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John_Ryan_Thesis_Final

  1. 1. Thesis Paper The use of Econometric Models in the Asset Selection of Actively Managed Commodity Portfolios John D. Ryan March 2011 - December 2012
  2. 2. TABLE OF CONTENTS 1. INTRODUCTION………...…………………………………………………………1 2. LITERATURE REVIEW…………………………………………………………...1 2.1 The Role of Commodities in the Asset Allocation Decision 2.2 Unique Features of Commodities as an Asset Class 2.3 Commodities vs “Commodity Stocks” 2.4 Commodities and Inflation 2.5 The Utility of Commodities in the Asset Allocation Decision 3. DATA…………………………………………………………………………………6 4. METHODOLOGY…………………………………………………………………..6 4.1 Momentum 4.2 Term Structure 4.3 Double Sort 4.4 Autoregressive Model 4.5 Autoregressive Moving Average Model 4.6 Auto Regressive Conditional Heteroskedasticity Model 4.7 Generalized Auto Regressive Conditional Heteroskedasticity Model 4.8 Model Averaging Model 5. EMPIRICAL RESULTS…………………………………………………………..14 5.1 Entire Sample 5.2 Pre-Crisis Sample Results 5.3 Post-Crisis Sample Results 6. CONCLUSION……………………………………………………………………..17 7. AREAS FOR FURTHER RESEARCH…………………………………...………17 8. REFERENCE LIST………………………………………………………………...19 9. APPENDIX………………………………………………………………………….22
  3. 3. 1. INTRODUCTION The focus of this study is to explore various econometric time series forecasting models with the ultimate goal of constructing actively managed commodity portfolios with better absolute and risk adjusted performance. The main inspiration and foundation for this paper is the research conducted by Fuertes, Miffre, Rallis , as in their paper Combining momentum and term structure signals (2010). Thus a universe of 24 commodities’ spot and roll returns, from 2000 to 2010 will serve as the universe from which to construct the actively managed portfolio. The four main econometric models which will be used in this study include the Autoregressive Model (AR), Auto Regressive Moving-Average Model (ARMA), Auto Regressive Conditional Heteroskedasticity Model (ARCH), Generalized Auto Regressive Conditional Heteroskedasticity Model (GARCH), and a Model Average as described by Granger of the above four classic methods. The results obtained will then be compared across the entire time series sample as well as a pre-crisis and post-crisis sub sample. 2. LITERATURE REVIEW 2.1 The Role of Commodities in the Asset Allocation Decision Strategic asset allocation may be thought to have evolved from Modern Portfolio Theory and Markowitz’s mean-variance analysis (Campbell & Viceira, 2002). In the past strategic asset allocations were traditionally composed using the 3 classic asset classes; those being stocks, bonds and cash, but would not include commodities (Ibbotson, 2006). Modern Portfolio Theory and mean-variance analysis reduces the investment decision to a tradeoff between risk and reward where the investor will seek to maximize the expected return for a given amount of risk and vice versa (Markowitz, 1952). In order to achieve the aforementioned portfolio optimization the investor will diversify a portfolio between these assets based on expected return, risk or standard deviation, and the correlation between the different available assets. There are several reasons why commodities have been excluded from the asset allocation decision. First, there are several different ways to define the overall commodity asset class thus making it difficult to construct an agreed upon commodity index. The existing commodity indices have markedly different risk-return characteristics. Other reasons that commodities have been excluded from the asset allocation decision include “a limited number of implementation
  4. 4. 2 vehicles; the major commodity industries have short histories that have been backfilled; ambiguity over what constitutes an asset class and an investment strategy; the role of commodities in the market portfolio is undefined; the lack of an accepted commodity pricing model; and the lack of an understanding of the inherent returns of commodities.” (Ibbotson, 2006). More recently, investors have begun to include commodities as a separate and unique asset class to be included in the investment universe the strategic asset allocation process (Ibbotson, 2006), (Fuertes, Miffre, & Rallis, 2010), (Gorton & Rouwenhorst, 2006). 2.2 Unique Features of Commodities as an Asset Class Commodities are a unique asset class that offers several benefits to a traditional strategic asset allocation of debt and equity. Commodities are a real return asset that may provide a “natural hedge against inflation” and thus improve the characteristics of a traditional portfolio that will usually lose value during unexpected inflation (Erb & Harvey, 2006), (Georgiev, 2001), (Bodie & Rosansky, 1980), (Ibbotson, 2006), (Fabozzi, Fuss, & Kaiser, 2008). Furthermore, commodity futures behave differently than other traditional assets over the course of the business cycle, and thus often have a negative correlation to stocks and bonds (Gorton & Rouwenhorst, 2006). Greer (1997) identifies commodities as being unique and of great value due to the diversification that they provide to a strategic asset allocation. According to Greer, there exist three broad “super-classes” of assets; Capital Assets which include stocks and bonds, Consumable/Transformable Assets such as commodities, and Store of Value Assets such as currency. He concludes that Consumable/Transformable assets, or commodities, are underutilized in the asset allocation decision, and that by including these assets into a diversified portfolio overall volatility can be reduced. Greer attributes the negative correlation of commodities with Capital Assets to the value of the two being driven by different economic factors (Greer, 1997).
  5. 5. 3 2.3 Commodities vs “Commodity Stocks” It is important to note that equities that may be considered “commodity stocks” (commodity producing companies), neither provide the same diversification benefits as commodities, nor offer very good exposure to the underlying commodities. The prices of these stocks are influenced by numerous idiosyncratic company specific factors such as management and operational risk. Commodity producing companies are actually more closely correlated to the equity market than to commodity futures, and thus serve as a poor substitute investment. The management may actually decide to be hedged against fluctuations in the currency market, which even decreases the dependency of the companies’ valuation towards commodities (Gorton & Rouwenhorst, 2006), (Georgiev, 2001). 2.4 Commodities and Inflation Being real assets whose prices depend not only on their intrinsic value but also on the currency value in which they are quoted, commodities have embedded inflation-hedging properties. Most commodities are quoted in USD, and as they have a positive correlation to inflation, they can be seen as natural hedge against US inflation (Fabozzi, Fuss, & Kaiser, 2008), (Gorton & Rouwenhorst, 2006), (Greer, 1997). “First, because commodity futures represent a bet on commodity prices, they are directly linked to the components of inflation. Second, because futures prices include information about foreseeable trends in commodity prices, they rise and fall with unexpected deviations from components of inflation” (Gorton & Rouwenhorst, 2006). This is extremely relevant in the context of strategic asset allocation. Generally, when inflation is increasing, so are commodity prices. In contrast, stocks and bonds are negatively correlated with inflation. When inflation is rising, bonds and stocks are likely to be dropping. The reason for the inflation stock/bond relationship is simple. Rising prices and rising interest rates (to limit inflation) make industrial expansion more costly, lower earnings ratios, and lower bond prices which ultimately exerts downward pressure on the economy. A naïve approach consisting of comparing GSCI and US CPI monthly changes over time confirms that a positive correlation exists between commodities and inflation, and that this correlation is particularly high when big swings occur (mid-2008 subprime crisis for instance). Several studies have reported positive and significant correlation levels, both statistically and
  6. 6. 4 economically. Depending on the adopted methodology, authors find levels in the 20%-50% range for standard maturities. Main differences between studies are the chosen commodity categories (indices vs. futures, global indices vs. basic indices, 1st nearby vs. back-end future contracts) and the way they are used (lagged log-returns, drifted vs. non-drifted series, or series length). Linear regressions run on the US CPI non-seasonally adjusted on 2 different lags or more of the US unemployment rate, S&P GSCI and Gold (yearly returns of the monthly average prices) lead to high R2 , above 80%, with t-stats significant at the 1% level. F-statistics are also higher than the 1% F-critical value, and 50% R2 if we take only the US CPI (Spierdijk & Umar, 2010). In addition to basic correlation considerations, one concept that has been introduced as another useful tool to evaluate the link between commodities and inflation is cointegration. Correlation quantifies the statistical relationship between several observed data values whereas cointegration measures the propensity of different variables to move back to a mean value (i.e. two series are deemed to be cointegrated if they share the same kind of trend toward their mean distance). Thus commodities being cointegrated with inflation would mean that both variables cannot drift away from each other for a long period of time. With regards to hedging, cointegration may be considered a more appropriate tool as correlation tends to change over time. Some recent studies dealing with the cointegration between commodities and inflation, show increasing levels not only between oil or gold and (expected and realized) inflation, but also between benchmark indices and inflation (Spierdijk & Umar, 2010). In all countries, the Consumer Price Index (CPI) includes direct energy and food prices. However, the proportions differ from one country to the next. In other words there is a certain allocation to commodities in order to hedge against inflation in one country, and different allocation for another country. As the exchange rate between the local currency and the US dollar is most of the time a floating rate, it also has an implication on the commodity allocation used to hedge against inflation.
  7. 7. 5 Adding commodities to a classic mix of bonds and equities can increase the risk adjusted return of the entire portfolio as is demonstrated by Proelss and Schweizer. The aforementioned authers state that “an efficient mixed commodity portfolio can be a promising investment, not only on its own but as a good portfolio diversifier (Proelss & Schweizer, 2008). To illustrate this point two mean variance efficient frontiers were calculated with MATLAB using the given dataset, one traditional bond equity portfolio and one bond / equity / commodities portfolio (weekly returns of Standard & Poors 500 Total Return Index, Vanguard Total Bond Market Index and Goldman Sachs Commodity Index). The chart is given in at the end of the paper in appendix XXI, and clearly shows that portfolios including commodities have a better mean variance performance. The portfolio with commodities is strongly superior to any bond and equity combination. 2.5 The Utility of Commodities in the Asset Allocation Decision Adding commodities to a classic bond and equity mix also improves the properties of the portfolio on the down side. Thus, when it is needed most during market downturns, the negative correlation between equities and commodities helps to improve the downside properties of a classic portfolio (Chong & Miffre, 2006). The issue of higher order risk was highlighted by Gorton & Rouwenhorst (2006). The authors criticize the negative skewness of stock returns and the therefore inherent downside risk. On the other hand commodities futures tend to have positive skewness (the mean having a greater value than the median). The inclusion of commodities in a portfolio may therefore mitigate extreme risks, and eliminate some of the negative skewness exhibited by an equities only portfolio. For the stated reasons, it is clearly beneficial to add commodities to a strategic asset allocation. An investor may invest directly in the physical good. This is prohibitively expensive and should be avoided. Indirect investments in commodity companies do not necessarily reflect the performance of commodities and therefore are to be avoided as well. Passive investments in long only commodity indices like the GSCI are substantially inferior to active strategies (Please see appendix I, column titled Benchmark). Alternatively an investor can get exposure to commodities via a structured commodity fund like a CTA.
  8. 8. 6 3. DATA The dataset consists of spot and roll returns of 24 commodities from the beginning of 2000 to the start of 2010. The interval is weekly. In the cross section there are several lacking data points. In order to have a dataset with a persistent number of commodities, four of them (Frozen Pork Bellies, Palladium, Platinum, Unleaded Gas) which lack data are excluded entirely. This leaves a set of 20 commodities with a complete time series. Those being cocoa, coffee, copper, corn, cotton, crude oil, feeder cattle, frozen concentrated orange juice, gold, heating oil, lean hogs, live cattle, natural gas, oats, Silver, Soybeans, Soybean meal, Soybean oil, sugar and wheat. 4. METHODOLOGY In his book, Introductory Econometrics for Finance, Chris Brooks defines econometrics as “measurements in economics” and “the application of statistical techniques to problems in finance” (Brooks, 2008). Although financial econometrics can have several uses such as measuring the relationships between observable phenomena, and confirming hypotheses, in the context of this study econometrics will be used primarily in time series analysis to forecast asset returns, measure risk, and for active portfolio management / asset selection decision making. A time series may be defined as “a set of observations on a variable’s outcomes in different time periods” (DeFusco, McLeavey, Pinto, & Runkle, 2004). As stated earlier, the purpose of this study was to reexamine a time series of commodity returns (both spot and roll) and to apply several econometric time series analysis methods (to the series of spot returns) with the aim of generating better risk adjusted returns than the naïve Momentum used Fuertes, Miffre, Rallis (2010). The 5 econometric models introduced to this study are the Autoregressive Model (AR), Auto Regressive Moving-Average Model (ARMA), Auto Regressive Conditional Heteroskedasticity Model (ARCH), Generalized Auto Regressive Conditional Heteroskedasticity Model (GARCH), and a Model Average of the above 4 methods. The above 4 econometric models will be implemented using MATLAB with all model parameters left in their default settings. Should a better momentum signal be devised, than this can be used to improve the Double Sort method by combining it with the Term Structure strategy found Fuertes, Miffre, Rallis (2010). Thus a starting point for this study was to replicate the Term Structure / Moment Double Sort Strategy found in Fuertes, Miffre, Rallis, and to use this as a base on which to expand and work forward with the aim of improving financial performance.
  9. 9. 7 4.1 Momentum The momentum strategy as in Fuertes, Miffre, Rallis (2010) is based on the idea that commodities exhibit momentum effects. In their highly cited paper Asness, Moskowitz and Pederson observe that “momentum ubiquitously generate abnormal returns for individual stocks within several countries, across country equity indices, government bonds, currencies, and commodities” (Asness, Moskowitz, & Pedersen, 2012). The authors observe that assets with positive moment (defined as high 12-month past returns) outperform assets with low momentum. If commodities are likely to show a consistent behavior in their returns it would be worthwhile to go long in commodity futures which appreciated strongly over a period R and go short in futures which depreciated strongly over the same period. If the forward curve realizes itself in contango (the condition where the futures contract price is trading above the expected spot price) then it is more interesting to be invested further along the curve (roll further along). The momentum strategy tends to maximize the spread between roll yields. The predominant source of returns of the strategy is extracted from differences in roll yields within the commodity portfolio. Even in situations where all commodities are in contango, according to the signal, it is possible to extract some value from the difference in the contango amplitudes. The strategy is invested on rolling 2nd nearby contracts, which optimizes the roll methodology using a static curve spread enhancement. Investing in the 2nd nearby contract allows one to handle the liquidity issue and avoid prohibitive replication costs. This is implemented via sorting the cumulative return of all commodities in the last R weeks in the dataset by their respective values. In the following a cut-off percentile is chosen. The percentile is symmetrical, so there is one low percentile and one high percentile. In between, there are commodities which do not exhibit strong positive or negative momentum and thus will be discarded from the investment universe for the given period. The percentile was chosen to be 33%. This relatively high value was necessary to still have an at least partly diversified portfolio when later applying the double sorting strategy. Thus from a universe of 20 commodities there are seven securities in the top 33 percentile and seven in the bottom 33 percentile (rounding to the nearest integer). Commodities which exhibit a momentum above the high percentile value are labeled “Winners” and those which show lower momentum than the low percentile value “Losers”. The winner-
  10. 10. 8 portfolio is an equally weighted sum of the Winners; the loser-portfolio is an equally weighted sum of the losers. Each portfolio has a gross exposure of 100%. In order to construct the long short portfolio, the investor now goes long the winner portfolio and shorts the loser portfolio. Thus the portfolio has net 0% directional exposure (the long and short positions cancel out which leads to a net exposure of zero for the investor). This portfolio is held for a holding period H, after which the ranking and investment process starts anew. Return series of the past R weeks are again sorted and the process continues. For the purpose of this study, the parameter ranking period R was chosen to be 52 weeks and the holding period H 1 week. However, the nearest 4 weeks (weeks 49, 50, 51, 52) are excluded from the parameter ranking period R as is done in Fuertes, Miffre, Rallis (2010), and Moskowitz (“MOM2-12”) due to “contrarian effect in returns at the one month level which may be related to liquidity or microstructure issues”(Asness, Moskowitz, & Pedersen, 2012). The rationale for this specific holding period is based on the results of Fuertes, Miffre, Rallis (2010), which indicate that shorter holding periods tend to produce better reults. Increasing the holding period above four weeks decreased the alpha of the applied momentum strategy in most of the cases. The ranking period was chosen to be 52 weeks to be consistent with the Double Sorting methodology. The estimation window of one year implies that returns for this strategy may only be calculated from 2001 on, as the first 12 months or 52 weeks are used for estimation. 4.2 Term Structure Signals originated from the term structure of the future curve may be used as means of generating abnormal returns as was shown by Erb & Harvey (2006). The concept is based on buying backwardated contracts (expected future spot price is greater than the futures price) and shorting contangoed contracts. This strategy makes use of the insight that the future curve has some predictive value on the roll returns of commodities. This insight is further developed by Gorton, Hayashi & Rouwenhorst (2008). As for implementing the strategy, the roll returns were used as buy or sell indicators. First a cut off percentile of 33 % was chosen. In each period the roll returns are sorted and buy and sell signals for the commodities with the highest and lowest roll returns are generated. This results in long portfolio of seven commodities and a short portfolio of seven commodities. The portfolios
  11. 11. 9 are held for four weeks after which the portfolio is rebalanced. The reported results stem from a back test starting in 2001 to be consistent in the time period across the different employed strategies. 4.3 Double Sort The double sorting strategy is a combination of the momentum and term structure model. First, trading signals are computed on the basis of the momentum strategy as illustrated beforehand with a cut off percentile of 33%. This results in one set of 7 “Winner” commodities and one set of 7 “Loser” commodities. Inside each of those sets, the Term Structure model is used to further restrict the futures in which to invest. Here again, a 33% cutoff percentile is employed. Therefore there are two double winners and two double losers. Those futures which are “double confirmed” are equally weighted. Going long the double winners is the Long portfolio, and conversely going short the double losers is the Short portfolio. The addition of going long the Long and short the Short portfolio results in the L/S Double Sort (Fuertes, Miffre, & Rallis, 2010). It is worth noting that a set of only 20 commodities restricts a double sort strategy substantially. The reason for this is that the amount of futures which are “Double Winners” and “Double Losers” is quickly converging towards one long and one short contract. In theory it would be advisable to choose a very small cut off percentile since that would mean investing only in commodity futures which show strong signals. However, small percentiles quickly lead to portfolios with only one long and one short position, whereby the investor loses the advantages of owning a diversified portfolio. With such a portfolio an investor would be exposed to high event and idiosyncratic risk when choosing to rely only on strong signals. One example could be a natural disaster which would quickly change the future curve of several commodities. This may lead to substantial losses since the investor is barely diversified with only one long and one short position. For real world applications it is consequently advisable to use a broader base of commodities in order to be more diversified and while only use strong signals. The ranking period R for the momentum signal and the holding parameter H are chosen to be 52 weeks and 4 weeks. This makes the results more comparable to the pure momentum strategy where the same parameters are employed. However, this means that the first year of data is
  12. 12. 10 exclusively used for the estimation of momentum. In their paper Fuertes, Miffre, Rallis (2010), show that a Mom52-4-TS1 has the highest reward to risk ratio among six other Double Sorting strategies. Therefore the ranking period R was chosen to be 52 weeks. The returns of the strategy can therefore only be calculated from 2001 on. 4.4 Autoregressive Model The first linear econometric model that will be applied to this study is a simple Autoregressive model (AR). An Autoregressive model (AR) may be defined as “one where the current value of a variable, y, depends upon only the values that the variable took in previous periods plus an error term. An autoregressive model of order q, denoted AR(q), can be expressed as, 𝑦𝑡 = 𝜇 + ∅𝑦𝑡−1 + ∅𝑦𝑡−2 + ⋯ + ∅ 𝑞 𝑦𝑡−𝑞 + 𝑢 𝑡 where 𝑢 𝑡 is white noise disturbance term” (Brooks, 2008). Thus q past values of the dependent variable 𝑦𝑡 are used to estimate the current value of 𝑦𝑡 (Brooks, 2008), (DeFusco, McLeavey, Pinto, & Runkle, 2004). It should be noted that a key assumption required for the validity of the model is that the time series being modeled is “covariance stationary” or “weakly stationary.” The basic concept of covariance stationary is that over the course of the time series, the main properties of the series (𝑠𝑢𝑐ℎ 𝑎𝑠 𝑓𝑖𝑟𝑠𝑡 𝑎𝑛𝑑 𝑠𝑒𝑐𝑜𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑠 𝜇 𝑎𝑛𝑑 𝜎2 ) remain somewhat constant (Brockwell & Davis, 2006), (Brooks, 2008), (DeFusco, McLeavey, Pinto, & Runkle, 2004). Should one encounter the pitfall of modeling a non-covariance stationary time series using an autoregressive function than the results will be meaning less or biased and one will have produced “spurious results” (DeFusco, McLeavey, Pinto, & Runkle, 2004). A second key assumption for the statistical validity of the autoregressive model is that the expected value of the error term in the times series model is equal to 0 ( E[𝑢 𝑡] = 0 ). 4.5 Auto Regressive Moving-Average Model The second linear econometric model that is used in this study is the Auto Regressive Moving- Average Model (ARMA). An Auto Regressive Moving-Average Model is the combination of an Autoregressive Model and a Moving Average Model (Brockwell & Davis, 2006). “By combining the AR(p) and MA(q) models, an ARMA(p, q) model is obtained” (Brooks, 2008). The model defines the forecasted value y of a time series as a linear combination of lagged
  13. 13. 11 values of the time series (𝑦𝑡−1) and both current and lagged values of the error term (𝑢 𝑡 , 𝑢 𝑡−1) (Brockwell & Davis, 2006), (Brooks, 2008), (DeFusco, McLeavey, Pinto, & Runkle, 2004). An Auto Regressive Moving-Average Model can be defined as, 𝑦𝑡 = 𝜇 + 𝜑1 𝑦𝑡 − 1 + 𝜑2 𝑦𝑡 − 2 +· · · +𝜑𝑝 𝑦𝑡 − 𝑝 + 𝜃1𝑢𝑡 − 1 + 𝜃2𝑢𝑡 − 2 +· · · +𝜃𝑞𝑢𝑡 − 𝑞 + 𝑢𝑡 where E( 𝑢𝑡 ) = 0; 𝐸( 𝑢𝑡 2 ) = 𝜎2 ; 𝐸( 𝑢𝑡 , 𝑢 𝑠) = 0 , 𝑡 ≠ 𝑠 as is written in Introductory Econometrics for Finance (Brooks, 2008). Many “advocates of ARMA models argue that these models may fit the data better and provide better forecasts than do plain autoregressive (AR) models” (DeFusco, McLeavey, Pinto, & Runkle, 2004). Again, when modeling a time series using an ARMA function the economic validity of the results rest upon the assumption that the time series is weakly stationary. 4.6 Auto Regressive Conditional Heteroskedasticity Model The third econometric process that is used to model the time series of commodity spot returns is the Auto Regressive Conditional Heteroskedasticity Model (ARCH). This model is a departure from the two previous models, the AR and ARMA models, as these assume homoscedasticity while the Auto Regressive Conditional Heteroskedasticity Model assumes heteroskedasticity. Homoskedasticity can be defined as “the dependence of the error term variance on the independent variable,” and conversely heteroskedasticity is defined as “the independence of the error term variance from the independent variable” (DeFusco, McLeavey, Pinto, & Runkle, 2004). In Robert F. Engle’s seminal 1982 work, for which he later won the Nobel Prize in Economics in 2003, he defined a model in which the error term in one period is dependent on the variance of error terms in previous periods (DeFusco, McLeavey, Pinto, & Runkle, 2004) (Engle, 1982). Thus under the Auto Regressive Conditional Heteroskedasticity Model “large shocks tend to be followed by another large shock,” or periods that exhibit low variance are followed by periods of low variance, which may also be called “volatility clusterings” (Brooks, 2008). If we allow 𝜀𝑡 to represent the error terms of a mean process of the time series than 𝜀𝑡 can be defined as 𝜀𝑡 = 𝜎𝑡 𝑧𝑡. Thus the error terms 𝜀𝑡 are composed of a random process and time varying standard deviation. The variance of a time series (𝜎𝑡 2 ) under this model is defined as “a linear function of past squared values of the process,
  14. 14. 12 𝜎𝑡 2 = 𝜔 + ∑ 𝛼𝑖 𝜖 𝑡 − 𝑖 2 𝑞 𝑖=1 = 𝜔 + 𝛼(𝐿)𝜀𝑡 2 where ω > 0 and αi ≥ 0, and L denotes the lag operator” (Bollerslev, Chou, & Kroner, 1992). Note that 𝜔 and 𝛼𝑖..𝑞 are independent and identically distributed (i.i.d.) from a uniform mean 0, standard deviation 1 distribution [0,1] (Bollerslev, Chou, & Kroner, 1992). Thus the utility of the ARCH model is in its ability to forecast variance or volatility. In summation, in order to execute the ARCH estimation procedure first one estimates the best fitting Auto Regressive model AR(q), as is described in the AR section above, and next takes the squares of the error terms 𝜀𝑡 2 , and regresses these on a constant and q lagged values such as 𝜀̂𝑡 2 = 𝛼̂0 + ∑ 𝛼̂ 𝑖 𝑞 𝑖=1 𝜀̂𝑡 − 𝑖 2 where q denotes the number of lags ARCH(q) (Engle, 1982), (Bollerslev, Chou, & Kroner, 1992). As the aim of this study is to forecast t+1 returns of a time series of commodity spot returns in order to better allocate assets in the selection of an actively managed commodities portfolio, one might question the utility of modeling volatility. In order to capture the information in the volatility estimation from the ARCH model a Volatility Ratio was defined as VolRatio = SigmaForecast/HistoricalVol, where the SigmaForecast is the in-sample (52-4 week) period sigma forecast from the ARCH model and the HistoricalVol is the realized volatility over the same period. In the next step the ARCH model estimated mean (𝜇) is multiplied by the Volatility Ratio. The concept is that if the estimated volatility is greater(less) than the historical realized volatility than the estimated mean in the next time step could be greater(less) than predicted, and to account for this the mean should be adjusted by the anticipated volatility. Thus, 𝜎 𝑅𝑎𝑡𝑖𝑜 = 𝜎𝐴𝑅𝐶𝐻/𝜎 𝐻𝑖𝑠𝑡 , and 𝜇̂ 𝑡 = 𝜇 𝐴𝑅𝐶𝐻 ∗ (𝜎𝐴𝑅𝐶𝐻/𝜎 𝐻𝑖𝑠𝑡) One of the most obvious weaknesses of the model is that the conditional variance has no sign as it is dependent on the square of previous shocks. A second weakness of the ARCH model for the purpose of forecasting is that they often over-predict volatility (Brooks, 2008).
  15. 15. 13 4.7 Generalized Auto Regressive Conditional Heteroskedasticity Model The fourth econometric model that will be used in this study was developed by Tim Bollerslev in 1986 and is a generalization of Engle’s 1982 Auto Regressive Conditional Heteroskedasticity Model, thus the model is dubbed the Generalized Auto Regressive Conditional Heteroskedasticity Model (GARCH). Bollerslev’s objective was to take Engle’s ARCH model and modify it much in the same way that the general ARMA model modifies the AR model, with the aim of creating a “more parsimonious description” of time series processes (Bollerslev, 1986). Thus, Bollerslev assumes an autoregressive moving average model (ARMA) for the error variance. The model therefor allows for the conditional variance to be dependent on previous lags of itself (Brooks, 2008). The GARCH(p,q) model, where p is the number of GARCH lags, and q is the number of ARCH lags estimates a conditional variance 𝜎𝑡 2 , which may be defined as, 𝜎𝑡 2 = 𝛼0 + ∑ 𝛼𝑖 𝜖 𝑡 − 𝑖 2 𝑞 𝑖=1 + ∑ 𝛽𝑖 𝜎𝑡 − 𝑖 2 𝑝 𝑖=1 where it is “effectively an ARMA model for the conditional variance” (Brooks, 2008). For the purposes of this study, as was done with the ARCH model, the additional information of the conditional variance estimation 𝜎𝑡 2 is attempted to be incorporated into the mean estimation via a volatility-ratio-multiplier. In a similar fashion as was done previously with the ARCH model, the VolRatio is defined as, 𝜎 𝑅𝑎𝑡𝑖𝑜 = 𝜎 𝐺𝐴𝑅𝐶𝐻/𝜎 𝐻𝑖𝑠𝑡 , and 𝜇̂ 𝑡 = 𝜇 𝐺𝐴𝑅𝐶𝐻 ∗ (𝜎 𝐺𝐴𝑅𝐶𝐻/𝜎 𝐻𝑖𝑠𝑡) so that if the estimated volatility is greater(less) than the historical realized volatility than the estimated mean will be increase(decreased) by the volatility ratio.
  16. 16. 14 4.8 Thick Model Averaging vis-à-vis Granger Finally, all four of the aforementioned econometric models estimated means (𝜇̂) where averaged in an iterative process as an attempt to exploit unique information provided by the models with the aim of increasing financial performance measured in returns. Thus at the end of each 52-4 week in sample forecasting period, each of the four estimated mean values where summed and divided by four, giving an equally weighted, naïve average of the models. In their paper, Thick Modeling, the authors find that very often a simple, equally weighted average of models produces results comparable to optimized weighting vectors via highly complex optimization routines (Granger & Jeon, March 2004). 5. EMPIRICAL RESULTS The summary statistics and risk-adjusted statistics from the empirical results for the entire sample (2001 to 2010), the pre-crisis (2001 to 2006) and the post-crisis (2007 to 2010) samples, for the 4 econometric forecasting methods introduced (AR, ARMA, ARCH, and GARCH) are given in Appendices II, V, VIII, and XI respectively. For the Naïve Momentum and Model Averaging models summary statistics and risk-adjusted metrics are available for the entire sample only, and may be found in Appendix I, and XIII respectively. The replication of the Mom-TS strategy (with naïve momentum) as produced by paper Fuertes, Miffre, Rallis (2010) appears to have been replicated accurately. The correlation matrices for the entire sample period, for all 6 forecasting methods are given in Appendices XV to XX. As previously shown by Fuertes, Miffre, Rallis , the Double Sort using the Term Structure signal first, and then using the momentum signal second outperforms performing the Double Sort in the reverse order (Mom then TS) (Fuertes, Miffre, & Rallis, 2010). In this study the aforementioned relationship was found to be true, across all 6 momentum signal methods (naïve Momentum, AR, ARMA, ARCH, GARCH, and Model Average) using the 20 previously named commodities from 2000 to 2010. It should also be noted that the addition of the Volatility Ratio multiplier used in the ARCH and GARCH models improved performance compared to the ARCH and GARCH models’ simple mean estimates. Thus information from the variance or conditional variance estimates were able to aid the models in better forecasting t+1 mean returns.
  17. 17. 15 5.1 Entire Sample (2001 to 2010) The overall best performing signal combination (over the entire sample period) was the TS-Mom Double Sort (ranked by arithmetic and geometric returns) using the TS-ARMA pair, which produced annualized arithmetic and geometric means of 23.14% and 22.48% respectively. All 6 of the momentum signal methods’ arithmetic mean performance is listed below. As is evident in the above table, none of the newly introduced momentum signal methods were able to outperform the original naïve momentum (i.e Moskowitz) / Term Structure Double Sort when sorted by Mom then TS. It is interesting that the order in which the two signal generaters are used effects the ranking across the different moment signal methods. This phenomenon is perhaps due to the relative information that the different momentum signal methods contain. 5.2 Pre-crisis Sample Results For the sub sample periods of pre and post financial crisis only the Mom first, TS second portfolios (Mom12-1-TS1 ) were implemented thus the resulting universes will be used as a basis for comparison. For the pre-crisis period of 2001 to 2006 the Naïve, AR, and ARCH models remained in the same ranking order as in the entire sample period (Mom12-1-TS1 ) results, ranking 1st , 2nd , and 3rd respectively. However, the other 4 strategies relative-to-cohort performance changed. Below is a table containing the performance and ranking of the 6 different momentum signal strategies during the pre-crisis period. Mom12-1 -TS1 Rank TS1 -Mom12-1 Rank Naïve 20.78% 1 21.93% 4 AR 18.48% 2 20.53% 6 ARMA 17.65% 4 23.14% 1 ARCH 18.11% 3 21.66% 5 GARCH 16.67% 6 22.75% 2 Model AVG 17.24% 5 22.41% 3 Double Sort Performance and Rank *Annualized Arithmetic Mean
  18. 18. 16 An interesting observation that can be made regarding the pre-crisis data in the table above, is that the signal methods’ absolute return rank is not identical to their relative return rank. Although the naïve signal and the AR signal rank 1st and 2nd, respectively, for both performance metrics this does not hold for the remaining 4 signals. Perhaps the most interesting is the comparative absolute and relative ranking of the ARCH signal. The ARCH model signal ranks 3rd in absolute terms and last or 6th in risk adjusted terms, as measured by ( 𝑟−𝑅 𝑓 𝜎 𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 ) , the Sortino Ratio. This serves as evidence to that the higher order moments of skew and kurtosis are significant in the returns distributions. The pre-crisis ARCH has a skewness of 0.08, and kurtosis of 4.04, or has excess kurtosis of 1.04. Econometric result tables are available for all 3 time-period subsamples for each signal, and may be found in the appendix. 5.3 Post-crisis Sample Results The post-crisis period from 2007 to 2010 yielded markedly momentum signal performance rankings from the pre-crisis sample period. For the post-crisis sample period the Model Average, GARCH and Naïve Momentum signal strategies ranked 1st , 2nd and 3rd respectively as measured by arithmetic mean return. Although the Naïve Momentum signal marginally outperformed the GARCH Model in terms of arithmetic mean, the GARCH massively outperformed the Naïve signal on a risk adjusted basis. The Naïve signal produced an arithmetic mean of 20.86% p.a. with a Sortino ratio of 2.90, while the GARCH model returned 20.86% with a Sortino ratio of 3.72. Please see post-crisis results in the table below. Arithmetic R Rank Sortino Rank Naïve 20.75% 1 2.94 1 AR 19.11% 2 2.58 2 ARMA 14.78% 5 1.99 4 ARCH 17.69% 3 1.47 6 GARCH 15.01% 4 2.16 3 Model AVG 14.26% 6 1.93 5 Mom12-1 -TS1 *Annualized Arithmetic Mean & Sortino Ratios **pre-crisis (2001-2006)
  19. 19. 17 6. CONCLUSION The results presented in this study are quite interesting for several reasons. First, should someone only consider the first moment as a means of profiling and ranking these different signals, than clearly between the Naïve Momentum and GARCH signal for example, the Naïve Momentum appears to be the better choice. However, when also considered the relative risk adjusted performance (taking the second moment into consideration) it should be clear that if forced to choose between the two models the GARCH is the more prudent decision. Another fascinating observation is the performance of the Model Average, and GARCH signals between the pre- crisis and post-crisis periods. The GARCH and Model Average ranked 4th and 6th in the pre- crisis sample, and 2nd and 1st respectively in the post-crisis sample. The most obvious explanation for the drastic changes in ranked performance could be caused by the differing levels of volatility in the two sample periods. The annualized volatility of an equally weighted portfolio of the 20 commodities used in this study was 10.77% p.a. pre-crisis, and 20.90% p.a. post-crisis. The volatility of the underlying assets nearly doubled from pre to post-crisis. Thus, one conclusion that can be drawn is that the GARCH and Model Average signals perform better in subsamples characterized by high volatility, while the Naïve Momentum and AR signals perform best in periods characterized by low volatility. Said more generally, it is evident that different econometric models perform differently depending on the sample or subsample for which they are being used to model. 7. AREAS FOR FURTHER RESEARCH One question that arises as a result of this study is why did the ARMA model perform rank so poorly relative to its peers (across all samples) in the Mom12-1-TS1 frame work, yet outperform Arithmetic R Rank Sortino Rank Naïve 20.86% 2 2.90 3 AR 17.22% 5 2.50 5 ARMA 14.78% 6 1.99 6 ARCH 18.95% 4 2.74 4 GARCH 20.03% 3 3.72 2 Model AVG 23.27% 1 3.78 1 Mom12-1 -TS1 *Annualized Arithmetic Mean & Sortino Ratios **post-crisis (2007-2010)
  20. 20. 18 all other models in the TS1-Mom12-1 ? Further in-depth study regarding this question could prove to give interesting answers. Based on the results of this study, and the observation that different models perform much differently across different samples, or different volatility regimes is another area which could be explored further. Some more advanced econometric have been widely written about recently, such as state-space models, neural networks, and kalman filters, and appear to provide excellent results for forecasting one time-step ahead forecasts. Of course there are countless more econometric models which could be back tested, with the same data as was used for this study, however this could lead to the perils of data mining and "back fitting" the model to suit the desired results of the researcher. A last area of further research could be to conduct similar test as the ones done in this study but to use more sophisticated Model Averaging Processes as are described by Granger (Granger & Jeon, March 2004). Acknowledgements: I would like to thank Professor Joëlle Miffre for teaching an inspiring course on commodities which serves as the foundation for this paper; Brian Burke for teaching me that ‘the reward for a job well done is having done it’; My colleagues at Assenagon Credit Management for making nearly every work day enjoyable; and a special thanks to my EDHEC colleagues Daniel Jakubowski and Alex Immler, for their support and without whom many programming feats would not be possible.
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  23. 23. 21 National Science Foundation:The President's National Medal of Science. (n.d.). Retrieved Feb. 1, 2012, from National Science Foundation: http://www.nsf.gov/od/nms/recip_details.cfm?recip_id=5300000000455 Proelss, J., & Schweizer, D. (2008). Efficient Frontier of Commodity Portfolios. In F. J. Fabozz, R. Fuss, & D. G. Kaiser, The Handbook of Commodity Investing (pp. Chapter 19, (454 - 778) ). New Jersey: John Wiley&Sons, Inc. Spierdijk, L., & Umar, Z. (2010). Are Commodities a Good Hedge Against Inflation? A Comparative Approach. Network for Studies on Pensions, Aging and Retirement.
  24. 24. i APPENDIX Benchmark TS1 -Mom12-1 L S L-S L S L-S L S L-S L-S Panal A: Summary Statistics Annualized arithmetic mean 21.03% -20.54% 20.78% 5.69% 1.80% 2.51% 28.91% -21.45% 25.66% 2.27% 21.93% Annualized geometric mean 18.62% -22.46% 20.06% 3.64% 0.10% 2.03% 27.28% -22.81% 25.30% -1.35% 21.14% Annualized volatility 21.81% 19.53% 12.07% 20.10% 18.37% 9.86% 18.07% 16.43% 8.50% 26.64% 12.65% Annualized downside volatility 16.75% 14.17% 8.61% 15.75% 13.46% 7.21% 13.51% 12.17% 5.89% 20.37% 8.88% Reward/risk ratio 0.85 -1.15 1.66 0.18 0.01 0.21 1.51 -1.39 2.98 -0.05 1.67 Sortino ratio (0%) 1.37 -1.31 2.93 0.37 0.13 0.36 2.48 -1.56 6.26 0.11 3.01 Omega (0%) 1.42 0.67 1.83 1.11 1.04 1.09 1.81 0.61 2.97 1.03 1.86 Skewness -0.83 -0.28 -0.04 -1.08 -0.48 -0.19 -0.52 -0.47 0.27 -0.75 0.33 Kurtosis 5.08 5.32 3.43 6.14 6.58 3.42 5.38 5.49 4.71 5.10 6.15 Annualized 95% VaR (Cornish-Fisher) -27.25% -32.60% -16.75% -25.23% -26.28% -15.29% -22.35% -27.11% -10.73% -37.07% -18.07% Annualized 99% VaR (Cornish-Fisher) -51.95% -56.89% -26.24% -52.95% -54.68% -22.89% -44.69% -47.97% -20.44% -67.55% -37.20% % of positive weeks 59.40% 45.51% 60.47% 56.73% 52.16% 55.34% 64.32% 44.23% 65.77% 53.42% 61.32% Max runup (consecutive) 32.60% 4.10% 21.72% 21.26% 14.47% 11.35% 25.69% 8.56% 15.79% 21.64% 19.49% Runup length (weeks) 9 7 10 9 7 9 13 8 15 8 10 Maximum drawdown 52.24% 88.89% 15.94% 53.02% 54.34% 18.62% 43.56% 89.21% 5.57% 70.79% 13.54% Drawdown length (weeks) 22 423 26 22 155 98 22 452 6 33 13 Valley to recovery (weeks) - - 24 - - 91 38 - - - 13 Max 52 week rolling return 90.80% 28.11% 55.58% 58.54% 33.66% 21.98% 92.13% 3.97% 57.76% 67.70% 67.50% Min 52 week rolling return -40.98% -60.71% -1.48% -48.32% -48.21% -13.72% -17.02% -59.44% 9.49% -62.66% -1.77% Panel B: Risk-Adjusted Performance Annualized α 20.87% -19.67% 19.80% 2.02% -3.14% 0.27% 30.11% -20.48% 24.70% 0.00% P-Value 0.00 0.00 0.00 0.68 0.50 0.93 0.00 0.00 0.00 0.98 β Bonds -0.09 -0.28 0.10 -0.18 -0.11 -0.03 0.07 -0.28 0.20 0.00 P-Value 0.65 0.12 0.46 0.25 0.49 0.76 0.63 0.06 0.02 0.67 β Market -0.01 0.08 -0.04 0.01 0.04 -0.02 -0.01 0.02 -0.01 0.00 P-Value 0.86 0.03 0.12 0.83 0.20 0.46 0.83 0.40 0.57 0.40 β Commodities 0.47 0.39 0.04 0.49 0.42 0.03 0.44 0.34 0.06 1.00 P-Value 0.00 0.00 0.06 0.00 0.00 0.05 0.00 0.00 0.00 0.00 Adjusted R² 0.33 0.30 0.01 0.45 0.39 0.00 0.43 0.32 0.04 1.00 Momentum Table I: Period 2001-2010 (whole sample) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only. Mom12-1 -TS1 Mom12-1 TS1
  25. 25. ii Benchmark TS1 -Mom12-1 L S L-S L S L-S L S L-S L-S Panal A: Summary Statistics Annualized arithmetic mean 15.47% -21.50% 18.48% 1.62% 2.08% -1.12% 28.91% -21.45% 25.66% 2.27% 20.53% Annualized geometric mean 12.68% -23.38% 17.78% -0.87% 0.53% -1.56% 27.28% -22.81% 25.30% -1.35% 19.87% Annualized volatility 23.42% 19.32% 11.83% 22.10% 17.59% 9.45% 18.07% 16.43% 8.50% 26.64% 11.51% Annualized downside volatility 18.20% 13.94% 8.53% 17.25% 12.72% 6.88% 13.51% 12.17% 5.89% 20.37% 8.34% Reward/risk ratio 0.54 -1.21 1.50 -0.04 0.03 -0.17 1.51 -1.39 2.98 -0.05 1.73 Sortino ratio (0%) 0.90 -1.38 2.55 0.09 0.17 -0.16 2.48 -1.56 6.26 0.11 2.97 Omega (0%) 1.27 0.67 1.75 1.03 1.05 0.96 1.81 0.61 2.97 1.03 1.90 Skewness -0.92 -0.26 -0.24 -0.99 -0.30 -0.26 -0.52 -0.47 0.27 -0.75 -0.21 Kurtosis 5.25 4.11 4.85 5.61 6.30 4.00 5.38 5.49 4.71 5.10 3.83 Annualized 95% VaR (Cornish-Fisher) -29.57% -33.01% -15.69% -29.13% -26.09% -14.86% -22.35% -27.11% -10.73% -37.07% -15.26% Annualized 99% VaR (Cornish-Fisher) -56.87% -51.18% -29.05% -56.79% -52.31% -23.48% -44.69% -47.97% -20.44% -67.55% -25.30% % of positive weeks 58.97% 43.80% 61.54% 55.77% 49.04% 49.79% 64.32% 44.23% 65.77% 53.42% 62.61% Max runup (consecutive) 44.78% 17.07% 22.89% 9.11% 15.99% 9.28% 25.69% 8.56% 15.79% 21.64% 12.82% Runup length (weeks) 13 6 15 7 6 10 13 8 15 8 11 Maximum drawdown 57.41% 90.19% 16.52% 61.93% 39.03% 27.76% 43.56% 89.21% 5.57% 70.79% 11.40% Drawdown length (weeks) 22 452 22 22 40 160 22 452 6 33 8 Valley to recovery (weeks) - - 14 - - - 38 - - - - Max 52 week rolling return 75.87% 19.13% 43.35% 42.16% 27.25% 15.65% 92.13% 3.97% 57.76% 67.70% 45.56% Min 52 week rolling return -42.98% -56.35% 0.68% -57.11% -38.77% -23.65% -17.02% -59.44% 9.49% -62.66% 4.82% Panel B: Risk-Adjusted Performance Annualized α 14.23% -20.36% 16.97% -3.74% -1.59% -3.37% 30.11% -20.48% 24.70% 0.00% P-Value 0.01 0.00 0.00 0.33 0.75 0.20 0.00 0.00 0.00 0.98 β Bonds -0.03 -0.36 0.17 -0.18 -0.21 0.02 0.07 -0.28 0.20 0.00 P-Value 0.85 0.06 0.15 0.16 0.22 0.86 0.63 0.06 0.02 0.67 β Market 0.03 0.00 0.02 0.01 -0.04 0.03 -0.01 0.02 -0.01 0.00 P-Value 0.41 0.93 0.49 0.61 0.25 0.14 0.83 0.40 0.57 0.40 β Commodities 0.66 0.28 0.19 0.68 0.29 0.19 0.44 0.34 0.06 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Adjusted R² 0.56 0.15 0.18 0.71 0.21 0.29 0.43 0.32 0.04 1.00 AR Model Table I: Period 2001-2010 (whole sample) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only. Mom12-1 -TS1 Mom12-1 TS1
  26. 26. iii Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 19.60% -18.61% 19.11% 7.90% 3.44% 2.23% 30.13% -19.43% 24.78% 7.12% Annualized geometric mean 17.26% -19.98% 18.38% 6.27% 2.55% 1.81% 28.96% -20.30% 24.45% 4.29% Annualized volatility 21.57% 16.52% 12.09% 18.04% 13.34% 9.19% 15.32% 13.16% 8.15% 23.67% Annualized downside volatility 16.06% 11.53% 8.72% 13.37% 9.31% 6.53% 11.07% 9.56% 5.88% 17.70% Reward/risk ratio 0.80 -1.21 1.52 0.35 0.19 0.20 1.89 -1.54 3.00 0.18 Sortino ratio (0%) 1.33 -1.43 2.58 0.62 0.38 0.35 3.36 -1.75 5.90 0.41 Omega (0%) 1.39 0.67 1.78 1.17 1.10 1.09 2.02 0.59 2.90 1.11 Skewness -0.56 0.09 -0.24 -0.39 0.09 -0.12 -0.22 -0.13 -0.14 -0.56 Kurtosis 4.78 2.94 5.22 3.37 3.71 4.15 3.30 2.76 3.58 4.04 Annualized 95% VaR (Cornish-Fisher) -28.79% -30.19% -15.94% -26.58% -21.60% -14.31% -20.03% -23.93% -9.57% -33.94% Annualized 99% VaR (Cornish-Fisher) -52.10% -41.33% -30.72% -39.92% -33.20% -23.16% -31.31% -31.92% -16.21% -55.07% % of positive weeks 56.55% 43.13% 62.62% 55.27% 48.88% 49.84% 63.58% 44.73% 67.09% 53.67% Max runup (consecutive) 44.78% 17.07% 22.89% 9.11% 15.99% 9.28% 25.69% 8.56% 15.79% 21.64% Runup length (weeks) 13 6 15 7 6 10 13 8 15 8 Maximum drawdown 21.55% 72.92% 13.93% 26.82% 21.38% 18.00% 13.46% 71.74% 5.00% 36.48% Drawdown length (weeks) 9 297 10 36 79 60 12 297 2 50 Valley to recovery (weeks) 38 - 30 - 39 - 13 - 23 55 Max 52 week rolling return 75.87% 19.13% 43.35% 31.79% 22.20% 15.65% 63.99% 3.97% 46.22% 59.81% Min 52 week rolling return -14.79% -41.99% 0.68% -24.99% -14.83% -15.11% 12.41% -38.37% 9.49% -34.89% Panel B: Risk-Adjusted Performance Annualized α 15.39% -18.67% 16.12% 6.96% 1.27% 0.23% 29.54% -19.81% 23.90% 0.00% P-Value 0.02 0.00 0.00 0.36 0.82 0.95 0.00 0.00 0.00 0.22 β Bonds -0.11 -0.47 0.18 -0.41 -0.06 -0.17 -0.01 -0.36 0.18 0.00 P-Value 0.60 0.04 0.22 0.11 0.75 0.20 0.97 0.04 0.11 0.21 β Market 0.07 0.04 0.02 -0.04 -0.03 -0.01 0.02 0.04 -0.01 0.00 P-Value 0.17 0.48 0.66 0.55 0.58 0.86 0.73 0.31 0.63 1.00 β Commodities 0.65 0.14 0.26 -0.07 -0.03 -0.02 0.37 0.21 0.08 1.00 P-Value 0.00 0.00 0.00 0.12 0.28 0.47 0.00 0.00 0.00 0.00 Adjusted R² 0.52 0.05 0.24 0.01 0.00 0.00 0.32 0.16 0.05 1.00 Mom12-1 -TS1 Mom12-1 TS1 AR Model Table II: Period 2001 - end 2006 (pre-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  27. 27. iv Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 7.12% -27.32% 17.22% -7.47% 1.31% -4.39% 26.46% -25.51% 25.98% -7.51% Annualized geometric mean 3.43% -30.24% 16.58% -11.12% -1.38% -4.79% 23.88% -27.87% 25.57% -12.70% Annualized volatility 26.81% 24.04% 11.32% 26.64% 23.14% 9.03% 22.67% 21.61% 9.24% 31.84% Annualized downside volatility 21.83% 17.74% 8.15% 21.65% 17.08% 6.59% 17.43% 16.14% 5.86% 24.81% Reward/risk ratio 0.13 -1.26 1.46 -0.42 -0.06 -0.53 1.05 -1.29 2.77 -0.40 Sortino ratio (0%) 0.33 -1.38 2.50 -0.34 0.08 -0.64 1.68 -1.42 6.57 -0.30 Omega (0%) 1.10 0.66 1.71 0.90 1.02 0.84 1.55 0.64 3.00 0.91 Skewness -1.24 -0.41 -0.27 -1.25 -0.39 -0.28 -0.64 -0.52 0.95 -0.81 Kurtosis 5.06 3.83 3.84 5.58 4.99 3.78 5.17 4.76 6.42 4.99 Annualized 95% VaR (Cornish-Fisher) -33.20% -40.29% -15.23% -34.60% -34.55% -14.66% -28.80% -35.34% -13.29% -45.29% Annualized 99% VaR (Cornish-Fisher) -62.38% -60.79% -25.06% -67.08% -61.14% -22.38% -55.40% -58.66% -28.44% -80.64% % of positive weeks 63.87% 45.16% 59.35% 60.00% 50.32% 47.74% 65.81% 43.23% 63.23% 52.90% Max runup (consecutive) 13.92% 11.65% 18.05% 11.68% 10.95% 10.00% 13.44% 11.10% 17.72% 18.27% Runup length (weeks) 6 5 12 6 5 7 9 5 9 6 Maximum drawdown 57.41% 67.26% 16.52% 61.93% 39.03% 24.34% 43.56% 63.81% 5.57% 70.79% Drawdown length (weeks) 22 79 22 22 40 44 22 132 6 33 Valley to recovery (weeks) - - 14 - - - 38 - - - Max 52 week rolling return 56.16% 3.00% 38.31% 42.16% 27.25% 15.19% 92.13% 3.61% 57.76% 67.70% Min 52 week rolling return -42.98% -56.35% 1.83% -57.11% -38.77% -23.65% -17.02% -59.44% 16.41% -62.66% Panel B: Risk-Adjusted Performance Annualized α 11.66% -21.68% 17.03% -1.43% 4.24% -4.65% 33.24% -19.70% 26.27% 0.00% P-Value 0.27 0.04 0.01 0.85 0.69 0.32 0.00 0.01 0.00 1.00 β Bonds 0.09 -0.24 0.17 -0.28 -0.19 -0.04 0.16 -0.19 0.18 0.00 P-Value 0.77 0.49 0.38 0.23 0.52 0.79 0.54 0.46 0.26 0.94 β Market 0.00 -0.03 0.01 -0.01 -0.08 0.04 -0.02 0.02 -0.02 0.00 P-Value 1.00 0.64 0.65 0.87 0.13 0.12 0.69 0.66 0.51 0.88 β Commodities 0.66 0.43 0.11 0.73 0.47 0.13 0.53 0.49 0.02 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.42 0.00 Adjusted R² 0.60 0.32 0.09 0.75 0.43 0.19 0.55 0.52 0.00 1.00 Mom12-1 -TS1 Mom12-1 TS1 AR Model Table III: Period 2007 - 2010 (post-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  28. 28. v Benchmark TS1 -Mom12-1 L S L-S L S L-S L S L-S L-S Panal A: Summary Statistics Annualized arithmetic mean 17.66% -17.64% 17.65% 3.66% 0.74% 1.25% 28.91% -21.45% 25.66% 2.27% 23.14% Annualized geometric mean 14.96% -19.54% 16.96% 1.38% -0.81% 0.85% 27.28% -22.81% 25.30% -1.35% 22.48% Annualized volatility 23.18% 19.43% 11.74% 21.23% 17.51% 8.90% 18.07% 16.43% 8.50% 26.64% 11.51% Annualized downside volatility 17.34% 14.18% 8.32% 16.13% 13.18% 6.27% 13.51% 12.17% 5.89% 20.37% 8.16% Reward/risk ratio 0.65 -1.01 1.45 0.06 -0.05 0.10 1.51 -1.39 2.98 -0.05 1.95 Sortino ratio (0%) 1.09 -1.14 2.49 0.23 0.06 0.20 2.48 -1.56 6.26 0.11 3.54 Omega (0%) 1.32 0.71 1.74 1.07 1.02 1.05 1.81 0.61 2.97 1.03 2.05 Skewness -0.44 -0.31 0.00 -0.64 -0.75 0.20 -0.52 -0.47 0.27 -0.75 0.08 Kurtosis 6.03 4.47 5.91 5.97 6.25 5.61 5.38 5.49 4.71 5.10 4.39 Annualized 95% VaR (Cornish-Fisher) -31.52% -32.21% -16.17% -29.53% -24.06% -14.48% -22.35% -27.11% -10.73% -37.07% -15.64% Annualized 99% VaR (Cornish-Fisher) -64.22% -52.14% -32.85% -58.72% -49.22% -26.61% -44.69% -47.97% -20.44% -67.55% -27.65% % of positive weeks 58.33% 44.66% 58.33% 55.77% 51.20% 52.78% 64.32% 44.23% 65.77% 53.42% 63.89% Max runup (consecutive) 33.59% 15.39% 17.78% 19.76% 11.53% 4.08% 25.69% 8.56% 15.79% 21.64% 20.49% Runup length (weeks) 9 6 13 9 8 8 13 8 15 8 14 Maximum drawdown 51.62% 84.63% 13.38% 57.48% 49.29% 22.79% 43.56% 89.21% 5.57% 70.79% 9.22% Drawdown length (weeks) 22 452 12 40 61 138 22 452 6 33 6 Valley to recovery (weeks) - - 22 - - - 38 - - - 12 Max 52 week rolling return 67.01% 42.45% 50.28% 49.55% 34.73% 20.84% 92.13% 3.97% 57.76% 67.70% 75.57% Min 52 week rolling return -40.73% -60.14% -4.46% -54.04% -43.89% -13.84% -17.02% -59.44% 9.49% -62.66% 2.39% Panel B: Risk-Adjusted Performance Annualized α 16.41% -17.26% 15.86% -1.30% -3.44% -1.22% 30.11% -20.48% 24.70% 0.00% P-Value 0.01 0.00 0.00 0.76 0.46 0.66 0.00 0.00 0.00 0.98 β Bonds 0.05 -0.33 0.19 -0.08 -0.21 0.07 0.07 -0.28 0.20 0.00 P-Value 0.80 0.08 0.12 0.54 0.17 0.45 0.63 0.06 0.02 0.67 β Market -0.01 0.02 -0.01 0.02 -0.03 0.02 -0.01 0.02 -0.01 0.00 P-Value 0.82 0.67 0.63 0.50 0.39 0.20 0.83 0.40 0.57 0.40 β Commodities 0.59 0.33 0.13 0.61 0.35 0.13 0.44 0.34 0.06 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Adjusted R² 0.46 0.21 0.09 0.62 0.30 0.15 0.43 0.32 0.04 1.00 ARMA Table I: Period 2001-2010 (whole sample) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only. Mom12-1 -TS1 Mom12-1 TS1
  29. 29. vi Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 20.14% -9.43% 14.78% 7.74% 5.29% 1.22% 30.13% -19.43% 24.78% 7.12% Annualized geometric mean 17.97% -10.80% 14.10% 6.29% 4.34% 0.86% 28.96% -20.30% 24.45% 4.29% Annualized volatility 20.76% 16.56% 11.69% 17.03% 13.83% 8.50% 15.32% 13.16% 8.15% 23.67% Annualized downside volatility 15.34% 11.69% 8.45% 12.61% 9.99% 6.11% 11.07% 9.56% 5.88% 17.70% Reward/risk ratio 0.87 -0.65 1.21 0.37 0.31 0.10 1.89 -1.54 3.00 0.18 Sortino ratio (0%) 1.45 -0.76 1.99 0.64 0.55 0.20 3.36 -1.75 5.90 0.41 Omega (0%) 1.41 0.82 1.59 1.17 1.15 1.05 2.02 0.59 2.90 1.11 Skewness -0.52 0.03 -0.26 -0.37 -0.22 -0.15 -0.22 -0.13 -0.14 -0.56 Kurtosis 4.99 3.05 5.48 3.32 4.02 4.43 3.30 2.76 3.58 4.04 Annualized 95% VaR (Cornish-Fisher) -27.68% -28.65% -15.78% -25.17% -20.93% -13.24% -20.03% -23.93% -9.57% -33.94% Annualized 99% VaR (Cornish-Fisher) -51.33% -40.17% -30.81% -37.59% -33.66% -22.00% -31.31% -31.92% -16.21% -55.07% % of positive weeks 56.55% 46.65% 56.55% 55.91% 52.08% 52.08% 63.58% 44.73% 67.09% 53.67% Max runup (consecutive) 33.59% 15.39% 17.78% 19.76% 11.53% 10.51% 25.69% 8.56% 15.79% 21.64% Runup length (weeks) 9 6 13 9 8 8 13 8 15 8 Maximum drawdown 27.00% 53.68% 13.38% 24.84% 20.21% 14.62% 13.46% 71.74% 5.00% 36.48% Drawdown length (weeks) 35 283 12 66 79 29 12 297 2 50 Valley to recovery (weeks) 27 - 22 - - - 13 - 23 55 Max 52 week rolling return 67.01% 42.45% 44.43% 28.09% 34.73% 17.88% 63.99% 3.97% 46.22% 59.81% Min 52 week rolling return -16.04% -41.21% -4.46% -19.19% -13.71% -10.80% 12.41% -38.37% 9.49% -34.89% Panel B: Risk-Adjusted Performance Annualized α 16.11% -11.21% 11.51% 6.64% 3.07% -0.81% 29.54% -19.81% 23.90% 0.00% P-Value 0.02 0.07 0.01 0.36 0.59 0.82 0.00 0.00 0.00 0.22 β Bonds 0.01 -0.38 0.20 -0.38 -0.02 -0.18 -0.01 -0.36 0.18 0.00 P-Value 0.98 0.09 0.20 0.12 0.94 0.14 0.97 0.04 0.11 0.21 β Market 0.08 0.08 0.00 -0.04 0.02 -0.03 0.02 0.04 -0.01 0.00 P-Value 0.13 0.13 0.98 0.48 0.75 0.34 0.73 0.31 0.63 1.00 β Commodities 0.57 0.19 0.19 -0.05 -0.04 0.00 0.37 0.21 0.08 1.00 P-Value 0.00 0.00 0.00 0.22 0.19 0.92 0.00 0.00 0.00 0.00 Adjusted R² 0.42 0.08 0.14 0.00 0.00 0.00 0.32 0.16 0.05 1.00 Mom12-1 -TS1 Mom12-1 TS1 ARMA Table II: Period 2001 - end 2006 (pre-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  30. 30. vii Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 12.67% -34.21% 23.44% -2.75% -7.70% 2.48% 26.46% -25.51% 25.98% -7.51% Annualized geometric mean 8.88% -37.15% 22.75% -6.22% -10.24% 2.06% 23.88% -27.87% 25.57% -12.70% Annualized volatility 27.49% 24.11% 11.83% 26.18% 22.38% 9.22% 22.67% 21.61% 9.24% 31.84% Annualized downside volatility 20.80% 17.83% 8.05% 20.30% 17.11% 6.29% 17.43% 16.14% 5.86% 24.81% Reward/risk ratio 0.32 -1.54 1.92 -0.24 -0.46 0.22 1.05 -1.29 2.77 -0.40 Sortino ratio (0%) 0.64 -1.69 3.68 -0.13 -0.44 0.41 1.68 -1.42 6.57 -0.30 Omega (0%) 1.19 0.59 2.11 0.96 0.87 1.11 1.55 0.64 3.00 0.91 Skewness -0.33 -0.36 0.51 -0.70 -0.79 0.72 -0.64 -0.52 0.95 -0.81 Kurtosis 5.98 4.15 6.59 5.76 5.18 7.95 5.17 4.76 6.42 4.99 Annualized 95% VaR (Cornish-Fisher) -39.42% -41.51% -16.95% -37.09% -32.18% -15.66% -28.80% -35.34% -13.29% -45.29% Annualized 99% VaR (Cornish-Fisher) -78.10% -64.19% -36.37% -71.55% -58.14% -34.14% -55.40% -58.66% -28.44% -80.64% % of positive weeks 61.94% 40.65% 61.94% 57.42% 48.39% 55.48% 65.81% 43.23% 63.23% 52.90% Max runup (consecutive) 11.47% 7.31% 15.11% 8.24% 12.50% 3.69% 13.44% 11.10% 17.72% 18.27% Runup length (weeks) 8 4 8 6 5 6 9 5 9 6 Maximum drawdown 51.62% 70.42% 6.95% 57.48% 49.29% 15.61% 43.56% 63.81% 5.57% 70.79% Drawdown length (weeks) 22 135 9 40 61 43 22 132 6 33 Valley to recovery (weeks) - - 7 - - 26 38 - - - Max 52 week rolling return 66.10% 1.38% 50.28% 49.55% 33.45% 20.84% 92.13% 3.61% 57.76% 67.70% Min 52 week rolling return -40.73% -60.14% 11.25% -54.04% -43.89% -13.84% -17.02% -59.44% 16.41% -62.66% Panel B: Risk-Adjusted Performance Annualized α 17.00% -26.43% 23.56% 2.72% -4.21% 1.54% 33.24% -19.70% 26.27% 0.00% P-Value 0.17 0.01 0.00 0.76 0.64 0.77 0.00 0.01 0.00 1.00 β Bonds 0.10 -0.28 0.19 -0.26 -0.22 -0.02 0.16 -0.19 0.18 0.00 P-Value 0.77 0.39 0.34 0.32 0.41 0.90 0.54 0.46 0.26 0.94 β Market -0.07 -0.03 -0.02 -0.01 -0.05 0.02 -0.02 0.02 -0.02 0.00 P-Value 0.22 0.63 0.53 0.86 0.21 0.34 0.69 0.66 0.51 0.88 β Commodities 0.61 0.48 0.07 0.67 0.51 0.08 0.53 0.49 0.02 1.00 P-Value 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.42 0.00 Adjusted R² 0.50 0.40 0.02 0.66 0.52 0.07 0.55 0.52 0.00 1.00 Mom12-1 -TS1 Mom12-1 TS1 ARMA Table III: Period 2007 - 2010 (post-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  31. 31. viii Benchmark TS1 -Mom12-1 L S L-S L S L-S L S L-S L-S Panal A: Summary Statistics Annualized arithmetic mean 17.44% -18.77% 18.11% 2.22% 1.24% 0.38% 28.91% -21.45% 25.66% 2.27% 21.66% Annualized geometric mean 14.90% -20.62% 17.44% 0.15% -0.29% 0.01% 27.28% -22.81% 25.30% -1.35% 21.05% Annualized volatility 22.44% 19.12% 11.63% 20.26% 17.41% 8.68% 18.07% 16.43% 8.50% 26.64% 11.11% Annualized downside volatility 16.85% 13.92% 8.15% 15.43% 13.07% 6.20% 13.51% 12.17% 5.89% 20.37% 7.89% Reward/risk ratio 0.66 -1.08 1.50 0.01 -0.02 0.00 1.51 -1.39 2.98 -0.05 1.89 Sortino ratio (0%) 1.11 -1.23 2.67 0.15 0.10 0.06 2.48 -1.56 6.26 0.11 3.42 Omega (0%) 1.33 0.70 1.73 1.04 1.03 1.02 1.81 0.61 2.97 1.03 2.00 Skewness -0.54 -0.37 0.04 -0.74 -0.74 -0.10 -0.52 -0.47 0.27 -0.75 -0.04 Kurtosis 4.76 4.76 3.60 5.36 6.51 4.29 5.38 5.49 4.71 5.10 4.11 Annualized 95% VaR (Cornish-Fisher) -30.47% -31.50% -16.61% -28.05% -23.79% -13.76% -22.35% -27.11% -10.73% -37.07% -14.89% Annualized 99% VaR (Cornish-Fisher) -54.67% -52.42% -26.36% -52.58% -49.97% -22.44% -44.69% -47.97% -20.44% -67.55% -25.56% % of positive weeks 57.48% 43.80% 55.98% 54.81% 52.40% 50.85% 64.32% 44.23% 65.77% 53.42% 60.90% Max runup (consecutive) 24.58% 13.37% 10.96% 9.02% 19.10% 7.02% 25.69% 8.56% 15.79% 21.64% 9.48% Runup length (weeks) 9 8 9 6 9 7 13 8 15 8 10 Maximum drawdown 50.48% 87.41% 13.15% 51.61% 47.37% 19.30% 43.56% 89.21% 5.57% 70.79% 10.91% Drawdown length (weeks) 22 450 11 22 22 107 22 452 6 33 22 Valley to recovery (weeks) - - 20 - - 139 38 - - - 13 Max 52 week rolling return 63.42% 24.28% 45.36% 37.46% 33.52% 17.67% 92.13% 3.97% 57.76% 67.70% 61.49% Min 52 week rolling return -37.02% -54.85% 0.73% -48.32% -44.42% -13.94% -17.02% -59.44% 9.49% -62.66% 3.70% Panel B: Risk-Adjusted Performance Annualized α 16.35% -18.45% 16.66% -2.77% -3.32% -2.02% 30.11% -20.48% 24.70% 0.00% P-Value 0.00 0.00 0.00 0.49 0.46 0.46 0.00 0.00 0.00 0.98 β Bonds 0.03 -0.25 0.14 0.00 -0.12 0.07 0.07 -0.28 0.20 0.00 P-Value 0.88 0.18 0.25 0.98 0.42 0.47 0.63 0.06 0.02 0.67 β Market 0.05 -0.02 0.03 0.07 -0.03 0.05 -0.01 0.02 -0.01 0.00 P-Value 0.13 0.66 0.16 0.01 0.36 0.01 0.83 0.40 0.57 0.40 β Commodities 0.60 0.34 0.13 0.58 0.37 0.11 0.44 0.34 0.06 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Adjusted R² 0.51 0.23 0.09 0.62 0.34 0.11 0.43 0.32 0.04 1.00 ARCH Table I: Period 2001-2010 (whole sample) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only. Mom12-1 -TS1 Mom12-1 TS1
  32. 32. ix Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 18.99% -16.39% 17.69% 7.33% 5.35% 0.99% 30.13% -19.43% 24.78% 7.12% Annualized geometric mean 16.94% -17.69% 17.03% 6.00% 4.43% 0.68% 28.96% -20.30% 24.45% 4.29% Annualized volatility 20.25% 16.13% 11.57% 16.30% 13.56% 7.93% 15.32% 13.16% 8.15% 23.67% Annualized downside volatility 14.69% 11.49% 8.06% 11.93% 9.59% 5.55% 11.07% 9.56% 5.88% 17.70% Reward/risk ratio 0.84 -1.10 1.47 0.37 0.33 0.09 1.89 -1.54 3.00 0.18 Sortino ratio (0%) 1.42 -1.28 2.63 0.64 0.58 0.18 3.36 -1.75 5.90 0.41 Omega (0%) 1.41 0.70 1.72 1.17 1.15 1.04 2.02 0.59 2.90 1.11 Skewness -0.27 -0.02 0.08 -0.23 0.00 0.16 -0.22 -0.13 -0.14 -0.56 Kurtosis 4.02 2.82 4.04 2.94 3.40 3.71 3.30 2.76 3.58 4.04 Annualized 95% VaR (Cornish-Fisher) -28.81% -28.76% -16.56% -24.84% -21.44% -13.13% -20.03% -23.93% -9.57% -33.94% Annualized 99% VaR (Cornish-Fisher) -47.35% -39.00% -27.58% -35.36% -32.04% -20.09% -31.31% -31.92% -16.21% -55.07% % of positive weeks 55.91% 45.05% 55.59% 56.23% 53.35% 51.44% 63.58% 44.73% 67.09% 53.67% Max runup (consecutive) 24.58% 13.37% 10.96% 9.02% 19.10% 7.02% 25.69% 8.56% 15.79% 21.64% Runup length (weeks) 9 8 9 6 9 7 13 8 15 8 Maximum drawdown 18.05% 66.47% 12.05% 27.40% 19.52% 19.30% 13.46% 71.74% 5.00% 36.48% Drawdown length (weeks) 7 281 10 94 89 107 12 297 2 50 Valley to recovery (weeks) 16 - 31 - 17 - 13 - 23 55 Max 52 week rolling return 59.22% 24.28% 45.36% 36.58% 33.52% 15.85% 63.99% 3.97% 46.22% 59.81% Min 52 week rolling return -2.87% -41.41% 0.73% -24.42% -12.80% -13.94% 12.41% -38.37% 9.49% -34.89% Panel B: Risk-Adjusted Performance Annualized α 14.34% -17.49% 14.76% 6.15% 3.42% -1.20% 29.54% -19.81% 23.90% 0.00% P-Value 0.03 0.00 0.00 0.37 0.55 0.71 0.00 0.00 0.00 0.22 β Bonds 0.18 -0.26 0.22 -0.39 -0.13 -0.13 -0.01 -0.36 0.18 0.00 P-Value 0.41 0.23 0.15 0.10 0.49 0.27 0.97 0.04 0.11 0.21 β Market 0.05 0.05 0.00 -0.05 0.04 -0.05 0.02 0.04 -0.01 0.00 P-Value 0.35 0.35 1.00 0.35 0.36 0.08 0.73 0.31 0.63 1.00 β Commodities 0.57 0.20 0.18 -0.04 -0.05 0.01 0.37 0.21 0.08 1.00 P-Value 0.00 0.00 0.00 0.36 0.17 0.77 0.00 0.00 0.00 0.00 Adjusted R² 0.43 0.09 0.14 0.00 0.00 0.00 0.32 0.16 0.05 1.00 Mom12-1 -TS1 Mom12-1 TS1 ARCH Table II: Period 2001 - end 2006 (pre-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  33. 33. x Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 14.32% -23.58% 18.95% -2.22% -7.95% 2.86% 26.46% -25.51% 25.98% -7.51% Annualized geometric mean 10.79% -26.52% 18.26% -5.43% -10.53% 2.39% 23.88% -27.87% 25.57% -12.70% Annualized volatility 26.40% 24.10% 11.79% 25.10% 22.56% 9.75% 22.67% 21.61% 9.24% 31.84% Annualized downside volatility 20.53% 17.80% 8.36% 19.75% 17.36% 7.17% 17.43% 16.14% 5.86% 24.81% Reward/risk ratio 0.41 -1.10 1.55 -0.22 -0.47 0.25 1.05 -1.29 2.77 -0.40 Sortino ratio (0%) 0.73 -1.21 2.74 -0.11 -0.45 0.41 1.68 -1.42 6.57 -0.30 Omega (0%) 1.23 0.69 1.75 0.97 0.87 1.12 1.55 0.64 3.00 0.91 Skewness -0.75 -0.50 -0.02 -0.94 -0.84 -0.40 -0.64 -0.52 0.95 -0.81 Kurtosis 4.71 4.53 2.78 5.08 5.36 4.82 5.17 4.76 6.42 4.99 Annualized 95% VaR (Cornish-Fisher) -35.30% -38.98% -16.75% -34.26% -32.11% -14.24% -28.80% -35.34% -13.29% -45.29% Annualized 99% VaR (Cornish-Fisher) -62.89% -63.63% -24.11% -62.38% -59.22% -25.02% -55.40% -58.66% -28.44% -80.64% % of positive weeks 60.65% 41.29% 56.77% 55.48% 50.97% 52.26% 65.81% 43.23% 63.23% 52.90% Max runup (consecutive) 30.21% 8.93% 7.61% 11.25% 9.52% 2.64% 13.44% 11.10% 17.72% 18.27% Runup length (weeks) 8 6 7 6 6 5 9 5 9 6 Maximum drawdown 50.48% 63.20% 13.15% 51.61% 47.37% 12.42% 43.56% 63.81% 5.57% 70.79% Drawdown length (weeks) 22 79 11 22 22 14 22 132 6 33 Valley to recovery (weeks) - - 20 - - - 38 - - - Max 52 week rolling return 63.42% 5.06% 44.12% 37.46% 19.18% 17.67% 92.13% 3.61% 57.76% 67.70% Min 52 week rolling return -37.02% -54.85% 5.03% -48.32% -44.42% -7.54% -17.02% -59.44% 16.41% -62.66% Panel B: Risk-Adjusted Performance Annualized α 21.76% -18.17% 19.57% 3.24% -4.53% 1.96% 33.24% -19.70% 26.27% 0.00% P-Value 0.04 0.06 0.01 0.70 0.62 0.72 0.00 0.01 0.00 1.00 β Bonds -0.21 -0.27 0.03 -0.20 -0.21 0.01 0.16 -0.19 0.18 0.00 P-Value 0.45 0.38 0.87 0.41 0.45 0.97 0.54 0.46 0.26 0.94 β Market 0.05 -0.05 0.05 0.06 -0.05 0.06 -0.02 0.02 -0.02 0.00 P-Value 0.25 0.28 0.09 0.14 0.23 0.03 0.69 0.66 0.51 0.88 β Commodities 0.65 0.50 0.07 0.65 0.50 0.07 0.53 0.49 0.02 1.00 P-Value 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.42 0.00 Adjusted R² 0.61 0.44 0.04 0.68 0.50 0.07 0.55 0.52 0.00 1.00 Mom12-1 -TS1 Mom12-1 TS1 ARCH Table III: Period 2007 - 2010 (post-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  34. 34. xi Benchmark TS1 -Mom12-1 L S L-S L S L-S L S L-S L-S Panal A: Summary Statistics Annualized arithmetic mean 18.55% -14.80% 16.67% 3.68% 2.45% -0.01% 28.91% -21.45% 25.66% 2.27% 22.75% Annualized geometric mean 16.01% -16.70% 16.04% 1.60% 0.81% -0.40% 27.28% -22.81% 25.30% -1.35% 22.15% Annualized volatility 22.59% 19.40% 11.30% 20.33% 18.01% 8.79% 18.07% 16.43% 8.50% 26.64% 11.04% Annualized downside volatility 16.18% 14.33% 7.66% 15.04% 13.58% 6.09% 13.51% 12.17% 5.89% 20.37% 7.57% Reward/risk ratio 0.71 -0.86 1.42 0.08 0.04 -0.05 1.51 -1.39 2.98 -0.05 2.01 Sortino ratio (0%) 1.25 -0.96 2.58 0.25 0.18 0.00 2.48 -1.56 6.26 0.11 3.81 Omega (0%) 1.36 0.76 1.75 1.07 1.05 1.00 1.81 0.61 2.97 1.03 2.15 Skewness 0.27 -0.43 0.59 -0.36 -0.81 0.44 -0.52 -0.47 0.27 -0.75 0.48 Kurtosis 8.88 4.51 7.24 7.12 7.19 7.88 5.38 5.49 4.71 5.10 6.82 Annualized 95% VaR (Cornish-Fisher) -33.54% -31.16% -17.08% -29.32% -23.88% -14.62% -22.35% -27.11% -10.73% -37.07% -15.57% Annualized 99% VaR (Cornish-Fisher) -83.23% -51.02% -37.57% -63.79% -53.95% -31.86% -44.69% -47.97% -20.44% -67.55% -34.29% % of positive weeks 56.84% 45.51% 58.97% 55.29% 52.40% 49.36% 64.32% 44.23% 65.77% 53.42% 61.54% Max runup (consecutive) 30.44% 13.21% 10.78% 11.49% 15.04% 4.60% 25.69% 8.56% 15.79% 21.64% 17.05% Runup length (weeks) 9 7 10 7 11 7 13 8 15 8 11 Maximum drawdown 46.04% 81.37% 10.46% 50.90% 54.71% 17.62% 43.56% 89.21% 5.57% 70.79% 9.00% Drawdown length (weeks) 22 426 58 40 40 66 22 452 6 33 6 Valley to recovery (weeks) 22 - 12 - - 146 38 - - - 12 Max 52 week rolling return 70.69% 26.78% 51.77% 36.34% 34.64% 16.74% 92.13% 3.97% 57.76% 67.70% 65.75% Min 52 week rolling return -29.22% -61.52% -3.90% -48.36% -50.59% -11.84% -17.02% -59.44% 9.49% -62.66% 6.63% Panel B: Risk-Adjusted Performance Annualized α 18.04% -14.56% 14.82% -1.69% -1.60% -2.34% 30.11% -20.48% 24.70% 0.00% P-Value 0.00 0.01 0.00 0.67 0.75 0.39 0.00 0.00 0.00 0.98 β Bonds -0.12 -0.44 0.17 -0.14 -0.17 0.02 0.07 -0.28 0.20 0.00 P-Value 0.52 0.02 0.16 0.28 0.29 0.83 0.63 0.06 0.02 0.67 β Market 0.00 0.03 -0.02 0.03 0.02 0.00 -0.01 0.02 -0.01 0.00 P-Value 0.98 0.40 0.50 0.33 0.60 0.80 0.83 0.40 0.57 0.40 β Commodities 0.58 0.33 0.13 0.59 0.35 0.12 0.44 0.34 0.06 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Adjusted R² 0.46 0.21 0.08 0.64 0.29 0.12 0.43 0.32 0.04 1.00 GARCH Table I: Period 2001-2010 (whole sample) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only. Mom12-1 -TS1 Mom12-1 TS1
  35. 35. xii Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 18.72% -11.31% 15.01% 9.50% 7.55% 0.98% 30.13% -19.43% 24.78% 7.12% Annualized geometric mean 16.81% -12.72% 14.40% 8.26% 6.62% 0.67% 28.96% -20.30% 24.45% 4.29% Annualized volatility 19.58% 16.81% 11.08% 15.78% 13.67% 7.83% 15.32% 13.16% 8.15% 23.67% Annualized downside volatility 14.14% 12.16% 7.99% 11.60% 9.64% 5.65% 11.07% 9.56% 5.88% 17.70% Reward/risk ratio 0.86 -0.76 1.30 0.52 0.48 0.09 1.89 -1.54 3.00 0.18 Sortino ratio (0%) 1.47 -0.87 2.16 0.87 0.83 0.18 3.36 -1.75 5.90 0.41 Omega (0%) 1.40 0.79 1.64 1.23 1.22 1.04 2.02 0.59 2.90 1.11 Skewness -0.19 -0.19 -0.15 -0.26 0.04 -0.14 -0.22 -0.13 -0.14 -0.56 Kurtosis 3.36 3.19 3.89 2.98 3.15 3.47 3.30 2.76 3.58 4.04 Annualized 95% VaR (Cornish-Fisher) -28.47% -28.29% -15.50% -23.55% -21.55% -12.39% -20.03% -23.93% -9.57% -33.94% Annualized 99% VaR (Cornish-Fisher) -43.26% -40.24% -25.40% -33.85% -31.43% -18.57% -31.31% -31.92% -16.21% -55.07% % of positive weeks 54.63% 45.69% 61.02% 57.83% 53.04% 50.16% 63.58% 44.73% 67.09% 53.67% Max runup (consecutive) 30.44% 13.21% 10.78% 11.49% 15.04% 4.02% 25.69% 8.56% 15.79% 21.64% Runup length (weeks) 9 7 10 7 11 7 13 8 15 8 Maximum drawdown 17.14% 54.11% 10.46% 22.31% 13.91% 13.44% 13.46% 71.74% 5.00% 36.48% Drawdown length (weeks) 5 298 58 21 75 25 12 297 2 50 Valley to recovery (weeks) 35 - 12 - 31 - 13 - 23 55 Max 52 week rolling return 59.80% 26.78% 34.92% 22.79% 32.49% 12.72% 63.99% 3.97% 46.22% 59.81% Min 52 week rolling return -13.09% -41.81% -3.90% -16.55% -10.59% -11.06% 12.41% -38.37% 9.49% -34.89% Panel B: Risk-Adjusted Performance Annualized α 14.75% -12.43% 11.62% 8.30% 5.58% -1.23% 29.54% -19.81% 23.90% 0.00% P-Value 0.02 0.04 0.01 0.22 0.33 0.70 0.00 0.00 0.00 0.22 β Bonds -0.04 -0.59 0.28 -0.29 -0.06 -0.11 -0.01 -0.36 0.18 0.00 P-Value 0.86 0.01 0.06 0.20 0.75 0.32 0.97 0.04 0.11 0.21 β Market 0.09 0.09 0.00 -0.05 0.03 -0.04 0.02 0.04 -0.01 0.00 P-Value 0.08 0.11 0.96 0.32 0.49 0.12 0.73 0.31 0.63 1.00 β Commodities 0.54 0.19 0.18 -0.05 -0.05 0.00 0.37 0.21 0.08 1.00 P-Value 0.00 0.00 0.00 0.16 0.11 0.98 0.00 0.00 0.00 0.00 Adjusted R² 0.43 0.09 0.14 0.00 0.00 0.00 0.32 0.16 0.05 1.00 Mom12-1 -TS1 Mom12-1 TS1 GARCH Table II: Period 2001 - end 2006 (pre-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  36. 36. xiii Benchmark L S L-S L S L-S L S L-S Panal A: Summary Statistics Annualized arithmetic mean 18.20% -21.86% 20.03% -3.85% -4.48% 0.31% 26.46% -25.51% 25.98% -7.51% Annualized geometric mean 14.41% -24.73% 19.36% -7.17% -7.27% -0.17% 23.88% -27.87% 25.57% -12.70% Annualized volatility 27.77% 23.82% 11.76% 25.68% 23.43% 9.93% 22.67% 21.61% 9.24% 31.84% Annualized downside volatility 19.71% 17.83% 6.95% 19.03% 18.15% 6.44% 17.43% 16.14% 5.86% 24.81% Reward/risk ratio 0.52 -1.04 1.65 -0.28 -0.31 -0.02 1.05 -1.29 2.77 -0.40 Sortino ratio (0%) 0.99 -1.13 3.72 -0.20 -0.24 0.05 1.68 -1.42 6.57 -0.30 Omega (0%) 1.29 0.71 2.02 0.94 0.93 1.01 1.55 0.64 3.00 0.91 Skewness 0.58 -0.50 1.83 -0.31 -0.93 1.05 -0.64 -0.52 0.95 -0.81 Kurtosis 10.16 4.34 12.25 6.71 5.99 11.71 5.17 4.76 6.42 4.99 Annualized 95% VaR (Cornish-Fisher) -43.44% -38.38% -20.09% -38.76% -31.98% -17.31% -28.80% -35.34% -13.29% -45.29% Annualized 99% VaR (Cornish-Fisher) -114.39% -61.62% -57.76% -79.71% -63.70% -47.01% -55.40% -58.66% -28.44% -80.64% % of positive weeks 61.29% 45.16% 54.84% 53.55% 52.90% 46.45% 65.81% 43.23% 63.23% 52.90% Max runup (consecutive) 10.77% 12.89% 8.50% 8.04% 15.04% 1.54% 13.44% 11.10% 17.72% 18.27% Runup length (weeks) 7 6 6 6 11 4 9 5 9 6 Maximum drawdown 46.04% 62.34% 8.59% 50.90% 54.71% 14.84% 43.56% 63.81% 5.57% 70.79% Drawdown length (weeks) 22 53 20 40 40 24 22 132 6 33 Valley to recovery (weeks) 22 - - - - - 38 - - - Max 52 week rolling return 70.69% 11.37% 51.77% 36.34% 34.64% 16.74% 92.13% 3.61% 57.76% 67.70% Min 52 week rolling return -29.22% -61.52% 8.08% -48.36% -50.59% -6.55% -17.02% -59.44% 16.41% -62.66% Panel B: Risk-Adjusted Performance Annualized α 25.54% -16.64% 20.30% 1.85% -0.92% -0.58% 33.24% -19.70% 26.27% 0.00% P-Value 0.05 0.09 0.01 0.83 0.93 0.92 0.00 0.01 0.00 1.00 β Bonds -0.26 -0.27 0.01 -0.33 -0.14 -0.09 0.16 -0.19 0.18 0.00 P-Value 0.44 0.40 0.96 0.19 0.63 0.59 0.54 0.46 0.26 0.94 β Market -0.06 0.00 -0.03 0.00 0.01 0.00 -0.02 0.02 -0.02 0.00 P-Value 0.27 0.99 0.37 0.94 0.87 0.86 0.69 0.66 0.51 0.88 β Commodities 0.62 0.48 0.07 0.66 0.51 0.07 0.53 0.49 0.02 1.00 P-Value 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.42 0.00 Adjusted R² 0.50 0.40 0.02 0.68 0.48 0.04 0.55 0.52 0.00 1.00 Mom12-1 -TS1 Mom12-1 TS1 GARCH Table III: Period 2007 - 2010 (post-crisis) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only.
  37. 37. xiv Benchmark TS1 -Mom12-1 L S L-S L S L-S L S L-S L-S Panal A: Summary Statistics Annualized arithmetic mean 17.61% -16.87% 17.24% 3.49% 3.18% -0.32% 28.91% -21.45% 25.66% 2.27% 22.41% Annualized geometric mean 14.67% -18.69% 16.56% 1.03% 1.64% -0.75% 27.28% -22.81% 25.30% -1.35% 21.74% Annualized volatility 24.20% 18.99% 11.76% 22.04% 17.46% 9.32% 18.07% 16.43% 8.50% 26.64% 11.59% Annualized downside volatility 17.95% 13.73% 8.21% 16.61% 13.10% 6.49% 13.51% 12.17% 5.89% 20.37% 8.06% Reward/risk ratio 0.61 -0.98 1.41 0.05 0.09 -0.08 1.51 -1.39 2.98 -0.05 1.88 Sortino ratio (0%) 1.05 -1.12 2.47 0.21 0.25 -0.05 2.48 -1.56 6.26 0.11 3.45 Omega (0%) 1.31 0.72 1.72 1.06 1.07 0.99 1.81 0.61 2.97 1.03 2.03 Skewness -0.33 -0.23 0.15 -0.59 -0.70 0.29 -0.52 -0.47 0.27 -0.75 0.32 Kurtosis 6.99 4.12 5.99 6.55 6.50 7.42 5.38 5.49 4.71 5.10 6.09 Annualized 95% VaR (Cornish-Fisher) -33.32% -32.00% -16.71% -30.71% -23.79% -15.26% -22.35% -27.11% -10.73% -37.07% -16.24% Annualized 99% VaR (Cornish-Fisher) -73.62% -49.96% -33.82% -64.36% -50.07% -32.31% -44.69% -47.97% -20.44% -67.55% -33.59% % of positive weeks 56.84% 45.30% 58.33% 53.85% 51.44% 51.50% 64.32% 44.23% 65.77% 53.42% 61.75% Max runup (consecutive) 28.83% 18.30% 9.65% 14.50% 6.68% 7.34% 25.69% 8.56% 15.79% 21.64% 22.57% Runup length (weeks) 9 8 9 6 7 8 13 8 15 8 16 Maximum drawdown 49.87% 84.83% 13.82% 51.98% 49.08% 17.90% 43.56% 89.21% 5.57% 70.79% 10.28% Drawdown length (weeks) 22 443 12 22 22 65 22 452 6 33 22 Valley to recovery (weeks) 26 - 32 - - 89 38 - - - 19 Max 52 week rolling return 97.56% 36.65% 66.05% 38.25% 33.31% 20.13% 92.13% 3.97% 57.76% 67.70% 73.04% Min 52 week rolling return -40.67% -55.20% -2.32% -50.25% -44.43% -14.34% -17.02% -59.44% 9.49% -62.66% 1.02% Panel B: Risk-Adjusted Performance Annualized α 16.34% -16.64% 15.39% -2.10% -0.80% -2.94% 30.11% -20.48% 24.70% 0.00% P-Value 0.01 0.00 0.00 0.62 0.87 0.29 0.00 0.00 0.00 0.98 β Bonds 0.05 -0.32 0.19 0.04 -0.21 0.13 0.07 -0.28 0.20 0.00 P-Value 0.79 0.08 0.12 0.77 0.18 0.17 0.63 0.06 0.02 0.67 β Market -0.01 0.02 -0.02 0.03 0.00 0.02 -0.01 0.02 -0.01 0.00 P-Value 0.72 0.53 0.46 0.23 0.93 0.31 0.83 0.40 0.57 0.40 β Commodities 0.65 0.32 0.16 0.65 0.34 0.15 0.44 0.34 0.06 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Adjusted R² 0.51 0.21 0.13 0.65 0.29 0.19 0.43 0.32 0.04 1.00 Model Average Table I: Period 2001-2010 (whole sample) Panel A reports summary statistics for the weekly returns of three Double-Sort, three Momentum and three Term-Structure models. A lacking value for "Valley to recovery" indicates that the portfolio has not recovered yet. Panel B reports coefficient estimates of the risk adjusted performance. α measures the abnormal performance, β Bonds, β Market and β Commodities measure the senstivities of returns to the Vanguard Total Bond ETF, the S&P500 Total Return composite index and the S&P GSCI respectively. The last row reports the adjusted goodness of fit statistic. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H refers to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Benchmark refers to the S&P GSCI long only. Mom12-1 -TS1 Mom12-1 TS1
  38. 38. xv Mom12-1-TS1 L Mom12-1 -TS1 S Mom12-1-TS1L-S Mom12-1 L Mom12-1S Mom12-1 L-S TS1 L TS1 S TS1 L-S Bonds S&P500TR S&PGSCI Mom12-1-TS1 L 1.00 0.32 0.64 0.91 0.36 0.56 0.82 0.46 0.43 -0.05 0.00 0.58 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.24 0.95 0.00 Mom12-1 -TS1 S 0.32 1.00 -0.52 0.40 0.88 -0.41 0.41 0.81 -0.35 -0.10 0.09 0.54 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.06 0.00 Mom12-1 -TS1 L-S 0.64 -0.52 1.00 0.49 -0.39 0.84 0.41 -0.25 0.67 0.03 -0.07 0.08 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.50 0.12 0.07 Mom12-1 L 0.91 0.40 0.49 1.00 0.45 0.58 0.80 0.60 0.27 -0.08 0.01 0.67 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.79 0.00 Mom12-1 S 0.36 0.88 -0.39 0.45 1.00 -0.47 0.55 0.78 -0.17 -0.07 0.05 0.63 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.14 0.27 0.00 Mom12-1 L-S 0.56 -0.41 0.84 0.58 -0.47 1.00 0.29 -0.13 0.43 -0.02 -0.03 0.09 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.67 0.46 0.04 TS1 L 0.82 0.41 0.41 0.80 0.55 0.29 1.00 0.52 0.56 -0.03 0.00 0.65 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.56 0.94 0.00 TS1 S 0.46 0.81 -0.25 0.60 0.78 -0.13 0.52 1.00 -0.42 -0.11 0.04 0.56 P-Value 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.02 0.43 0.00 TS1 L-S 0.43 -0.35 0.67 0.27 -0.17 0.43 0.56 -0.42 1.00 0.08 -0.04 0.15 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.40 0.00 Bonds -0.05 -0.10 0.03 -0.08 -0.07 -0.02 -0.03 -0.11 0.08 1.00 -0.02 -0.06 P-Value 0.24 0.03 0.50 0.07 0.14 0.67 0.56 0.02 0.10 0.00 0.68 0.16 S&P 500 TR 0.00 0.09 -0.07 0.01 0.05 -0.03 0.00 0.04 -0.04 -0.02 1.00 0.01 P-Value 0.95 0.06 0.12 0.79 0.27 0.46 0.94 0.43 0.40 0.68 0.00 0.88 S&P GSCI 0.58 0.54 0.08 0.67 0.63 0.09 0.65 0.56 0.15 -0.06 0.01 1.00 P-Value 0.00 0.00 0.07 0.00 0.00 0.04 0.00 0.00 0.00 0.16 0.88 0.00 Average Correlation 0.46 0.19 0.18 0.46 0.24 0.16 0.45 0.26 0.15 -0.04 0.00 0.36 The table reports Pearson correlations and their P-Values. TS1 uses the fron-end of the term structure to measure roll-returns, MomR-H referes to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Bonds, S&P 500 TR and S&P GSCI represent, respectively, the Vanguard Total Market Bond Index, the Standard & Poors 500 Total Return Index and the Goldman Sachs Commodity Index. Bold denotes significant values at the 5% level or better. The last row represents the arithmetic average of the correlations, excluding the correlation of each asset with itself. Momentum Whole Sample
  39. 39. xvi Mom12-1 -TS1 L Mom12-1 -TS1 S Mom12-1-TS1L-S Mom12-1 L Mom12-1S Mom12-1 L-S TS1 L TS1 S TS1 L-S Bonds S&P500TR S&PGSCI Mom12-1-TS1 L 1.00 0.40 0.66 0.90 0.46 0.61 0.82 0.50 0.39 -0.05 0.03 0.75 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.24 0.51 0.00 Mom12-1 -TS1 S 0.40 1.00 -0.42 0.47 0.83 -0.22 0.41 0.84 -0.37 -0.10 0.00 0.39 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.99 0.00 Mom12-1 -TS1 L-S 0.66 -0.42 1.00 0.51 -0.23 0.78 0.47 -0.19 0.69 0.03 0.03 0.42 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.52 0.52 0.00 Mom12-1 L 0.90 0.47 0.51 1.00 0.54 0.65 0.78 0.65 0.21 -0.09 0.02 0.84 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.68 0.00 Mom12-1 S 0.46 0.83 -0.23 0.54 1.00 -0.29 0.63 0.80 -0.10 -0.08 -0.04 0.46 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.08 0.35 0.00 Mom12-1 L-S 0.61 -0.22 0.78 0.65 -0.29 1.00 0.32 0.02 0.33 -0.03 0.06 0.54 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.71 0.00 0.51 0.19 0.00 TS1 L 0.82 0.41 0.47 0.78 0.63 0.32 1.00 0.52 0.56 -0.03 0.00 0.65 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.56 0.94 0.00 TS1 S 0.50 0.84 -0.19 0.65 0.80 0.02 0.52 1.00 -0.42 -0.11 0.04 0.56 P-Value 0.00 0.00 0.00 0.00 0.00 0.71 0.00 0.00 0.00 0.02 0.43 0.00 TS1 L-S 0.39 -0.37 0.69 0.21 -0.10 0.33 0.56 -0.42 1.00 0.08 -0.04 0.15 P-Value 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.10 0.40 0.00 Bonds -0.05 -0.10 0.03 -0.09 -0.08 -0.03 -0.03 -0.11 0.08 1.00 -0.02 -0.06 P-Value 0.24 0.03 0.52 0.05 0.08 0.51 0.56 0.02 0.10 0.00 0.68 0.16 S&P 500 TR 0.03 0.00 0.03 0.02 -0.04 0.06 0.00 0.04 -0.04 -0.02 1.00 0.01 P-Value 0.51 0.99 0.52 0.68 0.35 0.19 0.94 0.43 0.40 0.68 0.00 0.88 S&P GSCI 0.75 0.39 0.42 0.84 0.46 0.54 0.65 0.56 0.15 -0.06 0.01 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.16 0.88 0.00 Average Correlation 0.50 0.20 0.25 0.50 0.27 0.25 0.47 0.29 0.13 -0.04 0.01 0.43 The table reports Pearson correlations and their P-Values. TS1 uses the front-end of the term structure to measure roll-returns, MomR-H referes to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Bonds, S&P 500 TR and S&P GSCI represent, respectively, the Vanguard Total Market Bond Index, the Standard & Poors 500 Total Return Index and the Goldman Sachs Commodity Index. Bold denotes significant values at the 5% level or better. The last row represents the arithmetic average of the correlations, excluding the correlation ofeach asset with itself. AR Model Whole Sample
  40. 40. xvii Mom12-1 -TS1 L Mom12-1 -TS1 S Mom12-1-TS1L-S Mom12-1 L Mom12-1S Mom12-1 L-S TS1 L TS1 S TS1 L-S Bonds S&P500TR S&PGSCI Mom12-1-TS1 L 1.00 0.40 0.65 0.90 0.47 0.59 0.81 0.49 0.38 -0.04 0.00 0.68 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.45 0.94 0.00 Mom12-1 -TS1 S 0.40 1.00 -0.43 0.50 0.86 -0.24 0.43 0.82 -0.34 -0.10 0.02 0.46 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.64 0.00 Mom12-1 -TS1 L-S 0.65 -0.43 1.00 0.47 -0.25 0.78 0.44 -0.19 0.66 0.05 -0.02 0.29 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.30 0.64 0.00 Mom12-1 L 0.90 0.50 0.47 1.00 0.56 0.62 0.79 0.67 0.19 -0.07 0.02 0.79 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.13 0.59 0.00 Mom12-1 S 0.47 0.86 -0.25 0.56 1.00 -0.30 0.65 0.77 -0.05 -0.09 -0.03 0.55 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.28 0.06 0.53 0.00 Mom12-1 L-S 0.59 -0.24 0.78 0.62 -0.30 1.00 0.29 0.04 0.26 0.00 0.06 0.39 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.94 0.23 0.00 TS1 L 0.81 0.43 0.44 0.79 0.65 0.29 1.00 0.52 0.56 -0.03 0.00 0.65 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.56 0.94 0.00 TS1 S 0.49 0.82 -0.19 0.67 0.77 0.04 0.52 1.00 -0.42 -0.11 0.04 0.56 P-Value 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.02 0.43 0.00 TS1 L-S 0.38 -0.34 0.66 0.19 -0.05 0.26 0.56 -0.42 1.00 0.08 -0.04 0.15 P-Value 0.00 0.00 0.00 0.00 0.28 0.00 0.00 0.00 0.00 0.10 0.40 0.00 Bonds -0.04 -0.10 0.05 -0.07 -0.09 0.00 -0.03 -0.11 0.08 1.00 -0.02 -0.06 P-Value 0.45 0.03 0.30 0.13 0.06 0.94 0.56 0.02 0.10 0.00 0.68 0.16 S&P 500 TR 0.00 0.02 -0.02 0.02 -0.03 0.06 0.00 0.04 -0.04 -0.02 1.00 0.01 P-Value 0.94 0.64 0.64 0.59 0.53 0.23 0.94 0.43 0.40 0.68 0.00 0.88 S&P GSCI 0.68 0.46 0.29 0.79 0.55 0.39 0.65 0.56 0.15 -0.06 0.01 1.00 P-Value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.16 0.88 0.00 Average Correlation 0.48 0.22 0.22 0.49 0.28 0.23 0.46 0.29 0.13 -0.03 0.00 0.41 The table reports Pearson correlations and their P-Values. TS1 uses the fron-end of the term structure to measure roll-returns, MomR-H referes to a momentum strategy with R-month ranking period and H-month holding period. L, S and L-S stand for long, short and long-short, respectively. Bonds, S&P 500 TR and S&P GSCI represent, respectively, the Vanguard Total Market Bond Index, the Standard & Poors 500 Total Return Index and the Goldman Sachs Commodity Index. Bold denotes significant values at the 5% level or better. The last row represents the arithmetic average of the correlations, excluding the correlation ofeach asset with itself. ARMA Whole Sample

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