Stefano Giglio Bryan Kelly

University of Chicago and NBER

**March 4, 2015**

We document a form of excess volatility that is irreconcilable with standard models of prices, and in particular

cannot be explained by variation in the discount rates of rational agents. We compare behavior of prices to claims on the same stream of cash ows but with different maturities. Prices of long-maturity claims are dramatically more variable than justified by the behavior of short maturity claims. Our analysis suggests that investors pervasively violate the \law of iterated values.” The violations that we document are highly significant both statistically and economically, and are evident in all asset classes we study, including equity options, credit default swaps, volatility swaps, interest rate swaps, ination swaps, and dividend futures.

^{*}We are grateful to Drew Creal, Lloyd Han, Lars Hansen, and Stavros Panageas for many insightful.

comments.

The field of modern financial economics is in large part organized around the notion of excess volatility in asset prices. As Shiller (1981) and others famously document, price uctuations are \excessive” relative to predictions from the constant discount rate model. A potential resolution of the puzzle is to recognize that discount rates are variable. The leading frameworks of modern finance indeed center on descriptions of discount rate variation in models of rational expectations.

In this paper we document a form of excess volatility that is irreconcilable with standard models of prices, and that in particular cannot be explained by variation in the discount rates of rational agents. We compare behavior of prices to claims on the same stream of cash ows but with different maturities. Our analysis suggests that investors pervasively violate the “law of iterated values” (Anderson, Hansen, and Sargent (2003)). That is, the law of iterated expectations dictates that prices of long maturity claims reect investors’

expectations about the future value of short maturity claims. This imposes consistency requirements on the joint behavior of prices across the term structure.

We document an internal inconsistency in the price behavior of short and long maturity claims. In particular, prices on the long end of the curve are dramatically more variable than justified by the behavior of the short end. These violations are highly significant both statistically and economically. Excess volatility of long maturity prices is evident in all asset classes we study, including claims to equity volatility, sovereign and corporate credit default risk, interest rates, in ation, and corporate dividends.

A simple example illustrates the essence of our approach. Consider an asset that yields an uncertain cash coupon of

x_{t} each period. At time t, an n-maturity claim receives the cash flow x_{t+n} in period n. In the absence of arbitrage, the prices of claims across all maturities are coupled by the dynamics of x_{t}.

No-arbitrage implies the existence of a pricing measure,Q, that subsumes all risk pre-mia, and their potentially time-varying dynamics, by construction. Under Q, prices are expectations of future cash flows:

The convenience of representing prices as Q-expectations is that any excess volatility that we find in prices cannot

be due to a time-varying risk premium, by definition. If we can

^{1}For exposition we assume here that the risk free rt is 0, or, alternatively, that all cash flows in the contract are exchanged at maturity, as in the case of swaps. Later sections address the role of time varying interest rates in detail.

Importantly, a failure of the Q-dynamics extracted from the short end of the curve to explain prices at the long end of the curve can be directly linked to a violation of the law of iterated values. Under Q, iterated expectations bind together prices of claims across maturities. For example, the price of the n-period claim is coupled with the price of 1-period claim:

A violation of this equation { for example, by inconsistent variances of the left-hand and right-hand sides { constitutes a failure of the law of iterated values.

We develop a general methodology for measuring and testing excess volatility that re- quires minimal modeling assumptions and exploits the information contained in the term structure of cash flow claims on any asset. Our methodology extends the preceding example to any setting in which the cash- flow variable x_{t}follows a factor structure with linear dy-namics under Q and an arbitrary number of factors. This assumption is valid in standard term structure models, where it is typically derived from a linear factor structure under the physical measure together with an affine stochastic discount factor. It also describes many general-equilibrium asset pricing models, such as the long-run risks model of Bansal and Yaron (2004), the rare disaster model of Wachter (2013), and the cash flow duration model of Lettau and Wachter (2006). Furthermore, our setting allows for time-varying risk prices and stochastic volatility.

Our tests of excess volatility are remarkable for what they do not require. We require no data other than asset prices, obviating the need for data on the underlying cash flows x_{t} . Nor do we require a model of discount rates. By definition, the behavior of x_{t} under the pricing measure implicitly captures all relevant discount rate variation. In our approach, any surplus (or deficit) of p^{n}_{t} variance relative to its predicted value already accounts for discount rate effects. Finally, we require a minimal amount of time series information (used to compute the covariances of prices at different maturities). It is well known from the term structure literature that the dynamics under Q are precisely estimated with short time series via cross-sectional regressions of prices along the term structure. We leverage this fact and focus exclusively on Q dynamics, avoiding the dificulties of estimating physical dynamics and stochastic discount factors.

We consider a number of potential explanations for the excess volatility of long maturity claims such as liquidity and omitted factors. The evidence is inconsistent with both of these explanations. For most asset classes, we have detailed liquidity information across the term structure, we study only those maturities that transact regularly each day, and we demonstrate that observed price volatility is unassociated with bid-ask bounce or stale prices. Nor are omitted factors likely to explain our findings. First, parsimonious linear factor structures provide an extremely accurate description of every term structure that we study, with R^{2} values generally exceeding 98% based on one to three factors. Second, we project long maturity prices onto the short end of the curve, and show that even the variance of the projected prices (which are 100% explained by short prices) remain highly significantly excessive. Finally, we perform a range of robustness tests allowing for richer factor structures, and our findings remain essentially unchanged.

Our paper is related to Stein (1989) in terms of economic intuition and implementation, who studies the pricing of volatility of the S&P 100 using the term structure of implied volatility of options. Stein compares Ρ^{Q} with the persistence of volatility under the physical measure Ρ. He finds that Ρ^{Q}> Ρ , and interprets it as evidence of overreaction. We build on Stein (1989) in three ways. First and foremost, we do not compare physical and risk- neutral dynamics of the underlying cash flow process (ΡandΡ^{Q}). This is a conceptually crucial difference. As the term structure literature (developed largely subsequent to Stein’s analysis) has pointed out, Ρ^{Q} will generally be different from Ρ in the presence of time-varying discount rates, even in a rational model. In fact, in standard affine models Ρ^{Q}-Ρ exactly measures the time-varying component of risk premia. In other words, time varying discount rates provide a natural potential explanation for Stein’s facts, in the same way that discount rate variation helps resolve Shiller’s original excess volatility puzzle.

In contrast, we use only information in the cross-section of prices, solely focusing on dynamics under the pricing measure. In essence, we compare the estimates of Ρ^{Q} from the short end of the curve to Ρ^{Q} implied by the long end. Because our analysis is entirely conducted under the pricing measure, time-variation in risk premia cannot play a role in our analysis. Any overreaction (excess volatility) that we find is not mechanically resolved by a time-varying discount rate explanation. The second contribution of our paper is to show that this excess volatility phenomenon is not merely a feature of the options market. It holds across diverse asset classes and across countries. Finally, we propose a general methodology that allows for an arbitrary factor structure (as opposed to the single factor model in Stein) and derive a test statistic for excess volatility.

In Section 2 we present our general modeling framework, of which the preceding one- factor example is a special case. We present our approach to estimation and inference in Section 3. In Section 4 documents the our central empirical facts for excess volatility across many asset classes. Section 5 discusses a number of extensions and Section 6 concludes.

In this section we specify the dynamic structure of the economy under a probability mea- sure denoted Q , the pricing measure. 2 The state space under Q is described by a vector autoregressive process for the factors, F^{t} :

The intercept is an inconsequential constant function of remaining model parameters that drops out from all variance calculations. Note also that no risk-free rate adjustment appears in equation (4), a point we discuss below. All assumptions are discussed brie y in the next section and in detail, asset class by asset class, in the appendix.

How variable are prices in this model? What drives the comovement across claims of different maturity? Price uctuations for all claims to x_{t} are entirely driven by uctuations in the state factors, but they differ with maturity depending on the state persistence matrix,

^{2} The pricing measure is a transformation of the objective statistical measure that scales physical prob- abilities by investors’ marginal utilities state-by-state. This carries the implication that asset prices are martingales under Q , a feature that we exploit in our development. Such a measure is guaranteed to exist under the minimal assumption of no-arbitrage (Harrison and Kreps (1979), Harrison and Pliska (1981)).

Ρ^{Q} . Consider the stationary case in which all eigenvalue of Ρ^{Q} are strictly less than one in modulus. At the short end, the claim receives a single cash ow, and has sensitivity to state uctuations given byδ’_{1}(I+Ρ^{Q}). As the maturity rises to n , the claim receives a total of n cash ows. Because x_{t} is persistent, this claim is more sensitive to the state, represented by the additional Ρ^{Q} term in (12). The increasing powers of Ρ^{Q} also indicate that more distant cash ows matter less for price uctuations today. Thus, price volatility is an increasing but concave function of maturity. Moreover, the model structure places restrictions on the exact price variances that are admissible. If there are K factors, but n > K maturities, then the variance of any n – K prices in the term structure are entirely pinned down by the other K price variances, through Ρ^{Q}.

Given that the results of the paper are derived under the assumptions about the term structure described above, it is important to underline which are the key assumptions in that setup, and which assumptions play instead a minor role.

The two fundamental assumptions of the paper are:

1. Cash flows x_{t} follow a factor structure, and

2. Factors obey linear dynamics.

The first assumption of the model can be easily verified in the data. The term structure model we described above has an extraordinarily high degree of explanatory power for asset prices in a wide variety of asset classes. For traded claims with a term structure of maturities, we typically find that a small number of latent factors (typically one to three, many fewer than the number of traded maturities) explains close to 100% of the price variation throughout the term structure. The most well known example is the US treasury bond term structure, though we find the same feature among derivatives in credit, equity, currency, and other markets. This strongly supports the assumption of a factor structure.

The second main assumption of the paper is that state dynamics are linear under Q . The vast majority of models in the asset pricing literature assume linear dynamics for the underlying state variables, both in the structural general equilibrium asset pricing models and in the reduced-form term structure literature. In addition, the Wold decomposition applies in our setting as long as the factors are stationary under Q (stationarity is typically assumed in term structure models and is supported by our empirical evidence). This implies that state dynamics can be represented as a vector moving average process (with possible infinite lags), for which our exible finite-order VAR can be viewed as an approximation. Finally, Le, Singleton and Dai (2010) discuss conditions for a model to posses a linear affine structure under Q even when objective dynamics are non-affine.

The specification above also imposes some other, non-critical assumptions that we brie y discuss here and explore more in detail in the Appendix.

. We assumed above that the term structures we consider have prices that depend exponentially on the cash flows x_{t} . This is a natural assumption for some asset classes, such as bonds, in which the prices are exponentially affine functions of x_{t} = -r_{t}In other term structures, payoffs are linear in the cash ow variable:

In this section we develop a methodology for testing excess volatility across the term structure given the model specification of the Section 2. We start by discussing parameter identification for factor dynamics under

Q . We then show how to infer dynamics under Q from the comovement of prices at the short end of the term structure. Finally, we propose a test for \excess volatility” of prices at the long end of the term structure.

Note that each element i of Ρ^{Q} needs to satisfy this equation: Ρ^{Q} can therefore be computed by finding the roots of this polynomial equation. This structure has the convenient feature that we can estimate state dynamics from the yields without any maximization (as is typical in term structure models).

One consideration is that there will generally be n_{H+1} roots of this polynomial (some of them complex), while we only seek K parameters. This equation shows that the Q- measure dynamics and the comovements of prices only identify the eigenvalues of Ρ^{Q} up to the set of roots of this polynomial. It does not tell us which roots to choose, as they imply the same covariance among prices (while a full MLE procedure that exploits both information about the P and the Q dynamics will be able to choose among them). We use the following selection procedure. First, we only consider non-explosive roots. This is motivated by the unambiguous empirical fact that price variances are concave in maturity for all the markets we study, especially at the short end of the curve where our estimation is coming from. If prices rise less than linearly with horizon, the system is best described by stationary dynamics. Second, among the non-explosive roots, we select the K most persistent ones. This ensures that our excess volatility findings will be the most conservative (they will suggest the least excess volatility) of all of the covariance-equivalent roots we could have reported. Finally, we choose real roots whenever possible, since complex roots imply regression coefficients of prices on the factors p_{t} at maturities above n_{H+1} that display cycles across maturities, an implication that is strongly counterfactual.

In this section we briey describe the asset classes for which we test for excess volatility. In each case, the pricing and factor structure described in section XXX applies with small modifications, all of which are reviewed in detail in the Appendix. We also leave for the appendix a more in-depth description of the data.

US government bond prices are among the most well studied data in all of economics. Our data US bond data comes from Guryanak, Sack and Wright (2006). The data consists of zero-coupon nominal rates with maturities of 1 to 30 years for the period 1985 to 2014, and is available at the daily frequency. The term structure is bootstrapped from coupon bonds and uses only interpolation, not extrapolation (so that a maturity will only be present if enough coupon bonds are available for interpolation at that maturity). Given the high liquidity of the Treasuries market, we use all available maturities starting in 1985.

Credit default swaps (CDS) are the primary security used to trade and hedging default risk of corporations and sovereigns. As of December 2014, the notional value of single-name CDS outstanding was $10.8 trillion. Our CDS data is from MarkIt. The CDS data includes maturities of 1, 3, 5, 7, 10, 15, 20, and 30 years for the period 2001 to 2013 and is available at the daily frequency. While not all maturities are equally traded, we focus on the most liquid single-name and sovereign CDS. It is useful to remember that our test will be conducted and reported separately at all available maturities; so it will be easy to assess to what extent the results are driven by maturities for which liquidity is high or low.

Among the different CDS contracts written on the same reference entity, we choose those with highest liquidity. In particular, we choose CDS written on senior bonds, with modified- restructuring (MR) clause, and denominated in US dollars.^{4} Since there was little CDS activity before the financial crisis and most of these contracts had low liquidity, we focus on the period from January 2007 onwards. We choose the three most traded sovereigns (Italy, Brazil, Russia) and the three most traded corporates (JP Morgan, Morgan Stanley, Bank of America) during 2008^{5} , a year in which CDS trading volume and CDS spread variability were particularly high. The Appendix describes how CDS prices can be represented in the framework of Section XXX.

We obtain in action swaps data from JP Morgan. We observe the full term structure between 1 and 30 years, at the daily frequency, between 2004 and 2014. As reported in Fleming and Sporn (2013), \Despite a low level of activity and its over-the-counter nature, the U.S. in action swap market is reasonably liquid and transparent. That is, transaction prices for this market are quite close to widely available end-of-day quoted prices, and realized bid- ask spreads are modest.”. In addition, there is significant trading volume at all maturities, including the very top ones (20 to 30 years).

^{4} For sovereigns, we use contracts with the CR clause, as more data is available than for the MR contracts.

^{5} See Fitch (2009).

The term structure model for in action swaps follows closely the benchmark of Section XXX. We report the details in the appendix.

Markets for financial volatility, including options and variance swap markets, possess a rich term structure of claims. These markets allow participants to trade and hedge the price volatility of effectively any financial security, including equity indices and individual stocks, currencies, government and corporate bonds, etc.

The first market we study is that for variance swaps on the S&P 500 index, claims to future realized variance (the sum of squared daily returns of the index). The price of a variance swap corresponds on the expectation (under Q) of future realized variance:

and therefore it fits directly into our framework (with the only difference that the price depends on the payo variable x_{t} in levels, not in logs). As discussed in detail in Dew- Becker etal. (2015), the variance swap market is an over-the-counter market with a total outstanding notional of around $4bn vega at the end of 2013 (meaning that a movement of one point in volatility would result in $4bn changing hands). More importantly, the price of a variance swap is anchored to the price of a syntetic swap that can be constructed from option prices. Dew-Becker etal. (2015) show that the term structure of variance swap prices matches very closely the term structure of synthetic claims constructed from option (typically known as the VIX).

We also the term structure of at-the-money implied volatilities extracted from options in a variety of asset classes. In theory, synthetic prices of variance swaps (which follow exactly equation XXX) can be constructed in any market by combining the prices of puts and calls at different strikes: the price of these synthetic portfolio, commonly known as the VIX, is tied by arbitrage to the price of a variance swap. This would theoretically allow us to study the term structure of variance swap prices in any markets in which we can observe put and call prices, even if actual variance swaps are not trade. Unfortunately, for many asset classes not enough strikes are available to reliably construct the term structure of the VIX. We therefore rely on at-the-money (ATM) implied variances as a proxy for the VIX. This brings us closer to the original setup of Stein (1989), who was working with ATM implied volatilities, and is in part justified by the observation that ATM implied volatilities correspond { up to a first- order approximation { to the prices of claims to realized volatility (√RV), as demonstrated by Carr and Lee (2009). Using ATM implied variances allows us to study a large number of markets.

In addition to variance swap on the S&P 500, we study the term structure of ATM implied volatilities for the most liquid options available in OptionMetrics: three domestic indices (S&P 500, Nasdaq, Dow Jones), three international indices (Stoxx 50, FTSE, DAX), three individual names (Apple, Citigroup, IBM), and three currency options,

Currency forwards are the primary contracts (along with currency swaps) used to trade and hedge exchange rate risk. As of December 2014, the notional value outstanding of currency forwards and swaps was over $60 trillion. Our currency data is from JPMorgan Dataquery. We study six different currencies (versus the US dollar). For four of these we have maturities of 1, 3, 6, 9, and 12 months. For the Euro and the Mexican Peso we have maturities up to 15 years. Some data are available from 1996 to 2014, and all series have data at least as far back as 2002. We have daily data for all currencies.

We also study currency options, which allow investors to trade and hedge exchange rate volatility. Our data are from JPMorgan Dataquery and have maturities of XXX for the period 1990 to 2014 at the monthly frequency. These data focus on the term structure of Black-Scholes implied volatility.

We obtain Stoxx 50 dividend futures prices from Bloomberg and Eurex (using the latter prices whenever available). We obtain

weekly series from May 2009 to January 2015 (we use weekly data to reduce the impact of noise). Dividend futures data are obtained by interpolating contracts with fixed expiration dates in the December of each year. Since part of the dividends expiring in the first year are already accrued at the time of the transaction, we exclude the first maturity from our analysis. We therefore obtain contracts of maturity 2 to 7 years. Finally, we adjust all contracts by the risk-free rate to obtain spot prices rather than futures. This step is useful in comparing the prices of the dividend strips to those of the stock market.

The small set of maturities available does not allow us to really compare the short and the long end of the curve. However, in the case of dividends we actually observe the price of a claim to the whole infinite-horizon set of dividends: the Stoxx 50 itself. We can therefore perform the following exercise: we extract the Ρ^{Q} matrix using the time series of dividend strips, and compare them with the volatility of the price of the stock market.

Before describing the empirical results, we discuss here the procedure we used to identify the number of factors K , the number of prices at the short end of the curve that are used to extract Ρ^{Q} .

To choose the number of factors K , we use the panel of prices for all N available maturities to calculate the number of principal components necessary to explain at least 99% of the variance in the panel. This serves as our estimate of K .

We choose H (the prices that define the \short-end” of the term structure) for each asset based on the available maturities at which that cash ow is traded. In particular, for term structures for which claims are typically traded (and data are available) up to 24 months, we define the short end of the curve as composed of maturities up to 6 months. For term structure where the maturities traded are as high as 10 years, we define the short end as 3 years. Finally, for term structures that extend up to 30 years, we define the short and as maturities up to 5 years. The main theoretical justification for linking the definition of the \short end” of the term structure to the set of maturities traded is the following. The set of maturities n_{1};:::;n_{N} we observe to actually trade presumably span the maturities for which investors believe there would be significant price variation. Therefore, we define the short end of the term structure relative to the set of maturities the investors choose to trade.

We find pervasive evidence of excess volatility in each of the term structures we study. Our main findings are summarized in Figure 1. Clockwise from the upper left, the panels present results for claims to S&P 500 volatility (via options), Spanish sovereign default risk (via CDS), USD/GBP implied volatility and US treasury bonds.

Each figure reports variances of prices across a different term structure term structure. The x-axis shows the maturity of claims. The y-axis on the left side shows the volatility (in standard deviation terms) of prices, while the right axis reports the variance ratio.

The solid thin line in the figure reports the standard deviation of log prices at each maturity. Note that the total volatility at each maturity is a concave function of maturity. This is a first indication that the dynamics under the pricing measure Q cannot be explosive (in that case, the volatility would be an increasing and convex function of maturity).

The solid thick line reports the standard deviation of the component of log prices ex- plained by the covariance at the short end, √V(p^{n}_{t})^{unrestricted} . The difference between the two solid lines is the part of movements of prices that cannot be explained by comovement with the factors extracted from the short end of the curve: measurement error u^{n}_{t} . Note that the factors extracted from the short end of the curve have an extremely high explanation power for every maturity along the curve, with R^{2} close to 100% even at the very long end of the curve: the unrestricted factor model fits the whole term structure extremely well.

The dashed line reports the price volatility that the Q dynamics extracted from the short end of the curve imply at all maturities: √V(p^{n}_{t})^{restricted} . Note how in all cases the model- implied volatilities increase with maturity at a much slower rate than the actual volatilities of prices. This is precisely what we mean by \excess volatility”. The volatility of prices (and the comovement between the long and the short end of the curve) is entirely driven by Ρ^{Q} . However, the prices at the long end of the curve react to shocks to the factors driving the short end much more strongly than those dynamics would imply: long-term prices overreact to movements in the factors.

The shaded area encloses the 95th and 5th percentiles of the distribution of the volatility under the null of the model (obtained from a bootstrap procedure described in the Appendix). Under a one-sided test of the variance ratio, we reject the null of no overreaction whenever the think solid line (\total explained variance”) lies above the shaded area. In all the cases reported here, we find strongly significant evidence of excess volatility. Finally, two things are important to note. First, the excess volatility we find cannot be explained by movements in discount rates. Under the Q measure, all prices should just be equal to the expectation of future cash ows. Given the factor model we estimate, this expectation is fully pinned down by Ρ^{Q} (and the current level of the factors), and no other variation in expected cash flows or discount rates can affect prices. Second, the results of our paper are not about the fit of the factor model. The fact

For the case of dividend claims we perform a slightly different analysis, since we compare the term structure of dividend strips with the price of the entire stock market. We therefore report all the results in this section.

The term structure of log price-dividend ratios of dividend strips (pd^{n}_{t} ,n= 2:::7) has a strong factor structure. The first principal component explains 97.5% of the total variation, and the first two component explain 99.3%. We therefore employ a two-factor model. Given the limited number of maturities available, we extract the two factors using all available maturities.

We therefore document a significant violation of the law of iterated values when comparing the dividend strip curve and the infinite-claim. Interestingly, it points to a lower volatility of the stock market claim relative to the volatility of the corresponding-duration claim in the dividend strips market.

We believe there are two possible interpretation for this surprising result. First, it may reect different factors driving the dividend market and the infinite claim, for exmaple frictions or tax effects (see Boguth et al. ). If the two markets are not well integrated, and respond to different factors, it can look to the observer as a violation of the law of iterated expectations.

A second, maybe more interesting, possibility, is that given that the dividend strips market is actually a derivative market of the much-larger market for the stock market, what we are documenting here is overreaction and excess volatility in this market relative to the movements of the entire stock market. This may explain why dividend strips have been found to be extremely volatile (for example by Binsbergen et al.), though naturally is not enough to explain the declining term structure of Sharpe ratios across maturities documented by Binsbergen et al.

Key to the methodology of the paper is extracting Ρ^{Q} from the short end of the term structure and comparing the predicted and actual variance of prices of higher maturities. To do so, in our main analysis we extracted K factors from the short end of the curve, where K was chosen as the number of factors that explain 99% of the variation of the whole term structure, and the \short end” was defined to be the composed of the K shortest maturities available. In this section we consider several robustness tests. All the results are reported in the Appendix Figures.

First, we extract K factors by looking at more than K maturities at the short end of the curve. We follow the methodology presented in Section 2 to extract the K principal components out of the first H > K maturities. In the appendix we consider H = K + 1 and H = K + 2 maturities. Doing so has two effects. On the one hand, it potentially reduces measurement error in extracting the factors from the short end of the curve. On the other hand, it increases the maturities used as a \short end” of the curve, in some cases getting very close to the \long end” of the curve. Expanding the maturities used H therefore makes our test less sharp, because it often uses much longer maturities. While in most cases doing so results in a lower estimate of excess volatility, for the vast majority of cases the results are still strong and statistically significant.

In a third robustness test, we set again H = K as in the main text, but now require enough factors K to explain 99.9% of the variation in the term structure. This greatly increases the number of factors required, and again, using longer maturities, makes our test less sharp. Except in a couple of cases, all of our results still hold.

<
p>Finally, we report a version of our main results where we use asymptotic standard errors (instead of bootstrapping them).

We document excess volatility and a violation of the law of iterated values in a large cross- section of asset classes. Our test of excess volatility exploits the overidentification restrictions offered by observing a term structure of claims on the same cash flows.

We use the short end of the term structure to learn investor’s implied dynamics of cash flows under the pricing measure Q . This gives us a model of expectations under Q at all maturities, which are linked by the law of iterated expectations and the implied dynamics of the factors driving the cash flows. We find that prices of long-maturity claims are dramatically more variable than justified by the behavior of short maturity claims. This excess volatility cannot be explained by time variation in discount rates, because that is already accounted for by the risk-neutral expectations we extract from the short end of the term structure. Our results therefore show that the excess volatility puzzle first highlighted by Shiller (1982) cannot be fully accounted for via rational variation in discount rates.

Before moving to other asset classes, we take advantage of the well-researched interest rate setting to discuss a few considerations in more detail. We begin with a discussion of the pricing measure, Q .

Up to now, we have assumed the absence of arbitrage, which ensures the existence of a measure Q under which prices are given by equation (11). Q-measure event probabilities are distorted versions of the objective probability measure, P , that is observed by the econometrician and that describes the evolution of cash flows. The distortion of Q has a natural economic interpretation. In particular, the Q -probability of a given state of the world arises