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Systematic Volatility of Unpriced Earnings Shocks – Internet Appendix – Timothy C. Johnson and Jaehoon Lee∗ September 5, 2013

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Valuation of Unlevered Equity

The following matrix representation summarizes the dynamics of the three state variables in the economy.      σx x,t+1 ρ 0 0 0       = wc  + 0 αc 0  Xt +  sc ηc,t+1  sd ηd,t+1 0 0 αd wd

(1)

= µ + Φ Xt + ut+1

(2)



Xt+1

h i> where Xt ≡ xt vc,t vd,t . Let Ω denote the covariance matrix of the shocks, Ω ≡   Et ut+1 u> t+1 . The log pricing kernel of the Epstein–Zin preference is given as mt+1 = θ log δ −

θ ∆ct+1 + (θ − 1) rc,t+1 ψ

(3)

where rc,t+1 denotes the log gross return on an asset that delivers aggregate consumption 1−γ as its dividends each period. The parameter θ ≡ 1−1/ψ , with γ ≥ 0 being the risk-aversion parameter and ψ ≥ 0 the intertemporal elasticity of substitution (IES) parameter. ∗

Timothy Johnson: University of Illinois, Urbana-Champaign (e-mail: [email protected]). Jaehoon Lee: University of New South Wales (e-mail: [email protected]).

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Following Bansal and Yaron (2004), rc,t+1 is log-linearly approximated as rc,t+1 = κ0 + κ1 zc,t+1 − zc,t + ∆ct+1

(4)

where zc,t is the log price-dividend ratio of the claim to aggregate consumption, and κ0 and κ1 are approximating constants that both depend only on the average level of z c . κ1 =

ez¯c 1 + ez¯c

(5)

κ0 = log (1 + ez¯c ) −

ez¯c z¯c 1 + ez¯c

(6)

By plugging zc,t = A0 + A> Xt and the formulas above into the Euler equation, Et [emt+1 erc,t+1 ] = 1, we can derive     1
−1  1  A= 1− I − κ1 Φ>  12 (1 − γ) ψ 0     1 1 2 > > 1 A0 = log δ + κ0 + κ1 A µ + 1 − µc + 2 θ κ1 A Ω A 1 − κ1 ψ

(7)

(8)

Similarly, the log gross return on a firm’s underlying asset (unlevered equity) can be approximated as rd,t+1 = κ2 + κ3 zd,t+1 − zd,t + ∆dt+1 (9) z ¯

e d ¯d where zd,t represents the firm’s log price-dividend ratio, and κ2 = log (1 + ez¯d ) − 1+e z ¯d z ez¯d > and κ3 = 1+ez¯d . By plugging zd,t = B0 +B Xt again into the Euler equation, its solutions can be derived as   φ − ψ1  o   n 2 > −1  1 1  (10) B = I − κ3 Φ  2 (γ − φ) − (γ − 1) γ − ψ  1 2

B0 =

 1  θ log δ + (φ − γ) µc + (θ − 1) κ0 + κ1 (A0 + A> µ) − A0 1 − κ3 +κ2 + κ3 B > µ +

1 2

{(θ − 1)κ1 A + κ3 B}> Ω {(θ − 1)κ1 A + κ3 B}

Note that B3 is not zero because of Jensen’s inequality.

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i

(11)

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Analytic Solution of Debt and Levered Equity

Let’s begin with a lemma that is critical to derive the analytic solution of debt and levered equity. Lemma 1. If i is a (2 × 1) constant vector and X = (X1 , X2 )> is a bivariate normal with mean vector µ and covariance matrix Σ, then Z

a i + 1 i> Σ i 2

P (a < X2 < b ; µ∗ , Σ )

(12)

where P ( · ; µ∗ , Σ ) is the bivariate density function with mean vector µ∗ = µ + Σ i and covariance matrix Σ. In this internet appendix, we assume the traditional Merton (1974)’s payoff functions.  R (V + D ) T T FT = K

,

 0 ST = VT + DT − K

if VT + DT < K

(13)

if VT + DT ≥ K

where FT denotes the payoff to corporate debts at maturity, ST the payoff to levered equity, VT the value of underlying assets, DT dividends, K the principal amount of debts, and R the recovery rate in case of bankruptcy. Suppose the maturity comes in one time period, T = t + 1. Today’s price of these assets can be derived as Ft = Et [emt+1 Ft+1 ] = Ke−rf,t G (ω − σr ) + R Vt G (−ω)

(14)

St = Et [emt+1 St+1 ] = Vt G(ω) − Ke−rf,t G(ω − σr )

(15)

where ω ≡ {ln (Vt /K) + rf,t + 12 σr2 } /σr , σr2 ≡ vart (rd,t+1 ) = φ2 vc,t + vd,t + κ23 B > Ω B is the variance of unlevered asset returns, and g( · ) and G( · ) denote the pdf and cdf of standard normal random variable. Note that ω implies the firm’s leverage relative to its volatility. Its comparative statistics are 1 ∂ω = , ∂Vt σr Vt

∂ω ω =1− , ∂σr σr 3

∂ω 1 = ∂rf,t σr

(16)

In the rest of this section, it is assumed that the firm is not underwater, i.e., Vt ≥ K e−rf,t , thus ω > 0. The partial derivatives of the levered claims can be derived like those of Black and Scholes (1973), ∂St = G(ω) ∂Vt ∂St = Vt g(ω) ∂σr ∂St = K e−rf,t G(ω − σr ) ∂rf,t

∂Ft 1−R = R G(−ω) + g(ω) ∂Vt σr   ∂Ft ω = −Vt g(ω) R + (1 − R) ∂σr σr ∂Ft 1 = −K e−rf,t G(ω − σr ) + Vt g(ω)(1 − R) ∂rf,t σr

(17) (18) (19)

And the comparative statistics of the Black-Scholes inputs with regard to our state variables are ∂Vt = Vt B(1) ∂xt ∂σr =0 ∂xt 1 ∂rf,t = ∂xt ψ

∂Vt = Vt B(2) ∂vc,t ∂σr φ2 = ∂vc,t 2 σr ∂rf,t 1 − γ (1 + ψ) = ∂vc,t 2ψ

∂Vt = Vt B(3) ∂vd,t ∂σr 1 = ∂vd,t 2 σr ∂rf,t =0 ∂vd,t

(20) (21) (22)

where B(i) denotes the i-th element of the column vector B in equation (10). ∂Vt One interesting implications is that ∂v = Vt B(2) < 0 if 1 ≤ φ ≤ 2γ − 1 and ψ > 1, c,t that is, (i) (φ ≤ 2γ −1) if Jensen’s inequality is not large enough to overcome the increase in risk premium and (ii) (φ ≥ 1) the claim to dividends is not a hedge to consumption risk. In other words, the unlevered asset value Vt can actually increase with vc,t if φ is either negative or excessively large. We will assume that φ is within the given range so that B(2) < 0.

Now, let’s study St ’s comparative statistics with each of the state variables. (1) St and xt ∂St 1 = G(ω) Vt B(1) + K e−rf,t G(ω − σr ) > 0 ∂xt ψ

4

always

(23)

(2) St and vc,t ∂St φ2 1 − γ(1 + ψ) = G(ω) Vt B(2) + Vt g(ω) + K e−rf,t G(ω − σr ) {z } ∂vc,t | {z } | {z } | {z } 2σr | 2ψ |{z} | {z } (+) (+) (+) (−) (+)

2



(−) if γ>1



φ ≤ Vt G(ω)B(2) + g(ω) 2σr   φ2 √ ≤ Vt G(ω)B(2) + g(ω) 2κ3 B > Ω B ( ) φ2
≤ Vt G(ω)B(2) + g(ω) 2κ3 −sc B(2) ∴

(24)

φ if B(2) < − √ 2κ3 sc

∂St 0 always ∂vd,t 2σr

(29)

Comparative statistics of corporate debts, Ft , are more complicated as they depend on the firm’s leverage. For example, the long-run growth rate, xt , can move Ft in either direction since it raises not only the underlying asset value (Vt ) but also the riskfree interest rate (rf,t ). Let’s assume R = 1 for the sake of simplicity. Ft ’s comparative statistics are derived as (1) Ft and xt ∂Ft 1 = G(−ω) Vt B(1) − Ke−rf,t G(ω − σr ) ∂xt ψ   1 2 G(ω − σr ) 1 φ ψ − 1 −rf,t σr ω− 2 σr = Ke G(−ω) e − ψ 1 − κ3 ρ G(−ω) | {z }

(30) (31)

(+)

r) Since G(ω−σ ∈ (0, ∞) is monotonically increasing in ω and rises faster than eσr ω , G(−ω) t there exists ω ∗ such that ∂F (ω ∗ ) = 0. Thus, ∂xt

  ∂Ft > 0 ∂xt  ∂Ft < 0 ∂xt

if leverage is high, i.e., if leverage is low, i.e., 5

ln(Vt /K) < σr ω ∗ − rf,t − 12 σr2 ln(Vt /K) > σr ω ∗ − rf,t − 12 σr2

(32)

(2) Ft and vc,t ∂Ft φ2 = G(−ω) Vt B(2) − Vt g(ω) ∂vc,t 2σr 1 − γ (1 + ψ) − Ke−rf,t G(ω − σr ) 2ψ ∂Ft lim = Vt B(2) < 0 ω→−∞ ∂vc,t γ (1 + ψ) − 1 ∂Ft lim = Ke−rf,t >0 ω→∞ ∂vc,t 2ψ ∂Ft ∂Ft Since limω→−∞ ∂v < 0, limω→∞ ∂v > 0 and c,t c,t ∂Ft ∗ ∗ ω ) = 0. Therefore, ω ˆ such that ∂vc,t (ˆ

  ∂Ft < 0 ∂vc,t  ∂Ft > 0 ∂vc,t

if leverage is high, i.e., if leverage is low, i.e.,

∂Ft ∂vc,t

(33) (34) (35)

is continuous in ω, there exists

ln(Vt /K) < σr ω ˆ ∗ − rf,t − 12 σr2 ln(Vt /K) > σr ω ˆ ∗ − rf,t − 12 σr2

(36)

(3) Ft and vd,t Two opposite channels are effective: (i) vd,t ↑ ⇒ Vt ↑ (Jensen’s inequality) ⇒ Ft ↑ and (ii) vd,t ↑ ⇒ σr ↑ ⇒ Ft ↓. 1 ∂Ft = G(−ω) Vt B(3) − Vt g(ω) ∂vd,t 2σr   g(ω) 1 = Vt G(−ω) B(3) − 1 − G(ω) 2σr Since ∂Ft (˜ ω ∂vd,t

∴

g(ω) 1−G(ω) ∗

(37) (38)

∈ (0, ∞) is monotonically increasing in ω, there exists ω ˜ ∗ such that

) = 0. However, since

g(ω) 1−G(ω)

> 0.79 for ω > 0 as assumed previously,

∂Ft 0.79 < 0 if the Jensen’s inequality is not dominant, i.e., B(3) ≤ ∂vd,t 2σr

(39)

In contrast to the previous two cases, the comparative statics of credit spreads, cr ≡ − ln(Ft /K) − rf,t , are much simpler.

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(1) cr and xt ∂cr Vt G(−ω) =− ∂xt Ft

  1 B(1) + 0  B(2) +  G(−ω) −  ∂vc,t Ft  2ψ 2σr |{z}    | {z }   | {z }  (−) 

(41)

∂cr 1 ∂Ft =− >0 ∂vd,t Ft ∂vd,t

(42)

(−)

(−)

(3) cr and vd,t

Now let’s move on to expected excess returns. The excess returns of unlevered assets are determined by the covariance of its return with the pricing kernel, − σmr ≡ − covt (mt+1 , rd,t+1 ) = φ γ vc,t + constant

(43)

Those of levered equity can be derived as 

 Et [St+1 ] eerS ≡ ln − rf,t St     σmr −rf,t Vt e−σmr G ω − σσmr − Ke G ω − σ − r σr r exp(eerS ) = −r f,t Vt G (ω) − Ke G (ω − σr ) Vt ∴ eerS ≈ −σmr G(ω) by the Taylor approximation S | t {z } leverage effect

7

(44)

(45) (46)

And below are the excess returns of corporate bonds. 

 Et [Ft+1 ] eerF ≡ ln − rf,t Ft    −rf,t Vt e−σmr G −ω + σσmr + Ke G ω − σr − r exp(eerF ) = Vt G (−ω) + Ke−rf,t G (ω − σr ) Vt G(−ω) ∴ eerF ≈ −σmr Ft

(47) σmr σr

 (48) (49)

Note that the value-weighted average of stock and corporate bond excess returns is equal to the risk premium of the unlevered asset. St Ft eerS + eerF = −σmr Vt Vt

(50)

Let’s define the leverage factor of stocks as follows. Vt 1 G(ω) = St 1 − lS (ω) 1 2 G(ω − σr ) lS (ω) ≡ e−σr ω+ 2 σr G(ω)

LS (ω) ≡

(51) (52)

Both LS (ω) and lS (ω) are monotonically decreasing in ω. They will turn useful in the derivation of the comparative statistics of eerS that follow. (1) eerS and xt ∂ lS (ω) 1 ∂ lS (ω) · = ∂xt | ∂{zω } |σr (−)

  1 B(1) + 0 ∂vd,t

9

(63)

References Bansal, Ravi, and Amir Yaron, 2004, Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles, The Journal of Finance 59, 1481–1509. Black, Fischer, and Myron Scholes, 1973, The Pricing of Options and Corporate Liabilities, The Journal of Political Economy 81, 637–654. Merton, Robert C., 1974, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, The Journal of Finance 29, 449–470.

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