Is asset-pricing pure data-mining? If so, what happened to theory?

Is asset-pricing pure data-mining? If so, what happened to theory? Michael Wickens Cardiff Business School, University of York, CEPR and CESifo Lisbon ICCF 4-8 September 2017 Lisbon ICCF 4-8 September 2017 1 /

You can torture the data how you like. The problem is to give an economic explanation of the results The essence of scientific knowledge is to build a body of theory supported by evidence In finance - especially asset pricing - we have a body of theory that is not supported by evidence And a mountain of evidence - due to data-mining - not consistent with theory. In this lecture we reflect on this uncomfortable state of affairs Lisbon ICCF 4-8 September 2017 2 /

Whether one considers the pricing of bonds, equity or FOREX theoretical asset-pricing models perform poorly. As a result the finance industry relies on data-mining: statistical models chosen to fit the data often with little theoretical justification Is this because the theoretical models are poor, or because investors don t understand, and hence employ, the theories? Lisbon ICCF 4-8 September 2017 3 /

Deductive or inductive knowledge? One defence of this state of affairs is that knowledge is inductive and not deductive According to Karl Popper - Deductive knowledge starts with a conjecture which one then tests and tries to refute - Inductive knowledge is acquired by generalising from particular evidence Statistics - and data-mining - has encouraged the concept of inductive knowledge You might think that this is all too high-minded and that finance is just pragmatic - its the bottom line that counts - the world is to complicated to be explained by these simple theories Lisbon ICCF 4-8 September 2017 4 /

Asset pricing theory There are three common elements to different asset pricing theories 1 An asset s price is the expected discounted value of future pay-offs 2 The no-arbitrage condition 3 The measurement of a risk premium The differences between asset pricing theories are 1 The pay-off 2 The choice of discount factor 3 The information set that expectations are based on Lisbon ICCF 4-8 September 2017 5 /

The stochastic discount factor (SDF) approach This is the fundamental common element in all asset pricing approaches If an asset s price is the discounted value of future pay-offs then P t = E t [M t+1 X t+1 ] X t+1 P t = 1 + r t+1 This gives the no-arbitrage condition for the excess return E t r t+1 f t = (1 + f t )Cov t (M t+1, r t+1 f t ) The stochastic discount factor is usually a linear function of n factors z i,t+1 so that M t+1 = Σ n i=1β i z i,t+1 This gives the no-arbitrage condition E t r t+1 f t = (1 + f t )Σ n i=1β i Cov t (z i,t+1, r t+1 f t ) Key theoretical prediction: β i should be the same for all assets Lisbon ICCF 4-8 September 2017 6 /

The factors can be observable or unobservable variables or a mixture of the two If observable, the factors can be derived from - Absolute or general equilibrium asset pricing theories - Relative means pricing one asset off another as in bond pricing - Data mining If unobservable or latent, the factors are statistical artifacts and hence again the result of data mining Lisbon ICCF 4-8 September 2017 7 /

Estimating the SDF model To estimate the no-arbitrage condition we also require equations for the factors Often these are a VAR in z t = (z 1,t,..., z n,t ) r t+1 f t = (1 + f t )Σ n i=1β i Cov t (z i,t+1, r t+1 f t ) + ξ t+1 z t+1 = A t z t + ε t+1 (ξ t+1 ε t+1 ) {0, Σ t } Σ 1i,t = Cov t (z i,t+1, r t+1 f t ) Lisbon ICCF 4-8 September 2017 8 /

What the evidence shows General equilibrium asset pricing performs poorly The main problems are The equity premium puzzle Mehra and Prescott (1985) - too low variablity of the IMRS compared with excess returns The observable factors are macroeconomic variables and can only be observed at low frequencies whereas asset prices are high frequency and respond quickly to events As a result asset returns tend to have different properties from the observable factors seeking to explain the movements or returns such as - volatility clustering - excess kurtosis - often have short memory There is a principal-agent problem - eg FOREX trader has different aims from the pension fund and the pensioner Lisbon ICCF 4-8 September 2017 9 /

The academic literature has sought to deal with the poor performance of GE asset pricing by - adding "bells and whistles" to the theory to raise the IMRS - using more general statistical forms such as conditional volatility models, fractional integration, network models, chaos theory etc How different is this from data-mining? Lisbon ICCF 4-8 September 2017 10 /

For the rest of this lecture I will examine the problems of pricing equity and bonds that have been thrown up by my own research over the years. I don t have time for a similar analysis of FOREX I have also written extensively on macroeconomics and its disconnect with finance It is interesting that macroeconomic theory - the basis of GE theories - usually considers only the first moment and uses linear approximations. It ignores risk premia: to explain asset prices we need theories that have time-varying higher moments Lisbon ICCF 4-8 September 2017 11 /

Pricing Equity C-CAPM The SDF for power utility is M t+1 = βu (c t+1 ) U (c t ) β (1 σ t ln c t+1 ) where σ t = c t U t > U t 0 is the CRRA and c t r t W t. The no-arbitrage condition is E t (r t+1 f t ) = β(1 + f t )σ t Cov t ( ln c t+1, r t+1 f t ) The equity premium puzzle is that σ t has to be very large as ln c t+1 has low volatility The "bells and whistles" are ) σ ( - habit persistence M t+1 = β Ct+1 λx t+1 C t λx t [ ( ) 1 ] 1 σ - non-separable utility M t+1 = β Ct+1 γ 1 γ 1 C t 1 σ (Rt+1 m 1 )1 γ 1 Expectations here are conditional and not unconditional (sample Lisbon ICCF 4-8 September 2017 12 /

CAPM E t (r i,t+1 f t ) = β it E t (r m t+1 f t ), β it = Cov t(r m t+1, r i,t+1) V t (r m t+1 ) E t (r m t+1 f t ) = σ t V t (r m t+1). CAPM is also an SDF pricing model as M t+1 = σ t (1 + r m t+1 ) Moreover CAPM is almost identical to C-CAPM as c t r m t W t and r m = W t+1 W t E t (r i,t+1 f t ) = σ t Cov t (r m t+1, r i,t+1 f t ) = σ t Cov t ( W t+1 W t, r i,t+1 f t ) Lisbon ICCF 4-8 September 2017 13 /

CAPM and data-mining There is a huge amount of evidence on the failure of CAPM Tests of found that α > 0 and β < 1 r i,t+1 f t = α + β(r m t+1 f t ) + ε t+1 Fama and French (1993) three-factor model added two factors not part of the theory - the returns of small-minus-big capitalizations (SMB) - high-minus-low returns (HML) They were found to be more important then the market return r m This led to a large number of articles that searched for other significant factors - data-mining had now replaced theory Lisbon ICCF 4-8 September 2017 14 /

Statistical issues with data-mining for significant factors There is a fundamental statistical problem with searching among a large number of possible variables to find those that are significant in CAPM. The probability of a Type I error (finding significance by chance when there is none) is much higher than the conventional values. If in a single test the probability that a variable is significant when it is not is α = 5% Then in 20 tests there is a 64% chance of observing at least one significant result P(at least one significant result) = 1 P(no significant results) = 1 (1 0.05) 20 0.64 Lisbon ICCF 4-8 September 2017 15 /

Use a significance level of α/n With n = 20 and α = 0.05 P(at least one significant result) = 1 P(no significant results) = 1 (1 0.05 20 )20 0.0488 A little bit conservative Campbell Harvey, Liu and Zhu (2014) examine the 315 different variables used in 312 studies of CAPM They recommend using a t value of 3 and not 2. This corresponds to a p value = 0.0027 Campbell Harvey and Yan Liu "Lucky factors" (2017) using panel data and weighted portfolios find that by far the dominant variable is the market factor. SMB is marginally significant and HML is next but is not significant. Nor is any other factor. Lisbon ICCF 4-8 September 2017 16 /

An SDF test of CAPM with Fama-French factors Abhakorn, Smith and Wickens (2015) apply an SDF test treating the Fama-French factors as multiple factors They use the fact that the factors are all asset returns and should be jointly modelled According to the SDF approach under log-normality for all nominal assets E t r t+1 f t = 1 2 V t(r t+1 )+Σ n i=1β i Cov t (z i,t+1, r t+1 f t ) +Cov t (π t+1, r t+1 f t ) (1) Lisbon ICCF 4-8 September 2017 17 /

Hence for r, SMB and HML E t (r t+1 f t ) = 1 2 V t(r t+1 ) + β 1 C t ( c t+1, r t+1 ) +β 2 C t (SMB t+1, r t+1 ) + β 3 C t (HML t+1, r t+1 ) +C t (π t+1, r t+1 ) E t (SMB t+1 ) = (β 2 1 2 )V t (SMB t+1 ) + β 1 C t( c t+1, SMB t+1 ) +β 3 C t (HML t+1, SMB t+1 ) + C t (π t+1, SMB t+1 ) E t (HML t+1 ) = (β 3 1 2 )V t (HML t+1 ) + β 1 C t( c t+1, HML t+1 ) C-CAPM H 0 : β 2 = β 3 = 0 +β 3 C t (HML t+1, SMB t+1 ) + C t (π t+1, HML t+1 ) This restriction is not imposed in the "three-factor model" The model is estimated for each of 25 Size and Book-to-Market Portfolios imposing the cross-equation restrictions on the β i and with Garch-in-Mean Lisbon ICCF 4-8 September 2017 18 /

(a) C CAPM (b) General Three Factor Model 1.1 12 15 1.1 0.9 0.7 0.5 11 21 31 51 41 42 52 53 54 22 32 43 33 13 23 45 34 44 14 25 24 35 0.9 0.7 0.5 45 34 44 22 33 32 4312 23 13 54 55 35 25 24 14 15 Predicted Risk Premium 0.3 55 Predicted Risk Premium 21 0.3 11 31 42 41 53 52 51 0.1 0.1 0.3 0.5 0.7 0.9 1.1 Average Excess Return 0.1 0.1 0.3 0.5 0.7 0.9 1.1 Average Excess Return Lisbon ICCF 4-8 September 2017 19 /

The findings are C-CAPM performs well for all except portfolios with low book-to-market ratios But the general three-factor SDF model performs better Consumption and HML, but not SMB, significantly determine equity returns HML improves the fit of low book-to-market portfolios HML helps explain SMB and HML returns Lisbon ICCF 4-8 September 2017 20 /

The term structure of interest rates 2.6 US Term Structure for each month of 1947 2.4 2.2 2 Pure discount rate % 1.8 1.6 1.4 1.2 1 0.8 0mo 1mo 2mo 3mo 4mo 5mo 6mo 9mo 1yr 2 yr 3yr 4yr 5yr 10yr 15yr 20yr Maturity Lisbon ICCF 4-8 September 2017 21 /

18 US Term Structure for each month of 1981 17 16 15 14 Pure discount rate, % 13 12 11 10 9 8 0mo 1mo 2mo 3mo 4mo 5mo 6mo 9mo 1yr 2 yr 3yr 4yr 5yr 10yr 15yr 20yr 25yr Maturity Lisbon ICCF 4-8 September 2017 22 /

US Yield Curve 31 Jan 2006 4.8 4.75 4.7 4.65 % per annum 4.6 4.55 4.5 4.45 4.4 4.35 0 5 10 15 20 25 31 Jan 06 Lisbon ICCF 4-8 September 2017 23 /

UK Yield Curve 15 Feb 2006 4.40 4.30 4.20 % per annum 4.10 4.00 3.90 3.80 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Lisbon ICCF 4-8 September 2017 24 /

Lisbon ICCF 4-8 September 2017 25 /

UK long and short rates 1688-1997 Lisbon ICCF 4-8 September 2017 26 /

Pricing bonds Two approaches to pricing bonds dominate - the Expectations Hypothesis - assumes investors are risk neutral - the SDF approach based on latent or on both latent and observable factors Apart from C-CAPM, all involve Relative Asset Pricing - pricing long rates off the short rate I will focus my remarks on the SDF approach Lisbon ICCF 4-8 September 2017 27 /

EH Assumes investors are risk neutral with a no-arbitrage condition (i.e.ρ n,t = 0) (n 1)[E t [R n 1,t+1 R n,t ] = (R n,t s t ) + ρ n,t This implies that R n,t = 1 n n 1 i=0 E ts t+i There is evidence that long rates both over and under react to short rates compared with theory I have shown that including a risk premium corrects for this Lisbon ICCF 4-8 September 2017 28 /

C-CAPM E t (h n,t+1 s t ) + 1 2 V t(h n,t+1 ) = σcov t (h n,t+1, ln c t+1 ) + Cov t (h n,t+1, π h n,t+1 = nr n,t (n 1)R n 1,t+1 The RHS is the risk premium I have shown that the restrictions do not hold Lisbon ICCF 4-8 September 2017 29 /

The standard factor model The multi-factor CIR model with independent factors z it and R nt = 1 n p n,t is p n,t = [A n + B ni z it ] i m t+1 = z it + λ i e i,t+1, i i z i,t+1 µ i = φ i (z it µ i ) + e i,t+1, It then follows that e i,t+1 = σ i zit ε i,t+1. Bn 1,i 2 σ 2 i z it + λ i B n 1,i σ 2 i z it, i E t p n 1,t+1 p nt + p 1,t = 1 2 i risk premium = λ i B n 1,i σ 2 i z it. i For the Vasicek model the risk premium is constant through time For the CIR model the term structure of the risk premium is fixed through time Lisbon ICCF 4-8 September 2017 30 /

The short rate is a linear function of the factors In the one factor model Hence s t = p 1,t = (1 1 2 λ2 i σ 2 i )z it. i s t = (1 1 2 λ2 σ 2 )z t. z t = s t 1 1 2 λ2 σ 2 If z t is generated by an AR(1) then s t is generated by the AR(1) s t+1 µ(1 1 2 λ2 σ 2 ) = φ[s t µ(1 1 2 λ2 σ 2 )] + (1 1 2 λ2 σ 2 )e t+1 Lisbon ICCF 4-8 September 2017 31 /

It also follows that R n,t = 1 n = 1 n n 1 E t (s t+i + ρ n j,t+j ) j=0 { n 1 j=0 E t z i,t+j = 1 φj i 1 φ i µ i + φ j i z i,t Hence [(1 1 2 λ2 i σ 2 i ) 1 i 2 B2 n 1,i σ 2 i + λ i B n 1,i σ 2 i ]E t z i,t+ n 1 R n,t = γ n,0 + i=1 γ n,i z i,t It follows that the m factors z i,t can be expressed as linear functions of any m yields Lisbon ICCF 4-8 September 2017 32 /

Implications for the yields As the m factors are generated by a VAR so are the m yields If a vector of m yields is R t and the vector of m factors is z t then R t = Γ 0 + Γ 1 z t z t µ = Φ(z t 1 µ) + e t This implies that the yields are also generated by a VAR R t = Ψ 0 + Ψ 1 R t 1 + Γ 1 e t Ψ 0 = (I Γ 1 ΦΓ1 1)Γ 0 + Γ 1 (I Φ)µ Ψ 1 = Γ 1 ΦΓ1 1 Lisbon ICCF 4-8 September 2017 33 /

Look at the principal components of the set of yields US - three factors explain at least 98% of the variation in all yields for the US - a shift, a slope and a curvature factor - the shift factor (short rate) explains 90% - the slope factor (spread) explains about 80% of the remaining variation - the curvature factor never explains more than 5% of the total variation UK 1970-2015 - one factor explains 99.45% - two factors 99.98% 2008-2015 - one factor explains 97.25% - two factors explain 99.94%. Ang and Piazzesi add observable variables as factors in the SDF model - find two macro variables explain 85% of the forecast variance at short and medium maturities - inflation is the dominant factor. Lisbon ICCF 4-8 September 2017 34 /

What about...? Monetary policy The macro economy Short rates are determined by monetary policy - a Taylor rule? So are short rates generated by a VAR or a Taylor rule? Lisbon ICCF 4-8 September 2017 35 /

Rudebusch et al. (2007) and Dew-Becker (2013) embed the term structure within a New Keynesian DSGE model with a Taylor rule determining the short rate Rudebusch et al calibrate and then simulate their model taking a third-order approximation to capture time-varying volatilty effects they find that shocks to the Fed Funds rate, government expenditures and technology do not generate suffi ciently large changes in the term premium Dew-Becker uses a mixture of calibration and Bayesian estimation, and allows the shocks in the model to be autocorrelated The model generates large and volatile term premium driven mainly by two factors: - a negative response of interest rates to positive technology shocks - variation in risk aversion. Lisbon ICCF 4-8 September 2017 36 /

However, just as the term structure reduces to a VAR, DSGE models also reduce to a VAR All DSGE models after linearisation can be written [ xt+1 E t y t+1 ] [ Axx A = xy A yx A yy ] [ xt y t ] + [ Cx C y ] z t x t+1 vector of predetermined variables, y t+1 vector of jump variables, z t exogenous. Solution is [ xt y t ] = M [ xt 1 y t 1 ] + NΣ s=0γ s yy P y E t z t+s +Jz t 1 + K ξ t If z t is generated by the VAR z t+1 = Rz t + ε t+1 then all of the variables are generated by the restricted VAR x t x t 1 ξ xt y t = F y t 1 + G ξ yt. z t z t 1 ε Lisbon t ICCF 4-8 September 2017 37 /

Conclusions Like DSGE and simultaneous macro models, all of these asset pricing models lead to a VAR. In the case of finance models the VAR errors may have time-varying second and higher moments The role of theory is to provide restrictions on these VARs Data-mining consists in ignoring these restrictions - and hence being able to give an economic interpretation of the results Lisbon ICCF 4-8 September 2017 38 /

A final thought A VAR extrapolates the past. It doesn t anticipate future structural changes. What might be a high standard deviation event based on the past might be a low standard deviation event once the VAR has been adjusted for structural change i.e. the new model This is why keeping theory in mind is so important and data-mining is so dangerous Lisbon ICCF 4-8 September 2017 /