Identifying Commodity Price Bubbles and the Potential Risk andRewards of Holding Commodities for Retail Investor Portfolios

Master of Finance

Antwerp Management School

Abbey Cavalero, David De Wolf, Justin Shaffer

It is now easier than ever for retail investors to gain exposure to commodity markets. This exposes investorsto the potentially large and violent price swings that commodity markets are frequently associated with. Thispaper attempts to construct investment strategies where investors can either take advantage of, or avoid thisvolatility. To construct these portfolios, we rely on two bubble detection methods to date stamp explosivebubble periods. This paper utilizes an existing bubble stamping method created by (Phillips, Wu, Yu), andthe most recent contribution to the field of bubble detection, the GSADF test proposed by (Phillips Shi,Yu). The paper successfully identifies multiple bubble periods, however fails to create an effective strategyby which to trade around this investment phenomenon.

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1 IntroductionThroughout the past 20 years, the prices of many commodities have experienced large and at times violentprice swings. Notably the rise in prices in the years leading up to the 2008 financial crisis, and more recentlythe collapse of many commodity prices starting in mid 2014. Commodity prices can have profound impactson company financial statements, and large price swings almost instantly affect the daily lives of average citi-zens. Large price moves quickly reverberate throughout the economy and can have lasting impacts. Increasesin the price of oil translate to higher prices of gasoline, diesel, jet fuel, and energy. Higher prices of corn andsoybeans are quickly reflected in the prices of numerous food products, such as chicken and cereal.

Additionally, because of the now prolific increase in commodity funds, investors can easily gain exposure tocommodity futures markets, without the potential risk of taking delivery of the underlying contract. Followingthe principals of modern portfolio theory first introduced by Harry Markowitz in 1952, investors continuallysearch for new ways to diversify their portfolios. These funds provide an easy way for investors to diversifyinto markets unavailable only a short time ago. The ease with which investors can gain exposure to these newmarkets, means that a greater percentage of wealth is potentially exposed to commodity markets. Thusly,retail investors are increasingly exposed to commodity market price volatility.

Because commodity prices affect nearly everyone, large or volatile moves in their prices do not go unnoticed.These events immediately affect end consumers and are very quickly picked up by the media. Over the pastseveral years, the word “bubble” has been frequently used when describing the large movements in variousstock and commodity prices. A universal definition of a bubble has yet to be established, however, all most allof them carry negative connotations. Why are commodity price bubbles important to detect? Investigatingthe existence of bubbles in economic literature has been of essential importance as bursting bubbles have theability to invoke financial crises. It was shown that bubbles were indeed the cause of the global financial crisisof 2007-2008. In order to fix a bubble, central banks must implement heavy monetary and fiscal policies.Due to the fact that bubbles are not perceived well by markets and can destabilize financial systems, theirstudy is of great importance. Commodities growing role in financial markets coupled with their impacts oneconomic activity around the world make the study of commodity bubbles of particular significance.

Figuerola (2012) suggests that bubbles in one commodity, such as oil, can generate bubbles in other com-modities through increased extraction and transportation costs. Commodity bubbles may affect capitalexpenditure projects for a multitude of firms. Prices that are fundamentally unjustified in the short termmay lead to over or under investment in a project. Because there is a lag effect in the production of manycommodities to prices, poor decisions can be felt many years after an investment decision is made. Anotherreason is policy and regulatory decisions. If government officials believe commodity bubbles develop toofrequently and of severe magnitude, it may cause them to pass regulations on industry behavior or proposelimits to market participation. If bubbles do appear frequently, increased regulation might be justified. How-ever, if bubbles do not appear with regularity and of a severity that is significantly detrimental to the generalpopulation, this may be a sign that markets are effective in policing themselves.

The detection of bubbles can be a challenging matter because a unified definition of a price bubble does notexist. Due to the price volatility of the past decade, research in the field of bubbles has become quite robust.Many methods have been proposed for detecting the presence of speculative commodity price bubbles. Inthis paper, we will utilize the Phillips, Wu, Yu (2011) Supremum Augmented Dickey Fuller (SADF) test andthe Phillips, Shi, and Yu (2015) Generalized Supremum Augmented Dickey Fuller (GSADF) test to definebubbles and pinpoint their existence. Our study examines the futures price behavior of several well knowncommodities: crude oil, gold, corn, and soybeans. The goal of this paper is to implement existing bubbledetection methods in order to definitively test whether or not bubbles in these global commodities havepresented themselves as popular culture might have us believe. Our contribution to this field of study will bein the form of a longer examination window, and the use of an array of commodities both hard and soft. Weseek to explain with what frequency, duration, and magnitude bubbles present themselves in our time series.

We will then construct simple retail investor portfolios to test wether or not bubble detection can be imple-mented in a profitable way. For this we will use the GSADF test statistic as it is the most recent contributionto the field of bubble detection, and believed to be the most sensitive and accurate means of detectingbubbles. Once we have detected and reported the commodity price bubbles, we construct various simple

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portfolios to test whether incorporating bubble detection can increase long run portfolio returns. Utilizingthe signals generated by the GSADF, we will adjust the weights of these simple portfolios to capture anypossible benefits of bubble detection. Our hope is that this study may lead to further examination of bubbleperiods, and lead to more intelligent and strategic investment decisions.

2 Literature ReviewThe examination of asset price bubbles has been studied extensively in the past as their implications proveto significantly impact economic growth and financial institutions. Various methodologies and studies havebeen conducted on the topic, which we intend to analyze and replicate in this paper. Popular methodsused in past studies to time-stamp bubbles include the Supremum Augmented Dickey Fuller Test (SADF)test (Phillips, Wu, Yu, 2011), the Log-Periodic Power Law (LPPL) model (Johansen et al., 2000), and theMarkov regime switching model (Zhang and Wang, 2015). Although we do not perform all of these tests inthis paper, we will briefly go through the empirical findings of each test in order to get a idea of how researchon asset price bubbles has evolved in the past. Each method has unique elements and functions, however, allempirical research on the topic must begin with the definition of a “speculative bubble.”

Speculative bubbles are defined by the Commodity Futures Trading Commission (CFTC) as “a rapid run-upin prices caused by excessive buying that is unrelated to any of the basic, underlying factors affecting thesupply or demand for a commodity or other asset.” Sornette and Johansen (2003) define a positive (negative)bubble as an accelerating, ascending (descending) log-price ending at some future critical time. Unfortu-nately, no definitive consensus on the definition of a bubble has been determined in economic literature. OnSeptember 30th, 2002, Alan Greenspan stated “...We, at the Federal Reserve... recognized that, despite oursuspicions, it was very difficult to definitively identify a bubble until after the fact, that is, when its burstingconfirmed its existence.” Despite this statement, many models started to attract attention in academics indetecting bubbles prior to bursting.

The SADF test has been widely used in practice for detecting the existence of price bubbles. The empiricalmodel involves recursive implementation of a right-side unit root test for date-stamping explosive behavior.Phillips, Wu, and Yu (2011) were the first to successfully implement the model by performing the test onthe Nasdaq index for the period 1973 to 2005. Their aim was to discover whether price bubbles existed priorto famous historical remarks of researchers on market perceptions. Specifically, Alan Greenspan’s speechin 1996, which asked the controversial question “How do we know when irrational exuberance has undulyescalated asset values?” In order to answer this question, Phillips and Yu matched the possible origination ofa speculative bubble with the date of Greenspan’s remark. They found a period of explosive price movementsin 1995, giving true evidence and support to Greenspan’s claim.

Phillips and Yu extend their research in 2015 by developing a limit theory of the bubble detection algo-rithms stemming from 2011. They critique the original PWY model by demonstrating its lack in ability todetect the existence of multiple bubbles in a time series. As a solution, they introduce the Phillips, Shi, Yu(2015) PSY model, which provides better estimation of all bubbles in a time series. The corresponding testfor the model is the General Supremum Augmented Dickey Fuller (GSADF), which will be discussed in depthin the following sections. Phillips et al. (2013) verified through a Monte Carlo procedure that the PSY doesindeed outperform the PWY in the case of multiple bubbles. However, both the PWY and PSY show thatmore power comes from using rolling tests rather than standard ones. Therefore, we will perform both testsin this paper in order to more accurately detect bubbles. The PSY model was utilized by Etienne, Irwin,and Garcia (2014) to investigate bubbles from 2004-2013 in corn, soybean, and wheat futures markets. Theydiscovered that bubbles manifested only 2% of the time and are generally transient and small in magnitude.The longest bubbles detected only lasted 17 and 18 business days. They then use a multinomial logit model tocompare these bubble estimations in order to determine possible contributing drivers of these price explosiveperiods. From the model, they conclude that “commodity index traders do not significantly affect the proba-bility of positive bubbles occurring in grain futures markets.” This implies that overall speculation has littleimpact on explosive price fluctuations and grain futures markets are mainly driven by fundamental factors.Bettendorf and Shen (2013) perform both the SADF and GSADF to investigate the explosive behavior ofthe sterling-dollar exchange rate and discover that fluctuations stem from fundamentals rather than bubbles.

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Other means of identifying price bubbles include regime switching models. Zhang and Wang (2015) measurethe price of WTI crude oil and price bubbles with a multivariate regression, and then implement the Markovregime switching model in order to analyze the evolution of these crude oil bubbles. This involves separatingcrude oil price trajectory into two distinct regimes: the stable state and upheaval state. They find that crudeoil prices have a more persistent stable state vis-a-vis upheaval state, insinuating that crude oil prices arerelatively stable and change rather slowly. This suggests that the market-trading prices during January 2003to May 2012 were more volatile than fundamentals would indicate, affirming the existence of WTI crude oilbubbles.

Alternatively, the LPPL model has also proved to be sufficient for measuring dynamic asset price bub-bles. The model was proposed by Johansen et al. (2000) and Sornette (2003). They studied the historicalcrashes of October 1929 and October 1987 and found that the methodology successfully identified these crisesby evaluating prior bubbles. Therefore, the main motivation to use this procedure is the ability to identifybubbles ex-ante. The model examines super-exponential growth (growth in positive bubbles and decline innegative bubbles), which ultimately leads to a bursting bubble and log-periodic oscillations indicating systemfailure and significant investor sentiment swings.

In order to gain a better grasp on the theory of the LPPL model, it is necessary to explain certain componentsmore in depth. Johansen and Sornette (2001) came to the conclusion that the majority of speculative bub-bles have two similar characteristics: one being super-exponential growth in the price during a bubble periodand ending when the bubble seises, and two being oscillations with accelerating frequency. The idea thatencompasses the whole model is “log-periodicity”, which refers to a sequence of oscillations with progressivelyshorter cycles. Oscillations in this case, represent repetitive variations of commodity prices around a centralor equilibrium price. Zhang and Yao (2016) employ the LPPL model to detect oil bubbles during 2001-2015but also analyze the potential drivers of diesel and gasoline price shocks and whether they are similar tocrude oil price fluctuations.

3 Bubble Testing

3.1 Phillips, Wu, Yu (PWY) Bubble Test

We will perform the Supremum Augmented Dickey Fuller Test (SADF) proposed by Phillips, Wu, and Yu(2011) in order to detect and date-stamp commodity price bubbles. The SADF test is an extension of theRolling ADF (RADF) test where the ADF test statistic is calculated over a fixed rolling window. rw, r0,r1, and r2 will denote the size of the window, the fixed initial window, the window’s start point, and thewindow’s end point, respectively. rw is equal to the window end minus the window start (r2� r1). Recognizethat the windows in this test are overlapping with each rolling window in increments of one observation at atime. The following equation is used to perform this test:

Pt = µ+ �Pt+⇢X

i=1

�4Pt�i + "t (1)

where Pt is the daily log prices of the commodities, µ is the intercept, ⇢ is the maximum number of lags, and"t is the error term. In order to keep our window length consistent through both tests in this paper, the valueof rw is calculated from the following equation recommended by Phillips, Shi, Yu (2013): (0.01+1.8/

pT )× T

where T is equal to 6,544, the number of trading days in our study period. Therefore, the initial windowlength used in our tests is 211 days. The SADF window procedure is illustrated in Figure 1. The startingpoint (r1 = 0) corresponds to April 3, 1990, which is the starting date of our data sample. Each estimationwindow starts from this point and accumulates one extra day each time until the end of the window is reached(r2 = 6, 544). The entire procedure will end with a total of 6,333 windows (6,544-211). Each window yieldsan ADF test statistic denoted by ADFr2 . The SADF test statistic represents the supremum value of ADFr2 .The supremum represents the least upper bound of the sample, which is equivalent to the maximum valuewhen there is a finite sample period as we have in this case.

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Therefore, the supremum value, or SADF test statistic, is the maximum value of the ADFr2 statistics ineach estimation window. The SADF test statistic is described in the following formula:

SADF (r0) = sup{ADFr2} (2)r2✏[r0, 6544]

The origination date of a bubble period is when the test statistic surpasses the critical value and the enddate of the bubble is when the test statistic falls back below the critical value after the origination date. Theformulas to compute these values are as follows:

re = infr2✏[,r0,T ]�r2 : ADFr2 > cv⇥r2

(3)

rf = infr2✏[,r1e,+h,T ]

�r2 : ADFr2 < cv⇥r2

(4)

In this relationship, re denotes the origination of the bubble while rf denotes the termination point of thebubble. In Equations 3 and 4 above, cv⇥r2 are the 100⇥% critical values using ADF test statistics based onr2 observations and a minimum defined bubble length of h, which we calculated above as 8 days. ⇥ is thedesired level of significance.

Figure 1: SADF Estimation Window Procedure

3.2 Phillips, Shi, and Yu (PSY) Bubble Test

To test for explosive price episodes in the four commodity products, we also utilize the Phillips, Shi, Yu (PSY)bubble testing procedure. The PSY model is a modified and expanded version of the PWY. The noteworthycontribution of the PSY model is that it can better detect multiple bubbles within a time series. When thereis only a single bubble in a time series, the PWY and the PSY will perform equally. If multiple bubbles existin a time series, the detection of a second bubble by the PWY method will depend on the duration of thefirst bubble. If the duration of the first bubble is longer than the duration of the second bubble, the PWYmay fail to properly detect the anomaly and could potentially produce an inaccurate time stamp. Phillips,Shi, Yu (2015) test for bubbles uses the following equation:

�Pt = ↵r1,r2 + �r1,r2Pt+⇢X

i=1

�ir1,r2�Pt + "t (5)

The estimation start point is denoted r1, the estimation end point is denoted as r2, and ⇢ is the numberof lags. r0 is the minimum window size needed to estimate Equation 4 and r1 can vary between the firstobservation and observation r2 � r0 + 1. rw will denote the varying window size, which can vary betweenthe first observation and last observation such that rw = r2 � r1 + 1. Consistent with Etienne, Irwin,Garcia (2015), for each commodity we estimate an auto regressive model with a null hypothesis: �r1,r2 = 0,indicating that no bubble exists. The alternative hypothesis is as follows: �r1,r2 > 0, indicating the presenceof a speculative bubble in the time series. The standard test statistic is calculated as: ADF r1,r2 =

�r1,r2

SE(�r1,r2)and is used to test for significance.

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PSY then utilizes the supremum of the generated ADF test statistics to find BSADFr2(r0). The GSADFtest statistic used by PSY is based off a backward expanding window. This is another contribution of thePSY to PWY method as PWY uses a forward expanding window to calculate test statistics.

PSY allow the ending point, r2, to vary between the minimum window size r0 and T, representing thelast data point included in the estimation. This results in T � r0 +1 test statistics. These are then comparedwith the critical values. Equations 6 and 7 are used in order to estimate the origination and end dates ofbubble:

r1e = infr2✏[,r0,T ]�r2 : BSADFr2(r0) > cv⇥r2

(6)

r1f = infr2✏[,r1e,+h,T ]

�r2 : BSADFr2(r0) < cv⇥r2

(7)

where cv⇥r2 are the 100⇥% critical values using BSADF test statistics based on r2 observations and aminimum defined bubble length of h. ⇥ is the desired level of significance. In order to find the BSADF teststatistics, the following procedures are used. Using Equation 5, residuals "t are found for each commoditytime series. Using the "t residuals, we then generate bootstrap residuals "⇤t where "⇤t ="t⌘t. ⌘t is an i.i.d.sequence, which is N (0, 1). Using the residuals, recursive bootstrap samples P ⇤

t are generated using Equation8 for t = 1, 2, ...T.

P ⇤t = Pt+

⇢X

i=1

�ir1,r2�P ⇤t�i + "⇤t (8)

In the above regression, �ir1,r2 is the estimated autoregressive coefficient. Once the bootstrap sample isconstructed, PSY then calculate the BSADF values for the bootstrapped sample. This is conducted 200times to create a bootstrapped distribution of the BSADF critical values. The 95% quantile from thebootstrap distribution is then used to date stamp bubbles using Equations 6 and 7.

Due to the fact that the procedure is recursive it can be used to date stamp bubbles in real time. This meansif bubbles are correctly identified, there is the potential to make investment decisions based on the findings.An illustration of the estimation window procedure can be seen in Figure 2. The SADF test is repeated eachtime with an increased starting point: r1+1. Due to lack of computer computational power, for this test oursample had to be cut by 422 days, leaving us with an r2 value equal to 6,121. The GSADF test statistics arethe supremum values of the BSADF test statistics for each estimation window. The test statistic is describedin the following equation:

GSADF (r0) = sup{BSADF r2r1 } (9)

r2✏[r0, 6121]

r1✏[0, r2 � r0]

Figure 2: GASDF Window Estimation Procedure

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3.3 Data Descriptives

In our study, we use a continuous composite data series for each of the four commodities, which were found onDatastream. The four commodities being used in this study are NYMEX crude oil, COMEX gold, e-CBOTcorn, and e-CBOT soybeans.1 These continuous composite data series reflect daily futures prices for each ofthe four commodities spanning a period of 26 years, from April 3, 1990 to April 29, 2016. Note that all pricesare in U.S. dollars. The study examines 6,544 tradable days, and thusly 6,544 daily futures prices for eachcommodity. Figure 3 displays the daily log prices of crude oil, gold, corn, and soybeans from April 3, 1990 toApril 29, 2016. Table 1 presents summary statistics of the daily log price changes for the four commodities.

Table 1: Summary Statistics of Daily Log Price Changes

Table 1 displays summary statistics of daily log price changes in U.S. dollars for Crude Oil, Gold, Corn, andSoybeans futures prices. The prices are from the period April 3, 1990 to April 29, 2016.

Mean Median Standard Dev. Skewness Ex.Kurtosis RangeOil 0.000 0.000 0.024 -0.759 15.302 (-0.400, +0.164)

Gold 0.000 0.000 0.010 -0.219 7.743 (-0.098, +0.089)Corn 0.000 0.000 0.017 -0.567 12.703 (-0.024, +0.009)

Soybeans 0.000 0.000 0.015 -0.677 6.085 (-0.141, +0.008)

Crude oil was the commodity with the largest daily log price change range and had largest single downwardlog price move of -0.400. Corn prices showed the smallest range of log price differences. With regards toexcess kurtosis, soybeans produced the most normal distribution having an excess kurtosis of 6.085. Crudeoil had the least normal distribution with regards to excess kurtosis with a value of 15.302, which indicatesfat tails. This implies that extreme positive and negative returns happen more frequently.

Additionally, crude oil displayed the highest value of skewness of the four examined commodities. All fourof the commodities had negatively skewed log price changes. This demonstrates that losses hurt worse thangains feel good. Gold had the most normal distribution with regards to skewness. Crude oil had the high-est standard deviation of the commodities, which is in line with the commodities’ large log price differencedistribution and high excess kurtosis. Gold exhibited the most daily price swing stability throughout theperiod, producing the lowest standard deviation.

All four commodities had daily price difference means and medians of zero as commodity prices tend tofluctuate up and down with similar frequency. These summary statistics allow us to get acquainted with thedata and see trends and characteristics that may relate to the probability and magnitude of bubbles foundin our study.

Table 2: Commodity Log Price Correlation MatrixTable 2 displays the correlations between the daily log prices of Crude Oil, Gold, Corn, and Soybeans from the period

April 3, 1990 to April 29, 2016.

Oil Gold Corn SoybeansOil 1.000 0.853 0.710 0.766

Gold 1.000 0.870 0.899Corn 1.000 0.934

Soybeans 1.000

Table 2 displays the correlation coefficients between the daily log prices of the four commodities from thestudy sample period. The coefficients for the log prices of the commodities are all positive with correlationsranging from 0.710 to 0.934. Corn and soybeans are the most highly correlated commodities (0.934). Thisis consistent with Figure 3 as we can see the general price patterns are very similar for corn and soybeansand not as much for the other two commodities. Crude oil and corn are the commodity pair producing

1This data can be found on Datastream using the following mnemonics: NCLCS00 (Oil), NGCCS00 (Gold), CCFCS00(Corn), and CSYCS00 (Soybeans). These series represent perpetual futures prices, which roll forward on the final trading dayof the current front month’s contract, to record prices of the next sequential front month’s futures contract.

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the lowest value of correlation. The correlations between the commodities give an indication of how closelyrelated commodity price swings are. If commodities produce strong correlations, we might expect them toalso produce similar explosive behavior and bubble periods. Overall, the summary statistics, price graphs,and correlations can give us a quick glance into the nature of commodity price fluctuations and provide somesupporting evidence to the existence of bubbles.

Figure 3: Commodity Log PricesFigure 3 illustrates the daily log prices of Crude Oil, Gold, Corn, and Soybeans from the period April 3, 1990 to

April 29, 2016.

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3.4 Stationarity Testing

Before we test for price bubbles, it is necessary to test whether the commodity price series follow a stationaryprocess. A stationary process is stochastic with a mean and variance that are constant over time and do notfollow any trends. If the commodity prices follow a non stationary process, explosive characteristics could bepresent and tend to have a longer effect. The idea which encompasses the process is that increased volatilitywill lead to more volatility. Alternatively, in a stationary process, shocks to the system are more short livedas they fade quicker through time. Therefore, bubbles will not exist in a stationary process. In this section,we use the Augmented Dickey Fuller (ADF) and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests forstationarity testing. The Augmented Dickey Fuller (ADF) test is used to detect whether a time series datasethas a unit root. If the data does not have a unit root, it follows a stationary process. The test is describedby the following equation:

4Pt = µ+ �t� �Pt�1 +⇢X

i=1

i4Pt�i + "t (10)

where µ is a constant or drift term, t is the time index, � denotes the coefficient on the time trend, � denotesthe coefficient presenting the unit root, ⇢ represents the number of lags, and "t is the error term. The nullhypothesis (H0) being tested is � = 1, which infers existence of a unit root or a non-stationary process. Thealternative hypothesis (HA) is � < 0, implying a stationary process. The test statistic used is as follows:ADFt =

�

SE(�)where SE stands for Standard Error. The ADF test statistic is always a negative number.

The more negative it is, the stronger the rejection of the null hypothesis that there is a unit root at somelevel of confidence.

Table 3: ADF and KPSS Test ResultsTable 3 displays the test results from the Augmented Dickey Fuller (ADF) and Kwiatkowski-Phillips-Schmidt-Shin(KPSS) tests. Specifically, the resulting test statistics for Crude Oil, Gold, Corn, and Soybeans. The data used was

the commodity’s daily prices from the period April 3, 1990 to April 29, 2016.

ADF Test Statistic KPSS Test StatisticCrude Oil -2.198 1.213

Gold -1.630 2.516Corn -2.515 1.589

Soybeans -2.525 1.714

Table 3 displays the ADF test statistics for the four commodities. The critical values used for the 95% and99% confidence intervals are -3.41 and -3.96, respectively. We see that no commodity has a correspondingtest statistic that exceeds the critical values. Thusly, none of the commodity time series produced significantADF test statistics. Therefore, we are unable to reject the null hypothesis implying that a unit root mayexist. However, failure to reject the null hypothesis could occur either because the null was correct, or becausethere is insufficient information in the sample to enable rejection. One way to get around this problem is totest for both a constant and a trend when estimating the right econometric model. Unlike the ADF test,the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test has a null hypothesis of trend stationarity against thealternative of a unit root. The test is described by the following equations:

Pt = ⇠t+Rwt + "t (11)

Rwt = Rwt�1 + ut (12)

where Pt denotes the series of commodity prices, t is the deterministic trend, Rwt represents the randomwalk process, and "t is the error term. The variable Rwt is defined in Equation 12 where ut denotes the errorterm, which by assumption is a series of i.i.d random variables with expected value equal to zero. The nullhypothesis being tested is that the variance (�2) of the random walk process (Rwt) is equal to zero. When⇠ = 0, this means that the series Pt is stationary around a linear trend. If ⇠ 6= 0, then the series Pt is said tobe non stationary. The results of the KPSS test are displayed in Table 3. The test statistic was calculatedusing the following formula: T�2

P Sts2u

where St is the partial sum of the residuals and s2u is the estimate of

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the long-run variance of the residuals. The critical values used for the 95% and 99% confidence intervals are0.146 and 0.216, respectively. The test statistics for every commodity in question are far greater than thesethresholds. Thus we can reject the null hypothesis of stationarity and conclude that the price series of crudeoil, gold, corn, and soybeans all follow a unit root and therefore are non stationary.

3.5 Empirical Results

3.5.1 Phillips, Wu, Yu (PWY) Bubble Test: SADF

Table 4: SADF Explosive PeriodsTable 4 displays the test results of the SADF bubble test for the four commodities: Crude Oil, Gold, Corn, and

Soybeans. Results were computed using a minimum window of 211 calculated by the following equation recommendedby PSY: (0.01 + 1.8/

pT )× T . The minimum bubble size (also recommended by PSY) is set to 8 calculated by the

following equation: ln(T )/T . The Days column displays the length of each bubble in trading days. The % ChangeStart to Peak is the percent change in the daily log prices from the beginning day of the bubble to the peak or

maximum price in the bubble period. % Change Peak to End column show the percent changes in the daily log pricesfrom the peak or maximum price day to the last day in the bubble period.

Commodity Bubble Period Days % Change Start to Peak % Change Peak to End

Oil10/31/2007-11/27/2007 19 3.86 -3.8612/21/2007-1/10/2008 13 6.76 -5.932/19/2008-8/29/2008 136 45.28 -20.53

Gold

1/5/2004-01/14/2004 8 0.47 -1.1211/19/2004-12/7/2004 8 2.19 -0.0811/16/2005-9/9/2008 704 109.62 -21.539/15/2008-10/16/2008 24 15.37 -11.3212/24/2008-9/1/2009 22 4.28 -3.241/16/2009-3/17/2009 41 19.26 -8.483/19/2009-4/3/2009 12 0.00 -6.505/6/2009-6/19/2009 32 7.98 -4.897/31/2009-8/14/2009 11 1.49 -2.158/25/2009-2/12/2013 877 100 -12.71

Corn

2/14/1996-3/4/1996 13 4.50 -3.033/6/1996-6/26/1996 81 35.43 0.002/25/2008-3/19/2008 18 7.32 -7.873/24/2008-7/31/2008 92 44.95 -22.86

Soy

3/12/2004-4/19/2004 26 10.46 -8.244/23/2004-5/13/2004 15 6.93 -5.0312/7/2007-3/28/2008 76 39.23 -18.784/1/2008-8/4/2008 88 36.17 -22.07

8/20/2008-8/29/2008 8 4.17 -1.19

Using the PWY SADF test for bubbles with a minimum window of 211 days and a minimum bubble lengthof eight days, we find that gold demonstrated eight bubbles. This means gold was in a bubble 26.58%of the observation period. On average, these bubbles lasted 217 days, and caused a 32.58% rise in pricesstart to peak. From peak to conclusion of the bubble period, prices fell on average -14.40%. Oil incurredthree explosive price bubbles, occupying 2.57% of the 26 year window. On average, these bubbles lasted 56days. From start to peak, these episodes caused an average rise in price of 18%, from peak to conclusion, anaverage of -10.11%. Corn experienced four bubbles representing 3.12% of the observation period. The averageduration of a corn bubble lasted 51 days and rose start to peak an average of 23.05%. The average fall outof a corn futures bubble was -8.44%. Soybeans experienced five bubbles over the sample period occurring,3.26% of the time. Lasting, rising, and falling on average, 43 days, 19.34%, and -11.06%, respectively. Allfour of the observed commodities demonstrated widely varying bubble characteristics, lasting anywhere fromthe minimum defined bubble length of eight days to the monstrous bubbles produced in gold lasting up to877 days.

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Figure 4: SADF Test ResultsFigure 4 displays the test results of the SADF bubble test for the four commodities: Crude Oil, Gold, Corn, and

Soybeans for the time frame April 3, 1990 to April 29, 2016.

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Figure 4 above displays a graphical representation of the SADF results for crude oil, gold, corn, and soy-beans. The light grey line, dark grey line, and black line in each graph represents the commodity prices,the SADF critical values, and the SADF test statistics, respectively. The commodity prices were included inthese graphs as well in order to see price spikes in relation to actual bubbles.

A bubble is existent when the SADF test statistics surpass the critical values, or when the black line exceedsthe dark grey horizontal line, and last for at least eight days. We see that the graphs are consistent with theresults given in Table 4 with regards to the timing, duration, and magnitude of the explosive bubble periods.For example, looking at the crude oil graph, we can see about 3 bubbles during the time frame 2007-2008, thelatest one being the longest and most magnified, which is indeed consistent with the bubble periods noted inthe table.

3.5.2 Phillips, Shi, Yu (PSY) Bubble Test: GSADF

The results from the GASDF test seen in Table 5 are indeed very different from the results we obtainedfrom the SADF test. The GSADF reported nearly double the amount of explosive periods detected for eachcommodity throughout the observation period. This was expected as we discussed prior that the SADF testis less accurate when testing for multiple bubbles in a time series. Due to lack of computer computationalpower, only 200 iterations were able to be run in the bootstrapping setup due to our time frame of 6,544daily observations.

For oil, gold, corn, and soybeans, the test registered 15, 17, 12, and 11 explosive periods. This equated tothe following commodities being in a bubble 28.93%, 7.95%, 9.90%, and 7.41% of the observation period,respectively. Examining Table 5, we see nearly no similarities in bubbles across the four commodities andalmost no consistency in bubbles within a given commodity. Gold exhibited the widest range of bubbleduration from 8 days, to the multiyear 912 day bubble from 9/4/2009-4/18/2013.

All four commodities produced test results with wide ranges of bubble duration. The test also producedwide ranges in the start to peak percentage gains. Not surprisingly, gold again produced the largest bubblestart to peak of 89.77% during 9/4/2009-4/18/2013. Oil experienced the two largest fallouts of a bubbleperiod as defined by PSY. Oil fell from a bubble price peak to the bubbles conclusion 33.81% in the 73 daylong bubble at the end of 2014 through the beginning of 2015. The second largest bubble fall out came inthe wake of the financial crisis, where oil fell 30.35% in late 2008 to early 2009.

There are several periods where all four of the examined commodities are in a bubble state, accordingto the GSADF test. These periods are 12/14/2007-4/29/2008, 5/6/2008-7/7/2008, and 7/25/2008-8/5/2008.There are four periods where three of the four commodities are in a bubble state. These periods are2/6/2007-3/6/2007, 10/15/2007-12/3/2007, 12/13/2007-8/6/2008, and 12/23/2010-2/11/2011. These over-lapping commodity bubbles all took place during what is now referred to as the financial crisis, and the yearsimmediately following. Using these periods of overlapping bubbles will serve as a market timing indicatorfor which our potential trading strategy will be centered.

Figure 5 shows a graphical display of the GSADF test results. The black line, dark grey line, and lightgrey line represent the GSADF test statistics, the critical values, and the daily prices, respectively. The mo-ments where the test statistics (black lines) peak above the critical values (dark grey lines) denote the startof a bubble period. The point when the first test statistic passes back through the critical values representsthe end of the bubble period.

12

Tabl

e5:

GSA

DF

Test

Res

ults

Tabl

e5

disp

lays

the

test

resu

ltsof

the

GSA

DF

bubb

lete

stfo

rth

efo

urco

mm

oditi

es:

Cru

deO

il,G

old,

Cor

n,an

dSo

ybea

ns.

Res

ults

wer

eco

mpu

ted

usin

ga

min

imum

win

dow

of21

1ca

lcul

ated

byth

efo

llowin

geq

uatio

nre

com

men

ded

byPSY

:(0.01+

1.8/

pT)×

T.

The

min

imum

bubb

lesi

ze(a

lso

reco

mm

ende

dby

PSY

)is

setto

8ca

lcul

ated

byth

efo

llowin

geq

uatio

n:ln(T

)/T

.T

heD

ays

colu

mn

disp

lays

the

leng

thof

each

bubb

lein

trad

ing

days

.T

he%

Cha

nge

Star

tto

Pea

kis

the

perc

entch

ange

inth

eda

ilylo

gpr

ices

from

the

begi

nnin

gda

yof

the

bubb

leto

the

peak

orm

axim

umpr

ice

inth

ebu

bble

peri

od.

%C

hang

ePea

kto

End

colu

mn

show

the

perc

entch

ange

sin

the

daily

log

prices

from

the

peak

orm

axim

umpr

ice

day

toth

ela

stda

yin

the

bubb

lepe

riod

.

Oil

Gol

dBub

ble

Day

s%

Cha

nge

Star

tto

Pea

k%

Cha

nge

Pea

kto

End

Bub

ble

Day

s%

Cha

nge

Star

tto

Pea

k%

Cha

nge

Pea

kto

End

8/11/1999-8/26/1999

12

1.49

-4.08

6/16/1993-8/6/1993

37

9.79

-7.35

9/8/1999-10/8/1999

23

9.75

-15.96

1/16/1997-1/28/1997

90.64

-0.92

11/16/1999-12/1/1999

10

5.33

-7.65

1/30/1997-2/21/1997

16

2.22

0.00

2.14/2000-3/21/2000

26

12.83

-17.9

7/9/1997-7/18/1997

83.34

.0

8/2/2005-9/8/2005

27

12.80

-7.62

11/14/1997-1/29/1998

50

1.08

-1.56

7/5/2006-7/19/2006

11

2.45

-3.93

6/18/1999-9/27/1999

71

18.27

0.00

10/15/2007-12/3/2007

35

13.99

-9.03

1/23/2003-2/13/2003

16

5.45

-5.82

12/13/2007-9/12/2008

189

57.50

-30.35

12/9/2003-1/30/2004

34

4.48

-5.76

12/4/2008-12/31/2008

19

9.87

-7.04

12/5/2005-6/14/2006

132

41.30

-21.82

4/25/2011-5/4/2011

81.47

-4.12

7/5/2006-8/18/2006

33

6.08

-8.19

11/3/2014-2/18/2015

73

0.00

-33.81

9/22/2006-10/4/2006

92.54

-6.74

3/10/2015-3/26/2015

13

6.50

0.00

10/10/2006-9/11/2008

483

75.00

-26.11

7/24/2015-9/1/2015

28

2.20

-7.70

9/17/2008-10/22/2008

26

6.75

-18.0

9/3/2015-9/17/2015

11

.85

.53

5/21/2009-7/1/2009

29

3.42

-4.31

1/11/2016-3/1/2016

35

9.52

0.00

9/4/2009-4/18/2013

912

89.77

-26.59

5/1/2013-5/15/2013

11

0.43

-2.67

6/27/2013-7/19/2013

16

1.21

-1.00

Cor

nSoy

bea

ns

Bub

ble

Day

s%

Cha

nge

Star

tto

Pea

k%

Cha

nge

Pea

kto

End

Bub

ble

Day

s%

Cha

nge

Star

tto

Pea

k%

Cha

nge

Pea

kto

End

11/2/1993-1/28/1994

59

18.32

-6.45

7/2/1993-8/2/1993

21

10.38

-3.95

4/26/1995-5/30/1995

24

8.84

-1.48

7/11/2002-7/29/2002

13

7.03

-9.93

6/7/1995-6/29/1995

17

7.17

-5.63

1/6/2004-5/19/2004

94

32.00

-14.11

9/14/1995-6/4/1996

181

69.00

-11.83

2/6/2007-4/2/2007

39

6.09

-0.64

8/7/2002-8/20/2002

10

9.64

-5.13

6/7/2007-6/22/2007

12

2.76

-6.78

9/19/2006-3/6/2007

114

76.42

-2.76

7/3/2007-7/18/2007

11

7.71

-7.70

12/14/2007-7/7/2008

140

73.74

-4.34

9/11/2007-4/29/2008

160

66.67

-18.01

7/25/2008-8/6/2008

94.33

-15.61

5/6/2008-8/5/2008

64

29.13

-23.65

9/14/2010-10/14/2010

23

16.97

-2.07

11/5/2010-11/17/2010

94.28

-10.01

12/16/2010-2/11/2011

40

20.07

0.00

12/23/2010-2/15/2011

37

7.48

-5.72

3/8/2011-3/31/2011

18

0.00

1.84

9/11/2014-10/15/2014

25

0.00

-3.74

8/22/2014-9/10/2014

13

0.00

-5.21

13

Figure 5: GSADF Test ResultsFigure 5 displays the test results of the GSADF bubble test for the four commodities: Crude Oil, Gold, Corn, and

Soybeans for the time frame October 6, 1992 to April 29, 2016.

14

4 Portfolio OptimizationsThis section aims to use our commodity bubble test results by implementing them into investment strategies.Having the ability to detect whether we are in a bubble period may be keen and valuable information toinvestors looking to invest in various commodities. Do bubbles matter in the eyes of investors? That iswhat we hope to answer in this section. We look at the effects on multiple investment strategies for sixunique portfolios when designating full allocation to commodities during bubble periods and no allocation tocommodities during bubble periods. The results will indicate in a general way whether or not retail investorsshould consider the timing or magnitude of commodity bubbles when designing their investment strategies.

4.1 Data Descriptives

For the purpose of examining the benefits and potential consequences of exposure to commodity markets forretail investors, we create six simple portfolios. The portfolios consist of funds a typical retail investor wouldhave access to. Specifically, exposure to global equities, U.S. bonds, and commodities in varying allocations.The exposure to global equities is done through total returns of the Templeton Global Growth Fund. Forexposure to bonds, we use total returns for the SPDR Barclays U.S. Aggregate Bond ETF, which tracks theperformance of investment grade U.S. debt securities. It invests in fixed rate government, corporate, andmortgage backed securities with a duration longer than one year.

We use the S&P Goldman Sachs Commodity Index family of funds to incorporate commodity exposure.These funds mimic market movements in a variety of commodities. We utilize five of these funds in ourretail investor portfolio analysis, which include the GSCI Commodity, GSCI Oil, GSCI Gold, GSCI Corn,and GSCI Soybeans. The GSCI Commodity index carries weights of global commodities by their relativesocietal significance.2

Before moving on to the next section, summary statistics are listed in Table 6 for our seven investmentproducts. When looking to these statistics, we find that the equities and oil report the highest daily mean re-turns. Oil reports the highest standard deviation, followed by equities, corn, soybean, the overall commodityindex, gold and bonds. Overall, six out of seven commodities are left skewed while the return distributions ofboth equities and oil returns are heavily peaked. The high deviations in the return series of oil and equitiesare again reflected by the overall range. The largest price drop is detected in oil prices on January 17, 1991,while the highest price increase is allocated on August 6, 1990.

From the correlation matrix seen in Table 7, we learn that the Barclays Aggregate Bond Index is negativelycorrelated with the Franklin Templeton Global Growth Fund and that the Goldman Sachs Commodity Index(GSCI) for Gold has extremely high correlation to the overall Goldman Sachs Commodity Index. When aGranger test for causality is applied for these two indexes, we conclude that we fail to reject, given a p valueof 0.168, that prices of the crude oil index do not granger cause the prices in the commodity index.

In other words, we conclude that there is no strong indication for a bilateral causality effect for the entiresample. A reasonable explanation for these high correlation coefficients is shown in the Saghaian (2010)paper, which shows that high correlations could be explained by the fact that grains are directly linkedwith ethanol and that oil markets are related to the overall commodity market through the oil–ethanol–cornlinkages.

2This data can be found on Datastream using the following mnemonics: A:TGGX (Templeton Global Growth Fund), LH-GOVBD (Barclays Gov AGG), GSCITOT (Broad commodity index fund) GSCRTOT (Oil index fund), GSGCTOT (Gold indexfund), GSCNTOT (Corn index fund), and GSSOTOT (Soybean index fund). These funds, like the continuous futures seriesused in the bubble detection portion of this analysis, use prices of rolling front month futures contracts to mimic the price offront month futures contracts.

15

Table 6: Summary Statistics of Seven Investment Products

Table 6 displays the daily percent changes of the following seven Total Returns Indexes: Templeton GL GR FU,Barclays US AG BD, Goldman Sachs Commodity Index, Goldman Sachs Commodity Index Gold, Goldman Sachs

Commodity Index Oil, Goldman Sachs Commodity Index Corn, and Goldman Sachs Commodity Index Soybean. Thereturns are all denoted in U.S. dollars. Returns are from the period April 3, 1990 to April 29, 2016.

Mean Median Std Dev. Skewness Ex.Kurtosis RangeTempleton 0.037 0.023 1.860 -0.088 12.894 (-14.627, +12.894)Barclays 0.001 0.000 0.273 -0.163 1.929 (-1.731, +1.929)

GSCI 0.012 0.000 1.343 -0.361 7.900 (-16.833, +7.896)GSCI Gold 0.023 0.000 1.018 -0.071 9.244 (-9.344, +9.244)GSCI Oil 0.037 0.000 2.146 -0.453 14.606 (-31.875, +14.606)

GSCI Corn -0.011 0.000 1.539 0.154 9.057 (-7.807, +9.057)GSCI Soy 0.032 0.000 1.397 -0.059 6.963 (-7.074, +6.963)

Table 7: Correlations of Seven Investment Products

Table 7 displays the correlation coefficients of the daily percent changes of the following seven securities: TempletonGL GR FU, Barclays US AG BD, Goldman Sachs Commodity Index, Goldman Sachs Commodity Index Gold,

Goldman Sachs Commodity Index Oil, Goldman Sachs Commodity Index Corn, and Goldman Sachs CommodityIndex Soybean. The returns are all denoted in U.S. dollars. Returns are from the period April 3, 1990 to April 29,

2016.

Templeton Barclays GSCI GSCI Gold GSCI Oil GSCI Corn GSCI SoyTempleton 1.000 -0.035 0.111 0.110 0.071 0.061 0.075Barclays 1.000 -0.148 0.038 -0.141 -0.073 -0.096GSCI 1.0000 0.291 0.928 0.317 0.321

GSCI Gold 1.000 0.228 0.156 1.167GSCI Oil 1.000 0.179 0.191

GSCI Corn 1.000 0.656GSCI Soy 1.000

4.2 Simple Investment Strategies

For this section, we create four highly simplified investment strategies, which could be implemented by aretail investor who invested 100 dollars on April 3, 1990 and made no further contributions to the portfoliothrough out our observation period, running through April 29, 2016. For each of the four strategies, sixsimple subportfolios were created to test the bubble strategies. These portfolios hold the following mixand proportions of assets at their inception: Portfolio 1 consists of only investing in global equities via theTempleton Global Growth Fund, Portfolio 2 consists of 50% invested in Global Equities and 50% invested inU.S. fixed income securities via the SPDR Barclays U.S. Aggregate Bond ETF.

Portfolio 3 is the first to introduce commodity futures exposure holding 33.33% in global equities, 33.3% infixed income securities, and 33.33% in broad commodity futures exposure using the S&P Goldman SachsCommodity Index, Portfolio 4 consists of 33.33% global equities, 33.33% fixed income securities, and 33.33%exposure to the S&P Goldman Sachs Gold Index, Portfolio 5 consists of six assets all equally weightedinvested in global equities, fixed income, the S&P Goldman Sachs Gold Index, S&P Goldman Sachs OilIndex, S&P Goldman Sachs Corn Index, and S&P Goldman Sachs Soy Index, and lastly, Portfolio 6 containsonly exposure to the four commodities examined in this paper (oil, gold, corn, soybeans).

16

Note that rebalancing is done at the first trading day of the trading frequency. These six portfolios will beused to analyze the potential benefits of adding commodities to a portfolio, and the economic value of theproposed trading strategies. Before implementing the proposed trading strategies based on bubble detection,we examine how the portfolios would have performed over time using simple rebalancing frequencies. Thefirst naive strategy is a simple buy and hold strategy through out the 26 year period. The remaining naivestrategies consist of yearly, monthly, and daily equally weighted rebalancing. To ensure that the strategy isconsistent with real world conditions, a transaction cost of 50 basis points (50 bp) has been applied to alltransactions.

4.3 Performance Measurements

In order to evaluate portfolio performance, it’s necessary to calculate certain measures, such as Sharpe ratio,Sortino ratio, and Omega ratio.

4.3.1 Sharpe Ratio

Due to its simplicity, the Sharpe ratio is one of the most popular measures for return relative to risk. Theratio represents how much excess return is generated for every additional unit of risk. The higher the ratio,the better the performance of the portfolio is in terms of risk adjusted return. The formula is as follows:Rp�Rf

�pwhere Rp denotes the return on the portfolio, Rf is the risk free rate of return, and �p is the standard

deviation of the portfolio.

4.3.2 Sortino Ratio

The Sortino ratio is simply an extension of the Sharpe ratio. It differentiates normal volatility from harmfulvolatility by including the standard deviation of negative asset returns also known as downside deviation. Alarge ratio suggests that the portfolio has a low probability of large losses. The following formula is usedto calculate the ratio: R�Rf

�dwhere R denotes expected return and �d is the standard deviation of negative

asset returns.

4.3.3 Omega ratio

The omega ratio measures the likelihood of achieving a specific return (target return). It is the ratio ofthe cumulative probability of an outcome above an investor’s defined threshold level to the probability ofan outcome below an investor’s defined threshold level. Therefore, it represents the ratio of upside returns(gains) to downside returns (losses). The performance measure is calculated with the following equation:

⌦(r) =

b

r(1� F (x))dx

r

aF (x)dx

(13)

where r denotes the threshold return (investor’s defined return level) and F represents the cumulative den-sity function of returns. The higher the ratio, the more likely the investor’s required return will be met orexceeded. We assume r to be equal to the mean 26 year risk free rate on the one year U.S. Treasury Bill(T-bill), which is equivalent to 3.18%. The final back testing results can be found in Table 8.

17

4.4 Portfolio Back Test

Table 8: Portfolio Performance MeasuresTable 8 displays summary statistics and performance measures of each of our portfolios based on the end of the yearportfolio values and daily percent changes in the seven investment products. Std Dev. denotes the standard deviation.Portfolio 1 consists of only investing in global equities via the Templeton Global Growth Fund, Portfolio 2 consists of

50% invested in Global Equities and 50% invested in U.S fixed income securities via the SPDR Barclays USAggregate Bond ETF, Portfolio 3 is the first to introduce commodity futures exposure holding 33.33% global equities,

33.3% fixed income securities, and 33.33% broad commodity futures exposure using the S&P Goldman SachsCommodity Index, Portfolio 4 consist of 33.33% global equities, 33.33% fixed income securities, and 33.33% exposure

to the S&P Goldman Sachs Gold Index, Portfolio 5 consists of six assets all equally weighted invested in globalequities, fixed income, the S&P Goldman Sachs Gold Index, S&P Goldman Sachs Oil Index, S&P Goldman Sachs

Corn Index, S&P Goldman Sachs Soy Index, and Portfolio 6 contains only exposure to the four commoditiesexamined in this paper. Note that rebalancing is done at the first trading day of the trading frequency.

Buy and Hold: Equally WeightedPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

1 5.499 5.881 12.987 1.373 4.896 0.179 0.344 1.6852 3.363 3.323 7.093 0.895 3.057 0.026 0.041 1.0743 3.061 3.838 6.506 -0.705 0.712 -0.018 -0.023 0.9524 3.426 4.129 5.567 0.042 -0.203 0.044 0.064 1.1205 3.449 4.949 7.057 -0.211 -0.793 0.038 0.055 1.0956 3.748 6.601 10.268 -0.043 -0.572 0.055 0.081 1.140

Daily Rebalanced: Equally WeightedPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

1 5.499 5.881 12.987 1.373 4.896 0.179 0.344 1.6852 -170.321 -109.700 367.628 -2.019 5.142 -0.472 -0.437 0.1663 -121.189 -105.617 251.412 -1.195 3.425 -0.495 -0.467 0.2224 16.728 -65.541 491.389 2.594 10.864 0.028 0.057 1.1115 -97.744 -78.382 885.711 -2423 12.513 -0.114 -0.130 0.6006 1252.509 -97.532 5317.665 3.530 12.164 0.235 1.820 6.950

Monthly Rebalanced: Equally WeightedPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

1 5.499 5.881 12.987 1.373 4.896 0.179 0.344 1.6852 2.515 1.834 17.455 0.339 0.790 -0.038 -0.054 0.9033 0.723 1.961 65.415 1.486 7.446 -0.038 -0.059 0.8814 2.933 2.339 28.785 -0.055 -0.572 -0.009 -0.012 0.9795 87.646 -0.133 588.516 1.976 9.299 0.144 0.306 2.0736 -47.374 -16.734 474.106 -2.234 13.673 -0.107 -0.123 0.574

Yearly Rebalanced: Equally WeightedPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

1 5.499 5.881 12.987 1.373 4.896 0.179 0.344 1.6852 5.185 5.535 13.255 1.321 4.599 0.151 0.283 1.5503 5.255 5.252 13.156 0.017 0.851 0.158 0.249 1.5244 6.128 5.599 13.341 0.956 1.619 0.221 0.442 1.8525 8.735 9.323 22.254 0.493 -0.018 0.250 0.486 1.8676 9.443 11.181 29.153 0.781 1.080 0.215 0.423 1.746

Results from the back tests are listed in Table 8. From this table, we conclude that transaction costs play animportant role in choosing the optimal portfolio strategy. This is clearly shown by the results from the dailyand monthly rebalanced equally weighted portfolios. It is shown that yearly rebalanced portfolios manage todeliver the highest Sharpe, Sortino and Omega ratio due to its right skewness. Therefore, we conclude thatthe yearly rebalanced and equally weighted portfolios manage to give the best protection against downside

18

deviation. When looking to the Omega ratio, we observe that the yearly rebalanced portfolios deliver thehighest likelihood out of our four strategies to beat the mean risk free rate of the one year U.S. treasury billover our 26 year sample period. We additionally see that the benefits of rebalancing our portfolio each year donot outweigh the transaction costs (50bp per transaction), which is not the case for both monthly and dailyrebalanced portfolios. From the yearly rebalancing strategy, Portfolio 5 (Franklin Templeton Global GrowthFund + Barclays US Aggregate + GSCI Commodities, GSCI Gold, GSCI Oil, GSCI Corn and GSCI Soybean)and Portfolio 4 (Franklin Templeton Global Growth Fund + Barclays US Aggregate + GSCI Commodities+ GSCI Gold) managed to give us the best result.

From the Buy and Hold strategy, we keep in mind that introducing gold in the portfolio reduced the overallstandard deviation. Additionally, we see that the yearly rebalanced Portfolio 6 (GSCI Gold, GSCI Oil,GSCI Corn and GSCI Soybean) outperforms the yearly rebalanced Portfolio 1 (Franklin Templeton GlobalGrowth Fund) based on the Sharpe ratio, which is not the case in the Buy and Hold Strategy set up. Incontrast, when the Buy and Hold strategy is applied, we see that Portfolio 1 (equities only) outperforms themixed commodity Portfolio 6. We thus conclude that there is a diversification effect and that the timing ofrebalancing plays an important role in the equally weighted portfolios that we back tested.

4.5 Bubble Portfolio Back Testing

Because only portfolios three through six contain commodities, the remainder of the analysis performed will beconducted on these four portfolios. Strategies one and two are based on instances when all four commoditieswere in a bubble at the same time. This has occurred three times during the observation period. Strategiesthree and four are based on any period where three of the four examined commodities have been in a bubbleat the same time. This has occurred four times during the observation period. Once all four commoditiesenter a bubble, Strategy 1 (Risk Off) sells all commodity assets in the portfolio and reallocates those fundsto an equally distributed portfolio of global equities and fixed income.

At the time all four commodities are no longer all in a bubble, the portfolio rebalances to equal weightedweights among all assets in the original portfolio. Strategy 2 (Risk On) does the opposite and invests fullyinto all commodity assets in the portfolio, and sells the portfolios assets in global equities and fixed income.When all four commodities are no longer all in a bubble, the portfolio reallocates to equally weighted weightsamong all assets in the original portfolio. Strategies three and four perform the same transactions respectivelybut are based on periods when three of the four commodities are in a bubble period. All portfolios (excludingPortfolio 6) remain fully invested in the market at all times with cash never being held as an asset.

The primitive intuition behind these strategies is that if multiple commodities are in bubbles at the sametime an investor may be more or less inclined to take on additional commodity risk. The rule is written asfollows: if a bubble is detected, portfolio reallocation will take place in the following period. This is becausethough the PSY method can provide bubble signals in real time, action could not be taken until the followingbusiness day. The same applies to the conclusion of bubbles. It would be ideal to create a strategy whichcould attempt to capture the start to peak price bubble move, however, given the wide range of bubbledurations and magnitudes, it is challenging to create a perfect strategy without falling victim to look aheadbias. The results of the four strategies on Portfolios 3 through 6 can be found in Table 9.

The bubble investment strategies applied to portfolios 3-6 used in this paper is most similar to a B&Hstrategy of each respective portfolio. Comparing the strategies in Table 9 to the simple portfolios in Table8, we are able to see if there are any benefits to the simple strategies described in this paper. The loweststandard deviation for Portfolio 3 comes from the Three Commodity Risk On Strategy. Not surprisingly thelower level of risk is accompanied by lower mean and median returns, compared with buying and holdingPortfolio 3 over the entire observation period. The lowest level of risk for Portfolio 4 comes again from theThree Commodity Risk On Strategy.

19

Table 9: Performance Measures: Bubble Investment StrategiesTable 9 displays summary statistics and performance measures of each of our bubble portfolios based on the end ofthe year portfolio values and daily percent changes in the seven investment products. Std Dev. denotes the standarddeviation. Portfolio 1 consists of only investing in global equities via the Templeton Global Growth Fund, Portfolio 2consists of 50% invested in Global Equities and 50% invested in U.S fixed income securities via the SPDR Barclays

US Aggregate Bond ETF, Portfolio 3 is the first to introduce commodity futures exposure holding 33.33% globalequities, 33.3% fixed income securities, and 33.33% broad commodity futures exposure using the S&P Goldman SachsCommodity Index, Portfolio 4 consist of 33.33% global equities, 33.33% fixed income securities, and 33.33% exposure

to the S&P Goldman Sachs Gold Index, Portfolio 5 consists of six assets all equally weighted invested in globalequities, fixed income, the S&P Goldman Sachs Gold Index, S&P Goldman Sachs Oil Index, S&P Goldman Sachs

Corn Index, S&P Goldman Sachs Soy Index, and Portfolio 6 contains only exposure to the four commoditiesexamined in this paper. Note that rebalancing is done at the first trading day of the trading frequency.

Four Commodity Bubble Strategy: Risk OffPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

3 2.603 4.232 8.184 -1.787 5.215 -0.070 0.325 0.8094 3.086 3.930 6.353 -0.767 1.391 -0.015 -35.164 0.9605 2.636 5.634 9.809 -1.778 5.126 -0.055 -14.798 0.8576 3.040 6.696 11.829 -0.599 0.324 -0.012 1.483 0.971

Four Commodity Bubble Strategy: Risk OnPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

3 3.088 3.797 6.303 -0.547 0.289 -0.015 -0.019 0.9624 3.324 4.150 5.688 -0.264 -0.086 0.025 0.035 1.0685 3.009 5.429 8.177 -0.949 1.411 -0.021 -0.026 0.9486 3.746 6.601 10.243 -0.039 -0.551 0.055 0.081 1.140

Three Commodity Bubble Strategy: Risk OffPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

3 2.362 4.197 7.809 -1.146 2.206 -0.105 -0.123 0.7474 2.892 4.122 6.104 -0.223 -0.312 -0.047 -0.062 0.8865 2.121 5.046 9.022 -0.777 0.557 -0.117 -0.139 0.7466 2.653 6.447 11.039 -0.145 -0.532 -0.048 -0.063 0.893

Three Commodity Bubble Strategy: Risk OnPortfolio Mean Median Std Dev. Skewness Ex. Kurtosis Sharpe Sortino Omega

3 3.150 3.538 5.862 -0.167 -0.434 -0.005 -0.007 0.9874 3.321 3.622 5.303 0.083 -0.316 0.027 0.039 1.0715 2.928 5.046 7.048 -0.134 -0.651 -0.036 -0.048 0.9186 3.746 6.601 10.243 -0.039 -0.551 0.055 0.081 1.140

The strategy which would have produced the highest mean return was the Four Commodity Risk On Strategy.The mean returns do not out perform the buy and hold strategy of the same portfolio, however, median returnsdo. Portfolio 5 under all four investment strategies achieves lower returns than if the same portfolio was heldin a B&H strategy. In all but one instance, the investment strategies add additional risk to the portfolio.Portfolio 6, which only has exposure to commodities, is not affected by either the 3 or 4 Commodity Risk Onstrategies. Both the 3 or 4 Commodity Risk Off strategies produce lower average returns and higher risk. ForPortfolio 6, the two commodity investment strategies, which apply produce negative sharp ratios, where theB&H strategies of the same portfolio produce a Sharpe ratio of 0.055. The highest Sharpe ratio which can beachieved by selecting a portfolio and applying one of the investment strategies is, Portfolio six combined withthe Three Commodity Risk On strategy. The portfolio which was the least efficient at generating returnsper unit of risk was Portfolio 5 combined with the Three Commodity Risk Off strategy. The portfolio andinvestment strategy which produced the highest Sortino and Omega ratios was Portfolio 6 combined withthe 3 and 4 Commodity Risk On Strategy.

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Figure 6: Cumulative Portfolio PerformanceFigure 6 displays the cumulative portfolio performance of the Buy and Hold, Yearly Rebalanced, Four CommoditiesBubble Strategy: Risk Off, and Three Commodities Bubble Strategy: Risk On strategies for all of our six portfolios.The values reflect the daily portfolio values of each strategy if 100 dollars was invested on April 3, 1990. The values

span from April 3, 1990 to April 29, 2016.

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All of the investment strategies and and portfolio combinations fail to beat the returns and risk rewardtradeoffs which could be achieved by a simple yearly rebalancing approach. Figure 6 displays the cumulativeportfolio performance (in total returns) of the Buy and Hold, Yearly Rebalanced, Four Commodities BubbleStrategy: Risk Off, and Three Commodities Bubble Strategy: Risk On strategies for all of our six portfolios.The values reflect the daily portfolio values of each strategy if 100 dollars was invested on April 3, 1990.

5 ConclusionFor the past few decades, there has been noticeably high price volatility in commodity markets. Playing asignificant role in the daily lives of individuals, commodity price swings are pivotal to monitor. Specifically,the existence of speculative commodity price bubbles. Due to the fact that bubbles are not perceived well bymarkets and can destabilize financial systems, their study is of great importance. Ample research has beenconducted in the past to time-stamp bubbles and explain their causes, which we aimed to replicate in thispaper. We used the Phillips, Wu, Yu (PWY) and Phillips, Shi, Yu (PSY) bubble tests in order to detectthe frequency, duration, and magnitude of explosive price behavior in the following four commodity markets:crude oil, gold, corn, and soybeans.

To differentiate ourselves, we utilize a longer examination window (April 3, 1990-April 29, 2016), and theuse of an array of commodities both hard and soft. The results from the PWY test show oil, gold, corn,and soybeans being in a bubble for 2.57%, 26.58%, 3.12%, and 3.26%, respectively, of the entire observationperiod. All four of the observed commodities demonstrated widely varying bubble characteristics, lastinganywhere from the minimum defined bubble length of eight days to the monstrous bubbles produced in goldlasting up to 877 days.

The PSY test produced significantly more bubble periods for every commodity. This is not surprising asPhillips, Shi, Yu proved that their extension of the PWY test has more power and does a considerably betterjob at detecting multiple bubbles in a period. In our 26 year study period, the PSY test detected 15, 17, 12,and 11 explosive periods for oil, gold, corn and soybeans, respectively. The combined days of these periodsconstitute 7.95%, 28.93%, 9.90%, and 7.41% of the observation period. Like the PWY, all four commoditiesproduced test results with wide ranges of bubble duration.

According to our PSY test results, we see several periods where all four of the the examined commodi-ties are in a bubble as well as periods where three of the four commodities are in a bubble. We utilized theseoverlapping bubble periods as market timing indicators in order to see whether the timing and or magnitudeof commodity price bubbles positively influence portfolio performance. We do this by first evaluating perfor-mance when portfolios include exposure to commodities. From the portfolios and investments strategies wecreated, we find that commodity exposure has the potential to lower overall portfolio risk.

When implementing strategies that either completely reduce commodity exposure (risk off) or go 100%into commodities (risk on) during the overlapping bubble periods, we find that no strategy maximizes port-folio net worth. This does not mean, however, that investment decisions will never have positive effects whencommodity bubbles are taken into consideration. We are only able to conclude that our simple investmentstrategies do not benefit from alterations due to explosive price episodes in the commodity markets. Althoughwe do not establish investment benefits regarding bubbles, the study of price bubbles will still be a popularphenomenon in the years to follow and could potentially provide valuable and noteworthy benefits in thefuture.

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