Large Capitalization ETF Style Rotation Using Entropy ... · rotation strategies between value and...

Large Capitalization ETF Style Rotation Using Entropy

Measures of the VIX

AUTHORS

Levan Efremidze

Graziadio School of Business and Management

Pepperdine University, Malibu, California

James A. DiLellio



Darrol J. Stanley



CORRESPONDING AUTHOR

Levan Efremidze

Department of Finance and Accounting


Pepperdine University

24255 Pacific Coast Highway

Malibu, California 90263

Phone: 310-531-3302

Email: [email protected]

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Large Capitalization ETF Style Rotation Using Entropy

Measures of the VIX

Abstract

We examine the feasibility of market timing between large capitalization value and growth

portfolios with the use of entropy measures as compared to previously tested methods of

market timing using stock market volatility (VIX). Including transaction fees, style rotations

using entropy measures appear to provide superior risk-adjusted returns and may offer a

desirable alternative strategy to risk-averse investors seeking equity exposure.

JEL Classification: G11, G12, G17

Keywords: style rotation, entropy, approximate entropy, sample entropy, VIX, ETF, market

timing, value style, growth style

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I. Introduction

We study style rotation feasibility utilizing two new indicators. These indicators are Sample

Entropy (SaEn) and Approximate Entropy (ApEn) calculated from the VIX time series. The VIX

level and percentage changes appear to indicate the expected size of volatility changes, but they

do not inform us about the level of randomness within the same series. Entropy measures this

level of uncertainty, independently from predictable parts of the volatility changes, which could

impact the market risk premium and discount rates of value and growth stocks and thus lead to a

state in which value outperforms growth portfolio. These two entropy based signals produce

better performing value minus growth (or growth minus value) portfolios than trading strategies

based on the VIX percentage change signals. The performance results account for reasonable

transactions costs. The implementation of the strategy could be conducted with large

capitalization value and growth futures positions with minimal transactions costs.

The economic validity of the effectiveness of style rotation has attracted considerable

interest. In years when value outperforms growth, value managers are handsomely rewarded

while growth managers suffer and vice versa. There are sizable daily return differentials between

value and growth portfolios (see Figure I) and there is increasing evidence that market timing

and/or style rotation is appropriate in various time periods. Below et al. (2009) provided

evidence that style rebalancing significantly outperforms naïve buy and hold approaches. Others

have suggested the same general view. These include Arshanapalli et al. (2004); Holmes and

Faff (2007); Puttinon and Seppa (2007); and Maio (2013). The challenge is how to decide when

to make a switch between styles.

Merton (1980) suggests that the market risk premium, measured as the difference

between the market return and the risk-free rate, is directly related to the market variance.

Assuming that expected cash flows are unaffected, the rise in expected volatility will increase the

market risk premium and discount rate, and stock prices will decline. Changes in expected

4

volatility, measured by the VIX (the index of implied volatility), have been used to time the style

rotation strategies between value and growth portfolios, but the performance of this indicator

across different time periods has been mixed. Copeland and Copeland (1999) suggested that

portfolios of value stocks outperformed (underperformed) portfolios of growth stocks following

an increase (decrease) in the VIX, and the VIX is a better leading indicator for market timing

than macro factors that are released to the public with substantial time delay. They particularly

noted that trading strategies based on shorter periods were economically significant. Boscaljon et

al. (2011) closely followed the similar methodology and found, however, that the results were

only statistically significant for holding periods longer than 30 days. We revisited this strategy

by introducing two new indicators of uncertainty called Sample Entropy and Approximate

Entropy.1 As discussed in Pincus (2008), entropy can detect the extent of irregularity, rather than

simply the deviations from a mean, as it is done in volatility measures. We find that these two

entropy based signals produce better performing value minus growth (or growth minus value)

portfolios than trading strategies based on the VIX percentage change signals.

II. Entropy and Application to the VIX

There have been numerous studies that have found compelling evidence that implied volatility

measurements such as the VIX can be effective in predicting future market volatility especially

in the short-run (e.g., Chang et al., 2011; Low, 2004). Becker et al. (2006) suggest that

forecasting can improve with the use of the VIX. On the other hand, there is considerable debate

if the VIX is the best method or even a good method as a predictor to time the market. Arak and

1 The theory of entropy was initially developed in the field of thermodynamics as a way to measure the level of

randomness in the late 1850s. In that context it was used to characterize the amount of energy in a system that was

no longer available for doing work. Subsequently, the definition has been expanded to characterize a level of

randomness and disorder.

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Mijid (2006) showed that the old VIX did not forecast well the changes in volatility. Goldwhite

(2009) suggests the new VIX is useful in categorizing investments to enhance performance based

on unexpected changes in risk aversion.

FIGURE I HERE

The VIX represents the market’s expected volatility.2 Investors may respond to this

volatility as one measure of investment risk. As a consequence, increased volatility will alter

investors’ asset allocations by foregoing equity investments or reducing substantially their risk

exposure. While the VIX is often referred to as the “fear index”, it is really only an indicator of

market sentiment regarding the level of uncertainty in the equity markets. The VIX can have

multiple applications in the securities markets. One such application is the style rotation between

value and growth.3

In modern times, entropy has been applied to the study of financial markets (Pincus,

2008; Pincus and Kalman, 2004; Maasoumi and Racine, 2002; Molgedey and Ebeling, 2000).

Measuring the nonlinear dynamics of the financial time series is suggested to be achieved by

entropy concepts, in part due to the fact that entropy does not impose any restrictions on

theoretical probability distributions (Bentes et al., 2008). The basic idea is that more volatile

2 The history of the VIX is well documented. This market volatility index was first introduced by the Chicago Board

Options Exchange (CBOE) in 1993 as a measurement of implied volatility based on the Standard and Poor’s 100

call and put options. On September 23, 2003, the VIX was revised. The current index uses 30-calendar day out of

the money SPX calls and puts with weights that are inversely proportional to the squared strike price. The new VIX

is now based on the S&P 500 Index rather than the S&P 100. Further, the new index is based on the concept of fair

value of future variance rather than the Black-Scholes-Merton model.

3 One can make this more complex quickly by expanding the universe from the S&P 500 to the mid-cap S&P 400

and the small cap S&P 600. Actually, the combinations are numerous.

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securities have a greater entropy state than more stable securities. Two fundamentally different

phenomena exist in which time series data deviate from constancy. These two phenomena are

that series (1) exhibit larger standard deviations (can be expressed by VIX as implied volatility

measure) and (2) appear highly irregular (estimated by entropy).

These two phenomena are not mutually exclusive and as such they can be used to

characterize two very different features of uncertainty associated with the fluctuations in security

time series. The standard deviation measures the extent of deviation from centrality while

entropy provides a useful metric for categorizing the extent of irregularity or complexity of the

data set. Evaluating the subtle but complex shifts in data series is a primary goal for exploring

the potential information contained therein.

Pincus (2008) found that Approximate Entropy (ApEn) is both robust to outliers and can

be applied to time series with 50 observations or more. Another alternative measure of system

complexity that is often used in this regard is called Sample Entropy (SaEn). The literature is

replete with detailed discussions of these alternative measures of entropy (Thuraisingham and

Gottwal, 2006; Richman and Moorman, 2000). Additionally, other entropy measures, termed

superinformation (Bose and Hamacher, 2012), as well as Thannon, Renyi, and Tsallis entropy

(Bentes et al., 2008), have been applied to analyzing time series in the financial markets and their

underlying nonlinear dynamics. Within this study, we employed ApEn and SaEn models, which

consist of three inputs for (i) Time Series (TS), (ii) Matching Template Length (M), and (iii)

Matching Tolerance Level (R).

By employing the parameters listed in Table 1, we calculated both the ApEn and SaEn

from the VIX time series.

TABLE 1 HERE

Figure II compares ApEn and SaEn to time series of the percentage change in the VIX, as

originally proposed by Copeland and Copeland (1999) and also used by Boscaljon et al. (2011).

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The percentage change in the VIX was calculated against its prior 75-day moving average, while

SaEn and ApEn indicators used prior 120 days of the VIX data for July 14, 2003 to December

30, 2011. Since the first 120 daily observations were used in the calculation of SaEn and ApEn

series, we calculated the SaEn and ApEn series from January 2, 2004 to December 30, 2011.

FIGURE II HERE

III. Sample, Methodology and Empirical Results

To investigate empirical regularities and then the performance of three trading strategies using

either the percentage change in the VIX, SaEn or ApEn, we first conducted linear regression

analysis. The sample period used in regressions was January 2, 2004 to December 30, 2011. We

used the iShares large capitalization value ETF (IWD) and growth ETF (IWF) daily returns to

measure difference between value and growth portfolios. The following is the specification of

regression models:

(1)

Here, is the return of the value minus growth portfolio at time t, a is the

intercept, b is the slope, is the signal at time t-1, and is a normally distributed error term.

The regression results, based on employing each of our three different signals (Percentage

Change in the VIX, SaEn or ApEn), appear in Table II for various holding periods. Holding

period returns at time t for n number of days in the future represent the cumulative returns from t

to t+n.

Panel A of Table 2 suggests that the strong statistical significance in coefficients

previously reported by Boscaljon et al. (2011) no longer persists for our model’s intercept.

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Additionally, the test statistics for the slope are statistically significant at the 0.05 level or better

for all holding periods except for 40 and 50 days. This is in contrast to Boscaljon et al. (2011),

who reported significance at the 0.05 for 30 days, and 0.01 level at 40, 50 and 60 days. Also the

slopes turn negative for holding periods of 50 days or more. While not easily reconciled, we note

that we did not use the BARRA value and growth portfolios used in the previously

aforementioned study, and that our historical period is 2004-2011, which differs significantly

from the 1990-2008 period used by Boscaljon et al. (2011). In addition, correlations between

value and growth styles could change substantially over different sample periods.

The middle and lower panels of Table 2 are encouraging, suggesting that there is stronger

relationship between our measures of previous day entropy and return of the value-growth

portfolio. Indeed, statistically significant slope and intercept terms are obtained at the 0.01 level

for holding periods lasting five days or more. The other notable finding in Table 2 is that slope is

consistently positive across all holding periods and both magnitude and significance are an

increasing function of holding period length.4 Thus we see much stronger linear relationship

between value minus growth returns with our entropy indicators. The challenge will be how to

take advantage of this link, which will be described in the section of trading strategy

implementation.

TABLE 2 HERE

IV. Implementation of Style Rotation

We then implemented style rotation trading rules and evaluated their performance. After

generating the time-series of both SaEn and ApEn based on Table 1 inputs, we tested the time-

series properties of each entropy indicator against value minus growth returns. For the two

4 The reader should note that in the overlapping data samples of two of more days, where the same data enters the

sample more than once, the t-statistic can have an upward bias.

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entropy based trading strategies, we produced several signal thresholds based on a mean plus

multiple standard deviations of prior 140 days of each entropy series. We chose 140 days, about

7 months, in order to capture sufficient seasonality of the data. The signal was then lagged by 60

days (considering highest cross-correlations between the entropy and the value minus growth

return series). This means that there is a substantial time lag between entropy signal (it is a strong

leading indicator) and style performance. We tested several other lags between 1 and 50 days,

but overall their performance was inferior to a 60 day lag. The sample period for strategy

implementation was October 19, 2004 to December 30, 2011. 200 daily observations, from

January 2, 2004 to October 18, 2004, were needed to cover 140 days used in the calculation of

initial values of means and standard deviations and 60 days for the time lag.

We also created trading signal thresholds using the different levels of the percentage

change in the VIX. This strategy was used in previous literature (Copeland and Copeland, 1999;

Boscaljon et al., 2011) and we used it as a benchmark for the performance of our entropy based

trading strategies. In addition to the previously used criteria for the evaluation of trading rules,

we also calculated average transaction returns for each trading rule by dividing the cumulative

return on number of round-trip transactions. In real world trading, there are substantial

transaction costs which include trading fees and commissions, bid-ask spread, order execution

delays, and margin and shorting costs. It is not easy to estimate all these in advance, but we did

assume that these costs could add up to at least 0.5 percent for an average round-trip transaction.

The actual costs could be higher, so we multiply this number by 2. Thus, it is desirable that

average transaction return for a successful trading strategy exceed 1 percent per round-trip

transaction. The average transaction return is later used for the selection of potentially profitable

strategies and their further evaluation.

TABLE 3 HERE

TABLE 4 HERE

TABLE 5 HERE

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Tables 3, 4 and 5 present results of the value minus growth strategies (long value ETF,

short growth ETF) using three alternative signal indicators (percentage change in the VIX, SaEn

or ApEn). In Table 3, if a previous day percentage change in the VIX is higher than given

threshold, say 10 percent, then the investment is made in value minus growth. We calculated

cumulative returns, number of days a strategy was invested versus being in cash, number of

round-trip transactions, daily average returns and average transaction returns. For thresholds with

negative change, the strategy was to invest in growth minus value portfolio (long growth, short

value). The first columns in these three tables show the number of days the portfolio was held.

The entropy based trading strategies (Table 4 and 5) provide better performance than the

VIX percentage change based trading strategies (Table 3) according to the criteria of average

transaction returns. Overall, SaEn (Sample Entropy) signals produce 21 portfolios with greater

than 1 percent average transaction return (Table IV), ApEn (Approximate Entropy) signals lead

to 14 such portfolios (Table 5), while the signals based on the percentage change in the VIX only

deliver 7 such portfolios (Table 3). All other portfolios carry higher risk of negative returns when

transactions costs are taken into consideration. This leads us to argue that the entropy measures

of the VIX provide more accurate style rotation signals than the VIX percentage change

thresholds. Furthermore, Sample Entropy appears to be superior to Approximate Entropy within

the sample we studied.

TABLE 6 HERE

We conducted further evaluation of 22 strategies based on SaEn (Sample Entropy)

signals from Table 4, with the per-transaction-returns greater than 1 percent. The standard

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annualized performance measurements were computed against the following benchmarks5: S&P

500 Index, iShares Large Capitalization Value ETF (IVE), iShares Large Capitalization Growth

ETF (IVW) and iShares US Bond ETF (AGG). We included after transactions costs performance

for the 22 active strategies. Transactions costs were assumed to be 25 basis points per each

purchase or a sale, consistent with Balduzzi and Lynch (1999). The results are presented in Table

6. These strategies have strikingly low amount of risk, indicated by an average standard

deviation of 4.97% versus 18.49% for the group of benchmarks. Also, the average of maximum

drawdowns of these strategies is 9.41%, compared to 45.23% for the benchmarks.

Out of the 22 strategies, four of them have Sharpe ratios higher than 0.50, one of them

with the Sharpe ratio as high as 0.78. We decided to conduct further testing on these four

strategies by testing them on an extended sample period (extended the sample to May 17, 2013).

Table 7 includes results for two sample periods: October 19, 2004-December 31, 2011 and

October 19, 2004-May 17, 2013. Extending the time period does not alter results substantially.

Our four active strategies maintain high Sharpe ratios, only one of them drops below the Sharpe

ratio of one of the benchmarks, AGG. The standard deviations and maximum drawdowns of

these four strategies remain substantially low compared to the benchmarks (e.g. Figure III). This

additional testing provides further evidence that the selected entropy based strategies deliver

attractive risk-return characteristics, and could potentially enhance returns of other long-only

strategies while substantially improving overall risk characteristics.

TABLE 7 HERE

FIGURE III HERE

5The benchmark returns are based on buy-and-hold rule.

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V. Implications for Investment Management

The approach shown here provides a potentially beneficial alternative for investment managers

and investors to enhance returns and to achieve outstanding risk-reward results. This is supported

by the after-transaction cost Sharpe ratios and CVs in Table 7. The maximum drawdowns

indicate the low risk of this specific case of style based long-short strategy. Thus, this investment

strategy can be employed by even the most conservative of investors. However, our results need

to be taken with care, as these strategies may be sample specific and thus could benefit from

more backtesting on other time periods and sub-periods. For simplicity of illustration, in Table

VIII we provide purchase and sale dates and types of positions for one of the strategies (d1n15),

one day holding period strategy of investing in long growth-short value when the Sample

Entropy (lagged by 60 days) of the VIX falls below the threshold of its previous 140-day mean

minus 1.5 times standard deviation). There were 8 round-trip transactions during the sample

period of October 19, 2004-May 17, 2013; otherwise the strategy was invested in cash. It is

worth noting that our sample period included more than a full business cycle in it, with the worst

market decline since the Great Depression.

TABLE 8 HERE

VI. Conclusion

Changes in expected volatility, measured by the index of implied volatility (the VIX), has been

used to time the style rotation strategies between value and growth portfolios. Performance

results across different time periods have been mixed using this indicator as a signal. We

replicated this strategy on a new sample period of iShares large capitalization value and growth

ETFs and utilized two new, uncertainty related indicators. These indicators are Sample Entropy

and Approximate Entropy, which initially were developed in the fields of thermodynamics,

biology, physics, but were introduced only recently in the area of finance. They appear to be

more robust measures of changes in financial markets when analyzing the VIX series. The

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absolute VIX level and percentage changes appear to indicate the expected size of volatility

changes, but as it is argued in entropy studies, they don’t measure the level of randomness or,

more technically, level of non-linear persistence in volatility changes. Entropy measures this

level of uncertainty, independently from other predictable parts of the volatility changes, which

could impact the market risk premium and discount rates of value and growth stocks and thus

lead to a state in which value outperforms growth portfolio. These two entropy based signals

produce better performing value minus growth (or growth minus value) portfolios than trading

strategies based on percentage change of the VIX for the recent sample studied here.

Furthermore, Sample Entropy (SaEn) appears to be superior to Approximate Entropy

(ApEn) as a trading signal. These results are robust to a conservative amount of assumed

transactions costs. To keep transactions costs to a minimum, exchange traded value and growth

index funds or futures can be utilized. Both value minus growth and growth minus value

strategies could potentially be implemented with materially advantageous Sharpe ratios and CVs

of an investment portfolio. As a caution, robustness of our methodology would benefit from

testing it on other time periods.

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Becker, R., A. E. Clements, and S. I. White, 2006, On the international efficiency of S&P 500

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from broad market and equity style indexes, Financial Services Review 18, 231-248.

Bentes, S., R. Menezes, and D. A. Mendes, 2008, Long memory and volatility clustering: is the

empirical evidence consistent across stock markets? Physica A: Statistical Mechanics and its

Applications 387, 3826–3830.

Boscaljon, B., G. Filbeck, and X. Zhao, 2011, Market timing using the VIX for style rotation,

Financial Services Review 20, 35-44.

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markets, Physical Review E 86, 056112.

Chang, B., P. Christoffersen, K. Jacobs, and G. Vainberg, 2011, Option-implied measures of

equity risk, Review of Finance 16, 385-428.

Copeland, M. M., and T. E. Copeland, 1999, Market timing: style and size rotation using the

VIX, Financial Analyst’s Journal 55, 73-81.

Goldwhite, P., 2009, Diversification and risk management: what volatility tells us, Journal of

Investing 18, 40-48.

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Holmes, K. A., and R. W. Faff, 2007, Style drift, fund flow and fund performance: new cross-

sectional evidence, Financial Services Review 16, 55-71.

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Journal of Business 77, 527-546.

Maasoumi, E., and J. Racine, 2002, Entropy and the predictability of stock market returns,

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Figures and Tables

-.03

-.02

-.01

.00

.01

.02

.03

.04

04 05 06 07 08 09 10 11

Daily

Retu

rn D

iffe

rential, in

decim

als

Years

Figure I. Daily Return Differential Between iShares Large Capitalization Value and

Growth ETFs, January 2, 2004-December 30, 2011

17

0.0

0.5

1.0

1.5

2.0

2.5

-0.4

0.0

0.4

0.8

1.2

1.6

04 05 06 07 08 09 10 11

APEN VIXCH_PREV SEN

Figure II. Percentage Change in the VIX (top), SaEn (middle) and ApEn (bottom) Series

Based on the daily data of the VIX for January 2, 2004 to December 30, 2011. Percentage change in the VIX values

are on the right vertical axis in decimals. The left side axis values reflect ApEn and SaEn.

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-5%

0%

5%

10%

15%

20%

25%

30%

10

/19

/04

4/1

9/0

5

10

/19

/05

4/1

9/0

6

10

/19

/06

4/1

9/0

7

10

/19

/07

4/1

9/0

8

10

/19

/08

4/1

9/0

9

10

/19

/09

4/1

9/1

0

10

/19

/10

4/1

9/1

1

10

/19

/11

4/1

9/1

2

10

/19

/12

4/1

9/1

3

DATE

CU

MM

UL

AT

IVE

RE

TU

RN

d1n15

d2n15

d3n15

d10n15

Figure III. Cumulative Returns of the Four Best Strategies Based on SaEn

Cumulative returns are based on the daily returns of the four best strategies. The strategy names use following

method: d1n15 strategy has a 1 day holding period, with signal based on mean minus 1.5 stdev of entropy indicator.

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Table 1. Entropy Model Inputs

Time Series (TS) Matching Template

Length (M)

Matching Tolerance Level (R)

Running 120 days of daily

VIX series based on a

sample from July 14, 2003

to December 30, 2011

2 20% of the standard deviation of

the times series (TS) over the

matching template length (M)

20

Table 2. iShares Large Capitalization Value Minus Growth Returns Regressed

Either Against Percentage Change in the VIX(t-1), SaEn(t-1) or ApEn(t-1)

Holding Period Intercept t-stat Slope t-stat

Panel A: Percentage Change in the VIX(t-1) from 75 day moving average

1 0.0000 -0.24 0.0011 2.16 **

2 0.0000 -0.32 0.0016 2.46 **

5 -0.0001 -0.47 0.0036 3.60 ***

10 -0.0002 -0.66 0.0048 3.58 ***

20 -0.0004 -0.92 0.0058 2.98 ***

30 -0.0006 -1.11 0.0064 2.57 **

40 -0.0008 -1.19 0.0043 1.46

50 -0.0009 -1.22 -0.0026 -0.79

60 -0.0010 -1.23 -0.0167 -4.62 ***

90 -0.0014 -1.45 -0.0623 -14.73 ***

Panel B: SaEn(t-1) based regressions

1 -0.0006 -1.83 * 0.0006 1.93 *

2 -0.0012 -2.79 *** 0.0012 2.94 ***

5 -0.0030 -4.70 *** 0.0031 4.96 ***

10 -0.0064 -7.51 *** 0.0065 7.90 ***

20 -0.0130 -10.75 *** 0.0132 11.26 ***

30 -0.0202 -13.30 *** 0.0205 13.89 ***

40 -0.0287 -16.14 *** 0.0291 16.86 ***

50 -0.0383 -19.50 *** 0.0386 20.34 ***

60 -0.0484 -23.31 *** 0.0486 24.23 ***

90 -0.0660 -26.97 *** 0.0652 27.79 ***

Panel C: ApEn(t-1) based regressions

1 -0.0007 -1.08 0.0011 1.08

2 -0.0015 -1.68 * 0.0022 1.69 *

5 -0.0043 -3.02 *** 0.0061 3.04 ***

10 -0.0098 -5.28 *** 0.0140 5.29 ***

20 -0.0212 -7.92 *** 0.0302 7.92 ***

30 -0.0367 -10.88 *** 0.0524 10.88 ***

40 -0.0570 -14.49 *** 0.0814 14.51 ***

40 -0.0570 -14.49 *** 0.0814 14.51 ***

50 -0.0800 -18.54 *** 0.1143 18.56 ***

60 -0.1029 -22.60 *** 0.1467 22.60 ***

90 -0.1465 -27.62 *** 0.2074 27.54 ***

Statistical significance levels: * is for p<0.10, ** for p<0.05, *** for p<0.01

21

Table 3. Trading Strategy Results: Based on Percentage Changes in the VIX

Based on daily returns of iShares large cap value and growth ETFs between October 19, 2004-December 30, 2011.

Percentage changes in the VIX are calculated against previous 75-day moving average of the VIX and lagged by one

day.

Threshold Number of Daily Average Average

Holding Period Change in Cummulative Number of Round-Trip Return Transaction

(days) the VIX Return Days Transactions (basis points) Return

1 10% 11.81% 426 63 2.77 0.19%

1 20% 14.88% 267 44 5.57 0.34%

1 30% 15.65% 192 29 8.15 0.54%

1 40% 16.48% 132 29 12.49 0.57%

1 50% 7.62% 93 24 8.19 0.32%

1 60% 0.06% 67 20 0.08 0.00%

1 -10% 17.32% 610 91 2.84 0.19%

1 -20% 13.78% 135 40 10.21 0.34%

1 -30% -0.68% 7 3 -9.69 -0.23%

2 10% 6.34% 489 45 1.30 0.14%

2 20% 10.75% 311 35 3.46 0.31%

2 30% 14.59% 221 23 6.60 0.63%

2 40% 11.88% 161 21 7.38 0.57%

2 50% 10.66% 117 20 9.11 0.53%

2 60% 4.47% 87 17 5.13 0.26%

2 -10% 17.63% 700 65 2.52 0.27%

2 -20% 17.70% 174 31 10.17 0.57%

2 -30% -0.17% 10 3 -1.70 -0.06%

3 10% 8.37% 534 35 1.57 0.24%

3 20% 8.53% 346 29 2.46 0.29%

3 30% 9.88% 244 18 4.05 0.55%

3 40% 10.98% 182 17 6.04 0.65%

3 50% 7.05% 137 16 5.15 0.44%

3 60% 4.46% 104 14 4.28 0.32%

3 -10% 21.41% 764 46 2.80 0.47%

3 -20% 20.90% 204 25 10.24 0.84%

3 -30% -0.35% 13 3 -2.70 -0.12%

10 10% 2.56% 731 21 0.35 0.12%

10 20% 0.92% 519 19 0.18 0.05%

10 30% 5.72% 347 14 1.65 0.41%

10 40% 3.73% 254 7 1.47 0.53%

10 50% 5.70% 207 7 2.75 0.81%

10 60% 3.12% 164 7 1.90 0.45%

10 -10% 15.77% 975 22 1.62 0.72%

10 -20% 22.61% 338 17 6.69 1.33%

10 -30% 5.75% 27 3 21.28 1.92%

30 10% -7.66% 1041 12 -0.74 -0.64%

30 20% -12.38% 841 14 -1.47 -0.88%

30 30% -7.17% 599 12 -1.20 -0.60%

30 40% 2.21% 394 7 0.56 0.32%

30 50% 4.24% 347 7 1.22 0.61%

30 60% 0.73% 304 7 0.24 0.10%

30 -10% 14.46% 1303 13 1.11 1.11%

30 -20% 29.76% 606 12 4.91 2.48%

30 -30% 16.05% 67 3 23.96 5.35%

40 10% -13.23% 1156 11 -1.14 -1.20%

40 20% -15.05% 976 12 -1.54 -1.25%

40 30% -2.29% 719 11 -0.32 -0.21%

40 40% -4.27% 464 7 -0.92 -0.61%

40 50% -1.49% 417 7 -0.36 -0.21%

40 60% -2.57% 374 7 -0.69 -0.37%

40 -10% 4.94% 1407 10 0.35 0.49%

40 -20% 16.61% 716 12 2.32 1.38%

40 -30% 26.57% 87 3 30.54 8.86%

22

Table 4. Trading Strategy Results: Using SaEn (Sample Entropy)

Based Thresholds


SaEn thresholds are calculated on previous 140 days of SaEn data and then lagged by 60 days.

Multiple of Number of Daily Average Average

Holding Period Std. Deviation Cummulative Number of Round-Trip Return Transaction

(days) in Threshold Return Days Transactions (basis points) Return

1 0.25 8.31% 850 29 0.98 0.29%

1 0.50 5.01% 739 32 0.68 0.16%

1 0.75 5.68% 637 36 0.89 0.16%

1 1.00 6.68% 491 33 1.36 0.20%

1 1.25 9.87% 380 23 2.60 0.43%

1 1.50 11.50% 301 22 3.82 0.52%

1 -0.25 21.80% 802 21 2.72 1.04%

1 -0.50 13.08% 666 28 1.96 0.47%

1 -0.75 22.73% 541 19 4.20 1.20%

1 -1.00 13.79% 448 12 3.08 1.15%

1 -1.25 9.00% 348 16 2.59 0.56%

1 -1.50 23.35% 276 8 8.46 2.92%

2 0.25 8.09% 879 23 0.92 0.35%

2 0.50 5.64% 771 21 0.73 0.27%

2 0.75 2.42% 673 25 0.36 0.10%

2 1.00 9.24% 524 27 1.76 0.34%

2 1.25 9.47% 403 21 2.35 0.45%

2 1.50 12.51% 323 18 3.87 0.70%

2 -0.25 24.59% 822 18 2.99 1.37%

2 -0.50 12.68% 693 24 1.83 0.53%

2 -0.75 19.63% 559 18 3.51 1.09%

2 -1.00 12.63% 459 12 2.75 1.05%

2 -1.25 4.43% 363 12 1.22 0.37%

2 -1.50 20.75% 283 7 7.33 2.96%

3 0.25 8.51% 902 21 0.94 0.41%

3 0.50 4.25% 792 20 0.54 0.21%

3 0.75 3.01% 698 23 0.43 0.13%

3 1.00 11.74% 551 23 2.13 0.51%

3 1.25 6.97% 424 19 1.64 0.37%

3 1.50 11.75% 341 16 3.45 0.73%

3 -0.25 23.33% 839 15 2.78 1.56%

3 -0.50 19.23% 716 19 2.69 1.01%

3 -0.75 12.82% 576 16 2.23 0.80%

3 -1.00 13.99% 470 12 2.98 1.17%

3 -1.25 4.78% 374 11 1.28 0.43%

3 -1.50 21.94% 289 7 7.59 3.13%

10 0.25 5.51% 1008 12 0.55 0.46%

10 0.50 3.65% 911 12 0.40 0.30%

10 0.75 1.23% 826 15 0.15 0.08%

10 1.00 7.05% 697 20 1.01 0.35%

10 1.25 14.32% 531 13 2.70 1.10%

10 1.50 10.39% 440 10 2.36 1.04%

10 -0.25 14.67% 920 11 1.59 1.33%

10 -0.50 6.46% 814 13 0.79 0.50%

10 -0.75 2.57% 654 10 0.39 0.26%

10 -1.00 2.53% 537 10 0.47 0.25%

10 -1.25 7.30% 438 9 1.67 0.81%

10 -1.50 19.98% 331 7 6.04 2.85%

30 0.25 -4.74% 1216 10 -0.39 -0.47%

30 0.50 -7.80% 1140 10 -0.68 -0.78%

30 0.75 -3.79% 1051 10 -0.36 -0.38%

30 1.00 -3.73% 991 11 -0.38 -0.34%

30 1.25 3.55% 774 12 0.46 0.30%

30 1.50 7.18% 638 9 1.12 0.80%

30 -0.25 14.72% 1120 11 1.31 1.34%

30 -0.50 5.00% 1016 11 0.49 0.45%

30 -0.75 1.56% 807 8 0.19 0.20%

30 -1.00 -0.11% 698 9 -0.02 -0.01%

30 -1.25 -0.14% 598 9 -0.02 -0.02%

30 -1.50 17.03% 451 7 3.78 2.43%

40 0.25 -7.85% 1302 8 -0.60 -0.98%

40 0.50 -10.89% 1234 8 -0.88 -1.36%

40 0.75 0.77% 1142 8 0.07 0.10%

40 1.00 2.32% 1091 9 0.21 0.26%

40 1.25 15.08% 885 10 1.70 1.51%

40 1.50 8.19% 722 8 1.13 1.02%

40 -0.25 14.02% 1210 9 1.16 1.56%

40 -0.50 7.47% 1110 10 0.67 0.75%

40 -0.75 5.71% 874 7 0.65 0.82%

40 -1.00 0.20% 775 8 0.03 0.02%

40 -1.25 1.41% 677 8 0.21 0.18%

40 -1.50 15.83% 511 7 3.10 2.26%

23

Table 5. Trading Strategy Results: Using ApEn (Approximate Entropy)

Based Thresholds


ApEn thresholds are calculated on previous 140 days of ApEn data and then lagged by 60 days.

Multiple of Number of Daily Average Average

Holding Period Std. Deviation Cummulative Number of Round-Trip Return Transaction

(days) in Threshold Return Days Transactions (basis points) Return

1 0.25 5.62% 866 46 0.65 0.12%

1 0.50 -0.75% 745 46 -0.10 -0.02%

1 0.75 6.07% 635 35 0.96 0.17%

1 1.00 9.06% 453 51 2.00 0.18%

1 1.25 6.11% 278 40 2.20 0.15%

1 1.50 5.61% 193 26 2.91 0.22%

1 -0.25 23.25% 712 34 3.27 0.68%

1 -0.50 21.44% 612 39 3.50 0.55%

1 -0.75 24.81% 495 36 5.01 0.69%

1 -1.00 22.64% 415 32 5.46 0.71%

1 -1.25 16.83% 327 28 5.15 0.60%

1 -1.50 12.94% 254 17 5.09 0.76%

2 0.25 7.02% 912 31 0.77 0.23%

2 0.50 5.22% 791 34 0.66 0.15%

2 0.75 7.57% 670 27 1.13 0.28%

2 1.00 13.17% 504 38 2.61 0.35%

2 1.25 8.07% 318 32 2.54 0.25%

2 1.50 8.77% 219 19 4.00 0.46%

2 -0.25 21.93% 745 27 2.94 0.81%

2 -0.50 19.53% 650 26 3.00 0.75%

2 -0.75 18.07% 530 28 3.41 0.65%

2 -1.00 16.38% 446 24 3.67 0.68%

2 -1.25 18.08% 355 23 5.09 0.79%

2 -1.50 11.53% 271 13 4.26 0.89%

3 0.25 7.86% 943 26 0.83 0.30%

3 0.50 5.50% 825 27 0.67 0.20%

3 0.75 9.10% 697 23 1.31 0.40%

3 1.00 15.10% 542 32 2.79 0.47%

3 1.25 6.49% 350 28 1.85 0.23%

3 1.50 8.10% 238 18 3.40 0.45%

3 -0.25 26.45% 771 23 3.43 1.15%

3 -0.50 22.85% 675 20 3.38 1.14%

3 -0.75 13.26% 557 22 2.38 0.60%

3 -1.00 14.29% 469 21 3.05 0.68%

3 -1.25 19.13% 377 20 5.08 0.96%

3 -1.50 8.55% 284 13 3.01 0.66%

10 0.25 2.21% 1081 17 0.20 0.13%

10 0.50 -1.20% 971 17 -0.12 -0.07%

10 0.75 1.10% 832 17 0.13 0.06%

10 1.00 12.57% 694 16 1.81 0.79%

10 1.25 14.38% 498 13 2.89 1.11%

10 1.50 7.25% 349 15 2.08 0.48%

10 -0.25 19.62% 898 16 2.19 1.23%

10 -0.50 10.26% 795 15 1.29 0.68%

10 -0.75 9.08% 677 15 1.34 0.61%

10 -1.00 15.81% 571 13 2.77 1.22%

10 -1.25 24.85% 482 12 5.16 2.07%

10 -1.50 14.27% 367 12 3.89 1.19%

30 0.25 -4.51% 1367 10 -0.33 -0.45%

30 0.50 -8.49% 1257 11 -0.68 -0.77%

30 0.75 1.43% 1123 12 0.13 0.12%

30 1.00 5.23% 955 11 0.55 0.48%

30 1.25 5.25% 746 11 0.70 0.48%

30 1.50 8.88% 560 9 1.59 0.99%

30 -0.25 13.09% 1116 11 1.17 1.19%

30 -0.50 10.90% 994 10 1.10 1.09%

30 -0.75 6.31% 867 9 0.73 0.70%

30 -1.00 4.48% 765 10 0.59 0.45%

30 -1.25 11.85% 657 9 1.80 1.32%

30 -1.50 8.18% 538 8 1.52 1.02%

40 0.25 -11.07% 1448 7 -0.76 -1.58%

40 0.50 -5.14% 1352 7 -0.38 -0.73%

40 0.75 1.60% 1235 10 0.13 0.16%

40 1.00 5.84% 1062 10 0.55 0.58%

40 1.25 7.19% 854 10 0.84 0.72%

40 1.50 6.85% 650 9 1.05 0.76%

40 -0.25 15.40% 1203 9 1.28 1.71%

40 -0.50 11.18% 1077 9 1.04 1.24%

40 -0.75 1.94% 945 7 0.21 0.28%

40 -1.00 2.49% 855 10 0.29 0.25%

40 -1.25 11.96% 737 9 1.62 1.33%

40 -1.50 5.85% 608 8 0.96 0.73%

24

Table 6. Annualized Performance of 22 Selected Strategies

d30n15 means strategy with 30 day holding period, with signal based on mean minus 1.5 stdev of entropy indicator.

Average round-trip-transaction cost is assumed to be 50 basis points (0.5%).

Description

Mean

Annual

Return SD

Sharpe

Ratio CV Max DD HPY

Number

of Days

# of

Round

Trip

Trans.

Per Trans.

Return Net HPY

Net

Annual

HPY

Net

Sharpe

Ratio

Net

CV

Sample Period (Oct 19,2004-Dec 30,2011)

Benchmarks (buy and hold) Rf=2% (for buy and hold strategies)

S&P500 1.84% 22.90% -0.01 12.45 56.78% 13.99%

iShares Large Cap Value (IVE) 2.06% 23.99% 0.00 11.65 62.13% 15.79%

iShares Large Cap Growth (IVW) 4.79% 21.15% 0.13 4.42 49.22% 40.01%

iShares US Bond ETF (AGG) 5.30% 5.90% 0.54 1.11 12.80% 44.68%

SaEn Strategies Rf=0 (for zero investment strategies)

d1n025 2.78% 6.38% 0.43 2.29 11.73% 21.80% 802 21 1.0% 11.30% 1.50% 0.23 4.26

d1n075 2.88% 5.32% 0.54 1.85 9.89% 22.73% 541 19 1.2% 13.23% 1.74% 0.33 3.06

d1n1 1.81% 4.75% 0.38 2.62 9.47% 13.79% 448 12 1.1% 7.79% 1.05% 0.22 4.54

d1n15 2.96% 3.20% 0.92 1.08 3.41% 23.35% 276 8 2.9% 19.35% 2.48% 0.78 1.29

d2n025 3.10% 6.45% 0.48 2.08 10.00% 24.59% 822 18 1.4% 15.59% 2.03% 0.31 3.18

d2n075 2.52% 5.40% 0.47 2.14 9.27% 19.63% 559 18 1.1% 10.63% 1.41% 0.26 3.82

d2n1 1.66% 4.81% 0.35 2.90 10.66% 12.63% 459 12 1.1% 6.63% 0.89% 0.19 5.38

d2n15 2.65% 3.29% 0.81 1.24 3.41% 20.75% 283 7 3.0% 17.25% 2.23% 0.68 1.47

d3n025 2.95% 6.53% 0.45 2.21 10.68% 23.33% 839 15 1.6% 15.83% 2.06% 0.32 3.17

d3n050 2.47% 6.25% 0.40 2.53 10.33% 19.23% 716 19 1.0% 9.73% 1.30% 0.21 4.82

d3n1 1.83% 4.84% 0.38 2.64 9.86% 13.99% 470 12 1.2% 7.99% 1.07% 0.22 4.51

d3n15 2.79% 3.31% 0.84 1.19 3.41% 21.94% 289 7 3.1% 18.44% 2.38% 0.72 1.39

d10p125 1.88% 3.52% 0.53 1.87 9.28% 14.32% 531 13 1.1% 7.82% 1.05% 0.30 3.35

d10p15 1.38% 2.89% 0.48 2.09 5.61% 10.39% 440 10 1.0% 5.39% 0.73% 0.25 3.95

d10n025 1.92% 6.82% 0.28 3.55 12.77% 14.67% 920 11 1.3% 9.17% 1.23% 0.18 5.57

d10n15 2.56% 3.69% 0.69 1.44 6.93% 19.98% 331 7 2.9% 16.48% 2.14% 0.58 1.72

d30n025 1.93% 7.15% 0.27 3.70 11.32% 14.72% 1120 11 1.3% 9.22% 1.23% 0.17 5.80

d30n15 2.21% 4.54% 0.49 2.05 7.76% 17.03% 451 7 2.4% 13.53% 1.78% 0.39 2.56

n40p125 1.97% 4.65% 0.42 2.36 16.97% 15.08% 885 10 1.5% 10.08% 1.34% 0.29 3.47

n40p15 1.10% 3.57% 0.31 3.25 14.12% 8.19% 722 8 1.0% 4.19% 0.57% 0.16 6.26

d40n025 1.84% 7.28% 0.25 3.96 11.65% 14.02% 1210 9 1.6% 9.52% 1.27% 0.17 5.74

d40n15 2.06% 4.77% 0.43 2.32 8.43% 15.83% 511 7 2.3% 12.33% 1.63% 0.34 2.93

After Transactions Performance

25

Table 7. Annualized Performance of the Four Best Strategies Based on SaEn

d30n15 means strategy with 30 day holding period, with signal based on mean minus 1.5 stdev of entropy indicator.

Average round-trip-transaction cost is assumed to be 50 basis points (0.5%).

Description

Mean

Annual

Return SD

Sharpe

Ratio CV Max DD HPY

Number

of Days

# of

Round

Trip

Trans.

Per

Trans.

Return Net HPY

Net

Annual

HPY

Net

Sharpe

Ratio

Net

CV

Sample Period (Oct 19,2004-Dec 30,2011)


S&P500 1.84% 22.90% -0.009 12.45 56.78% 13.99%





d1n15 2.96% 3.20% 0.925 1.08 3.41% 23.35% 276 8 2.92% 19.35% 2.48% 0.78 1.29

d2n15 2.65% 3.29% 0.807 1.24 3.41% 20.75% 283 7 2.96% 17.25% 2.23% 0.68 1.47

d3n15 2.79% 3.31% 0.843 1.19 3.41% 21.94% 289 7 3.13% 18.44% 2.38% 0.72 1.39

d10n15 2.56% 3.69% 0.695 1.44 6.93% 19.98% 331 7 2.85% 16.48% 2.14% 0.58 1.72

Extended Sample Period (Oct 19,2004-May 17,2013)


S&P500 4.94% 21.57% 0.13 4.37 56.78% 51.14%





d1n15 2.39% 2.94% 0.81 1.23 4.05% 22.58% 277 8 2.8% 18.58% 2.01% 0.68 1.46

d2n15 2.13% 3.03% 0.70 1.42 4.08% 19.96% 285 7 2.9% 16.46% 1.79% 0.59 1.69

d3n15 2.22% 3.05% 0.73 1.37 4.31% 20.91% 292 7 3.0% 17.41% 1.89% 0.62 1.61

d10n15 1.97% 3.41% 0.58 1.73 6.93% 18.49% 341 7 2.6% 14.99% 1.64% 0.48 2.08

After Transactions Performance

26

Table 8. Trading Dates and Positions of the Best Performing Strategy

This table is for the best performing strategy of d1n15 based on SaEn. The strategy names use following method:

d1n15 strategy has a 1 day holding period, with signal based on mean minus 1.5 stdev of entropy indicator.

Purchases and sales are done at closing time of trading and we assume that prices are not affected by these

transactions.

Date Description Position

10/19/2004 Start of Sample Period Cash

2/25/2005 Purchase Growth-Value

4/28/2005 Sale Growth-Value

Cash



Cash



Cash



Cash



Cash



Cash



Cash



5/17/2013 End of Sample Period Cash

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