresight will outperform those who don’t. Furthermore, the investing, particularly derivatives, are referred to as zero-sum games. The profits are zero-sum; however, ex ante, all of these trades are positive sum in terms of utility. These trades show ex post profits or losses depending on the skill in forecasting of the entrepreneur.

Finally, as Hayek notes, there’s no such thing as perfect knowledge, as is assumed by the actors in CAPM. Knowledge is dispersed throughout society and the purpose of the market is to organize that knowledge. Market prices reflect the knowledge of all market participants.

Combined these three factors make tactical asset allocation an attractive prospect. Since there are cycles that can be observed by students of the Austrian School, it makes little sense to buy and hold equities when there are lengthy periods of time where you can lose a significant sum of money. Also, not only is there a purpose to being an entrepreneur which is ignored in the EMH, but paying attention to what happens to prices can reveal knowledge about other market participants’ knowledge and opinions in the market. Michael Covel notes in his book about trend following that trend followers tend not to be in the business of predicting trends, they have imperfect knowledge and as a group they have no opinions on the market. However, if the market is going up, they would be more than happy to buy and vice versa to sell. The logic is essentially the same for TAA. Mr. Faber isn’t providing a service in predicting the market, he’s trying to improve on the buy and hold passive strategy by staying out of the market when market participants have a negative outlook. Nothing wrong with that.

Component Tactical Asset Allocation: Part 2

This is the continuation to the previous Component TAA: Part 1 post.

First I will present the results with a single momentum strategy comparing the AA, TAA, and Momentum TAA strategies with 0 leverage and with 2-1 leverage. Then, I will present alternate momentum strategies using different js and ks, but investing in a constant number of ETFs followed by a constant j and k with a different % of ETFs available. I will conclude with work in progress to improve it further. I might add an additional post describing why the economist in me prefers Tactical Asset Allocation as an investment philosophy to the Efficient Markets Hypothesis and the Capital Asset Pricing Model.

Before I begin, I should note that the Domestic ETFs I invest in a separated into two groups, sector and style. Whichever one I can invest in earlier (sector), I will use that return and then later, I average the two's returns. I might get a better return without doing this, but that particular market is so broad that I wanted to investigate the combined effect. Not only do some sectors out perform, but sometimes value outperforms growth and large-cap outperforms small-cap. I wanted to be able to include this relationship as well, I'm just not sure how much stronger this effect is compared to the sectors. I also had a longer list of sectors that I cut down on prior to running these returns, so the ranking since I updated the data can actually choose from more sectors and gets better returns. I'm only reporting a sector basket with the 9 Spider select ETFs.

The first table represents a unlevered comparison of the AA, TAA, and Momentum TAA (with j=4 and k=2 investing in the ETFs ranking in the top 25%) with equal asset allocation between the 5 asset classes and the Risk Parity weights discussed in the previous part. The TAA beats the AA which is the conclusion reached by Faber. However, the Sharpe Ratio (@ 6% for all) increases with a more normal kurtosis (3=normal, greater than 3 indicates fat tails) by using the risk parity weights. There are similar results comparing the TAA and Momentum TAA, however it seems like the Kurosis for the TAA portfolio is relatively constant. This makes sense since the data is cut off prior to 1998 and excludes some of the larger price movements.


The next table is the same strategies and comparison as above, but with 100% leverage. It is more for general interest than comparison. The method used in the paper by Panagora was to use the Risk Parity weight and then lever the portfolio to a desired return (such as the S&P500's average return) so that variance would be minimized. In this case, the return on the TAA without Risk Parity Weights would be greater than the standard deviation on something like the S&P500. You could use roughly 20% leverage to increase the return of the TAA Risk Parity to roughly the return on the normal TAA (this same argument works to target the standard deviation as well). However, the Sharpe ratio in this case would be less than if you had not used leveraged. The return is the same, but the variance is actually greater (the same holds true for AA, TAA, and Momentum TAA). So even without the large 2-1 leverage reported below, if you measure your investment success by your Sharpe ratio, then it won't make sense to use leverage. However, relatively speaking, the risk parity weights outperform the equal weighted portfolio. If you're an investor seeking to maximize profit or would be willing to accept more risk in exchange, then you should use the risk parity instead of the equal weight portfolios.



The third table presents the CAGR, Standard Deviation, and Sharpe Ratio comparing different momentum strategies. Recall from the previous article that j represents the number of periods to look back to and k represents the number of periods to hold (since k can be greater than 1, then even if you hold 6 ETFs when k=1, it will be variable for k>1). Using more complete data, it is clear that the Sharpe Ratios increase as j comes to 3 or 4 and declines after that. However, there is no clear trend on what happens with k. It usually increases to 2 and declines after that, but it is not consistent. If at all possible, I would prefer a larger k to a smaller k since it guarantees that I will have less turnover.



Finally, the last table shows the returns with j=4 and k=2, but investing in a different percentage of the ETFs that have sufficient return histories. The trend in this case is clear, return increases as you increase the percentage until it tops out between 25 and 33.3%. However, these returns are all gross and the others could relatively increase if transactions costs are included. Furthermore, I would suspect the tax consequences are greater. Instead of picking the best sectors, at 75% you're getting out of the worst. For portfolios with less than half a million dollars, there might be too many ETFs to be able to use the 75% or 50% to make it worth it. However, I should also note that the benefit of the original TAA model is that each position can be approximated with futures contracts which could possibly reduce costs and provide an easier method to use leverage.



To conclude, gross returns and gross Sharpe ratios are greater using the Momentum TAA with risk parity portfolios. However, there are still additional ways that it could be improved. This strategy can be considered a component in a larger overall strategy. For example, Mr. Faber discusses alternative strategies such as mean reversion and following hedge fund managers that produce significant returns. I think that there are strategies in options, distressed debt, value investing, macro investing, mean reversion, and statistical arbitrage (or investing in hedge funds that specialize in stat, risk, or convertible arb) that can add to this return while not being correlated with the TAA or Momentum TAA. Unfortunately, with the exception of mean reversion, these strategies are either not quantitative (macro, distressed debt, value) or are difficult to backtest (options - competence, and arbitrage are arbitraged away).

Next, there are additional beta factors that can be considered or thought about in different ways. For example, a recent paper indicates that the returns for currency managers are largely Beta. Those returns could be an additional asset class that could be added with little correlation to the others. Also, there is evidence that investing in commodities based on their term structure (buy most backwardated positive roll-return commodities, short most negative roll-return contango commodities). These two strategies, combined with mean reversion of the five assets used in TAA and the Momentum TAA risk parity weights, could be particularly strong and they could be included in a broader portfolio using the risk parity weights.

Finally, I recently discussed a probit model I use to forecast recessions. I am considering linking that model (and augh converting it to Matlab) to this program so that I choose margin based on the probability of a recession. The returns of this strategy outperform the S&P500 during the bad times, but they still underperform compared to the remainder of the period. I'm going to consider increasing leverage when the probability estimates are low and cut off leverage when the probability begins to increase. I believe this can improve returns.

edit: There was a slight discrepancy with the interest rate data in the original results that has been corrected.

On to Part 3.

Component Tactical Asset Allocation: Part 1

Mebane Faber published an article in the Journal of Wealth Management in the Spring of 2007 called a Quantitative Approach to Tactical Asset Allocation. The thrust of the paper is that if you invest in U.S. stocks, foreign stocks, bonds, commodities, and REITs when they are above their respective 200 day moving averages and invest in commercial paper otherwise, you can achieve returns similar to equity investments with significantly lower volatility. Mr. Faber has graciously provided the monthly returns from the strategy as well as much more information on his website, World Beta. Based on the data on the website (more up to date than the original paper), the timing model returned 12% since 1972 with a standard deviation of 6.43% (.93 Sharpe) compared to an 11.5% return on the buy and hold asset allocation strategy (20% in each asset mentioned above) with a 9.78% standard deviation (.56 Sharpe). I programmed his strategy into Matlab using the same data and found similar results (slightly different due to the vagaries of Matlab rounding and computing returns statistics based on monthly data instead of yearly data).

Lately I have been interested in how to improve on this concept. First, I would like to discuss two additions I made and I will make an additional post to discuss the results of what I tested.

On his blog, Mr. Faber compares different methods that readers have requested to improve the returns (that he doesn't use). The first is to enter long positions above the 200 day MA and short positions below while the second is to enter each position "all in," equally weighted for each buy signal, no positions in cash . Each of these methods fails to improve the Sharpe ratio. I expect the L/S portfolio fails due to the fact on that most of the top 50 best and worst days are when the market is below the 200 day moving average. It is possible to profit by shorting the worst, but you can get burned on the best.

I believe the "all in" portfolio fails by ignoring the correlations between the assets (and would require more re-balancing costs than the traditional TAA). In order to test this, I followed a white paper by Panagora Research which describes the Risk Parity Portfolio. Their concept is to adjust the weights of a portfolio so that the amount you can risk on each position is equal. The traditional method to do this is to estimate the Value at Risk for a portfolio and break it into the component parts for each security. This method takes into account the correlations among each asset and the Beta. As a technical concern, I waited a year to create the Risk Parity weights (but used the initial 20% allocation during that period to make comparisons to Faber's paper) and brute forced the first weights using the covariance matrix and existed at the time of investment decisions and only changed the weights if the Component Value at Risk of an individual asset went outside predefined bounds. The weights stay relatively constant over time, but I could have created tighter bounds where they would change more often. As of the time of writing, bonds would have 34.8% weight, REITs 15.9%, Commodities 19.3%, Domestic Stocks 15.1%, Foreign Stocks 14.9%. In other words, REITs and Stocks would have their shares reduced and bonds would increase their weights in order to take into account the fact that they are more strongly correlated with each other than Commodities and Bonds and have higher variances. Based on the research provided by Panagora, I expected a slightly lower return, but a substantially reduced standard deviation.

The other method I used was investigating the j-k Momentum strategy proposed by Jegadeesh and Titman. In this paper, J and T investigate ranking stocks based on their returns from j periods ago and holding them for k periods forward. They used this model to show that stocks have a momentum factor like a size or value factor that helps determine their future returns. Within the context of the TAA model, I chose to test this strategy by choosing a proportion of the ETFs for an asset class and then applying the j, k methodology to a proportion of the ETFs with returns. I waited until a certain proportion of the total ETFs (in each classand that I considered representative of the asset class) began to trade to start the momentum strategy for that asset class. I will only report (and compare) the returns since the earliest strategy began to take effect (Select Spiders began trading in December of 1998, but it requires j months before the strategy can work). Since they have uneven start times, I used the returns from the normal TAA strategy when the Momentum strategy cannot work. I should emphasize that I am not using only a Momentum strategy on ETFs, but investing in a j-k Momentum strategy based on a TAA model. Within each Momentum category, the ETFs are equally weighted and I don't think it makes sense to use Risk Parity Portfolios in this context.

Furthermore, if I am not mistaken, Mr. Faber uses a method similar to this in actual practice, however, he does not report his results using this method. The most obvious reason is that he created his model in Excel which is substantially more cumbersome the model gets more complex. Also, ETFs have a short history that may not be indicative of the 35 years of returns where the TAA model has shown considerable strength. There's also no doubt that using ETFs in this strategy would require more trading costs and more taxes (unless in a tax-free account). Even if this strategy is not successful (it is), it is at least interesting to investigate and note the return characteristics for different levels of j and k.

The next post will compare the Equal Allocation (no TAA) model to the TAA and their Risk Parity equivalent portfolios, it will compare the TAA models with the Momentum TAA models (equal allocation and risk weighted), and some discussion about future additions I plan on testing.

On to Part 2.