The Option-Implied Probability of a Negative Return Konstantinos Metaxogloua,1,, Aaron D. Smithb a

b

Carleton University, Department of Economics, 1125 Colonel by Drive, Ottawa, K1S 5B6, Ontario, Canada.

University of California Davis, Department of Agricultural and Resource Economics, One Shields Avenue, Davis, CA 95616, United States.

Abstract We extract a state-price series from options prices and use it to forecast S&P 500 returns between 1990 and 2012. Our state price, the option-implied probability of a negative return in the next month, has two prominent features: (i) long swings that are correlated with the business cycle, and (ii) substantial “noise”, i.e., serially uncorrelated variation around the long swings. After smoothing out the noise, our state price predicts future returns with a positive coefficient. At the two-year horizon, it generates an out-of-sample R-squared of 0.33 and implies an annualized standard deviation of expected excess returns equal to 6%. The noise also contains information; it predicts short-horizon returns with a negative coefficient. Thus, long swings in the state price represent variation in aversion to downside risk, whereas a short-run spike indicates that the index may be overvalued and is likely to decline. Our findings can only be partially explained by financial and macroeconomic variables previously used in the literature. Keywords: Forecasting; Returns; Stochastic Discount Factor; State Prices. JEL classification: C5, G12, G13

1 Corresponding author. Carleton University, Department of Economics, 1125 Colonel by Drive, Ottawa, K1S 5B6, Ontario, Canada. Email: [email protected]

1. Introduction A typical paper on stock return predictability proceeds by first proposing a variable that may generate predictability. This proposal may be motivated by a particular utility function, a model of information processing, a specification for the dynamics of the state variables in the model economy, or an accounting identity. The researcher then assesses whether the variable of choice predicts returns—see Lettau and Ludvigson (2010) or Rapach and Zhou (2012) for a succinct, yet informative, set of examples. We take a different approach. Rather than imposing a model of economic behavior or specifying the state space, we investigate what options prices imply about future returns. To this end, we study a simple statistic—the option-implied probability of a negative monthly return on the S&P 500. This statistic, which we refer to as the state price of a negative return, is forward looking; it provides a monthly snapshot of the expectations and sentiment of the market. Moreover, it is model free, so it can be interpreted without conditioning on a model of behavior or the state space. We document two features of the state price of a negative return. The first feature is long swings in line with the business cycle, which we capture by exponentially smoothing the state price. Figure 1 shows that the smoothed state price was relatively low in the pre-recessionary periods of 1990, 1998–2001, and 2006–07, and it was relatively high in 1992–94, 2003–05, and 2009–10. The second feature is substantial variation around the long swings. This “noise,” which we measure as the the difference between the actual and smoothed state price, has negligible serial correlation and can be quite large. The state price of a negative return may increase in a particular month because traders foresee a higher probability of a negative return, or because traders become more sensitive to downside losses and so place a larger discount on a negative return (Ang et al. (2006)).2 These sources of variation have different implications for future returns; the first predicts lower returns, whereas the second implies a higher risk premium and, hence, predicts higher returns. Which of these forces dominates is an empirical question that we address in this paper. The smoothed state price exhibits strong predictive power at the one- and two-year horizons. It enters our predictive regressions with a positive coefficient indicating that it captures fluctuations in aversion to downside risk. This finding is consistent with Ang et al. (2006), who 2

See Section 2.4 in Estrada (2007) and the references there in for CAPM-like models based on downside risk measures.

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estimate a 6% premium for bearing downside risk per annum. At the two-year horizon, the smoothed state price generates an out-of-sample R-squared of 0.33 and implies an annualized standard deviation of expected excess returns around 6%. The return volatility during this period was about 13%, so this series accounts for a substantial fraction of price variation. The noise, on the other hand, is negatively correlated with future returns at horizons less than a year. The two features of the state price largely maintain their predictive power in the presence of financial and macroeconomic predictors previously used in the literature; the dividend yield explains part of the risk premium captured by our state price. Empirical exercises that focus on the predictability of stock returns, like the one in this paper, have attracted skepticism due to the use of highly persistent predictors. The combination of long horizons and small samples makes this skepticism even more severe. High persistence can bias coefficient and standard-error estimates in predictive regressions and generate spurious relations. In addition, standard corrections, such as the Newey-West, tend not to perform well (Ang and Bekaert (2007)). Following Ang and Bekaert, we use the standard errors developed by Hodrick (1992) because they retain the correct size, even in small samples and in the presence of multiple regressors. Furthermore, if predictive regressions are spurious, then one may expect poor out-of-sample forecasting performance. In a widely cited study, Welch and Goyal (2008) investigate this possibility and find that the standard predictors forecast worse out of sample than the historical mean. We show that our state-price series performs well out-of-sample. The use of the risk-neutral distribution to infer our state prices is a novelty since the existing literature has focused on the use of the risk-neutral density. While retrieving the distribution requires estimation of the first derivative of the options pricing curve, retrieving the density requires hinges on the estimation of the second derivative. Thus, the distribution is necessarily estimated more precisely (Figlewski (2010)). The greater precision generated by our approach sharpens our inference about variation in the risk premium captured by our state-price series. The remainder of the paper is organized as follows. Section 2 develops our state-price statistic and its connection to expected returns. Section 3 describes our data and estimated state prices. In Section 4, we present our forecasting approach and results follow in Section 5. Section 6 concludes.

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2. State Prices and Expected Returns 2.1. Background Before presenting our method, we develop the connection between the option-implied probability of a negative return and future returns. In dynamic equilibrium models, the price of a financial asset equals the expected value of discounted future payoffs on the asset (e.g., see Cochrane (2001)). The discount rate is stochastic, in the sense that it varies across the possible future states of the world. Payoffs in good states of the world, such as macroeconomic booms, are less valued by investors than payoffs in bad states of the world. Thus, potential payoffs in good states of the world receive a high discount relative to payoffs in bad states. The mapping from states of the world into discount rates is known as a pricing kernel or stochastic discount factor. The pricing kernel would be observable if we had options markets with payoffs that spanned the space of all possible outcomes and if we knew the probability of each outcome. This is not the case in the real world. Instead, a researcher can, in principle, estimate the pricing kernel using times series data if she is willing to make two strong assumptions: (i) that she knows the set of variables that define the state space, and (ii) that the pricing kernel and asset returns have a known stationary joint distribution. If the pricing kernel or the returns distribution varies over time in ways not captured in the model, then the researcher estimates the kernel with error. Following Engle and Rosenberg (2002), we sidestep the first assumption by projecting the state space onto what we can observe, which is the payoff on a single asset. Then, using options prices, we can compute the option-implied probability distribution of returns, which is often called the risk-neutral distribution (RND). The RND is the price of a digital option that pays a dollar in the event that returns lie in a specified range and zero otherwise. Because we observe a rich set of options contracts on outcomes in the projected space, the RND can be precisely estimated without model-specification assumptions on the utility function or the variables in the state space (Banz and Miller (1978), Breeden and Litzenberger (1978)). In this framework, the RND can be decomposed into two components: (i) the objective distribution of future returns, and (ii) the pricing kernel on the projected space—the so-called projected pricing kernel (PPK). Given a specified model for the conditional distribution of returns, one could extract the PPK from the RND. Standard asset-pricing models, such as those based on the power utility function with aggregate consumption being proportional to aggregate wealth (e.g., Brown and Gibbons (1985)),

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imply that the PPK is downward sloping and monotonic. In such models, an increase in the slope of the PPK implies an increase in expected returns. A steeper slope implies that investors discount good states more heavily, which lowers prices and raises expected returns.3 Put another way, a steeper PPK reflects higher aversion to downside risk. When risk aversion increases, investors are willing to pay less for the asset, and a higher expected return compensates them for the additional risk. A similar framework can be constructed from a behavioral perspective without appealing to dynamic equilibrium models (e.g., Han (2008)). Options prices reflect the subjective probabilities of investors about future returns. When investors become more pessimistic they are willing to pay more for out-of-the-money put options to hedge against negative returns. This increase in put option prices generates a steeper PPK. At the same time, investor pessimism depresses the price of the asset and raises average future returns. We do not attempt to extract the PPK from the RND. In principle, the econometrician could specify a model for the conditional distribution of returns, extract an implied PPK, and test it by ascertaining whether it forecasts returns. This approach faces a simultaneity problem. The researcher would obtain the PPK from an assumption about the conditional distribution of returns and then test whether this implied PPK predicts average returns. However, if she knew the return distribution, then she would already know its mean. Rather than specifying a model to handle this simultaneity problem, we focus on the median of the RND, i.e., the risk-neutral probability of a negative return. On the one hand, If the variation in the slope of the PPK dominates the variation in this statistic, the statistic will enter a predictive regression of future returns with a positive coefficient. On the other hand, if the variation in the objective probability of a negative return is the dominant one, the statistic will enter such a regression with a negative coefficient. Thus, we use a forecasting experiment to interpret variation in the RND, rather than specifying a model for the PPK a priori. We provide the details of this approach, which also has the advantage of being model free, in the next section. 2.2. Estimating the Risk-Neutral Distribution We rely heavily on the literature that originated with Banz and Miller (1978) and Breeden and Litzenberger (1978) to estimate the risk-neutral distribution. Let St denote the price 3

Several papers—Jackwerth (2000) is probably the earliest example—have found that the pricing kernel may be non-monotonic. This stylized fact has been termed the pricing kernel “puzzle.” In spite of this puzzle, the negative slope remains the dominant feature of the PPK.

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of a financial asset at time t, St+1 the payoff on that asset at t + 1, and Mt,t+1 denote the pricing kernel between periods t and t + 1. By definition, the pricing kernel has expected value equal to one. Suppose the state of the economy at time t can be described by a vector Xt , which implies that Mt,t+1 and St+1 are functions of Xt+1 . In equilibrium, the asset price is given by Et [Mt,t+1 St+1 ] , (1) St = f 1 + Rt+1 where the notation Et [·] is short hand for the expectation conditional on Xt and Rtf denotes the risk-free rate of return. At time t, the researcher observes St and the prices of any derivatives defined by payoffs on the asset, but not Xt . Thus, we focus on the risk-neutral distribution implied by the observed asset and derivative prices, which is the risk-neutral distribution of the asset return after integrating out the unobserved parts of the state space. Using the law of iterated expectations, equation (1) becomes 1

Et [Et [Mt,t+1 |Rt+1 ] St+1 ] f 1 + Rt+1 Z 1 ∗ Mt,t+1 St+1 dFt (Rt+1 ) , = f 1 + Rt+1

St =

(2)

∗ where Rt+1 ≡ St+1 /St − 1 denotes the asset return and Mt,t+1 ≡ Et [Mt,t+1 |Rt+1 ] is the PPK.

We can rewrite (2) as an expectation under the risk-neutral distribution Ft∗ St =

Z

1 1+

St+1 dFt∗ (Rt+1 ) ,

f Rt+1

(3)

with Ft∗ defined as follows Ft∗

Z

α

(α) ≡

∗ Mt,t+1 dFt (Rt+1 ).

(4)

−∞

For our analysis, it is useful to write the risk-neutral distribution as  ∗  Ft∗ (α) = Ft (α) Et Mt,t+1 |Rt+1 < α .

(5)

This expression reveals the two sources of variation in the risk-neutral distribution. The first source of variation is due to changes in the perceived probability of a return less than α captured by Ft (α). The second source of variation is due to changes in the discount applied

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 ∗  to returns less than α through steepening of the PPK captured by E Mt,t+1 |Rt+1 < α . We use Ft∗ (0) as one of our predictors in the return forecasting regressions that follow. Using X and ST to denote the strike and underlying price at expiration day T , respectively, the price of a European put option on the asset may be written as Pt (X, T ) =

1 1+

f Rt,T

Et [Mt,T × max (X − ST , 0)] =

Z

1 1+

f Rt,T

X

(X − ST )dFt∗ (ST ),

(6)

−∞

f where Rt,T denotes the return on holding a risk-free asset from t until T . We define the adjusted put option price f Pet (X, T ) ≡ (1 + Rt,T )Pt (X, T ) (7)

and take the derivative with respect to the strike price to get ∂ Pet (X, T ) = ∂X

Z

X

dFt∗ (ST ) = Ft∗ (X).

(8)

−∞

Put-call parity produces a parallel expression for call prices 

f Pet (X, T ) = 1 + Rt,T



Ct (X, T ) − St +

X f 1 + Rt,T

! ,

(9)

which leads us to define an adjusted call price as   et (X, T ) ≡ 1 + Rf (Ct (X, T ) − St ) + X, C t,T

(10)

such that et (X, T ) ∂C = Ft∗ (X). (11) ∂X As a result, we obtain the risk-neutral distribution Ft∗ (ST ) by estimating the first derivative et (X, T ) and Pet (X, T ), with respect to of the adjusted call and put option price curves, C X. We use a mixture of logistic distributions to approximate the risk-neutral distribution of the adjusted option prices in (7) and (10). We opt for mixture distributions because they offer flexible approximations to unknown distributions (Marron and Wand (1992)). By using a parametric distribution, we avoid the problems that other curve-fitting methods (e.g., splines) have in estimating the tails, and getting the estimated distribution to integrate to one—see Figlewski (2010). The logistic distribution is appealing because its integral exists in closed form, which enables us to work with the observed options pricing curve directly. Instead of fitting a distribution to the derivatives of the adjusted option prices, we fit the 7

integral of a distribution to the adjusted options prices themselves. Fitting the curve before differentiating the option pricing curve is important because it avoids arbitrary assignment of the point at which the derivative applies. Thus, by using a mixture of logistic distributions, we are able to fit a flexible function to the adjusted option prices, and simultaneously impose the restriction that its derivative maintains the properties of a distribution. Specifically, using XT to indicate the exercise price of an option that expires at T , we fit the following function Ft∗

(XT ) =

J X

ωjt Λjt (XT ; µjt , σjt ) ,

(12)

j=1

where Λjt (·) is the logistic distribution. We do so by fitting the options pricing curve to the integral of Ft∗ (XT ) given by Fet∗ (XT ) =

J X

e jt (XT ; µjt , σjt ) , ωjt Λ

(13)

j=1

Z e jt (XT ; µjt , σjt ) = Λ

     µjt XT + exp . Λjt dXT = σjt ln exp σjt σjt

(14)

P We impose Jj=1 ωjt = 1, with ωjt ≥ 0, j = 1, . . . , J. In our empirical exercise described below, we set J = 2, which implies that we fit the distribution using five parameters at each point in time. This number of parameters provides considerable flexibility when compared to other parametric distributions, such as the normal (two parameters), skew-normal and student-t (three parameters), as well as skew-t (four parameters; see Azzalini and Capitanio (2003)). We obtain estimates of the parameters for the logistic distribution in (14) using constrained non-linear least squares as follows. et (X, T ) and Pet (X, T ) in the Nt × 1 For each date t, we collect the adjusted options prices C vector of option prices ot , such that the following holds oit =

J X

e jt (XiT ; µjt , σjt ) + εit , ωjt Λ

(15)

j=1

where the subscript i denotes the ith element of ot and Xit . We then solve the constrained non-linear least squares (CNLS) problem     !2 Nt J X X 1 XiT µjt oit − ωjt σjt ln exp + exp , min ωt ,µt ,σt Nt σjt σjt i=1 j=1

8

(16)

s.t.

J X

ωjt = 1, ωjt ≥ 0, j = 1, . . . , J.

j=1

We estimate Ft∗ (X, T ) by taking the parameters estimated from (16) and plugging them into (12). We solve this CNLS problem for J = 2 separately for each date t that occurs one month ahead of an option expiration, such that µt and σt are two-dimensional vectors.4 Two practical difficulties arise in the setup described above. First, we don’t observe the exact closing options prices. Instead, we observe the end-of-day bid and ask quotes. We use the mid point between these quotes as our options price. This choice implies that we will have less than a perfect fit in our least squares problem, but we do not expect a systematic bias. To improve the accuracy of our option price data, we omit option prices implying mid quotes of less than 0.5 and those with no trading volume on day t following Figlewski (2010). Our second difficulty stems from the fact that option trading closes 15 minutes after the stock market at 3:15pm central time. This institutional fact creates an asynchronicity when we calculate the adjusted call price in (10) due to the presence of St . We address the asynchronicity issue by assuming that information in the last 15 minutes of options trading does not change the shape of the pricing curve, but may shift it up or down by an unknown constant. We can estimate this constant because we observe multiple call option prices shifted by the same constant. Thus, we adjust our CNLS problem to the following

min

at ,ωt ,µt ,σt

    !2 n J X XiT µjt 1X , oit − at dit − ωjt σjt ln exp + exp n i=1 σ σ jt jt j=1

s.t.

J X

(17)

ωjt = 1, ωjt ≥ 0, j = 1, . . . , J

j=1

where dit equals one for call prices and zero otherwise, while at is an unknown shift of the pricing curves that we estimate.5 The error term εit in (15) captures noise in the data that may result from stale quotes, our use of the bid-ask mid point rather than the unobserved efficient price, or from approximation error. We assess the fit of the logistic mixtures using the following R-squared measure Rt2

PNt

= 1 − Pi=1 Nt

(oit − obit )2

2 i=1 (oit − ot )

4

,

(18)

Throughout the paper, we use the term month to mean four weeks. eit (X, T ) = (1 + The adjusted call price that
addresses the timing discrepancy may be written as C rt,T ) Cit (X, T ) − (St + δ timing ) . The term at dit in our estimating equation plays the role of δ timing . 5

9

where Nt denotes the number of observations for the date under consideration and ot = P t (1/Nt ) N i=1 oit .

3. Preliminary Analysis 3.1. Data The data for the S&P 500 index are from the Commodity Research Bureau and span January 1990 to April 2012.6 The time-series plot of the daily closing values for the index in Figure 2 is characterized by two pairs of peaks and troughs during this period. The closing values of the index at each of these peaks are 1,527 on March 24, 2000, and 1,565 on October 9, 2007. The first of these dates corresponds to the peak of the dot-com boom, and the second date comes soon after the peak of the housing boom. The corresponding trough values are 777 on October 9, 2002, around the Enron and WorldCom scandals, and 696, on March 3, 2009. Other notable dips include September 21, 2001 (966), which is soon after the September 11 terrorist attacks, March 11, 2003 (801), which falls within the beginning of Gulf War II; and November 20, 2008 (752), soon after Lehman Brothers collapsed on September 15th of the same year. The data for the S&P 500 options are from the Chicago Board Options Exchange (CBOE). The market operates between 8:30am and 3:15pm central time and at any point in time, three near-term expiration months are trading along with three additional months from the March quarterly cycle (March, June, September and December). Currently, the strike price intervals are set at 5 points (25-point intervals for distant expiration months). The expiration date is the Saturday following the third Friday of the expiration month. Trading ceases on the business day preceding the day on which the exercise-settlement value is calculated. The options may be exercised only on the last business day before expiration (European style) with exercise resulting in delivery of cash on the business day following expiration.7 The data in hand contain information regarding trading volume, open interest, as well as closing bid and ask quotes, for calls and puts between January 1990 and April 2012. During this period of 268 months, we focus on the set of options trading on the Friday four weeks prior to expiration—there is one pair of trading and expiration dates for each 6 7

See http://www.crbtrader.com/. Additional information about the market is available at http://www.cboe.com/.

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of the 268 months. The first pair of trading and expiration dates corresponds to Friday, 01/19/1990, and Saturday, 02/17/1990, respectively. The last pair of trading and expiration dates corresponds to Friday, 04/20/2012, and Saturday, 05/19/2012. We measure the option price using the closing mid quote, defined as the average of the bid and ask quotes on the trading date after dropping observations with mid quotes below 0.5 and those with no trading volume following Figlewski (2010).8 Price information by position across the 268 trading dates is available in Table 1. The average call (put) strikes are 948 (1,083). Call mid prices are on average 167 with their put counterparts just above 70. On average, we observe 50 (45) strikes per trading date for calls (puts). Overall, the maximum number of strikes for both calls and puts on a given trading date increases over time from around 30 in the early 1990s to about 200 in the last two years of our sample, which is consistent with the increasing liquidity of the S&P 500 derivatives market over time. 3.2. Estimated Risk-Neutral Distribution We performed the curve-fitting exercise in (16) for each of the 268 trading dates in our full sample calculating the adjusted options prices in (7) and (10) using the LIBOR rate f 9 that covers the interval from t to T as our measure of Rt,T . We provide an example of the adjusted put and call option pricing curves in Figure 3. We constructed these pricing curves using the expressions in (7) and (10) for contracts trading on Friday, 10/24/2008, and expiring on Saturday, 11/22/2008. In Figure 3, we have 169 strikes with calls for 142 of them and puts for all but one of them. The strikes range from 300 to 1,700 with no calls for those between 1,175 and 1,500. The average mid-quote price is 8.7 for calls and 11.4 for puts. We estimate a weight of 0.52 on the most probable component of the logistic mixture. This component has location and scale parameters of 969 and 51, respectively. The second mixture component has a location parameter of 761 and a scale parameter of about 100. The estimated call dummy, which we include to allow for asynchronous closing times between the options and futures markets, equals -10.92, and the R2 is 0.99. We repeated the curve-fitting exercises just described for each of the remaining 267 trading 8

We exclude “mini” contracts identified by the following codes “SXZ”, “SPB”, “LSW”, “LSX”, “LSY”, “LSZ”, “XSC”, “XSE”, “XSK”, “XSL”, “XSO”,“XSP”. 9 We use the 1-Month London Interbank Offered Rate (LIBOR), based on U.S. Dollar, from FRED. See http://research.stlouisfed.org/fred2/series/USD1MTD156N. To check the robustness of this measure, f we repeated our analysis treating Rt,T as a parameter to be estimated when we fit the risk-neutral distribution to option prices. This alternative approach does not change our results, so we proceed with LIBOR.

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dates in our sample. As the bottom panel of Table 1 indicates, the minimum number of strikes across both positions for a given trading date is 12, while the maximum is 338. The median number of strikes for calls (puts) is 37 (35), which provides a sufficient number of degrees of freedom for our logistic fit. Table 2 provides summary statistics for the R2 when either a single logistic distribution or a mixture of two logistic distributions is used to infer the risk-neutral distribution. In the case of the mixture, we also report the weight associated with the most probable component of the mixture, which has a median value of 0.82. Fitting either a single logistic distribution or a mix of two logistic distributions leads to a median R2 value greater than 0.99. 3.3. State Prices Over Time We provide a time-series plot of the the state prices from January 1990 through to April 2012 in Figure 1. The vertical lines signify events contributing to the volatility of the U.S. stock market and the shaded areas correspond to the U.S. business cycles as identified by the National Bureau of Economic Research.10 In addition to the events highlighted in Bloom (2009), we also identify Alan Greenspan’s famous “irrational exuberance” speech in December 1996, the peak of the dotcom boom in March 2000, and the peak of the U.S. housing market in July 2006 using the S&P/Case-Shiller 20-City Composite Index.11 We show the values of state prices for each of the 268 option trading dates (i.e., 4 weeks before option expiration) during this period. The smoothed version of the state-prices series (thick black line) is based on an exponentially-weighted moving average (EWMA) of the actual state prices (light gray line) and captures the long swings in the state price. We smoothed the state-price series using an EWMA with a smoothing parameter of 0.8. More specifically, using zt to denote the state-price series, and ztw to denote the corresponding EWMA, the following holds  = z t=1 t ztw = (19) = 0.2z + 0.8z w t = 2, . . . , T t t−1 The state prices indicate the marginal investor’s willingness to pay for insurance against negative returns on the S&P 500. This insurance takes the form of a one-dollar payout in the event that the index return falls below zero. The state price can also be viewed as the price of a digital option that pays in the event of a negative return. The average state price is 44 cents and the standard deviation is 5 cents. The standard deviation of the smoothed series 10 11

www.nber.org/cycles.html. http://www.spindices.com/indices/real-estate/sp-case-shiller-20-city-composite-home-price-index.

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is 3 cents, and the standard deviation of the difference between the actual and smoothed series is 4 cents. The deviations from the long swings in the state price, captured by the difference between the smoothed and the actual series, which we term noise, are particularly notable between 1993 and 1995, in 1997, and in 2007. We observe the largest smoothed state prices, in excess of 47 cents, in 1992, 2003, 2004, and 2009, which are years following a recession. The smoothed series exhibits an upward trend from the beginning of the sample until the spring of 1992, a period that includes the recession between July 1990 and March 1991 and Gulf War I. This period of increase in state prices is replaced by an overall downward trend until the late 1996. The end of this second period is highlighted by Alan Greenspan’s irrational exuberance speech in December of 1996 and the beginning of the Internet bubble of the late 1990s. The upward trend resumes between the fall of 1998 and the summer of 2003, a period characterized by a relatively high return volatility. Events contributing to a volatile stock market this period include the Long-Term Capital Management (LTCM) collapse, the attacks on September 11, the Enron and WorldCom scandals, Gulf War II, and a recession. Recall that the NASDAQ peaked in March of 2002 and plunged in late September of 2002. A clear downward trend in the smoothed state prices characterizes the period between the summer of 2003 and late summer of 2007 when the signs of the most recent financial crisis emerged following the burst of the housing bubble a year earlier. This was an almost 4-year period of a steady increase in the value of S&P 500 along with an overall decrease in its volatility. The turmoil in the financial markets during the most recent crisis is accompanied with an increase in the state prices. They jump from 37 cents in August 2007 to 48 cents in January 2009, and remained at this level well into the slow recovery from the Great Recession.

4. Forecasting S&P 500 Returns 4.1. Preliminaries In this section, we compare the forecasting performance of our state-price series with that of other lagged predictors previously suggested in the literature. Our full sample consists of 247 monthly observations between January 19, 1990 and July 23, 2010. Using h to denote the forecasting horizon, we estimate linear regressions of the form rt+h − rtf = x0t β + εt

(20)

open rt+h − rtf ≡ 100 × ah × ln(SPt+h /SPtclose ) − rtf ,

(21)

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where rt+h is the log cumulative close-open return on the S&P 500 using the closing value of open . We use rtf the index on trading date t, SPtclose , and the opening value h days apart, SPt+h to denote the risk-free rate. Additionally, ah = 12, 4, 2, 1, 2/3, and 1/2 is an annualization factor. Since we consider up to five alternative horizons corresponding to 1 month, 1 quarter, half year, one year, 18 months, and two years, the values of h are: 28, 84, 168, 336, 504, and 672 days.12 We use the 3-month T-bill rate on date t as a proxy for the risk free return rtf and 247 observations for the full sample across all forecasting horizons. For brevity, we refer to annualized excess returns calculated using equation (21) as returns in the remainder of our discussion. We estimate various models, which differ in the elements of xt . More specifically, we estimate models that consist both of a single predictor and mulitple predictors aside from a constant. For both types of models, we follow Campbell and Diebold (2009) and standardize the predictors to have zero mean and standard deviation equal to one to ease interpretation of their coefficients for the models reported in Tables 5 through 8. We evaluate the forecasting ability of the models considered employing a rolling-window estimator while keeping fixed the window size, which may be written as follows βbt = argminβ

t−h X

(ys+h − x0s β)2 .

(22)

s=t−h−R+1

Given that we use a window size of R = 120 observations, our first out-of-sample forecast corresponds to December 1999. In what follows, we use n1 to denote the first 120 in-sample observations and n2 to denote the last 127 out-of-sample observations. We use these out-ofsample observations to assess the forecasting ability of our models using the metrics discussed in the next section. Correct full-sample inference requires care. Due to the use of overlapping data, even if oneperiod returns were iid, a regression of long-horizon returns on a constant would produce serially correlated residuals and proper inference should take into account the presence of such autocorrelation. There exist many cases in the literature in which the predictive ability of a variable vanishes once the econometrician accounts for the presence of autocorrelation. Moreover, standard corrections, such as the Newey-West, tend not to perform well (Ang 12

For example, the first observation for rt+h is constructed using the index closing value on Friday, 01/19/1990, and the opening value on Friday, 02/16/1990 in the case of the one month (h = 28) horizon. In the same spirit, the excess return in the case of the 6 month horizon (h = 168), is calculated using the closing value on 01/19/1990 and the opening value on Friday, 07/06/1990. Similar reasoning applies for the remaining observations and horizons.

14

and Bekaert (2007)). Following Ang and Bekaert, we use the standard errors developed by Hodrick (1992) because they retain the correct size, even in small samples and in the presence of multiple regressors. 4.2. Evaluating Predictive Performance We evaluate the models both within the estimation sample and out of sample. If our ultimate aim is to determine whether we can reject the null hypothesis that β = 0 in (20), then in-sample tests will typically be more powerful than out-of-sample tests; see Inoue and Kilian (2004) and Cochrane (2008). Intuitively, unlike in-sample tests, out-of-sample tests cannot utilize the entire available sample when estimating the parameters used to generate return predictions; in-sample tests inherently make more observations available for estimating parameters, thereby increasing estimation efficiency and test power. To check the robustness of our in-sample findings, we also evaluate the forecasting performance of the various models considered using the out-of-sample R2 calculated as follows:13 2 Rout = 1 − (M SP Ei /M SP E0 )

M SP E0 = (1/n2 )

n2 X

(23)

(rn1 +s − rn1 +s )2

(24)

(rn1 +s − rbi,n1 +s )2

(25)

s=1

M SP Ei = (1/n2 )

n2 X s=1

2 Rout captures the improvement in the mean squared prediction error (MSPE) that the predictive model i achieves relative to the historical mean, here labeled model 0. The statistic also allows protection against overfitting.14

We are also interested in testing the null hypothesis, M SP E0 ≤ M SP Ei , against the al2 ternative, M SP E0 > M SP Ei , which is equivalent to testing the null Rout ≤ 0 against the 2 alternative, Rout > 0 (see, e.g., Rapach and Zhou (2012)). We test this hypothesis using the adjusted MSPE-t statistic of Clark and West (2007), which has an asymptotic distribution well approximated by the standard normal when comparing nested models. Following Clark and West, in the first step, we calculate the following adjusted MSPE for each model i and 13

See Campbell and Thompson (2008), among others. See the discussion on page 654 of Lettau and Ludvigson (2010) regarding “protection against overfitting/data mining” for the proper use of out-of-sample tests in such framework. 14

15

forecasting horizon h   i = (rt+h − rb0,t+h )2 − (rt+h − rbi,t+h )2 − (b r0,t+h − rbi,t+h )2 fbt+h

(26)

i In the second step, we regress fbt+h on a constant and we use the resulting t-statistic for a zero coefficient rejecting the null if the t-statistic exceeds 1.645 at 5% level. We use Newey-West standard errors with the the number of lags equal to the forecasting horizon to calculate the t-statistic in the second-step regression to take into account any autocorrelation in the prediction errors.15

The overall intuition on whether regressions using predictors perform better out of sample relative to the historical mean is outlined in a very clear way in Lettau and Ludvigson (2010).16 Suppose that returns are predictable in a linear regression by some state variable xt , such that the historical mean is a biased predictor of the returns. This bias leads to a higher MSPE for forecasts that utilize only the mean relative to forecasts that incorporate the information in xt . However, estimation error in the the coefficient of xt increases the variance of the return forecasts, which increases the MSPE for predictive regressions that use xt relative to predictions based on the historical mean. As a result, the historical mean forecast is biased but has lower variance, while the forecasts based on xt are unbiased but have greater variance. Whether the forecast error is higher or lower than that for the historical mean, depends on how large the reduction in bias is relative to the increase in variance. 4.3. Predictors In addition to the state-price series, which we calculate by evaluating the risk-neutral distribution at zero, the remaining predictors fall within two broad groups. The first group consists of the volatility-related variables in Bollerslev et al. (2009). The second group consists of a series of popular macro/finance variables employed extensively in the return predictability literature.17 An in-depth discussion of these variables and how they relate to stock returns is beyond our scope; details regarding the underlying data for the various predictors are available in Appendix A. Starting with the variables in Bollerslev at al., our first predictor is the variance risk premium 15

See the discussion on page 27 of Rapach and Zhou (2012) and the references therein on the use of the adjusted MSPE t-statistic. 16 Our discussion here follows closely the material in their Section 2.7. 17 See e.g., Welch and Goyal (2008), Section 2.3. in Lettau and Ludvigson (2010), or Section 3.1 in Rapach and Zhou (2012), and the references there in.

16

defined as the difference between the implied and the realized variance. The authors use the squared VIX index from the CBOE as their measure of implied variance (IV). They estimate realized variance (RV) using the sum of the 78 within-day five-minute squared returns during the normal trading hours 9:30–4:00 p.m. along with the squared close-to-open overnight return. We include both IV and RV among our predictors. The log price-earnings ratio (PE) and the log dividend yield (DY) for the S&P 500 are the first two of our macro/finance predictors. We also include the default spread (DFSP), which is given by the difference between Moody’s BAA and AAA bond yields. Our next predictor is the 3-month T-bill rate (TBILL). Additionally, we use the term spread (TMSP), the difference between the 10-year and the 3-month Treasury yields, and the stochastically detrended 3-month T-bill (RREL). Finally, we follow Lettau and Ludvigson (2001a) and Lettau and Ludvigson (2001b) and include CAY, the residual of the cointegrating relation for log consumption (C), log asset wealth (A), and log labor income (Y), among our predictors. For the default spread, the 3-month T-bill rate, and the term spread, we use the values on the option trading date of the month prior to expiration, i.e., the same date for which we compute the state price.18 The remaining predictors are reported monthly, and we use the first lag of each one. Because we are forecasting returns approximately from mid-month to mid-month, the information in these predictors is about two weeks old.

5. Results 5.1. Summary Statistics We present summary statistics, including measures of unconditional skewness and excess kurtosis that are both moment and quantile based, for the annualized excess returns and the various predictors in Table 3. We use the expressions in White et al. (2010) for the quantilebased measures of the higher moments. The statistics are calculated using 247 monthly observations between January 1990 and July 2010. The long-horizon returns span a longer time period than the short-horizon returns; for example, the July 2010 observation on 24-month returns refers to the return from July 2010 to July 2012, whereas the 1-month return runs from July to August of 2010. These differences explain the variation in mean returns, which range between 1.4% (3 months) and 3.4% (1 18

For example, we use the value of the default spread on 01/19/1990 when we calculate the monthly returns between 01/19/1990 and 02/16/1990.

17

month). Consistent with conventional wisdom, the 24-month returns are the least volatile with a standard deviation of 13.4%, whereas the 1-month returns are the most volatile with a standard deviation of 54.3%. The most volatile year for returns of all horizons is 2008. This is largely due to the turmoil of the financial markets in the fall of the same year marked by the bankruptcy filing of Lehman Brothers, which was holding in excess of $600 billion in assets at the time. For example, the standard deviation for 1-month (3-month) returns in 2008 was 120% (62%). Both the moment- and the quantile-based measures of skewness are typically negative. The moment-based measure produces smaller absolute estimates for longer-horizon returns because the longer-horizon returns have fewer outliers than the one-month returns. The quantile-based skewness statistics, which are based on the distance of the median from the upper and lower quartiles, do not display the same pattern. As is the case with most financial series, the returns across all horizons exhibit excess kurtosis. The moment-based measures decline markedly with horizon, but the quantile-based measures show less of such a tendency. The longer-horizon returns are all highly persistent as indicated by the autocorrelation coefficient, AR(1), in the rightmost column of Table 3; AR(1) exceeds 0.9 for all horizons above a year. The majority of our predictors are also highly persistent. With the exception of the noise in the state price, the variance risk premium (IV-RV), and the realized variance (RV), the remaining predictors exhibit AR(1) values between 0.82 for implied variance (IV) and 0.99 for the 3-month T-bill rate (TBILL). 5.2. State Price as a Predictor Table 4 contains the results of a return forecasting regression of for 1- to 24-month horizons using our state-price series and a constant. We include the two components of the state price shown in Figure 1, namely, the exponentially smoothed series (STATE), and the noise, the difference between the actual and smoothed series (NOISE). We report Hodrick (1992) 2 standard errors, as well as the out-of-sample R-squared (Rout ). We also report the t-statistic for the adjusted MSPE of Clark and West (2007), for which the benchmark model is that of a rolling mean. The coefficient on the smoothed state price is positive for all horizons, but is only statistically significant for horizons of 12 months or longer. The magnitude of the coefficient is similar across the 12-, 18-, and 24-month horizons and notably smaller for shorter horizons. This result suggests that variation in the smoothed state price is driven by variation in aversion to 18

downside risk and, therefore, exhibits a positive relationship with future returns. Investors command a higher premium to hold the asset when the state price is high. Figure 4 illustrates this result by showing the smoothed state price alongside the subsequent two-year S&P 500 returns. The smoothed state price tracks subsequent returns closely, notably dropping in advance of the dotcom and housing crashes and then rebounding before the stock market recovery. The noise coefficient is negative for all horizons, and is statistically significant at the 10% level for all horizons between 3 and 18 months. The same coefficient attains its highest (absolute) value in the case of the 3-month horizon. In sharp contrast with the smoothed state price, the noise predicts negative returns and its predictive ability is stronger for shorter horizons. This result suggests that month-to-month variation in the state price reflects changes in the objective conditional distribution of short-horizon returns. Specifically, a short-run jump in the risk-neutral probability of a negative return tends to indicate that negative returns are more likely in the next few months. In contrast, a long-run increase in the risk-neutral probability of a negative return tends to indicate greater aversion to downside risk and, hence, higher expected returns in the next 1–2 years. Regarding the various model diagnostics, the full-sample R2 does not exceed 0.04 for horizons less than 6 months. For longer horizons, it is larger and reaches 0.21 for the 24-month horizon. The out-of-sample R2 increases nearly monotonically with the forecasting horizon reaching values of 0.37 and 0.33 for the 18- and 24-month horizons, respectively. The MSPE is also significantly better than that implied by a rolling mean across all horizons. For horizons exceeding 12 months the regressions imply a standard deviation of expected returns in excess of 6%. Boudoukh et al. (2008) caution that regression coefficients and R2 values increase approximately linearly with the forecasting horizon even for variables with no predictive ability. Thus, we do not emphasize the fact that these statistics are larger at longer horizons in our application. However, there are two reasons to believe that our state price does in fact have predictive power. First, because we use the Hodrick (1992) standard error estimator, our test for the null for no predictability has proper size (Ang and Bekaert (2007)). Second, we find significant out-of-sample predictability; our predictions have significantly smaller MSPE than the rolling mean. The regressions in Table 4 imply that the standard deviation of two-year returns is about 6%, which is large compared to the average return of about 2% during this period. It is slightly larger than the standard deviation of expected returns (5.5%) reported by Cochrane

19

(2011) in his one-year forecasting regressions of CRSP-value weighted excess returns using the dividend-price ratio as a predictor for the period 1947-2009. As Cochrane points out, these standard deviations are large and are similar in value to the average equity premium, which Mehra and Prescott (1985) estimated at 6%. 5.3. Single-Predictor Models We report the results from single-predictor forecasting regressions for 12- and 24-month horizons in Tables 5 and 6. To ease comparison of coefficients across models, all the predictors have been standardized to have zero mean and unit standard deviation. Similar to the results discussed in the previous section, STATE denotes the smoothed state price and NOISE denotes the difference between the actual and smoothed state price. Once again, any increase in the full-sample R2 and the coefficients’ magnitude as we expand the forecasting horizon should be interpreted with caution in the presence of highly persistent predictors and overlapping long-horizon returns. The standard deviation of expected returns is 5.6% for the 12-month and 6.2% for the 24month horizon and accounts for a notable fraction of the return volatility during this period in both cases. Furthermore, the MSPE is significantly better than that of the rolling mean for both horizons. Despite the fact that noise delivers an MSPE that is better than that of the rolling mean for the 12-month horizon, its full- and out-of-sample predictive performance is rather poor when compared to that of the other variables considered. This is consistent with the results in the previous section, which showed that the noise is a better predictor at short horizons. The dividend yield (DY) is the only other predictor with a significant coefficient (0.18) in the case of the 12-month regressions. It delivers a full-sample R2 of 0.10 and an out-of-sample R2 of 0.09. Its MSPE fails to be significantly better than that of the rolling mean and the standard deviation of the implied returns is around 5.8%. The significance of the dividendyield coefficient holds also in the case of the 24-month horizon regression—this time at the 5% level. Its magnitude is around 0.19 and is smaller by at least an order of magnitude than that of the state-price coefficient. The values for the full- and out-of-sample R2 increase to 0.21 and 0.17, respectively. The MSPE still fails to be significantly better than that of the rolling mean. The consumption-wealth ratio (CAY), and the term spread (TMSP) are the two other predictors that exhibit explanatory power in the case of the 24-month horizon with coefficients of 0.03 and 0.04, respectively. CAY delivers a full-sample R2 of 0.18 and an out-of-sample 20

R2 of just 0.06, failing, however, to deliver an MSPE that is significantly better than that of the rolling mean. The result for CAY implies a standard deviation of expected returns around 5.7%. The term spread gives rise to a higly comparable value of full-sample R2 (0.15) and to a notable out-of-sample R2 of 0.26. Its MSPE is also significantly better than that of a rolling mean, and it points to a standard deviation of about 5.3% for expected returns. None of the remaining seven predictors show significant evidence of full-sample predictive ability for either horizon. When it comes to out-of-sample performance, several variables give rise to an MSPE that is significantly better than that of a rolling mean. In the case of the 12-month horizon, this holds for the default spread (DFSP), the variance risk premium (IV-RV), and the realized variance (RV). For the 24-month horizon, the same holds for the implied variance (IV), the variance risk premium, the T-bill rate (TBILL), and the term spread. Differences between in- and out-of-sample performance can indicate that a predictor is spurious. In our application, if a predictor performs well in sample but not out of sample, then it performed worse in the 2000s than the 1990s. All else equal, it could be argued that such changes in performance weaken the evidence in favor of these predictors, i.e., it may indicate that the apparently good in-sample performance is spurious. On the other hand, many things about the 2000s were different from the 1990s. The 2000s exhibited a housing boom and bust followed by a global financial crisis, whereas the 1990s had a dotcom boom punctuated by a few localized crises. The significant out-of-sample performance of the volatility variables, for example, could thus be interpreted as an indicator that volatility risk mattered in the financial crisis in a way that it did not in the prior decade. Our analysis cannot distinguish between spurious in-sample relationships and true predictability that varies in strength over time. However, the fact that our state-price series performs well both in and out of sample provides evidence that expected returns vary over time. In the next section, we investigate the connection between the predictive ability of our state price with its counterpart of the other variables considered. 5.4. Models with Multiple Predictors The preceding section shows that our state price exhibits predictive power that is at least as strong as that of other variables previously used in the literature. However, our state prices do not reveal the economic variables underlying the changes in risk premia they capture. It is natural, therefore, to ask how much of the predictive ability of our state price is taken away in the presence of other state variables. We report the results from such multiple-predictor forecasting regressions for 12- and 24-month horizons in Tables 7 and 8, which provide the 21

same statistics with their counterparts for the single-predictor models with one difference. The benchmark model for the calculation of the adjusted MSPE statistic of Clark and West consists of a constant, the state price and noise (i.e., the model specification in Table 4). We opted for relatively parsimonious models that include at most 6 predictors, each of which is standardized to have zero mean and unit standard deviation. Other than the state price and noise, the predictors considered include CAY, the dividend yield (DY), the price-earnings ratio (PE), the detrended 3-month T-bill (RREL), and the term spread (TMSP). These are the variables that gave rise to the highest full-sample R2 values for either the 12- or 24month horizons in the single-predictor models of the previous section. If these macro/finance predictors capture components of the risk premium represented by the state prices, then we would expect them to be statistically significant and to reduce the magnitude of the stateprice coefficient as we move from the single- to the multiple-predictor specifications. In the case of the 12-month horizon, the full-sample R2 for the benchmark model is 0.12. When we include the five additional predictors one at a time, the full-sample R2 ranges from 0.14 to 0.24. The largest increase arises from the detrended T-bill followed in turn by CAY, the dividend yield, and the price-earnings ratio. For the 24-month horizon, the benchmark model has R2 equal to 0.22, and the largest increases come when we add CAY or the dividend yield. The increase in the full-sample R2 at both horizons for CAY and the dividend yield suggests that they may capture information for returns that is unrelated to that captured by our benchmark variables. In spite of the improvements in fit, only one of the estimated coefficients of the additional predictors in the two tables is significant at the 10% level, whereas the state-price coefficient remains significant in most cases. This finding reinforces our result that the state price has strong predictive power, and it suggests that the improvements in fit generated by the additional predictors could have been produced by chance. On the other hand, we do find significant improvements in out-of-sample MSPE for models including CAY and/or the price-earnings ratio. The addition of these two variables has little effect on the state-price coefficient, so any additional predictability they provide is mostly orthogonal to the state price. The dividend yield is the only variable with a noticeable effect on the state-price coefficient. Adding the dividend yield reduces the state-price coefficient from 2.1 to 1.5 in the 12-month horizon and from 2.2 to 1.6 in the 24-month horizon. These drops put the state-price coefficient below the threshold for significance at 5%, although it remains significant at 10% in the 24-month model. The out-of-sample MSPE is not improved significantly by adding

22

the dividend yield to the model. The predictive content of the dividend yield, therefore, has some overlap with that of our state-price variable. Augmenting our benchmark model with more than one predictor at a time produces large improvements in both the full and the out-of-sample R2 . In all three cases, the full-sample R2 exceeds 0.23 in the 12-month models and 0.39 in the 24-month models, while its out-ofsample analog exceeds 0.34 and 0.47 for the two horizons, respectively. All but one of the estimated coefficients are statistically insignificant, but the out-of-sample MSPE improves significantly over the benchmark model. In sum, the state prices maintain their predictive power when paired with CAY, the priceto-earning ratio, the term spread, or the stochastically detrended T-bill rate, especially for the 24-month horizon. There seems to be some overlap in their predictive content with that of the dividend yield in both the 12- and the 24-month horizons, and evidence that CAY and the price-earnings ratio provide some additional out-of-sample predictive power.

6. Conclusion We show that the option-implied probability of a negative return on the S&P 500 exhibits strong predictive power on the index returns between 1990 and 2012. This novel predictor, which we term the state price, captures the investors’ willingness to pay for insurance on the downside of the market and exhibits two prominent features—long swings in line with the business cycle and substantial short-term variation. The first feature captures aversion to downside risk and exhibits a strong positive relationship with index returns at the two-year horizon. The second feature captures an increase in the negative-return probability and exhibits a negative relationship with index returns at shorter horizons. Overall, the state price gives rise to an out-of-sample R-squared of about 0.33 and to a standard deviation for expected returns in the neighborhood of 6% at the two-year horizon. Among popular predictors previously used in the literature, the dividend yield exhibits some overlap with our state price in terms of predictive power but the remaining do not. This finding suggests considerable scope for new macro/finance models to improve understanding of risk premia and return predictability. In particular, the apparent correlation of the smoothed state price with the business cycle suggests that further research into the connection between risk and the business cycle could be fruitful.

23

Acknowledgements We thank the participants at the April 2011 Applied Econometrics Conference at the Federal Reserve Bank of St. Louis for useful comments.

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D. Brown, M. Gibbons, A Simple Econometric Approach for Utility-Based Asset Pricing Models, Journal of Finance 40 (2) (1985) 359–381. J. Jackwerth, Recovering Risk Aversion from Option Prices and Realized Returns, Review of Financial Studies 13 (2) (2000) 433–451. B. Han, Investor Sentiment and Option Prices, Review of Financial Studies 21 (1) (2008) 384–414. J. Marron, M. Wand, Exact Mean Integrated Squared Error, The Annals of Statistics 20 (2) (1992) 712–736. A. Azzalini, A. Capitanio, Distributions Generated by Perturbation of Symmetry with Emphasis on a Multivariate Skew t Distribution, Journal of the Royal Statistical Society Series B 65 (2) (2003) 367–389. N. Bloom, The Impact of Uncertainty Shocks, Econometrica 77 (3) (2009) 623–685. S. Campbell, F. Diebold, Stock Returns and Expected Business Conditions: Half a Century of Direct Evidence, Journal of Business and Economic Statistics 27 (2) (2009) 266–278. A. Inoue, L. Kilian, In-Sample or Out-of-Sample Tests of Predictability: Which One Should We Use?, Econometric Reviews 23 (3) (2004) 371–402. J. Cochrane, The Dog That Did Not Bark: A Defense of Return Predictability, Review of Financial Studies 21 (4) (2008) 1533–1575. J. Campbell, S. Thompson, Predicting Excess Stock Returns Out of Sample: Can Anything Beat the Historical Average?, Review of Financial Studies 21 (4) (2008) 1509–1531. T. Clark, K. West, Approximately normal tests for equal predictive accuracy in nested models, Journal of Econometrics 138 (2007) 291–311. T. Bollerslev, G. Tauchen, H. Zhou, Expected Stock Returns and Variance Risk Premia, Review of Financial Studies 22 (11) (2009) 4463–4492. M. Lettau, S. Ludvigson, Consumption, Aggregate Wealth, and Expected Stock Returns, Journal of Finance 56 (3) (2001a) 815–849. M. Lettau, S. Ludvigson, Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia are Time-Varying, Journal of Political Economy 109 (6) (2001b) 1238–1287.

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H. White, T. Kim, S. Manganelli, Modeling Autoregressive Conditional Skewness and Kurtosis with Multi-Quantile CAViaR, in: T. Bollerslev, J. Russell, M. Watson (Eds.), Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle, Oxford University Press, 2010. J. Boudoukh, M. Richardson, R. Whitelaw, The Myth of Long Horizon Predictability, Review of Financial Studies 21 (4) (2008) 1577–1605. J. Cochrane, Presidential Address: Discount Rates, Journal of Finance 66 (4) (2011) 1047– 1108. R. Mehra, E. Prescott, The Equity Premium Puzzle, Journal of Monetary Economics 15 (2) (1985) 145–161.

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Appendix A. Data in the Forecasting Regressions This Appendix provides the details behind the data used in the forecasting regressions of Section 4. 1. Implied Variance (IV): We use monthly data for the implied variance series from Bollerslev et al. (2009), which has been extended to 2012, and is readily available from Hao Zhou’s personal website.19 2. Realized Variance (RV): We use monthly data for the realized variance series from Bollerslev et al. (2009) which has also been extended to 2012, and is also readily available from Hao Zhou’s personal website. 3. CAY: Quarterly data for the CAY are available from Martin Lettau’s website.20 We use the CAY value of the previous quarter for the three months of the current quarter. For example, we use the 1989Q4 CAY value for January, February, and March 1990. 4. Default Spread (DFSP): We calculated the default spread as the difference between Moody’s BAA and AAA corporate bond yields for each of the trading dates in our sample using information from the FRED website.21 5. Dividend Yield (DY): We calculated the dividend yield for the S&P 500 using the index closing value on the last trading date of the previous quarter and the sum of the cash dividends per share for the 4 last quarters. For example, we calculated the dividend yield for January, February, and March 1990 using the index closing value on 12/31/1989, and the cash dividends per share on 12/31/1989, 09/30/1989, 06/30/1989, and 03/31/1989, as reported by Standard and Poor’s. 22 6. Price-Earnings ratio (PE): We calculated the price-earnings ratio for the S&P 500 using the index closing value on the last trading date of the previous quarter and the sum of the operating earnings per share for the 4 last quarters. For example, we calculated the price-earnings ratio for January, February, and March 1990 using the index closing value on December 12/31/1989, and the operating earnings per share on 12/31/1989, 09/30/1989, 06/30/1989, and 03/31/1989, as reported by Standard and Poor’s.23 19

https://sites.google.com/site/haozhouspersonalhomepage/. http://faculty.haas.berkeley.edu/lettau/data/cay_q_11Q3.txt. 21 For BAA yields, see http://research.stlouisfed.org/fred2/series/DBAA?cid=119. yields, see http://research.stlouisfed.org/fred2/series/DAAA?cid=119. 22 http://www.spindices.com/documents/additional-material/sp-500-eps-est.xlsx. 23 Ibid. 20

27

For AAA

7. T-bill Rate (TBILL): This is the 3-month yield from the U.S. Treasury, for each of the trading dates in our sample.24 8. Stochastically detrended T-bill rate (RREL): Using T B t to denote the monthly average of the T-bill rate for the month under consideration, we calculated RREL as follows: RRELt = P T B t − 12 τ =1 T B t−τ . For example, we calculated T B for January 1990 using the average TBILL between January 1989 and December 1989. 9. Term Spread (TMSP): This is the difference between the 10-year and the 3-month yields from the U.S. Treasury for each of the trading dates in our sample.25

24

http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/ TextView.aspx?data=yield. 25 http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/ TextView.aspx?data=yield.

28

Tables

Table 1: Summary statistics for the S&P 500 options

Postion Call Put Total

Number of trading dates 268 268 268

Postion Call Put Total

Strike Prices min mean max 50 948 1,800 250 1,083 3,000 50 999 3,000

mean 50 45 59

Average Prices bid ask mid 165 168 167 70 72 71 120 123 122

Number of strikes per trading date median std.dev. min 37 42 8 35 39 7 41 50 12

max 284 336 338

Note: The columns under the heading “Strike Prices” show the minimum, mean, and maximum observed prices across 268 trading dates, which are four weeks prior to option expiration for each month between Jan-1990 to Apr-2012. The columns under “Average Prices” show the mean bid and ask prices, and the mean of the midpoint between the bid and ask prices.

Table 2: Summary statistics for the the fit of the adjusted call and put option prices

Min Max Median

Weight Mixture 0.500 0.998 0.823

R-squared Single Mixture 0.969 0.973 1.000 1.000 0.999 1.000

Note: The table summarizes the results for the curve-fitting exercises we performed to estimate the risk-neutral distribution on each of the 268 trading dates, which are four weeks prior to option expiration for each month between Jan-1990 and Apr-2012. We show the minimum, maximum, and median estimates of the weight parameter when we use a mixture of two logistic distributions. The two rightmost columns summarize the fit of the single and mix models using the standard R2 .

29

Table 3: Summary statistics of variables in forecasting regressions

Variable

Mean

Std. dev.

Skew.

Ex. Kurt.

Skew.

Ex. Kurt.

(mom.)

(mom.)

(quant.)

(quant.)

AR(1)

1m excess return

3.402

54.280

-1.657

11.000

-0.129

1.889

-0.003

3m excess return

1.387

30.550

-1.398

8.020

-0.086

0.907

0.631

6m excess return

1.873

22.930

-1.618

8.310

0.076

1.050

0.795

12m excess return

2.049

17.560

-1.233

4.976

-0.012

1.441

0.912

18m excess return

2.029

14.640

-1.177

4.213

0.024

0.676

0.950

24m excess return

1.871

13.400

-0.991

3.504

-0.152

0.734

0.965

STATE

0.444

0.029

-0.184

2.105

0.000

-0.788

0.940

NOISE

0.001

0.040

0.248

3.799

0.102

0.390

-0.017

IV-RV

18.310

20.300

-2.906

41.400

0.405

0.438

0.318

IV

39.380

35.970

3.384

19.640

0.080

0.904

0.815

RV

21.070

38.580

7.923

86.220

0.264

2.825

0.653

CAY

0.000

1.000

-0.248

1.684

-0.274

-1.367

0.982

DFSP

0.953

0.452

3.243

15.730

0.118

3.531

0.950

DY

0.000

1.000

0.100

1.976

0.406

-0.910

0.988

PE

0.000

1.000

0.529

2.362

0.292

-0.436

0.965

RREL

0.000

1.000

-0.407

2.897

-0.195

0.376

0.972

TBILL

3.762

2.029

-0.227

2.249

-0.434

-0.588

0.992

TMSP

0.000

1.000

-0.054

1.725

0.143

-1.104

0.970

Note: The table summarizes the variables that we use in the forecasting regressions for Jan-1990 to Jul-2010. We measure excess returns as the annualized log change in the S&P 500 minus the 3-month T-bill rate. Each excess return is measured over a period beginning on the date on which we calculate the monthly state price, which is four weeks before each option expiration. STATE is the smoothed state price and NOISE is the difference between the actual and smoothed state price. IV and RV are the implied and realized variance, respectively, from Bollerslev et al. (2009). CAY is the consumption-wealth ratio from Lettau and Ludvigson (2001a). The default spread, DFSP, is the difference between Moody’s BAA and AAA corporate bond yields. DY and PE are the log price-earnings ratio and the log dividend yield for the S&P 500. TBILL is the 3-month T-bill rate and RREL is the stochastically detrended version of TBILL. The term spread, TMSP, is the difference between the 10-year and the 3-month Treasury yields. Additional details about the variables are available in Appendix A. We report both moment- and quantile-based measures of skewness and kurtosis.

30

Table 4: State-price forecasting regressions over multiple horizons

Model STATE NOISE Constant R2full 2

R out MSPE-adj sd(E[R])

Forecasting horizon in months 6 12 1.329 2.098 ** (0.988) (0.981) *** -0.960 ** -0.768 * (0.373) (0.413) -0.571 -0.911 ** (0.446) (0.452)

1 0.387 (1.059) -0.822 (0.808) -0.137 (0.472)

3 0.899 (1.004) -1.153 (0.444) -0.385 (0.451)

0.00

0.02

0.04

0.03 5.04 * 3.28%

0.06 7.94 * 4.89%

0.09 7.25 * 4.96%

18 2.434 ** (1.083) -0.648 ** (0.256) -1.061 ** (0.502)

24 2.224 ** (1.015) -0.294 (0.217) -0.969 ** (0.473)

0.12

0.23

0.22

0.21 7.35 * 6.33%

0.37 7.83 * 7.08%

0.33 9.61 * 6.34%

Note: The table shows the results of forecasting regressions over six horizons. For each horizon, we regress excess returns on the smoothed state price (STATE) and the difference between the actual and smoothed state price (NOISE). Both STATE and NOISE are measured in the beginning of the time interval over which we calculate excess returns. Hodrick (1992) standard errors are in parentheses below the coefficient estimates. For coefficient estimates, the asterisks indicate statistical significance as follows: 1%(***), 5%(**), 10%(*). Rf2 ull is the adjusted R2 over the full 2 sample and Rout is the out-of-sample R-squared from a prediction excercise using a rolling window of 120 observations. MSPE-adj is the t-statistic of the adjusted MSPE in Clark and West (2007) 2 2 > 0. The asterisk indicates rejection of the ≤ 0 against the alternative, Rout testing the null Rout null at 5%. The row denoted sd(E[R]) reports the standard deviation of the fitted values from the full-sample regression.

31

Table 5: Single-predictor forecasting regressions: 12 months

Model STATE NOISE CAY DFSP IV IV-RV DY PE RREL RV TBILL TMSP

Slope 1.928 ** (0.934) -0.524 (0.364) 0.023 (0.014) 0.030 (0.070) 0.001 (0.001) 0.001 (0.001) 0.177 * (0.098) -0.222 (0.167) 0.070 (0.046) 0.000 (0.001) -0.003 (0.015) 0.022 (0.024)

Constant -0.836 *** (0.431) 0.021 *** (0.034) 0.009 (0.038) -0.008 (0.063) -0.005 (0.033) 0.001 (0.039) 0.713 * (0.373) 0.678 (0.486) 0.034 (0.030) 0.015 (0.031) 0.031 (0.074) -0.019 (0.057)

R2full

R2out

0.10

MSPE-adj 0.15 2.25 *

sd(E[R]) 5.56%

0.01

0.04

1.70 *

2.08%

0.09

0.04

1.46

5.33%

0.00

0.13

1.71 *

1.36%

0.01

0.06

1.34

2.31%

0.01

0.03

1.90 *

2.12%

0.10

0.09

1.15

5.77%

0.06

0.06

1.56

4.49%

0.11

0.13

1.19

5.97%

0.00

0.07

1.85 *

1.04%

0.00

0.07

1.28

0.56%

0.02

0.05

1.30

2.77%

Note: The table shows the results of forecasting regressions at the 12-month horizon. Each row reports the results from a regression of 12-month excess returns on the predictor indicated in the “Model” column and a constant. The predictors are summarized in Table 3. The regression results are displayed as in Table 4 and are based on standardized (zero mean, unit standard deviation) predictors.

32

Table 6: Single-predictor forecasting regressions: 24 months

Model STATE NOISE CAY DFSP IV IV-RV DY PE RREL RV TBILL TMSP

Slope 2.159 (0.979) -0.035 (0.148) 0.025 (0.015) 0.034 (0.053) 0.000 (0.001) 0.001 (0.001) 0.189 (0.096) -0.189 (0.166) 0.022 (0.028) 0.000 (0.000) -0.012 (0.011) 0.043 (0.022)

**

*

**

**

Constant -0.940 *** (0.457) 0.019 *** (0.033) 0.007 (0.038) -0.013 (0.055) 0.004 (0.033) 0.006 (0.039) 0.758 ** (0.357) 0.578 (0.485) 0.023 (0.032) 0.016 (0.031) 0.066 (0.053) -0.056 (0.063)

R2full

R2out

0.21

MSPE-adj 0.33 3.09 *

sd(E[R]) 6.23%

0.00

0.00

0.75

0.14%

0.18

0.06

1.18

5.66%

0.01

0.03

1.33

1.53%

0.01

0.12

1.98 *

1.30%

0.01

0.04

1.90 *

1.41%

0.21

0.17

1.25

6.16%

0.08

0.05

1.04

3.82%

0.02

0.01

0.71

1.89%

0.00

0.10

1.36

0.47%

0.03

0.22

1.70 *

2.53%

0.15

0.26

2.58 *

5.32%

Note: The table shows the results of forecasting regressions at the 24-month horizon. Each row reports the results from a regression of 24-month excess returns on the predictor indicated in the “Model” column and a constant. The predictors are summarized in Table 3. The regression results are displayed as in Table 4 and are based on standardized (zero mean, unit standard deviation) predictors.

33

34

0.023 (0.020) 0.023 (0.015) 0.020 (0.019)

0.022 (0.015)

CAY

0.054 (0.160)

-0.052 (0.157)

0.120 (0.101)

DY

-0.033 (0.215)

-0.202 (0.174)

-0.179 (0.166)

PE

0.078 (0.048) 0.078 (0.052)

0.073 (0.045)

RREL

0.000 (0.033) 0.002 (0.034)

-0.026 (0.034)

TMSP

STATE 2.098 (0.981) 1.938 (0.968) 1.456 (0.967) 1.888 (0.964) 2.244 (1.014) 2.929 (1.396) 1.970 (1.066) 2.067 (1.211) 1.706 (1.252) *

*

**

**

*

**

**

NOISE -0.768 (0.413) -0.807 (0.427) -0.730 (0.416) -0.749 (0.415) -0.511 (0.305) -0.843 (0.456) -0.805 (0.450) -0.537 (0.332) -0.505 (0.349) *

*

*

*

*

*

*

Constant -0.911 (0.452) -0.850 (0.446) -0.156 (0.654) -0.289 (0.660) -0.962 (0.464) -1.234 (0.598) -0.471 (0.908) -0.894 (0.517) -0.425 (0.963) *

**

**

*

0.34 0.57

0.23 0.42

0.23 0.34

0.14 0.23

0.24 0.31

0.16 0.29

0.16 0.23

0.20 0.28

3.14 *

2.07 *

2.43 *

1.15

1.32

2.01 *

1.38

2.43 *

R2full R2out MSPE-adj ** 0.12 0.21

10.45%

10.28%

8.64%

6.71%

8.80%

7.26%

7.21%

8.00%

sd(E[R]) 6.33%

Note: The table shows the results of forecasting regressions at the 24-month horizon. Each row reports the results from a regression of 12-month excess returns on the set of predictors indicated in the “Model” column and a constant. The predictors are summarized in Table 3. The regression results are displayed as in Table 4 and are based on standardized (zero mean, unit standard deviation) predictors. MSPE-adj is the t-statistic of the adjusted MSPE in Clark and West (2007) with the benchmark model containing the smoothed state-price series, noise, and a constant.

CAY DY PE RREL STATE NOISE

CAY RREL TMSP STATE NOISE

CAY DY PE STATE NOISE

TMSP STATE NOISE

RREL STATE NOISE

PE STATE NOISE

DY STATE NOISE

CAY STATE NOISE

STATE NOISE

Model

Table 7: Multiple-predictor forecasting regressions: 12 months

35

0.022 (0.019) 0.023 (0.015) 0.022 (0.018)

0.022 (0.015)

CAY

-0.014 (0.124)

-0.009 (0.122)

0.126 (0.091)

DY

-0.142 (0.187)

-0.125 (0.151)

-0.141 (0.164)

PE

0.042 (0.032) 0.037 (0.035)

0.028 (0.029)

RREL

0.025 (0.021) 0.035 * (0.021)

0.012 (0.020)

TMSP

STATE 2.224 (1.015) 2.056 (0.981) 1.550 (0.935) 2.058 (0.998) 2.279 (1.034) 1.837 (1.029) 1.960 (1.037) 1.337 (0.897) 0.938 (0.802) *

*

**

**

*

**

**

R2full NOISE Constant -0.294 -0.969 ** 0.22 (0.217) (0.473) -0.334 -0.906 ** 0.36 (0.231) (0.459) -0.254 -0.177 0.29 (0.212) (0.578) -0.279 -0.479 0.26 (0.216) (0.671) -0.196 -0.988 ** 0.24 (0.159) (0.480) -0.259 -0.818 * 0.22 (0.230) (0.467) -0.323 -0.526 0.39 (0.244) * (0.882) -0.116 -0.622 0.39 (0.155) (0.409) -0.093 -0.098 0.45 (0.151) (0.816) 0.55

0.47

0.47

0.34

0.35

0.44

0.39

0.46

3.79 *

3.32 *

2.59 *

0.90

0.85

1.80 *

1.34

2.78 *

R2out MSPE-adj 0.33

9.10%

8.79%

8.47%

6.42%

6.75%

6.93%

7.30%

8.14%

sd(E[R]) 6.34%

Note: The table shows the results of forecasting regressions at the 24-month horizon. Each row reports the results from a regression of 24-month returns on the set of predictors indicated in the Model column and a constant. The predictors are summarized in Table 3. The regression results are displayed as in Table 4 and are based on standardized (zero mean, unit standard deviation) predictors. MSPE-adj is the t-statistic of the adjusted MSPE in Clark and West (2007) with the benchmark model containing the smoothed state-price series, noise, and a constant.

CAY DY PE RREL STATE NOISE

CAY RREL TMSP STATE NOISE

CAY DY PE STATE NOISE

TMSP STATE NOISE

RREL STATE NOISE

PE STATE NOISE

DY STATE NOISE

CAY STATE NOISE

STATE NOISE

Model

Table 8: Multiple-predictor forecasting regressions: 24 months

Figures

Credit Crunch

Housing Market Peak

Financial Crisis begins

2007

Gulf War II

September 11

Enron/WorldCom

2006

0.55

Dotcom Peak

0.60

Asian Crisis

Gulf War I

0.65

LTCM/Russia

Greenspan's IE speech

0.70

0.50 0.45 0.40 0.35

State price

2012

2011

2010

2009

2008

2005

2004

2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

1993

1992

1991

1990

0.30

State price smoothed

Figure 1: S&P 500 state price of a negative return. We smooth the state-price series using an exponentially weighted moving average (EWMA) with a smoothing parameter of 0.8. More specifically, using zt to denote the state-price w , t = 2, . . . , T . The series, and ztw to denote the corresponding EWMA, ztw = zt , t = 1, ztw = 0.2zt + 0.8zt−1 vertical lines indicate notable volatility events including events highlighted in Bloom (2009), Alan Greenspan’s famous “irrational exuberance” speech, the peak of dotcom boom, and the peak of the U.S. housing market. The shaded areas correspond to the U.S. business cycles as identified by the National Bureau of Economic Research. 1800

1600

index closing values

1400 1200 1000 800 600 400 200 0 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure 2: Daily S&P 500 closing values for 1990-2012.

36

1,000 900 800

option price

700 600

500 400

300 200

100 0

-100 0

200

400

600

800 1,000 1,200 1,400 1,600 1,800 2,000 strike price

adjusted put price CNLS fit

adjusted put price

1,000 900 800

option price

700 600

500 400

300 200

100 0

-100 0

200

400

600

800 1,000 1,200 1,400 1,600 1,800 2,000 strike price

adjusted call price CNLS fit

adjusted call price

Figure 3: Constrained nonlinear least squares fit of adjusted put and call options prices. The top panel of the figure shows the put option prices on the contracts traded on 10/24/2008 for expiration on 11/22/2008. Each put price is adjusted (discounted) by the risk-free rate, for which we use LIBOR. The bottom panel shows the parity-adjusted call prices. We use a dark line to indicate the fit achieved by the logistic mixture distribution.

37

20

0.55

10

0.50

0

0.45

-10

0.40

-20

0.35

-30

0.30

-40

0.25

-50

0.20

State Price

0.60

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Subsequent Excess Returns

30

Subsequent Excess Returns

State price

Figure 4: S&P 500 state prices and subsequent excess returns. We calculate the state prices once each month from January 1990 to July 2010 on the Friday four weeks before the next option expiration. At each point in time, the state price equals the risk-neutral evaluated at zero returns. The subsequent excess returns series is the annualized log change in the S&P 500 from the date on which the state price is measured until the date 96 weeks hence minus the 3-month T-Bill yield.

38