Dynamic asset allocation strategies based on unexpected volatility.

Zakamulin, Valerty

Volatility is widely used by both academics and practitioners as a measure of risk. In a rational expectation model, investors should be compensated for taking on more risk by receiving a higher expected excess return. Hence, the excess market return should be positively related to market volatility (see, for example, Merton [1980]). On the contrary, the contemporaneous relation between the excess market return and volatility is found to be negative. For example, using the monthly data on the S&P 500 index over the period from 1970 to 2012, the regression results are

[r.sub.t]- [r.sub.f,t] = 0.026-0.502[[sigma].sub.t], (1)

where [r.sub.t] is the return on the S&P 500 index in month t, [r.sub.f,t] is the risk-free rate of return, and [[sigma].sub.t] is the monthly volatility of the returns on the index. All regression coefficients are significantly different from zero at less than 1% level.

French et al. [1987] propose the resolution of the puzzling negative relationship between market risk and return. They decompose the total market volatility into the expected and unexpected components

[[sigma].sub.t]=[[^.[sigma]].sub.t|t-1.sup.e] + [[sigma].sub.t]" where the expected market volatility, [[^.[sigma]].sub.t|t-1.sup.e], is predicted using the generalized autoregres-sive conditional heteroskedasticity (GARCH) model of Bollerslev [1986], and the unexpected volatility is given by [[sigma].sub.t]" = [[sigma].sub.t] - [[^.[sigma]].sub.t|t-1.sup.e]. In words, unexpected volatility at time t is computed as the difference between the realized volatility at time t and expected volatility of time t that was forecasted at time t-1. Using the GARCH(1,1) model to predict the volatility and the same data as in regression (1), the contemporaneous relation between the excess market return and both the expected and unexpected volatility is estimated to be

[r.sub.t] - [r.sub.f,t]=0.021+0.624[[^.[sigma]].sub.t|t-1.sup.e] - 0.768[[sigma].sub.t]" (2)

Both the regression coefficients in front of independent variables are significantly different from zero at less than 1% level. Regression (2) advocates that the excess market return is positively related to expected volatility and negatively related to unexpected volatility.

French et al. (1987, page 27) note that the negative relation in regression (1) can be explained as follows: If the excess market return is positively related to expected volatility and volatility is highly persistent, then a positive unexpected change in volatility (and an upward revision in predicted volatility) increases future expected risk premiums and lowers. current stock prices (assuming that the future expected cash flows are unaffected). In other words, when volatility increases unexpectedly, investors anticipate higher risk in the near future and, therefore, require higher risk premiums, which causes current stock prices to fall.

There is a large strand of that explores the predictive abilities of historical and implied volatility, and proposes how historical and implied volatility can be used in dynamic asset allocation. For example, Whaley [2000]; Giot [2005]; and Bann:* et al. [2007] find that the CBOE volatility index (VIX) predicts returns on stock market indexes. Copeland and Copeland [1999] propose how to use VIX as a size and style rotation tool.

Even though there is no statistically significant evidence that historical volatility is able to predict future stock market returns, asset allocation strategies with volatility target mechanism are growing in popularity among practitioners (see, for example, Collie et al. [2011]; Butler and Philbrick [2012]; and Albeverio et al. [2013]). In particular, empirical studies usually find that active strategies that seek to keep volatility at a target level outperform passive buy-and-hold strategies.

A low volatility anomaly is documented by Ang et al. [2006] and by Blitz and van Vliet [2007]. These findings report that stocks with high past volatility have low expected future returns. Where the former authors focus on a very short-term (one-month) idiosyncratic volatility measure, the latter authors concentrate on long-term (past three years) total volatility.

To the best knowledge of this author, the predictive abilities of unexpected volatility are largely unexplored. Yet there are indications that at the individual stock level the low volatility anomaly is driven mainly by unexpected volatility. In particular, Chua et al. [2010] distinguish between expected and unexpected idiosyncratic volatility and find a strong negative relation between unexpected idiosyncratic volatility and expected future returns, while expected idiosyncratic volatility is positively related to expected future returns.

The first contribution of this article is to document that, at the aggregate stock market level, unexpected volatility is able to predict both future excess return and volatility. Specifically, unexpected volatility is negatively related to expected future returns and positively related to future volatility. This suggests that when volatility increases unexpectedly and investors require higher risk premiums, which causes current stock prices to fall, the adjustment of stock prices does not happen instantly, but takes some time.

In addition, since volatility is persistent, when volatility increases it has a tendency to remain increased for some time. This suggests that the sluggishness of both the processes can be exploited in dynamic asset allocation strategies.

The second contribution of this article is to test two strategies that dynamically reallocate between stocks and the risk-free asset, depending on the value of unexpected volatility. Whereas in the first strategy the weight of stocks is changed gradually between 0 and 1, the second strategy involves switching between stocks and the risk-free asset. We find that both the strategies deliver a substantial improvement in risk-adjusted performance as compared to traditional buy-and-hold strategies. In addition, we demonstrate that active strategies based on unexpected volatility outperform the popular active strategy with volatility target mechanism, and have some edge over the popular market timing strategy with 10-month simple moving average rule.

The rest of the article is organized as follows. The second section describes our data, while the third section outlines how we measure the actual and unexpected volatility. In the fourth section we investigate the predictive abilities of unexpected volatility. The fifth section explores the usefulness of this predictability for dynamic asset allocation between stocks and the risk-free asset. The final section presents conclusions.

DATA

In our study we use data on two stock market indexes: the S&P 500 and the Dow Jones Industrial Average. Our sample period begins January 3, 1950, and ends December 31, 2012. The data for these stock market indexes come in daily and monthly frequency.

The S&P 500 index is a value-weighted stock index which is intended to be a representative sample of leading companies in leading industries within the U.S. economy. Stocks in the index are chosen for market size, liquidity, and industry group representation. The daily capital appreciation return series for this index are obtained from Yahoo Finance.(1)

Dividends on this index are 12-month moving sums of dividends paid on the S&P 500 index. They are obtained from Robert Shiller's website.(2) Using capital appreciation return and dividend yield, we compute the continuously compounded monthly returns on the S&P 500 index, including dividends.

The Dow Jones Industrial Average index is a price-weighted stock index. Specifically, the DJIA is an index of the prices of 30 large U.S. corporations selected to represent a cross section of U.S. industry. The components of the DJIA have changed 48 times in its 1.17-year history. Changes in the composition of the DJIA are made to reflect changes in the companies and in the economy. The daily capital appreciation return series for this index over the total sample period and dividends for the period 1988 to 2012 are provided by S&P Dow Jones Indices LLC, a subsidiary of the McGraw-Hill Companies.(3)

The dividends for the period 1950 to 1987 are obtained from Barron's.(4) The dividends are smoothed using 12-month moving sum. Using capital appreciation return and dividend yield, we compute the continuously compounded total monthly returns on the DJIA index.

The monthly risk-free rate of return, proxied by the Treasury bill rate, comes from the data library of Kenneth French.(5) For the purpose of measuring the performance of alternative strategies, from the data library of Kenneth French we also obtain the monthly returns on the market portfolio (which is the value-weighted return on all NYSE, AMEX, and NASDAQ stocks), the monthly returns on the Small-Minus-Big (SMB) and High-Minus-Low (HML) Fama-French factors, and the monthly returns on the momentum (MOM) factor of Jegadeesh and Titnian [1993].

MEASURING ACTUAL AND UNEXPECTED VOLATILITY

The actual (realized) monthly variance of index returns is computed as the sum of squared daily returns. Specifically, monthly variance is calculated using the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where N, is the number of trading days in month t and [r.sub.it] is the return on the ith day of month t. It should be mentioned, however, that French et al. [1987] suggest using a correction of the formula above

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the second term (twice the sum of the products of adjacent returns) accounts for possible autocorrelation of daily returns. We tried both these methods of computations of monthly variance and these experiments showed that a method of computation had little effect on our results.

The volatility of monthly returns is assumed to evolve according to the GARCH(1,1) model.

In particular, the model is given by

[r.sub.t] = [mu] + [[epsilon].sub.t], [[epsilon].sub.t] ~ N (0, [[epsilon].sub.t.sup.2])

[[epsilon].sub.t.sup.2] = a + b[[epsilon].sub.t-1.sup.2] + c[[epsilon].sub.t-1.sup.2]

The reasons for this choice are the following: the GARCH(1,1) model is simple, robust, popular, and works very well in many practical applications (see Engle [2001]). The key features of this process are its mean reversion and its symmetry with respect to the magnitude of past returns. We also tried two asymmetric GAR.CH models, which are developed to capture the tendency for volatilities to increase more when past returns are negative. These models are the GJR-GARCH(1,1) model of Glosten et al. [1993] and the EGARCH(1,1) model of Nelson [1991]. Our experiments showed that the type of GARCH model had negligible effect on our results.

The unexpected volatility for month t is defined as the difference between. the expected volatility for month t (forecasted at the end of month t-1) and the actual volatility for month t (computed at the end of month t): [[^.[sigma]].sub.t]" = [[^.[sigma]].sub.t|t-1.sup.e]-[[sigma].sub.t]. Since the ultimate goal of this article is to analyze the real-life (that is, out-of-sample). performance of some dynamic asset allocation strategies based on unexpected volatility, in order to get rid of the look-ahead bias we employ the recursive out-of-sample forecasting using the expanding window estimation scheme to compute the expected volatility [[^.[sigma]].sub.t|t-1.sup.e].

Specifically, out-of-sample forecasting of [[^.[sigma]].sub.t|t-1.sup.e] is performed as follows. First of all, the historical data are segmented into in-sample and out-of-sample periods. The split point between the initial in-sample and out-of-sample periods is denoted by [tau], 1 < [tau] < T, where

T is the number of months in the total sample. The initial in-sample period of [1, [tau]] is used to estimate the parameters of the GARCH(1,1) process. Subsequently, the expected volatility for month [tau]+1 is forecasted using the estimated parameters of the GARCH(1,1) process.

We then expand the in-sample period by one month, perform the estimation of the parameters of the GARCH(1,1) process once again using the new in-sample period of [1, [tau] +1], and forecast the expected volatility for month + 2 using the re-estimated parameters of the GARCH(1,1) process. We repeat this procedure, pushing the endpoint of the in-sample period ahead by one month with each iteration of this process, until the expected volatility for the last month T is computed.

Our out-of-sample forecasting period starts in December 1969, which means that the first computation for the unexpected volatility [[sigma].sub.t]" is made at the end of this month. Thus, the composition of the dynamic portfolio is first determined at the end of December 1969 and, consequently, the first return observation on the active portfolio is over January 1970.

Observe that our initial in-sample period for forecasting the volatility is January 1950 through November 1969, which covers almost 20 years. This insures high forecast accuracy because the majority of studies report that GARCH forecast accuracy increases as the length of the in-sample period increases (see, for example, Figlewski [1997] and Brownlees et al. [2012]). In particular, Brownlees et al. [2012] find that GARCH models perform best using the longest available data series. Exhibit 1 plots the unexpected volatility of the returns on the S&P 500 index over our total out-of-sample period. The graph of the unexpected volatility for the DJIA index is basically the same.

[ILLUSTRATION OMITTED]

PREDICTIVE ABILITIES OF UNEXPECTED VOLATILITY

In this section we estimate the predictive abilities of unexpected volatility. Specifically, we regress the monthly excess returns on a stock index on the lagged unexpected volatility (6)

[r.sub.t] - [r.sub.f,t] = [alpha] + [beta][[sigma].sub.t-1]" + [[epsilon].sub.t] (3)

For example, if [beta]= 0 in (3), the future excess return is unrelated to the unexpected volatility. If [beta] <0, then the future excess return is negatively related to the unexpected volatility. In this case, a high unexpected volatility would predict a low excess return. If, on the other hand, [beta] > 0, then the future excess return is positively related to the unexpected volatility. If this is the case, a high unexpected volatility would predict a low excess return.

We also regress the monthly volatility of the returns on a stock index on the lagged unexpected volatility in order to find out whether there is a statistically significant relationship between these two variables

[[sigma].sub.t] = [alpha] + [beta][[sigma].sub.t-1.supll] + [epsilon], (4)

To check the robustness of our findings, we estimate each predictive regression not only for our total out-of-sample period 1.970-2012, but also for different subperiods. It is worth noting that our total out-of-sample period covers four decades, two that can be characterized as long bull markets (the 1980s and 1990s) and two that were marked by periods of severe bear markets (the 1970s and 2000s).

First, we split our total period into two (almost) equal subperiod (1970-1990 and 1991-2012) and estimate the predictive regressions. Each of these subperiods comprises a decade of a long bull market and a decade with severe bear markets. Second, we estimate the predictive regressions for each decade in our sample.

The results of our estimations are reported in Exhibit 2. In particular, for each predictive regression (3) and (4), stock market index (S&P 500 and DJIA), and historical period, Exhibit 2 reports the value of the coefficient in front of the predictor, the P-value of the coefficient, and the goodness of fit (as measured by R2-statistics). The P-values are computed using White's heteroskedasticity-consistent estimator for the asymptotic variance of the coefficient in front of the predictor (see White [1980]).

EXHIBIT 2

Results of the Estimations of the Predictive Regressions for
the Excess Return on a Stock Index and the Standard Deviation
of Returns on a Stock Index In both cases the predictor is the
lagged unexpected volatility ot returns on a stock index. Coef.
Is the value of the estimated coefficient in front of the
predictor (the value of [^.[beta]]). P-value is the p-value of
the coefficient in front of the predictor. R-square is the value
of the [R.sup.2]-statistics. Bold text indicates values that are
statistically significant at the 5% level.

 [r.sub.t] - [r.sub.f,t] [r.sub.t] - [r.sub.f,t]
 = [alpha] + [beta = [alpha] + [beta]
 [[sigma].sub.t-1.sup.u] [[sigma].sub.t-1.sup.u]
 + [[epsilon].sub.t] + [[epsilon].sub.t]

Period Coef. P-value R-squared Coef. P-value K-squared

Panel A: Standard and Poor's 500

1970-2012 -0.300 0.01 0.02 0.645 0.00 0.32

1970-1990 -0.346 0.01 0.02 0.327 0.00 0.09

1991-2012 -0.290 0.06 0.03 0.837 0.00 0.50

1970-1979 -0.498 0.30 0.02 0.653 0.00 0.32

1980-1989 -0.322 0.00 0.03 0.218 0.00 0.04

1990-1999 0.371 0.15 0.02 0.642 0.00 0.28

2000-2012 -0.371 0.02 0.05 0.815 0.00 0.50

Panel B: Dow Jones Industrial Average

1970-2012 -0.221 0.02 0.01 0.570 0.00 0.27

1970-1990 -0.298 0.00 0.02 0.315 0.00 0.09

1991-2012 -0.167 0.24 0.01 0.767 0.00 0.47

1970-1979 -0.386 0.27 0.01 0.655 0.00 0.34

1980-1989 -0.276 0.00 0.02 0.217 0.00 0.04

1990-1999 0.376 0.19 0.02 0.536 0.00 0.21

2000-2012 -0.248 0.09 0.02 0.763 0.00 0.49

The results for both the stock indexes are virtually the same, so we discuss the results for the S&P 500 index only. First, our results suggest that unexpected volatility is negatively related to future excess returns. In particular, the coefficient in front of the unexpected volatility is negative and statistically significant at the 1% level for the overall sample and the first half of the sample; for the second half of the sample the coefficient is negative, but it is statistically significant at the 6% only.

The analysis of the sign of the relation between the unexpected volatility and future excess returns for each decade in the sample reveals that the negative relation holds for each decade except the decade of 1990s. Specifically, during this decade the coefficient in front of the unexpected volatility is positive, though it is not statistically significant at the conventional statistical levels. This suggests that the negative relation between the unexpected volatility and future excess returns is not very robust; most often the relation is negative, but sometimes the relation can have the opposite sign.

It should be mentioned, however, that the decade of the 1990s was not quite a typical one for the stock market. Specifically, this decade was marked by the outset of the dot-com bubble. It is plausible to assume that the stock market dynamics during a bubble differ from "normal" dynamics.

For example, Hsu and Li [2013] study the performance of low-volatility strategies during the period from 1990 to 2012 and report that low-volatility strategies consistently outperformed passive benchmarks in all subperiods except for the second half of the 1990s. This suggests that during the second half of the 1990s the low-volatility anomaly was replaced by the high-volatility anomaly, where high-volatility stocks showed better risk-adjusted performance than low-volatility stocks. Our result for the decade of the 1990s is closely related to the findings reported by Hsu and Li [2013]. During this period, a high unexpected volatility predicted high future excess returns, where during the other periods the relation was quite the opposite.

When it comes to the relation between unexpected. volatility and future volatility, our results suggest that this relation is positive and highly statistically significant during the overall sample and each of the subsamples. That is, this relationship is very robust with respect to the sign of the relationship. Our results also suggest that, judging by the values of [R.sup.2]-statistics, the unexpected volatility can explain up to 50% variation in the future volatility.

PERFORMANCE OF DYNAMIC ASSET ALLOCATION STRATEGIES

The results reported in the previous section suggest that at the aggregate stock market level the unexpected volatility is able to predict both the future excess return and volatility. Specifically, the unexpected volatility is negatively related to expected future returns and positively related to expected future volatility. In this section, we explore the usefulness of this predictability for dynamic asset allocation between stocks and the risk-free asset.

The idea is to reduce exposure to stocks when unexpected volatility increases, and to raise that exposure when unexpected volatility decreases. The hypothetical advantage of such a strategy over the passive counterpart comes from two effects: a reduction in volatility and an increase in excess returns. Both these effects contribute to a better risk--return trade-off of a dynamic strategy. Even though the negative relationship between. the future excess return and unexpected volatility is not especially robust, nevertheless a reduction in volatility alone can improve the risk--return trade-off of an active strategy.

Denoting by [a.sub.t] the weight of stocks in the active portfolio (this weight depends on the value of unexpected volatility), the return on the active portfolio of stocks and the risk-free asset is given by

[r.sub.t.sup.active] = [a.sub.t][r.sub.l] + (1 - [a.sub.t])[r.sub.f,l]

We consider two different long-only dynamic asset allocation strategies based on unexpected volatility. Long-only means that we impose the borrowing and short-selling constraints that result in the following restriction on the weight of stocks in the active portfolio: 0 [less than or equal to] [a.sub.t] [less than or equal to] 1.

Unexpected Volatility Responsive Strategy

In the first strategy, the weight of stocks in the active portfolio is changed gradually between 0 and 1, depending on the value of unexpected volatility. In order to ensure tha.t the weight of stocks always lies between the two limits, we compute the weight in the following manner

[a.sub.t+1] = N([[sigma].sub.t]" - [E.sub.t][[[sigma].sub.t]"]/[Std.sub.t][[sigma].sub.t]") (5)

where [E.sub.t][[sigma]"] and [Std.sub.t][[sigma]"] are the mean and standard deviation ofunexpected volatility up to time t, and N(*) is the standard normal cumulative distribution function. That is, at the end of month t we compute the z-score of [[sigma].sub.t]" and the standard normal cumulative distribution function translates the z-score into a percentile value between 0 and 1.

As a result, if the value of unexpected volatility equals its historical mean, the weight of stocks equals 50%. An increase of, for example, one standard deviation in the value of unexpected volatility above its mean leads to a decrease in the weight of stocks in the active portfolio to 16%. By Contrast, a decrease of one standard deviation in the value of unexpected volatility below its mean leads to an increase in the weight of stocks to 84%.

Both the mean and standard deviation of unexpected volatility in (5) is computed using the data series beginning from January 1950. Since the first forecasted value for the volatility of stock returns refers to December 1969, over the period January 1950 to November 1969 the fitted volatility (instead of forecasted volatility) from the GARCH(1,1) model is used in the computation of unexpected volatility.

A natural passive benchmark for our active strategy is a fixed 50/50 allocation between stocks and cash. As an additional benchmark (this time active, not passive) we consider a popular dynamic asset allocation strategy with volatility target mechanism (see, for example, Collie et al. [2011], Butler and Philbrick [2012], and Albeverio et al. [2013]). This strategy is also a natural benchmark for our active strategy, since the principle that underpins volatility-target asset allocation is virtually the same: to reduce exposure to stocks when volatility is high and to increase that exposure when volatility is low.

A typical volatility-target strategy uses the trailing historical volatility, computed over periods that range from one day to one year, as a forecast for the future volatility. In addition, in a typical volatility-target strategy the weight of stocks is computed using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[sigma].sup.carget] is the target volatility, [[^.[sigma]].sub.t+1|t] is the forecasted volatility for month t + 1, and [a.sup.max] is the maximum allowable exposure to stocks (for example, Albeverio et al. [2013] use a[a.sup.max] = 200%). Since our active strategy is a long-only strategy, instead of a volatility-target strategy we implement a volatility-responsive strategy that performs similarly to a volatility-target strategy.

In a volatility-responsive strategy the weight of stocks is computed in a similar manner to (5), specifically

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [E.sub.t][[sigma]] and [Std.sub.t][[sigma]] are the mean and standard deviation of stock return volatility up to time t, and [[sigma].sub.t] is the historical volatility over month t. That is, in this strategy the historical volatility in month t is used A as a predictor of volatility in month t + 1: [[^.[sigma]].sub.t+1|t] = [[sigma].sub.t]. As a matter of fact, we varied the length of the trailing period (over which the historical volatility is computed) and our experiments showed that the use of one-month a target Ill

historical volatility produced the best performance of the volatility-responsive strategy.

To measure the performance ot alternative strategies, we use the following five measures: (1) the Sharpe ratio; (2) the alpha of the standard one-factor market model (CAPM); (3) the alpha of the Fama--French three-factor model (FF); (4) the alpha of the Fama--French--Carhart four-factor model (FFC); and (5) growth of initial investment of $100 produced by each of the alternative strategies. We test the null hypothesis that the Sharpe ratio of an alternative strategy equals the Sharpe ratio of the market portfolio. For this purpose we apply the Jobson and Korkie [1981] test with the Memmel [2003] correction. Specifically, given two portfolios, 1 and 2, with [SR.sub.1], [SR.sub.2], [rho] as their estimated Sharpe ratios and correlation coefficient over a sample of size T, the test of the null hypothesis: [H.sub.0] : [SR.sub.1]-[SR.sub.2]= 0 is obtained via the test statistic

z = [SR.sub.1] - [SR.sub.2]/[square root of (1/T[2(1-[[rho].sup.2]) + 1/2([SR.sub.1.sup.2] + [SR.sub.2.sup.2]-2[SR.sub.1][SR.sub.2][[rho].sup.2])]

which is asymptotically distributed as a standard normal. Also for each alpha we report the P-value of testing the null hypothesis that alpha is equal to zero..

To check the robustness of our findings, we measure the performances of alternative strategies not only during our total period from 1970 to 2012, but in each half of this period (1970-1990 and 1991-2012) as well. Exhibit 3 reports the descriptive statistics and performance measures of the three alternative strategies simulated using the returns on the S&P 500 and DJIA indexes. Exhibit 4 shows the growth of $100 (log scale) invested in each of the three alternative strategies.

EXHIBIT 3

Descriptive Statistics and Performance Measures of Three
Alternative Strategies; Means and Standard Deviations Are
Annualized and Given as Percentages

Passive Denotes die Passive 50/50 Portfolio of Stocks and Cash.
HistVol Denotes the Volatility-Responsive Active Strategy.
UnexVol Denotes the Unexpected-Volatility-Responsive Strategy.
The Sharpe Ratios are Annualized; The P-values of Testing the
Null Hypothesis of their Equality to the Sharpe Ratio of the
Market Portfolio are Reported in Brackets. The alpha Values
are Annualized and Given as Percentages; The Corresponding
P-values are Reported in Brackets. The bold Text Indicates
Values that are Statistically Significant at the 5% Level.

 Standard and Poor's 500

Statistics Passive HistVol UnexVol

Panel A: Period 1970-2012

Mean returns 8.05 7.60 9.00

Standard 7.79 5.38 6.51
deviation

Skewness -0.43 0.26 0.53

Sharpe ratio 0.36 0.43 0.87

 (0.95) (0.48) (0.04)

CAPM Alpha 0.04 0.93 1.94

 (0.81) (0.10) (0.00)

FF Alpha 0.05 0.92 1.72

 (0.65) (0.10) (0.01)

FFC Alpha 0.14 0.44 1.40

 (0.17) (0.43) (0.03)

Growth of $100 1,674 2,425 4,297

Panel R: Period 1970-1990

Mean returns 9.53 8.88 10.97

Standard 8.13 5.88 7.48
deviation

Skewness -0.29 0.39 0.58

Sharpe ratio 0.24 0.22 0.45

 (0.59) (1.00) (0.08)

CAPM Alpha 0.15 0.30 2.03

 (0.54) (0.71) (0.03)

FF Alpha 0.14 0.06 1.38

 (0.38) (0.94) (0.15)

FFC Alpha 0.21 -0.19 1.50

 (0.20) (0.82) (0.13)

Growth of $100 587 619 936

Panel C: Period 1991-2012

Mean returns 6.46 6.25 6.98

Standard 7.47 4.85 5.40
deviation

Skewness -0.63 -0.08 0.04

Sharpe ratio 0.45 0.66 0.72

 (0.66) (0.27) (0.12)

CAPM Alpha -0.07 1.63 2.06

 (0.80) (0.03) (0.01)

FF Alpha -0.00 1.72 2.06

 (0.98) (0.02) (0.01)

FFC Alpha 0.10 1.24 1.72

 (0.43) (0.10) (0.03)

Growth of $100 281 384 448

 Dow Jones Industrial Average

Statistics Passive Hist UnexVol
 Vol

Panel A: Period 1970-2012

Mean returns 8.29 7.88 8.94

Standard 7.68 5.64 6.65
deviation

Skewness -0.46 0.41 0.49

Sharpe ratio 0.39 0.46 0.55

 (0.51) (0.37) (0.08)

CAPM Alpha 0.48 1.22 1.94

 (0.27) (0.05) (0.01)

FF Alpha 0.07 1.17 1.72

 (0.86) (0.06) (0.01)

FFC Alpha 0.27 0.77 1.32

 (0.48) (0.22) (0.06)

Growth of $100 1,764 2,716 4,159

Panel R: Period 1970-1990

Mean returns 9.57 9.12 10.57

Standard 8.07 5.81 7.29
deviation

Skewness -0.39 0.70 0.66

Sharpe ratio 0.25 0.27 0.41

 (0.74) (0.79) (0.21)

CAPM Alpha 0.29 0.60 1.75

 (0.62) (0.49) (0.09)

FF Alpha -0.31 0.35 1.16

 (0.56) (0.69) (0.26)

FFC Alpha -0.17 0.09 0.87

 (0.77) (0.92) (0.41)

Growth of $100 590 650 863

Panel C: Period 1991-2012

Mean returns 6.90 6.61 7.25

Standard 7.31 5.48 5.98
deviation

Skewness -0.59 0.06 0.06

Sharpe ratio 0.52 0.65 0.70

 (0.54) (0.32) (0.18)

CAPM Alpha 0.70 1.87 2.25

 (0.28) (0.04) (0.02)

FF Alpha 0.53 1.91 2.20

 (0.31) (0.03) (0.02)

FFC Alpha 0.76 1.52 1.88

 (0.15) (0.09) (0.05)

Growth of $100 295 413 472

[ILLUSTRATION OMITTED]

The results for the two stock indexes are virtually similar, so we discuss the results for the S&P 500 index only. The evidence presented in Exhibit 3 suggests that the unexpected-volatility-responsive strategy outperforms both the passive strategy and (historical) volatility-responsive strategy, both over the total sample and in each half of the total sample. As compared to the 50/50 passive strategy, the unexpected-volatility-responsive strategy has higher mean returns and lower standard deviation.

While the skewness of the returns to the 50/50 passive strategy is negative, the skewness of the returns to the unexpected-volatility-responsive strategy is positive. Thus, the unexpected-volatility-responsive strategy has both lower total risk and lower downside risk compared to the passive benchmark. In addition, the unexpected-volatility-responsive strategy has not only substantially higher Sharpe ratio and long-term growth of wealth than the passive benchmark, but also delivers large and statistically significant alphas in the CAPM, FF, and FFC models (over the total sample and the second half of the sample). Over the total sample, the Sharpe ratio of the unexpected-volatility-responsive strategy is statistically significantly higher than the Sharpe ratio of the market portfolio.

It is worth noting that the volatility-responsive strategy, which serves as an active benchmark, has virtually no advantage over the passive buy-and-hold strategy over the first half of the sample. It could be argued, therefore, that the advantage of the volatility-responsive strategy over the passive buy-and-hold strategy appeared mainly over the decade of 2000s. In contrast, the unexpected-volatility-responsive strategy delivered consistent and stable outperformance when compared to the passive buy-and-hold strategy, in all decades in the sample but the 1990s.

Unexpected Volatility Switching Strategy

The second active strategy is the switching strategy. Specifically, this strategy prescribes investing either all in stocks or all in cash, depending on whether the unexpected volatility is below or above its historical average. In particular, the weight of stocks for month t + 1 is determined according to the following rule

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [E.sub.i][[sigma].sup.11] is the historical average of the unexpected volatility up to time t. Since the mean value of unexpected volatility is virtually zero, this strategy roughly prescribes investing all in stocks or all in cash, depending on the sign of unexpected volatility. For example, if the unexpected volatility is negative, all should be invested in stocks.

In essence, the unexpected-volatility-switching strategy closely resembles a market-timing strategy. Hence, besides a passive buy-and-hold strategy, a market-timing strategy can serve as an additional benchmark for the unexpected-volatility-switching strategy. The most popular market-timing strategy is based on using the 10-month simple moving average rule (SMA10), which prescribes buying stocks (moving to cash) when the stock price is above (below) the 10-month simple moving average (see Brock et al. [1992]; Siegel [2002, Chapter 171; Faber [2007]; Gwilym et al. [2010]; and Kilgallen [2012], among others). More formally, let ([P.sub.1] [P.sub.2], ... , [P.sub.T]) be the observations of the monthly closing prices of a stock price index. A 10-month simple moving average at month-end t is computed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The weight .of stocks for month t + 1 is determined according to the following rule.:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We use the 75/25 portfolio of stocks and cash as the passive buy-and-hold benchmark in this case, because over the total sample this passive portfolio has virtually the same risk (as measured by the standard deviation) as the active market timing strategy based on the SMA10 rule.

Exhibit 5 reports the descriptive statistics and performance measures of the three alternative strategies simulated using the returns on the S&P 500 and DJIA indexes. Exhibit 6 shows the growth of $100 (log scale) invested in each of the three alternative strategies. The evidence presented in Exhibit 5 suggests that, over the total sample, the unexpected-volatility-switching strategy outperforms both the passive strategy and the SMA1.0 strategy. This is valid for both the S&P 500 and DJIA stock market indexes. Yet the degree of outperformance varies over each half of the sample and for each stock market index. 0 if P, SMA, (10)

EXHIBIT 5

Descriptive Statistics and Performance Measures of Three Alternative
Strategies; Means and Standard Deviations Are Annualized and Given
as Percentages

Passive denotes the passive 75/23 portfolio of stocks and cash.
SIVIA10 denotes the market timing strategy that uses the 10-month
simple moving average strategy. Unexvol denotes the unexpected-
volatility-switching strategy. The sharpe ratios are annualized;
the P-values of testing the null hypothesis of their equality to
the sharpe ratio of the market portfolio are reported in brackets.
The alpha values are annualized and given as percentages: the
corresponding P-values are reported in brackets. The bold text
indicates values that are statistically significant at the 5% level.

 Standard and Poor's 500 Dow Jones Industrial Average

Statistics Passive SMAlu UnexVol Passive SMA10 UnexVol

Panel A: Period 1970-2012

Mean 9.44 10.74 10.64 9.80 9.92 10.41
returns

Standard 11,67 11.59 10.00 11.50 11.89 9.70
deviation

Skewness -0.43 -0.69 0.46 -0.46 -0.68 0.72

Number - 58 145 - 68 134
ol'trades

Sharpe 0.36 0.47 0.54 0.39 0.39 0.53
ratio

 (0.95) (0.30) (0.17) (0.51) (0.77) (0.23)

CAPM Alpha 0.07 2.42 3.12 0.72 1.70 3.19

 (0.81) (0.05) (0.01) (0.27) (0.20) (0.01)

FF Alpha 0.07 2.32 2.76 0.10 1.61 2.87

 (0.65) (0.06) (0.02) (0.86) (0.22) (0.02)

FFC Alpha 0.21 0.61 2.34 0.41 0.37 2.08

 (0.17) (0.61) (0.06) (0.48) (0.78) (0.10)

Growth of 3,084 7,364 7,656 3,477 5,126 7,029
SI 00

Panel B: Period 1970-1990

Mean 10.51 10.94 13.23 10.58 11.40 11.25
returns

Standard 12.22 12.74 11,36 12.13 13.02 10.87
deviation

Skewness -0.29 -0.82 0.66 -0.38 -0.82 0.89

Number of - 33 76 - 27 75
trades

Sharpc 0,24 0.26 0.50 0.25 0.29 0.34
raiio

 (0.59) (0.78) (0.11) (0.74) (0.66) (0.55)

CAPM Alpha 0.22 1.16 3.92 0.43 1.68 2.22

 (0,54) (0.51) (0.03) (0.62) (0.39) (0.24)

FF Alpha 0.21 0.76 3.37 -0.47 0.92 1.58

 (0.38) (0.67) (0.07) (0.56) (0.64) (0.41)

FFC Alpha 0.31 -1.92 3.20 -0.25 -1.35 0.70

 (0.20) (0.28) (0.10) (0.77) (0.50) (0.72)

Growth of 708 830 1,391 716 906 932
S100

Panel C: Period 1991-2012

Mean 8.14 10.53 8.16 8.80 8.50 9.60
returns

Standard 11.18 10.37 8.44 10.93 10.68 8.42
deviation

Skewness -0.62 -0.45 -0.25 -0.58 -0.47 0.27

Number of - 25 68 41 59
trades

Sharpe 0.45 0.72 0.60 0.52 0.51 0.77
ratio

 (0.66) (0.15) (0.52) (0.54) (0.83) (0.16)

CAPM Alpha -0.10 4.13 2.85 1.05 2.22 4.49

 (0.80) (0.01) (0.06) (0.28) (0.21) (0.00)

FF Alpha -0.00 4.27 2.47 0.80 2.64 4.20

 (0.98) (0.01) (0.11) (0.31) (0.13) (0.01)

FFC Alpha 0.16 3.09 2.21 1.14 1.95 3.67

 (0.43) (0.05) (0.15) (0.15) (0.27) (0.02)

Growth of 419 893 554 468 569 759
$100

[ILLUSTRATION OMITTED]

First, we consider the performance of the active strategies when the underlying passive benchmark is the S&P 500 index. In this case, over the first half of the sample the performance of the SMA10 strategy is similar to the performance of the passive benchmark, while the unexpected-volatility-switching strategy produces a Sharpe ratio that is about 10:0% higher than the Sharpe ratio of either the passive or SMA10 strategy. At the same time, all the alphas are positive and statistically significant at either the 5% or 10% level. In contrast, over the second half of the sample the unexpected-volatility-switching strategy performs better than the passive strategy, yet the SMA10 shows an even better performance.

The situation is different when the underlying passive benchmark is the DJIA index. In this case, over the first half of the sample the performance of the unexpected-volatility-switching strategy is only marginally better than the performance of the SMA10 strategy, which, in turn, performs only slightly better than the passive strategy. Over the second half of the sample, on the other hand, the unexpected-volatility-switching strategy significantly outperforms the passive benchmark, whereas the performance of the SM Al0 strategy is virtually the same as the performance of the passive benchmark.

All in all, while the unexpected-volatility-switching strategy demonstrates consistent long-term outperformance with respect to the passive benchmark, the SMA10 strategy produces clear evidence of outperformance only during the decade of the 20ons and when the passive benchmark is the S&P 500 index. As a matter of fact, over the past 50+ years the DJIA index is notorious for being practically impossible to beat using popular market-timing strategies with moving average and time-series momentum rules (see Zaka-mulin [2013]). In contrast, the unexpected-volatility-switching strategy works even when the DJIA is used as the passive benchmark.

Also observe that whereas the returns of the SMA10 are negatively skewed, the returns of the unexpected-volatility-switching strategy are most often positively skewed. This means that the return distribution of the unexpected-volatility-switching strategy has a fat right tail (higher variation on gains), while the return distribution of the SMA10 has a fat left tail (higher variation on losses). Thus, the return distribution of the unexpected-volatility-switching strategy most often has lower downside risk than the return distribution of the SMA10 strategy. It should be noted, however, that the number of trades in the unexpected-volatility-switching strategy is usually more than double the number of trades in the SMA10 strategy. That is, the implementation of the unexpected-volatility-switching strategy requires more frequent trading and, as a result, incurs more transaction costs.

CONCLUSION

Our research shows that there is usually a negative relationship between unexpected volatility and future excess returns. That is., most often increasing unexpected volatility predicts decreasing excess return on the market in the near term. In addition, our research demonstrates that there is a strong positive relationship between unexpected volatility and future volatility. Specifically, increasing unexpected volatility predicts increasing volatility in the future. These results hold for both the S&P 500 and IDJIA stock market indexes.

The uncovered relationships between unexpected volatility and future excess returns and volatility allow the use of unexpected volatility in an active asset allocation strategy. We test two dynamic asset allocation strategies based on unexpected volatility and find that both of them deliver a substantial improvement in risk-adjusted performance as compared to traditional buy-and-hold strategies. We demonstrate that active strategies based on unexpected volatility outperform the popular active strategy with volatility target mechanism and have some edge over the popular market timing strategy with SMA10 rule.

Using the period from 1970 to 2012, we find that both volatility targeting and market-timing strategies outperform their passive counterparts, mainly over the decade of the 2000s. In contrast, active strategies based on unexpected volatility outperformed the passive counterparts not only over the 2000s, but over the 1970s and the 1980s as well.

Last but not least, while market-timing strategies with moving average and momentum rules do not work when the underlying passive benchmark is the DjIA index, we find that unexpected volatility can be used as a timing indicator for the DJIA index as well.

ACKNOWLEDGEMENTS

The author is grateful to Hossein Kazemi (the editor), Steen Koekebakker, and Steve LeCompte for their insightful comments. The usual disclaimer applies.

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ENDNOTES

(1.) http://finance.yahoo.corn/

(2.) http://www.econ.yale.edu/--shiller/data.htm

(3.) http://www.djaverages.com

(4.) http://online.barrons.com

(5.) http://mba.tuck.dartmouth.edu/pagesifaculty/ken.french/data.library.html

(6.) Note that the unexpected volatility appearing in the two predictive regressions is a function of realized volatility at time t-1 and expected volatility of time t-1 that was calculated at time t-2.

VALERIY ZAKAMULIN is a professor of finance, School of Business and Law at the University of Agder, Kristiansand, Norway.

valeri.zakamouline@uia.no