ESTIMATING EXPECTED EXCESS RETURNS USING HISTORICAL AND OPTION-IMPLIED VOLATILITY

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The Journal of Financial Research • Vol. XXIX, No. 1 • Pages 95–112 • Spring 2006

ESTIMATING EXPECTED EXCESS RETURNS USING HISTORICALAND OPTION-IMPLIED VOLATILITY

Charles J. CorradoMassey University-Albany

Thomas W. Miller Jr.Saint Louis University

Abstract

We test the relation between expected and realized excess returns for the S&P 500

index from January 1994 through December 2003 using the proportional reward-

to-risk measure to estimate expected returns. When risk is measured by historical

volatility, we find no relation between expected and realized excess returns. In

contrast, when risk is measured by option-implied volatility, we find a positive

and significant relation between expected and realized excess returns in the 1994–

1998 subperiod. In the 1999–2003 subperiod, the option-implied volatility risk

measure yields a positive, but statistically insignificant, risk-return relation. We

attribute this performance difference to the fact that, in the 1994–1998 subperiod,

return volatility was lower and the average return was much higher than in the

1999–2003 subperiod, thereby increasing the signal-to-noise ratio in the latter

subperiod.

JEL Classification: G11, G14, C53

I. Introduction

A fundamental prediction of finance theory is that a positive relation exists betweenexpected risk and expected return on the market portfolio. Indeed, a large bodyof financial theory relies on the crucial role of the expected return on the marketportfolio. Unfortunately, the market portfolio is not directly observable (Roll 1977).Instead, empiricists have been forced to adopt a proxy for the unobservable marketportfolio to test the risk-return relation and other theoretical predictions.

Empirical tests of the risk-return prediction report mixed results. French,Schwert, and Stambaugh (1987) report a positive, but insignificant, relation between

We gratefully acknowledge comments and suggestions received from Maria Curelaru, John Golob,Michael Hemler, Kiesok Lee, Scott Lyden, Carolyn Moore, Rob Neal, Richard Pettway, John Scruggs,Glenn Tanner, Kenneth Yip, and seminar participants at the University of Missouri, the University of SouthCarolina, Deutsche Asset Management, and the Federal Reserve Bank of Atlanta. We are responsible forany remaining errors.

95

96 The Journal of Financial Research

the expected market risk premium and predictable volatility, whereas Harvey (1989),Turner, Startz, and Nelson (1989), and Baille and DeGennaro (1990) report a posi-tive and significant relation between the expected market risk premium and volatil-ity. Turner, Startz, and Nelson (1989) also report a negative relation. Other studiesthat report a negative relation include Campbell (1987), Glosten, Jagannathan, andRunkle (1993), Whitelaw (1994), and Lettau and Ludvigson (2003).

Several studies of the risk-return trade-off attempt to avoid assumptionsof the capital asset pricing model (CAPM) type. For example, Sheikh (1993) andGoyal and Santa-Clara (2003) suggest that total risk, not just systematic risk, is adeterminant of realized returns. Sheikh finds a significant relation between expectedtotal risk and realized returns in a sample of 30 blue-chip industrial firms. However,Sheikh does not find a significant relation between expected total risk and realizedreturns for the S&P 100 index. Using Center for Research in Security Prices (CRSP)data from 1928 through 2000, Goyal and Santa-Clara (2003) find a significant andpositive relation between average stock variance and market return.

Given the importance of the risk-return relation in finance theory, it isdisconcerting that empirical evidence supporting a positive risk-return relation forthe market portfolio is not more prevalent. One could argue that the relation be-tween expected returns and risk is more sophisticated than Merton’s (1980) pro-portional reward-to-risk model implies (Ferson, Heuson, and Su 2002). Or, per-haps more elaborate methods are required, such as Scruggs’s (1998) two-factorapproach that uses sophisticated econometric techniques (Nelson 1991). Nev-ertheless, it would be satisfying if the risk-return relation were sufficiently ro-bust so that its empirical validity could be confirmed using a relatively simplemodel.

In this article, we test the relation between expected and realized excessreturns for the S&P 500 index from January 1994 through December 2003. Weuse the proportional reward-to-risk measure in Merton (1980) to estimate expectedreturns. Merton provides the motivation when he states (p. 331): “a direct ‘ex ante’estimate for the variance rate on the market could be deduced from the price of anoption on the market portfolio. However, at the current time [late 1970s], no suchoptions are traded.” Fortunately, because options on the S&P 500 index have beentrading since 1983, a lengthy and potentially rich time series of forward-looking,market-based assessments of risk is available.

Our main contribution is comparing the respective abilities of an option-implied risk measure and historical risk measures to predict realized excess returns.We use two classes of volatility measures and a reward-to-risk estimator developedin Merton (1980) to generate a monthly time series of expected excess returns forthe S&P 500 index. Using a reward-to-risk estimator developed in Merton and eitherhistorical or option-implied volatility, we generate a monthly time series of expectedexcess returns for the S&P 500 index. Realized returns can then be compared withthese expected excess returns.

Estimating Expected Excess Returns 97

We find that if the risk measure is based on option-implied volatility, thereis a positive and significant relation between expected and realized excess returns.Tests support the hypothesis that the information contained in this forward-lookingvolatility measure results in rational forecasts of excess returns. In contrast, wefind no evidence of a relation between expected and realized returns using thecomposite extreme-value volatility estimator of Yang and Zhang (2000), or us-ing a generalized autoregressive conditional heteroskedasticity (GARCH)-in-meanvolatility specification.1

II. Forecasting Expected Returns: Theoretical Underpinnings

For a continuous-time economy in which asset prices and state variables followdiffusion processes, Merton (1973) derives an equation for the equilibrium expectedmarket risk premium, Et−1[RM,t − rf ,t ]:

Et−1[RM,t − r f ,t ] =[−JW W W

JW

]σ 2

M,t +[−JWF

JW

]σMF,t . (1)

In this intertemporal CAPM, Et−1 is the expectation operator at time t−1, RM,t isthe market return at time t, and rt,t is the risk-free rate at time t.

The basic assumption underlying equation (1) is the existence of a derivedutility of wealth function, that is, J (W (t), F(t), t), for a representative investor. Theassumed partial derivatives of this function are JW > 0 and JWW < 0, where W (t)represents wealth at time t and F(t) represents the set of investment opportunitiesin the economy at time t. If the market covariance with the state variable (σMF,t )is constant over time or if the cross-partial JWF = 0, then equation (1) becomesa proportional relation between the expected market risk premium and σ 2

M,t , themarket return variance:

Et−1[RM,t − r f ,t ] =[−JWW W

JW

]σ 2

M,t . (2)

Merton (1980) presents three theoretical economic models of reward-to-risk thatcan be used to forecast expected returns in this proportional relation. These threemutually exclusive models are expressed as follows:

Et−1[RM,t − r f ,t ] = Yi [σM,t ]3−i , (3)

1Guo and Whitelaw (2003) use S&P 100 implied volatility data from the Christensen and Prabhala(1998) study that span May 1983 through November 1995. Guo and Whitelaw find a positive relationbetween stock market risk and return over this period.


where Y i is a reward-to-risk parameter for each model indexed by i = 1, 2, 3. Fori = 1, Y 1 is an estimate of the representative investor’s relative risk aversion.2 Thatis,

Y 1 =[−JWW W

JW

](4)

in equation (2).Given a time series of values for σ and [RM,t − r f ,t ], Merton (1980) shows

that the appropriate least squares estimator for Y 1 is given by

Y 1 =

N∑t−1

(RM,t − r f ,t )

N∑t−1

σ 2M,t

+ 0.5. (5)

Then, given an estimate of the reward-to-risk measure, an expected excess returncan be estimated by

Et (RM,t+1) − r f ,t = Y 1,t

[σ 2

M,t

]. (6)

In practice, the values for σ 2 used in equations (5) and (6) are estimates of a volatilityparameter at time t. This fact in turn suggests that the method used to estimatemarket volatility could be important. Volatility estimates in this study are thereforeeither based on option-implied data or on historical data. We use implied volatilityfrom the S&P 500 index option market as forward-looking volatility estimates. Weemploy two historical methods to estimate volatility. First, we use weekly historicaldata to estimate volatility from a GARCH-in-mean specification. Second, overnonoverlapping return intervals, we estimate volatility using the composite extremevalue estimator developed by Yang and Zhang (2000).

III. Expected Volatility Estimation

Unconditional Historical Variance Estimators

Traditionally, based on a security’s closing prices, volatility is estimated by calcu-lating the variance of a time series of daily, weekly, or monthly returns. The varianceof this return series is then assumed to be a proxy for future volatility. Using closing

2For i = 2, Y 2 is a measure of the CAPM market price of risk. For i = 3, Y 3 is the reward-to-riskestimate that results by assuming a stable expected excess return on the market portfolio no matter themarket level of risk.


prices alone to estimate volatility ignores information contained in intradaily highand low prices. Using daily high and low security prices, Parkinson (1980) reports atheoretical relative efficiency gain ranging from 2.5 to 5 over the traditional close-to-close variance estimator. In contrast, the Garman and Klass (1980) and Yang andZhang (2000) variance estimators incorporate daily high, low, and closing prices.These two estimators have theoretical efficiency gains of 7.4 and 7.3, respectively,over the traditional close-to-close variance estimator.

The Yang and Zhang (2000) composite extreme-value estimator is unbi-ased, independent of any drift, and consistent in the presence of jumps in the open-ing price. Because of its potential efficiency gains, we use the Yang and Zhangextreme-value variance estimator in our empirical work.

Estimating Unconditional Historical Variances

In this article, historical return variances are calculated from daily returns overnonoverlapping intervals. Each of these intervals begins on a Tuesday following anoption expiration date and ends at the next option expiration date.

The Yang and Zhang (2000) variance estimator requires a daily open, high,low, and closing value for the S&P 500 index, which we obtained from Yahoo. Tobegin the estimation process, define the following variables in logs.

Ot = opening price of the index on day t ;Ht = high price of the index on day t ;Ht = high price of the index on day tLt = low price of the index on day t;Ct = closing price of the index on day t;ot = Ot − Ct−1;ht = Ht − Ot ;lt = Lt−O;ct = Ct − Ot ;

Given these variables, the Yang and Zhang (σ 2yz) variance estimator is calculated

as follows:

σ 2yz = 1

N

N∑i=1

(ot − o)2 + κ1

N

N∑i=1

(ct − c)2 + (1 − κ)σ 2rs . (7)

In equation (7),

o = 1

N

N∑i=1

oi (8)


and

c = 1

N

N∑i=1

ci . (9)

Yang and Zhang recommend that in practice the weight, κ , should equal

κ = 0.34

1.34 + N + 1

N − 1

. (10)

In equation (7), σ 2rs is the Rogers and Satchell (1991) extreme value variance esti-

mator, which is given by:

σ 2rs = 1

N

N∑i=1

[ht (ht − ct ) + lt (lt − ct )]. (11)

Each variance estimate is then annualized by multiplying it by the numberof trading days per annum divided by the number of days in the estimation interval(Hull 2003). Because interest is paid by the calendar day, periodic returns for theS&P 500 index are annualized by (365/d )ln(1+r), where r is the periodic return inexcess of the risk-free rate and d is the number of days in the period. In this article,we assume the risk-free rate to be the yield to maturity of a three-month Treasurybill. The Treasury-bill data are from FRED, freely available from the Web site ofthe Federal Reserve Bank of St. Louis (www.research.stlouisfed.org).

Forward-Looking Volatility

When using historical data to forecast future volatility, we implicitly assume that fac-tors generating past volatility will generate future volatility. This assumption couldbe problematic because the empirical finance literature suggests that past volatilityis often an unreliable predictor of future volatility (e.g., French, Schwert, and Stam-baugh 1987; Fama and French, 1992). The investigation of alternative methods forestimating future volatility is thus an area of ongoing interest to researchers.

An estimate of future volatility can be obtained from option-implied volatil-ity. The chief advantage of implied volatility over historical volatility is that impliedvolatility represents an ex ante market assessment of risk. For this reason, option-implied forecasts of return volatility have been frequently regarded as superior tovolatility estimates that are based on historical data. For example, Fleming (1998)finds that implied volatility from S&P 100 index options outperforms historicalvolatility in terms of ex ante forecasting power. Christensen and Prabhala (1998)also report that when they forecast future volatility using nonoverlapping periodsand S&P 100 index option data, implied volatility outperforms past volatility.


Corrado and Miller (2005) find that the ability of Chicago Board OptionExchange (CBOE) implied volatility series to forecast realized volatility has signifi-cantly improved in recent years. However, the ability of implied volatility to forecastrealized volatility is not the focus of this article. Instead, the focus is the ability ofimplied volatility, and other volatility measures, to forecast expected returns.

Option-Implied Estimates of Forward-Looking Volatility

The CBOE publishes an implied volatility index based on options on the S&P500 index. This daily time series, known by the ticker symbol VIX, is the im-plied volatility for a hypothetical option with 30 days to maturity. The CBOE Website (www.cboe.com/micro/vix/vixwhite.pdf ) freely distributes VIX data, whichare available from January 1990. As of September 22, 2003, the CBOE begancalculating the VIX index by a new method.3

The current VIX index is computed using implied volatility values corre-sponding to two nearby option expiration dates. Each day, the VIX Index representsa thirty calendar day measure of expected future volatility. Implied volatility valuesfrom each of two nearby expirations are computed according to the formula shownimmediately below.4

σ 2 = 2

T

∑i

�Ki

K 2i

erTM(Ki ) − 1

T

(F

K0

− 1

)2

, (12)

where

T = time to expiration, measured in minutes;Ki = strike price of the i th out-of-the money option: a call if Ki > F, a put

if Ki < F;�Ki = interval between strike prices;

r = risk-free interest rate to option expiration;F = forward index level, from index option prices;

K0 = first strike below the forward index level, F; andM(Ki) = midpoint of the bid-ask spread for each option with strike Ki.

To insure synchronicity with return periods used to calculate historical volatilities,VIX values are obtained on the Tuesdays following option expiration dates.

3The careful reader will note that the ticker symbol VIX was once affiliated with implied volatilityfrom S&P 100 index options. Now, an implied volatility index from S&P 100 index options is computedusing the pre-September 22, 2003 method. The new ticker symbol for this S&P 100 volatility series is VXO.

4At the following Web site, the CBOE provides a detailed numerical example of how the VIX volatilityindex is calculated: www.cboe.com/micro/vix/vixwhite.pdf. Also, see Demeterfi et al. (1999) for an in-depthdiscussion of the VIX calculation.


IV. Relation Between Realized Returns and Expected Returns

Using a Conditional Variance Estimator for Comparison

Intuition suggests that recent market returns contain the most information aboutthe current relation between return and risk. As one measure of risk for recentmarket returns, we estimate volatility in each calendar month from daily price datausing the unconditional Yang and Zhang (2000) volatility estimator. One potentialadvantage of this method is that it allows for large monthly volatility shifts.

Another way to emphasize recent market movements is to use one of theconditional heteroskedasticity models developed by Engle (1982) and generalizedby Bollerslev (1986). Although parameters for these models are estimated using arelatively long historical period, volatility forecasts from these models are respon-sive to the most recent price movements. One potential advantage of these modelsis their ability to account for recent volatility “spikes.”

The success of the original conditional heteroskedasticity model has led tomany subsequent variations. In particular, the GARCH-in-mean model provides anappealing method to estimate the relation between risk and return (Engle, Lilien,and Robins 1987).5

Given a time series of returns in excess of the riskless rate, RM,t − r f ,t , weestimate the following conditional heteroskedasticity model:

RM,t − r f ,t = β1σ2t−1 + εt , (13)

where, by assumption, εt ∼ N(0, σ 2t ). If σ 2

t is specified as α0 + α1σ2t−1 + α2ε

2t ,

the resulting conditional heteroskedasticity model is GARCH-in-mean. Once theconditional volatility parameters have been estimated, expected excess returns canbe estimated by the equation

RM,t+1 − r f ,t+1 = β1σ2t . (14)

One important motivation for using a GARCH-in-mean model is that thismodel also provides an estimate of relative risk aversion, represented by the es-timate of β in equation (14). It is thus possible to compare relative risk aversionestimates derived from a conditional heteroskedasticity model with those derivedfrom Merton’s (1980) proportional reward-to-risk model.

Estimates of a Relative Risk Aversion Coefficient

Merton (1980) suggests using a 36-month window to calculate a time series ofrelative risk aversion parameters for a representative investor. For our data sample,

5Bollerslev, Chou, and Kroner (1992) and Pagan (1996) discuss the appropriate applications of manyforms of conditional heteroskedasticity models.


we found that a 36-month window yielded some negative risk aversion parameterestimates, and we therefore used a 48-month window. All relative risk aversionparameter estimates obtained with a 48-month window were positive. The first 48-month window used data from January 1990 through December 1993 and yieldedthe first relative risk aversion parameter estimate, which was for January 1994.Rolling through the entire time series generated 120 monthly relative risk aversionparameter estimates, the last of which was for December 2003.

Table 1 presents the average of these estimated relative risk aversion coef-ficients for the S&P 500 index. The values were derived from equation (5) usingthe entire sample of monthly data. Clearly, the estimated relative risk aversion co-efficients are sensitive to volatility measures. When implied volatility is used toestimate relative risk aversion, the coefficient average is 4.08. When the Yang andZhang (2000) estimator is used to estimate relative risk aversion, the coefficientaverage is 7.78.6

Table 1 also presents a compilation of relative risk aversion estimates fromprior research. Our estimates of relative risk aversion using implied volatility arecomparable to those reported in classic, as well as recent, studies. For example,using a value-weighted portfolio of all New York Stock Exchange (NYSE) stocksfrom 1926 to 1978, Merton (1980) reports average relative risk aversion coefficientestimates of 4.80 and 4.05 for one-year and four-year periods, respectively. Mertonassumes relative risk aversion is constant over both periods. Pindyck (1988) usesmonthly returns on the combined NYSE and American Stock Exchange (AMEX)index from 1962 to 1983 and reports a relative risk aversion estimate of 3.35. Guoand Whitelaw (2003) estimate relative risk aversion using S&P 100 index datafrom 1983 to 1995 and report relative risk aversion estimates that range from 3.98to 4.40. Using S&P 500 index option data with a four-week horizon from 1983 to2001, Bliss and Panigirtzoglou (2004) estimate relative risk aversion coefficientsthat range from 4.08 to 6.33.

As discussed earlier, a GARCH-in-mean model also generates a relativerisk aversion estimate. Using weekly excess return data for the S&P 500 index fromJanuary 1990 through December 2003, our GARCH-in-mean estimate of relativerisk aversion is 3.22. By comparison, French, Schwert, and Stambaugh (1987)report an estimated relative risk aversion coefficient of 2.41 using a GARCH-in-mean model with daily excess returns from the S&P composite index from 1928to 1984. Also using a GARCH-in-mean model, Chou (1988) reports a relative riskaversion estimate of 4.50 using data for weekly returns on the NYSE value-weightedindex from 1962 to 1985.

6We also estimated the relative risk aversion parameter using the ordinary close-to-close varianceestimator and the Parkinson (1980), Garman and Klass (1980), and Rogers and Satchell (1991) extreme-value volatility estimators. Each of these generates a higher estimate of relative risk aversion than impliedvolatility.


TABLE 1. Summary of Empirically Estimated Relative Risk Aversion Parameters from TheoreticalEconomic Models, from GARCH-in-Mean, and from EGARCH-in-Mean, by Author.

Return Series Relative RiskUsed for Aversion (RRA)

Author(s) and Year Sample Period Estimation Estimate

Panel A. Estimates from Theoretical Economic Models

Merton (1980) 1926–1988 NYSE value-weighted 4.80(Constant RRA index, monthly

assumed, for one-year)Merton (1980) 1926–1988 NYSE value-weighted 4.05(Constant RRA index, monthly

assumed, for four-years)Pindyck (1988) 1962–1983 Combined NYSE and 3.35

AMEX index, monthlyGuo and Whitelaw (2003) 1983–1995 S&P 100 index, monthly 3.98–4.40Bliss and Panigirtzoglou (2004) 1983–2001 S&P 500 index, 4.08–6.33

four-week horizonGhysels, Santa-Clara, and Valkanov

(2005)1928–2000 CRSP value-weighted 1.55–3.75

index, monthlyCorrado and Miller (this article) 1990–2003 S&P 500 index, 4.08

(using CBOE VIX) monthlyCorrado and Miller (this article) 1990–2003 S&P 500 index, 7.78

(using Yang and Zhang 2000) monthly

Panel B. Estimates from GARCH-in-Mean

French, Schwert, and Stambaugh (1987) 1928–1984 S&P composite 2.41index, daily

Chou (1988) 1962–1985 NYSE value-weighted 4.50index, weekly

Corrado and Miller (this article) 1990–2003 S&P 500 index, weekly 3.22

Panel C. Estimates from EGARGH-in-Mean (Nelson 1991)

Scruggs (1998) 1950–1994 CRSP value-weighted 3.25index, monthly

Note: GARCH = generalized autoregressive conditional heteroskedasticity; EGARCH = exponentialGARCH; CBOE VIX = Chicago Board Option Exchange implied volatility index based on options on theS&P 500 index.

Interpreting the Value of a Relative Risk Aversion Coefficient

Friend and Blume (1975) show that the proportion of wealth, α, that an investorwill place into risky assets is

α = Et (RM,t+1) − r f ,t

σ 2M,t

· 1

Y 1

. (15)


Hansen and Jagannathan (1991) derive the following lower bound for therisk premium in terms of wealth:

Et (RM,t+1) − r f ,t

σM,t≤ −JWW W

JWσ (�W ). (16)

For investors who hold the market portfolio, σ (�W ) equals the standarddeviation of the stock return, σ M,t . For α to be less than one, equation (16) musthold.

Based on S&P 500 index return data and riskless rates for 1990 to 2003,the average annual risk premium was about 4.76%, and the annual S&P 500 returnstandard deviation was 15.54%. This implies a lower bound of 1.97 for the relativerisk aversion coefficient.

According to data gleaned from the Investment Company Institute Web site(www.ici.org), investor holdings among different mutual fund classifications weredistributed as follows as of April 2004: 51% to equity funds, 23% to bond funds,and 26% to money market funds. For 1990 to 2003, a relative risk aversion estimateof 3.86 results when assuming a risk premium of 4.76%, a standard deviation of15.54%, and an α coefficient of 0.51. To gain confidence in this estimate, notethat equity and bond allocations account for about 74% of worldwide mutual fundholdings. Using a risk premium of 4.76%, a return standard deviation of 15.54%,and an α of 0.74 yields a relative risk aversion coefficient of 2.66. These estimatesare largely consistent with the set of empirically estimated relative risk aversioncoefficients presented in Table 1.

These estimated ranges of relative risk aversion are not specific to oursample period. From Stocks, Bonds, Bills, and Inflation Yearbook (2003) data, theaverage annual large-cap stock return was 12.2% from 1926 through 2002, with astandard deviation of 20.5%. Given that the average annual risk-free rate was 3.8%over this period, this implies a lower bound for the relative risk aversion coefficientof 2.00. Combined with α coefficients of 0.51 and 0.74, these data yield relativerisk aversion coefficients of 3.92 and 2.70, respectively.

Forecasting Realized Returns with Expected Returns

To test the ability of Merton’s (1980) reward-to-risk measure to forecast out-of-sample realized returns, we generate a time series of expected returns as follows.First, we estimate reward-to-risk measures for each volatility measure using datafrom January 1990 through December 1993. Next, using equation (6), we forecastthe January 1994 annualized excess return using these estimated reward-to-riskmeasures, the inputs for risk, and an annualized risk-free rate. We then iteratethis procedure for each month through December 2003. Because the independentvariable is measured with error, parameter estimates are made using two-stage leastsquares.


In Table 2, we present results for three types of volatility measures: theforward-looking implied volatility, the most refined and efficient extreme-valuevariance estimator (Yang and Zhang 2000), and a conditional variance estimator(GARCH-in-mean). Table 2 contains estimated slope coefficients, p-values, andadjusted R2 for the two-stage least squares regression

RM,t − r f ,t = α + β[Et−1(RM,t − r f ,t )] + ηt , (17)

where RM,t − r f ,t is the realized market return in excess of the risk-free rate andEt−1[RM,t − r f ,t ] is the expected excess market return generated by the reward-to-risk measure corresponding to the coefficient of relative risk aversion. FollowingMerton’s (1980) recommendation to adjust for heteroskedasticity, we use White(1980) standard errors.7

Table 2 shows that a positive and significant relation between risk and returnis only obtained from a risk estimate based on the CBOE implied volatility index,VIX. However, this positive and significant relation only holds for the 1994–1998subsample. For the 1994–2003 entire sample and the 1999–2003 subsample, wefind no positive and significant slope coefficients.

In Panel A of Table 2, we report results for the entire period, 1994–2003.Results for two subperiods, 1994–1998 and 1999–2003, are reported in Panels B andC, respectively. The results for the entire sample indicate that the slope coefficientsare not significantly different from zero when historical volatility measures are usedin the regressions. The negative adjusted R2 values for these regressions suggestthat these historical volatility measures apparently have no explanatory power inforecasts of future expected returns.

Panel A reports an intercept of −0.0560 and a slope of 0.5780 for the CBOEVIX regression. In a test of the hypothesis H0: β = 1, we obtain a p-value of .2566;therefore, we cannot reject the null hypothesis of a slope coefficient equal to 1. Ina joint test of the hypotheses H0: α = 0 and β = 1, the p-value of .0464 indicatesthat the joint null hypothesis is rejected with a confidence level greater than 95%.

In Panels B and C, we again report that the slope coefficients are not signif-icantly different from zero when using historical volatility measures in the regres-sions. Again, the adjusted R2 values suggest almost no explanatory power in fore-casts of future expected returns. In marked contrast, by splitting the10-year sample into two 5-year subperiods, the significance of slope coefficientsfor implied volatility increases substantially. In the first subperiod, 1994–1998,Panel B reports an intercept of −1.6240 and a slope coefficient of 1.7927 for im-plied volatility, for which the joint hypothesis H0: α = 0 and β = 1 is not rejected. In

7Our unreported regression results using the Garman and Klass (1980), Parkinson (1980), and Rogersand Satchell (1991) extreme-value estimators do not differ substantively from the regression results basedon the Yang and Zhang (2000) estimator.


TABLE 2. Two-Stage Least Squares Regressions of Realized Periodic Returns on ExpectedPeriodic Returns for the S&P 500 Index.

Volatility Series(Squared) forEstimating RelativeRisk Aversion, p-value for H0 :Which Is Used toEstimate Expected αi = 0;

Returns αi βi R2 βi = 1 βi = 1

Panel A. 1994–2003

CBOE VIX −0.0560 0.5780 0.012 (.2566) (.0464)(.9661) (.1212)

Yang and Zhang (2000) 1.2394 0.0992 −0.007 (.0008) (.0002)(.2206) (.7043)

GARCH-in-mean 1.1312 0.1383 −0.008 (.0302) (.0052)(.3953) (.7253)

Panel B. 1994–1998

CBOE VIX −1.6240 1.7927 0.068 (.3191) (.6058)(.3581) (.0269)

Yang and Zhang (2000) 0.3070 0.6918 0.006 (.6067) (.7357)(.8108) (.2501)

GARCH-in-mean 0.8067 0.4081 −0.001 (.1548) (.2501)(.4255) (.3243)

Panel C. 1999–2003

CBOE VIX −2.6947 0.9965 0.025 (.9956) (.0377)(.3383) (.1166)

Yang and Zhang (2000) 0.9679 0.1118 −0.016 (.0303) (.0045)(.6272) (.7810)

GARCH-in-mean 0.8592 0.1493 −0.016 (.3311) (.0494)(.8081) (.8640)

Note: Results reported are from the following two-stage least squares regression:

RM,t − rf ,t = α + β[Et−1(RM,t − rf ,t )] + ηt ,

where RM,t − rf ,t is the realized market return in excess of the risk-free rate and Et−1[RM,t − rf ,t ] is theexpected excess market return generated by the reward-to-risk measure corresponding to the coefficientof relative risk aversion. Using the squares of the various monthly volatility estimates, the time seriesof Merton’s (1980) representative investor relative risk aversion parameter is calculated using a rolling48-month window. Thus, the first observation is for January 1994 and the last observation is for December2003. Once an estimate is obtained for this parameter, a squared value for volatility and a risk-free ratecan be used to generate an expected return, as specified by equation (6). The p-values, in parentheses, arecomputed using the heteroskedasticity-consistent covariance matrix (White 1980). CBOE VIX = ChicagoBoard Option Exchange implied volatility index based on options on the S&P 500 index; GARCH =generalized autoregressive conditional heteroskedasticity.


the second subperiod, 1999–2003, Panel C reports an intercept of −2.6947 and aslope coefficient of 0.9965 for implied volatility.

A test of the simple hypothesis H0: β = 1 is not rejected, as the slopecoefficient of is only trivially different from one. A test of the joint hypothesis H0:α = 0 and β = 1, however, is rejected with a confidence level greater than 95%.

In all three panels of Table 2, we see that the GARCH-in-mean modelperforms poorly. For the entire period and in both subperiods, the GARCH-in-meanmodel yields negative R2 values, which indicate an absence of any predictive power.Predictions of expected returns based on the Yang and Zhang (2000) compositeestimator also yield negative R2 values for the entire period (Panel A) and for the1998–2003 subperiod (Panel C).

As Merton (1980) notes, the volatility of realized returns is expected to bemuch higher than the volatility of expected returns. The expectation predicts a lowadjusted R2 value in a regression such as that shown in Table 2. Indeed, Merton’sprediction is confirmed in Table 2, as the highest adjusted R2 value reported is0.068.

Performance of Implied Volatility Across Subperiods

There are some noticeable performance differences of implied volatility duringthe two 5-year subperiods, 1994–1998 and 1999–2003. These differences seemto be explained by differences in the signal-to-noise ratio in the two periods. Asseen in Figure I, the dispersion of realized returns widens noticeably in the secondsubperiod relative to the first subperiod. Average returns across the two subperiods

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Figure I. Predicted and Realized Annual Returns for the S&P 500 Index, 1994–2003.


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Figure II. Relative Risk Aversion (RRA) Estimates Based on the Chicago Board Option ExchangeImplied Volatility Index, VIX, and Merton’s (1980) Estimator, 1994–2003.

are different. From a level of 466.65 in December 1993, the S&P 500 index reached alevel of 1,229.23 at the end of December 1998. The average annual return (withoutdividends) over this first subperiod was 19.38%. In strong contrast, at the endof December 2003, the S&P 500 index had fallen to 1,111.92. This yields anaverage annual return of −2.01% (without dividends) over the second subperiod.The pronounced shift from a bull market to a bear market appears to have affectedthe ex post statistical nature of the signal-to-noise ratio. This creates a differencein the difficulty of estimating the parameters of an ex ante model.

Figure II illustrates the time series of estimated relative risk aversion pa-rameters. These estimates suggest that market risk aversion was rising during thefirst three years of our sample, when market volatility was low. Thereafter, whenmarket volatility was high, relative risk aversion declined, descending precipitouslyduring the bear market of 2000–2002. These results are consistent with Bliss andPanigirtzoglou (2004), who find that estimated coefficients of risk aversion arehigh during periods of low volatility.

It is well known that upward spikes in market volatility, as proxied byimplied volatility, do not result in lower equity prices. Bliss and Panigirtzoglou(2004) conclude that this is evidence of an inverse relation between the equity riskpremium and the level of risk. They suggest that a possible explanation for thisinverse relation “lies in the proxy for consumption risk. If consumption risk is morestable than equity risk, as seems likely, then periods of high equity volatility willoverstate consumption risk and the representative investor will appear correspond-ingly less risk averse. Similarly, when equity volatility is low, it will more closelyapproximate consumption volatility and the representative investor will appear tobe correspondingly more risk averse (p. 434).”


V. Summary

Given the immense importance of the risk-and-return trade-off for the market port-folio, financial economists continue to test the empirical relation between expectedrisk and expected return. To make this theoretical relation supportable, empiricistsshould be able to uncover evidence of a relation between expected risk and expectedreturn.

In this article we investigate the ability of implied volatility and other riskmeasures to forecast realized returns of a widely used market index portfolio. Bytransforming these risk measures into proportional reward-to-risk measures, wecalculate an expected return that is compared with realized returns for the S&P 500index from January 1994 through December 2003.

We find that variance estimator efficiency plays an important role in testingthe relation between expected and realized excess returns. When risk is measuredusing historical variance estimators, we find no relation between expected andrealized excess returns in our 1994–2003 sample. In contrast, in the 1994–1998subsample, we find a positive and significant relation between expected and realizedexcess returns when the risk measure is option-implied volatility.

We estimate a proportional relation between the expected market risk pre-mium and the market return variance. Specifically, we implicitly assume that eitherσ MF,t in equation (1) is constant over time or the cross-partial JWF = 0. Our esti-mates are thus obtained without the theoretical hedge component suggested by Guoand Whitelaw (2003), who argue that, in the absence of a hedge component, relativerisk aversion estimates based on equation (1) might yield a severe downward bias.Although our estimation periods overlap each other by only 16 months, we obtainan estimated relative risk aversion coefficient of 4.08. Our estimate lies within therange of favored estimates, 3.98 and 4.40, from models 5 and 4, respectively, in Guoand Whitelaw. We, along with Guo and Whitelaw and others, find that althoughstock market volatility is positively priced, it explains only a small fraction of totalreturn variation.

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ESTIMATING EXPECTED EXCESS RETURNS USING HISTORICAL AND OPTION-IMPLIED VOLATILITY