# «A RESE ARCH REP ORT FROM S WEDISH INS TITUTE FOR FINANCIAL RESE ARCH C-CAPM without Ex Post Data PAUL SÖDERLIND N O 39 — D EC EMB ER 20 05 Swedish ...»

## A RESE ARCH REP ORT FROM S WEDISH INS TITUTE FOR FINANCIAL RESE ARCH

C-CAPM without Ex Post Data

## PAUL SÖDERLIND

N O 39 — D EC EMB ER 20 05

Swedish Institute for Financial Research (SIFR) is a private and independent non-proﬁt

organization established at the initiative of members of the ﬁnancial industry and actors

from the academic arena. was launched in January 2001 and is situated in the center SIFR of Stockholm. Professor Magnus Dahlquist serves as director of the Institute. The mission

**of SIFR is to:**

• Conduct and stimulate high quality research on issues in ﬁnancial economics, where there are promising prospects for practical applications,

• Disseminate research results through publications, seminars, conferences, and other meetings, and

• Establish a natural channel of communication about research issues in ﬁnance between the academic world and the ﬁnancial sector.

The activities of are supported by a foundation based on donations from Swedish SIFR ﬁnancial institutions. Major contributions have been made by: AFA, Alecta, Alfred Berg, AMF Pension, Brummer & Partners, Carnegie, Handelsbanken, Kapitalmarknadsgrup¨ a ¨a ¨ ¨ pen, L¨ nsfors¨ kringar, Nordea, Svenska Fondhandlareforeningen, and Ostgota Enskilda Bank.

Sveriges Riksbank funds a position as visiting professor at SIFR.

SIFR also gratefully acknowledges research grants received from Bankforskningsinstitutet, ¨ Foreningsbankens Forskningsstiftelse, Jan Wallanders och Tom Hedelius Stiftelse, Riks¨ bankens Jubileumsfond, and Torsten och Ragnar Soderbergs stiftelser.

Swedish Institute for Financial Research, Saltm¨ targatan 19A 11, SE-113 59 Stockholm, Sweden a Phone: +46-8-728 51 20, Fax: +46-8-728 51 30, E-mail: info@sifr.org, Web: www.sifr.org C-CAPM without Ex Post Data ¨ Paul Soderlind C-CAPM without Ex Post Data Paul S¨ derlind∗ o November 2005 Abstract Survey and option data are used to take a new look at the equity premium puzzle.

Survey data on equity returns (Livingston survey) shows much lower expected excess returns than ex post data. At the same time, option data (CBOE’s VIX) indicates that investors overestimate the volatility of equity returns. Both facts reduce the puzzle.

However, data on beliefs about output volatility (Survey of Professional Forecasters) shows marked overconﬁdence. On balance, the equity premium is somewhat less of a puzzle than in ex post data.

Keywords: equity premium puzzle, Livingston survey, CBOE VIX, Survey of Professional Forecasters JEL Classiﬁcation Numbers: G12, E130, E320 ∗ University of St. Gallen, CEPR, and SIFR. Address: s/bf-HSG, Rosenbergstrasse 52, CH-9000 St. Gallen, Switzerland. E-mail: Paul.Soderlind@unisg.ch. I thank Michael Fischer for excellent research assistance and Anna Cieslak for comments.

1 Introduction This paper studies if the consumption based asset pricing model is compatible with survey data on subjective beliefs.

Asset pricing models are statements about how expected returns are related to the perceived risk exposure of different assets. This is seldom tested directly. Instead, the models are typically tested by comparing theoretical pricing implications with the properties of historical (ex post) data. The results can therefore only be given a clear interpretation under the maintained hypothesis that the historical sample is a good representation of subjective beliefs. For instance, the ﬁnding of an ”equity premium puzzle” relies on the assumption that investors have really expected an excess return of 6%–8% on U.S. equity.

There are several reasons to believe that the moments of historical data can be quite poor approximations of investors’ expectations. Early empirical evidence from survey data suggests that expected returns may deviate from historical averages (for instance, Lakonishok, 1980) and more recent evidence suggests that investors may underestimate the uncertainty of risk (for instance, Thaler, 2000, and Giordani and S¨ derlind, 2003).

o These ﬁndings could be driven by either small sample problems or some sort of distorted expectations.

The small sample problems are quite likely, since equity returns are highly volatile and sometimes exposed to unusually large shocks. It can easily happen that a fairly long sample has sample moments that deviate substantially from the true values (and subjective, ex ante, beliefs).

Even in the absence of small sample problems, historical data may be poorly suited for testing asset pricing models. Recent ﬁndings in behavioural economics (for instance, Hirshleifer (2001)) often point to overconﬁdence among economic agents. Research on learning (for instance, Lewellen and Shanken, 2002) and on robust decision making (for instance, Tornell, 2000, and Anderson, Hansen, and Sargent, 2003) emphasise that there may be good theoretical reasons for why ex ante beliefs deviate systematically from ex post data.

The approach in this paper is to evaluate the consumption based asset pricing model by using survey data on expected returns and the volatility of the risk factors. This circumvents the problems with expectations errors—and is therefore a way to test the model more directly.

Under some simplifying assumptions, the key implication of the consumption based asset pricing model is that expected excess equity returns are a product of a risk aversion coefﬁcient, the standard deviations of consumption growth and equity returns, and their correlation.

To get a reasonably long sample with high-quality data, I combine several data sources.

The Livingston survey and the Survey of Professional Forecasters are used to measure subjective beliefs of expected equity returns and consumption volatility. Both surveys are focused on the beliefs of economists close to the ﬁnancial markets and are administered by the Federal Reserve Bank of Philadelphia. The CBEO volatility index (VIX) is used as a measure of equity return uncertainty.

The plan of the paper is as follows. Section 2 summarises the standard consumption based asset pricing model. Section 3 presents the survey data and the empirical results.

Section 4 sums up. There are several appendices giving details on data and derivations of some analytical results.

**2 The Standard Consumption Based Asset Pricing Model**

This section gives a brief summary of the standard CRRA model and the equity premium puzzle.

The standard consumption based asset pricing model assumes a utility function with 1−γ constant relative risk aversion, Ct /(1 − γ ), where Ct is consumption and γ the risk aversion coefﬁcient. The Euler equation for optimal portfolio choice can be written as E[Rte (Ct /Ct−1 )−γ ] = 0, where Rte is the excess return on an asset—and E() denotes the expectations of the (representative) investor.

To simplify the analysis, assume that the investor thinks that the excess return and consumption growth have a bivariate normal distribution. Use Stein’s lemma1 to rearrange the Euler equation as

ct is the growth rate of consumption, ln(Ct /Ct−1 ). It is worth emphasising where 1 Stein’s lemma says that if x and y have a bivariate normal distribution and h(y) is a differentiable function such that E[|h (y)|] ∞, then Cov[x, h(y)] = Cov(x, y) E[h (y)]. See Cochrane (2001).

that this equation is part of the investor’s decision process—and therefore holds only for his/her beliefs. It really does not matter where those beliefs come from, if they are rational, or if they are the result of a learning process (as in, for instance, Brennan (1998)).

We can relax the assumption that the excess return is normally distributed: (2) holds also if Rte and ct has a bivariate mixture normal distribution—provided ct has the same mean and variance in all the mixture components (see Appendix B). This restricts consumption growth to have a normal distribution, but allows the excess return to have a distribution with fat tails and skewness.

The gain from assuming a normal distribution of consumption growth is that the unknown relative risk aversion, γ, enters multiplicatively in (1)–(2). This allows us to work with correlations and standard deviations of returns and consumption growth—for which it might be possible to ﬁnd survey data. In contrast, if we did not assume normally distributed consumption growth rates, then we would need information about, for instance, the standard deviation of (Ct /Ct−1 )−γ. There is no survey data on such things.2 The equity premium puzzle (see Mehra and Prescott (1985)) is that (2) does not hold for ex post data unless γ is very high. To illustrate this, I use quarterly US data for 1957Q1–2003Q4. Consumption growth is measured as the growth rate of per capita consumption of nondurables and services (see Appendix A) and the return is the real return on the S&P 500 in excess of the real return on a 3-month T-bill rate.

In annualised terms (quarterly growth rates are multiplied by 4 and quarterly standard deviations by 2) the ex post data gives E(Rte ) = Corr(Rte, ct )σ (Rte )σ ( ct )γ. (3) 0.01 0.06 0.14 or 0.31 0.16 Two numbers are given for the correlation: the lower is when consumption growth (a ﬂow variable) is measured as the growth between the current and lagged quarter; the higher when it is measured as the growth between the next and current quarters. In any case, even if the correlation was perfect (one), then we need a relative risk aversion, γ, of 38 to make (3) hold.

2 The general expression is E(R e ) = − Cov[Rte, (Ct /Ct−1 )−γ ]/ E[(Ct /Ct−1 )−γ ]; the covariance can be t expanded in terms of the correlation and the two standard deviations.

3 Evidence from Survey Data and Options The survey and option data used in this paper come from different sources, covere different periods, and have different sampling frequencies. There really are not many alternatives, at least not if we want to work with reasonably long samples.

3.1 Relation Between Survey Data and the Asset Pricing Equation Survey data gives information on conditional moments of subjective beliefs of heterogeneous agents. In contrast, the asset pricing equation (3) is expressed in terms of unconditional moments of a representative investor. This section discusses how the survey data may still be useful to understand the properties of the asset pricing equation.

The evidence from surveys and options is on conditional moments—both data sets essentially contain answers to a question like “based on your information today, what do you believe about x in the near future?” We therefore need a rule for transforming back to unconditional moments. This is straightforward for the expected return, E(Rte ), since the unconditional expectation is best estimated by the average conditional expectation (by the law of iterated expectations, E[Et (Rte )] = E(Rte )). For the standard deviations in (3), matters are slightly more complicated. But, if the excess returns and the consumption growth rates are unpredictable (which is a good approximation), then the unconditional variance is the average conditional variance. The unconditional standard deviations are then the square roots of the average conditional variances—which is the approach used below.

The Euler equation (2) is valid for each investor—provided we are careful enough to use the moments of his/her beliefs about the return and his/her own consumption. In (3) it is further assumed that all investors are identical: they share the same beliefs and it is aggregate consumption that matters.

The survey data has a different structure: it asks for his/her beliefs about the return and aggregate output. To bridge the gap between data and the asset pricing equation, we need strong assumptions. First, that results on over/underestimation of the volatility of output (which we have data on) carries over to consumption (which enters the asset pricing equation). Second, that equity premia are not affected by non-insurable idiosyncratic risk (since we have no information on this risk). Third, that aggregation of beliefs (since the respondents in the survey disagree) is unimportant.

None of these assumptions is likely to hold exactly, but they may well be reasonable approximations. First, the time series properties of output and consumption are very similar in many respects: they are strongly correlated over all horizons and so are the forecast errors. In addition, most macroeconomic theory would suggest a very strong link. Second, the evidence on (the importance for asset pricing of) idiosyncratic risk is mixed (see, for instance, Lettau (2002) and Cogley (1998).) Third, aggregation of heterogenous beliefs may well suggest that an average belief is a good approximation of the representative investor’s belief—in particular when the relative risk aversion is high. For instance, Giordani and S¨ derlind (2005) show (by extending a model by Varian (1985)) that a repo resentative investor should be assigned a mean equal to the cross-sectional mean, and a variance equal to a combination of the (cross sectional average of) individual uncertainty and the cross-sectional disagreement. It turns out that the relative weight on disagreement is 1/(γ + 1): with log utility disagreement is as important as individual uncertainty (this is the case in Rubinstein (1974) and Detemple and Murthy (1994)), but a more realistic value of the relative risk aversion gives a very small role to disagreement. This suggest that it might be reasonable to disregard the aggregation issue.

**3.2 Expected Stock Return**

The Livingston survey summarizes the forecasts of economists from industry, government, banking, and academia. It is the oldest continuous survey of economists’ forecasts, started in 1946 by the ﬁnancial columnist Joseph Livingston. It was taken over by Federal Reserve Bank of Philadelphia in 1990. It has questions about the expected Standard & Poor (S&P) index level 6 and 12 months ahead in time.