What Ties Return Volatilities to Price Valuations and Fundamentals? On-Line Appendix

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1 Wha Ties Reurn Volailiies o Price Valuaions and Fundamenals? On-Line Appendix Alexander David Haskayne School of Business, Universiy of Calgary Piero Veronesi Universiy of Chicago Booh School of Business, CEPR, andnber December 9, 1

2 This online appendix conains he complee proof o Proposiion 1 (a), as well as addiional maerial ha is excluded from he aricle. 1. Proof of par (a) of Proposiion 1 The proof follows he seps of Proposiion 1 in Veronesi (). The pricing equaion is [ M τ D [ τ M τ E τ P = D E dτ = D E dτ = D E e (mτ m)+(eτ e) dτ M D M E where we assume ha he usual ransversaliy condiion holds and hus all inegrals are well defined. 1 From Io s Lemma ( dm = k i σ ) Mσ M d σ M dw ( de = θ i σ ) Eσ E d + σ E dw Denoe V ( ; ν i) = E e (mτ m)+(eτ e) dτ ν = ν i Because he sochasic variable ν is righ-coninuous, we can selec a small inerval Δ such ha ν τ = ν i for τ [, + Δ), which yields V ( [ ; ν i) +Δ = E e (mτ m)+(eτ e) dτ ν = ν i + E e (mτ m)+(eτ e) dτ ν = ν i +Δ [ +Δ «(τ )+( σ M +σ E )(W τ W ) dτ ν = ν i = E +Δ e k i +θ i σ M σ M σ E σ E +E e (mτ m)+(eτ e) dτ ν = ν i The firs expecaion is given by [ +Δ E e = = +Δ +Δ e k i +θ i σ M σ M σ E σ E k i +θ i σ M σ M σ E σ E e ( ki +θ i σ M σ E)(τ ) dτ = e( ki +θi σ M σ E )Δ 1 ( k i + θ i σ M σ E ) «(τ )+( σ M +σ E )(W τ W ) dτ ν = ν i «(τ )+ 1 ( σ M +σ E )( σ M +σ E ) (τ ) dτ 1 A sufficien condiion is ha he ransversaliy condiion holds for each regime, i.e. under he assumpion of no regime shifs. In his case, he inegral is finie if k i + θ i σ M σ E <. 1

3 The second erm can also be compued as [ E e (mτ m)+(eτ e) dτ ν = ν i = E e (m +Δ m )+(e +Δ e ) +Δ The firs erm in he las expression is and he second erm can wrien as E e (mτ m +Δ)+(e τ e +Δ ) dτ ν = ν i +Δ +Δ = E [ e (m +Δ m )+(e +Δ e ) ν = ν i E e (mτ m +Δ)+(e τ e +Δ ) dτ ν = ν i +Δ E [ e (m +Δ m )+(e +Δ e ) ν = ν i = e ( ki +θ i σ M σ E)Δ +Δ e (mτ m +Δ)+(e τ e +Δ ) dτ ν = ν i = E e (mτ m +Δ)+(e τ e +Δ ) dτ ν +Δ = ν j λ ij Δ j i [ ( +E e (mτ m +Δ)+(e τ e +Δ ) dτ ν +Δ = ν i 1 ) λ ij Δ +Δ j i ) = j i Using a Taylor expansion on ime, we see ha for each i V ( +Δ;v i) = V ( ; v i) + V ( ; v i) Δ+o (Δ) V ( +Δ;v j) λ ij Δ+V ( +Δ;v i) ( 1 j i Puing all he erms ogeher, and eliminaing erms o (Δ n )forn>1, we obain λ ij Δ V ( ; v i) = e( ki +θi σ M σ )Δ E 1 ( k i + θ i σ M σ E [ ) +e ( ki +θ i σ M σ E)Δ V ( ; v i) + V ( ; v i) Δ j i λ ij V ( ; v i) Δ+ j i λ ij V ( ; v j) Δ Rearranging, dividing by Δ, and recalling λ ii = j i λ ij, weobain V ( ( ) [ ; v i) 1 e ( ki +θi σ M σ E )Δ = e( ki +θi σ M σ )Δ E )Δ+e( ki 1 +θi σ M σ E )Δ V ( ; v i) + Δ ( k i + θ i σ M σ E Taking he limi as Δ, and rearranging V ( ; v i) n = 1 λ ij V ( ; v j) + ( ) ( k i θ i + σ M σ ) E V ; v i j=1 Define A = Λ + diag ( ) k i θ i + σ M σ E and wrie in vecor form V () =AV () 1 Being he model ime homogeneus, V () = and we obain he resul V = A 1 1 n. n λ ij V ( ; v j) j=1

4 . Addiional Maerial This secion conains some addiional maerial excluded from he aricle. Firs, Table 1 repors he momens from he SMM procedure, described ( in he ) appendix. Recall ha our esimaion procedure compues momens as ε() = e(), ˆL,wheree() Ψ collecs he differences beween daa observed financial quaniies and heir model s counerpar, which in urn depend on probabiliies. The second erm ˆL are he scores of he Ψ likelihood funcion. Deails of he procedure are in he appendix of he paper. Second, Figure 1 repors a plo of inflaion and earnings uncerainy compued from he model and proxies from he Survey of Professional Forecasers probabiliies. In boh cases, he uncerainy is compued as he condiional variance of he fuure inflaion or economic growh. Deails are in Secion 4.1 of he paper. Third, Figure plos he fied asse prices from our model wih six composie regimes, and compares hem o he fied asse prices from a simpler model, ha is, as model wih wo earnings drifs and wo inflaion regimes. As i is apparen, he model, which has hree sae variables (beliefs), does no fi he asse pricing daa well, especially on he dimension of bonds yields, volailiy, and, mos imporanly, he covariance beween socks and bonds. The model is srongly rejeced. Because he 4-regime model does no produce a dynamics of asse prices ha is vaguely comparable o he daa, we canno hope o learn much from he model abou he relaion beween fundamenals, volailiies,and price valuaions. In conras, he model wih six composie regimes allow us o obain numerous new predicions, ha we can es in he daa. 3

5 Table 1: Momens and Mean Absolue Errors from SMM A: Pricing Errors Variable Mean Error MAE P/E M Yield (%) Y Yield (%) S. Vol (%) Y B Vol (%) y B Vol (%) CS-Y (%) CS-1Y (%) C1Y-Y (%) Sharpe Raio B: Scores of Likelihood Funcion Variable Mean Error (Scaled) MAE β β β β 4 1.8E E- θ θ θ E-4 σ Q, σ Q, σ E, σ S,1-6.E- 4.8E-4 σ S, -.3E- 4.13E-4 σ C, E- σ C, E- σ N E-4 λ 1 1.8E-7.E-6 λ E E-6 λ E E- λ 1 -.8E- 1.6E-4 λ 3 -.3E-6 9.1E- λ E- 6.36E- λ -1.6E-7.7E-6 λ 6 1.4E-7 3.7E-6 λ 3-1.9E-.9E-4 λ E-7 3.3E-6 λ E-6 6.9E- λ 4 4.1E-6 4.8E- λ 1 -.8E-7.47E-6 λ 3.9E E-6 λ E- 7.8E- Noes: Panel A repors he pricing errors from he SMM procedure, while Panel B repors he momens from he scores of he likelihood funcion. The deails of he esimaion procedure are conained in he Appendix of he paper. 4

6 Figure 1: Model Uncerainy versus SPF Uncerainy corr. = 39% A: Inflaion Uncerainy Model SPF Model SPF B: Earnings Uncerainy corr. = 43% Noes: Panel A plos he condiional variance of fuure inflaion compued using he model s probabiliies (solid line) or he inflaion probabiliies exraced from he Survey of Professional Forecasers (doed line). Panel B repors he condiional variance of fuure earnings growh using model s probabiliies (solid line), and he condiional variance of economic growh using he Survey of Professional Forecasers probabiliies of real GDP decline. Deails on he compuaion are in Secion 4.1 of he paper.

7 Figure : Comparison wih Four-Regime Model A. P/E Raio B. Sock Volailiy C. Shor erm Yield 1 D. Long erm Yield E. Year Bond Volailiy. F. Sock year Bond Covariance Daa Model Model 4 regimes Noes: Panel A - F compare he fied asse prices from he six composie regimes presened in he main aricle o he resriced case wih only four regimes. The four regimes have wo inflaion regimes, and wo real earnings growh regimes. 6