Long Term Risk

Long Term Risk

Lectures at Coll`ege de France

Objective

In financial economics risk-return tradeoffs show how expected

rates of return and consequently asset prices are altered in response to changes in the exposure to the underlying shocks that impinge in the economy. In these lectures we will:

(a) Present some of the recent literature that is concerned with the effect of long run risk on returns and prices.

(b) Develop an analytical structure that reveals the long-run

risk-return relationship in nonlinear continuous time Markov environments.

Motivation

• Evaluation of economic models of preferences and technologies using asset prices.

– How long run-risks affect short term returns.

∗ 6% equity premium ∗ low risk-free rate

– What are the long run implications of a model

∗ Market microstructure, transaction costs... may make it hard to evaluate these models using

short run data. ∗ Behavioral biases

∗ Long run risk-return frontier

• How risk averse agents value the risks of permanent shocks. • How should we discount risky projects.

Utility • C a consumption process in [0, ∞] 1. Standard additive utility

• Us = Es[  _∞ s eb(s−t)u(Ct)dt], b > 0 • u(C) = C 1−a _{− 1} 1 − a , a > 0 – If a = 1, U (C) = log C.

2. Habit Formation

• An external (to the individual) habit process H •

Us = Es[

 _∞ s

eb(s−t)dsu(Ct, Ht)dt], b > 0

• Campbell and Cochrane (1999), Menzly, Santos and Veronesi (2004)

u(C, H) = (C − H)

1−a

3. Kreps-Porteus utility

• A probability space with a d dimensional Brownian motion and associated (completed) filtration Fs _{and Es}(·) = E(·|Fs) • ¯Wγ the continuation utility for a consumption path c from

t on (conditional on current information) • ¯Wγ solves ¯ Wsγ = Es[  T s ¯ fγ(Ct, ¯Wtγ)dt + 1 2A γ_{( ¯} Wtγ)||σγ(t)||2 + ¯W γ T] • + Transversality condition

– Aγ is a variance multiplier applying a penalty to the volatility of continuation utility σγ.

• γ = (ρ, a), ρ ≤ 1, a ≥ 0, a = 1. • ¯fγ(C, x) = β_ρ C_xρρ−1−xρ , Aγ(x) = −a_x

• ρ = 1 − a standard additive utility • Transformation Wγ

= φγ( ¯_Wγ) of Duffie-Lions (1991) to eliminate variance multiplier A

• Wsγ = Es[  T s fγ(Ct, W_tγ)dt + W_Tγ] • fγ(C, W ) = b ρ Cρ − [(a − 1)W ]a−1ρ (a − 1)a−1ρ _W(a−1ρ −1) • For ρ = 0,

f0,a(C, x) = −bx{(a − 1) log C + log[(1 − a)x]} • Duffie-Epstein (1992).

Stochastic discount factor

• {Xt : t ≥ 0} a Markov process in (Ω, F, P r) and Ft the associated (completed) filtration.

• A Stochastic Discount Factor S is a strictly positive adapted process with S0 = 1 such that if s ≤ t

E [StΠ_t|Fs] Ss

(1) is the price at time s of a claim to the payoff Πt at t.

•

Stψ(x) = E [Stψ(Xt)|X0 = x] , is the time-zero price of payoff ψ(Xt).

• Law of one price with intermediate trading dates • S0 = I and St+u = StSu

• Let (θtX)s = Xt+s

• Su is a function of the realization of the Markov process X that only depends on the history of X between dates 0 and u. Thus Su(θt) only depends on the history of X between dates t and t + u.

• Consider payoffs at t + u that are indicator functions of sets of histories observable at t + u, i.e. sets B ∈ Ft+u, and again using intermediate trading dates and the law of one price one obtains: E[St+u1B|X0] = E[StE[Su(θt)1B|Ft]|X0] = E[StSu(θt)1B|X0] • S0 = 1 and St+u = StSu(θt).

Example • dXtf = ξf(¯xf − X f t )dt +  XtfσfdB f t , dXto = ξo(¯xo − Xto)dt + σodBto

with ξi > 0, ¯xi > 0 for i = f, o and 2ξfx¯f ≥ σ_f2 where B = (Bf, Bo) is a bivariate standard Brownian motion. • Per-capita consumption dct = X_todt +  XtfϑfdB f t + ϑodB o t where ct = log(Ct) • Interesting case:

– σo > 0, ϑ0 ≥ 0 (positive Bo’s are unambiguously good) f

Breeden model

• Representative investor preferences are given by: E  _∞ 0 exp(−bt)Ct 1−a _{− 1} 1 − a for a and b strictly positive.

• In the standard case St = e−bt_u(Cu(Ct)

0)

• Ito’s Lemma implies that the stochastic discount factor in the Breeden model in this example is St = exp(As_t) where

Ast = −a  t 0 Xsds − bt − a  t 0  XsfϑfdB_sf − a  t 0 ϑodB o s.

Kreps-Porteus with ρ = 0 • Utility aggregator satisfies:

f0,a(C, W ) = −bW {(a − 1) log C + log[(1 − a)W ]} .

• Supress dependence of W on a

• Guess a continuation value process of the form: Wt = 1 1 − a exp  (1 − a)(wf_Xtf + wo_Xto _{+ log Ct} + ¯w) 

• The coefficients satisfy:

−ξf wf + (1 − a)σ2f

2 (wf )2 + (1 − a)ϑf σf wf +

(1 − a)ϑ2_f

2 = bwf

−ξowo + 1 = bwo

ξf ¯xf wf + ξo ¯xowo + (1 − a)σ2o

2 (wo)2 + (1 − a)ϑoσowo +

(1 − a)ϑ2_o

2 = b¯w.

ξf + b ≥ 2(a − 1)|ϑfσf|. In either case wf < 0 and we are

interested in root with smallest absolute value.

• From Duffie-Epstein St = e

[R₀t fW(Cs,Ws)ds]_f

C(Ct, Wt) fC(C0, W0)

• The stochastic discount factor is the product of two functionals. One is the exponential of:

ABt = −  t 0 X o sds − bt −  t 0  XsfϑfdB_sf −  t 0 ϑodB o s. The other is a martingale that is the exponential of:

AKPt = (1 − a) »Z _t 0 q Xsf(ϑf + wfσf)dBsf + Z _t 0 (ϑo + w_oσo)dBso – − (1 − a)2 2 Z _t 0 X f s (ϑf + wfσf)2ds − (1 − a) 2_(ϑ o + woσo)2 2 t

• For ρ = 0 expand the continuation value for ρ small

– Expand stochastic discount factor for ρ small (Hansen et al. 2008) – For instance, ABt = −(1 − ρ)[ Z _t 0 X o sds + bt + Z _t 0 q XsfϑfdBsf + Z _t 0 ϑodB o s].

Heterogeneity • Production.

• When consumers are heterogeneous individual consumption is determined in equilibrium.

• With complete markets, first welfare theorem guarantees that stochastic discount factor can be calculated using an “artificial” consumer who is a weighted average of actual consumers.

• Case where tradeable claims are a subfield of Ft.

– There are tradeable claims on aggregate uncertainty but not

on individual risk. • Heterogeneous beliefs.*

The basic Markov process

• {Xt : t ≥ 0} be a continuous time Markov process on a state space D0 ⊆ Rn. The sample paths of {Xt : t ≥ 0} are

continuous from the right and with left limits. Let Ft be

completion of the sigma algebra generated by {Xu : 0 ≤ u ≤ t}. • Often treat the case where X is stationary.

• Semimartingale X = Xc

+ Xj

• An Ft n-dimensional Brownian motion {Bt} • Xc t = X0 +  t 0 ξ(Xu)du +  t 0 σ(Xu)dBu • (ξ, Σ, η), Σ = σσ

• Xj _{with a finite number of jumps in any finite interval and} compensator η[dy|x]dt. – dXtj =  Rn yζ(dy, dt)

where ζ = ζ(·, ·; ω) is a random counting measure. That is, for each ω, ζ(b, [0, t]; ω) gives the total number of jumps in [0, t] with a size in the Borel set b in the realization ω.

– Finite number of jumps implies there is a finite measure

η(dy|x)dt that is the compensator of the random measure ζ.

– The (unique) predictable random measure, such that for each

predictable stochastic function f (x, t; ω), the process  t 0  Rn f (y, s; ω)ζ(dy, ds; ω) − t 0  Rn f (y, s; ω)η[dy|Xs−(ω)]ds is a martingale

Building non-stationary processes

• A functional: A real-valued process {Mt : t ≥ 0} adapted, (with a version that is) right continuous with left limits.

• The functional {Mt : t ≥ 0} is multiplicative if M0 = 1, and

Mt+u = Mu(θt)Mt.

• Product of multiplicative processes is multiplicative.

• If M strictly positive log(M) will satisfy an additive property. • A functional is additive if A0 = 0 and At+u = Au(θt) + At, for

each nonnegative t and u. • Parameterize log(M)

Multiplicative semigroups

• A family of linear operators {Tt : t ≥ 0} in a Banach space L is a one-parameter semigroup if T₀ = I and Tt_+s = TtTs for all s, t ≥ 0.

Proposition 1. Let M be a a multiplicative functional such that

for each ψ ∈ L, E [Mtψ(Xt)|X0 = x] ∈ L. Then Mtψ(x) = E [Mtψ(Xt)|X0 = x] is a semigroup in L.

Proof. For ψ ∈ L, M0ψ = ψ and:

Mt+uψ(x) = E [E [Mt+uψ(Xt+u)|Ft]|X0 = x]

= _{E [E [MtMu(θt)ψ[θtXu}]|F_t]|X_{0 = x]} = _{E [MtE [Mu(θt)ψ[θtXu}]|Xt]|X_{0 = x]} = _{E [Mt}Muψ(Xt)|X_{0 = x] = Mt}Mu_ψ(x).

Parameterization of multiplicative processes • (β, γ, κ) that satisfies:

a) β : D0 → R and ₀t β(Xu)du < ∞ for every positive t;

b) γ : D0 → Rm and ₀t |γ(Xu)|2du < ∞ for every positive t;

c) κ : D0 × D0 → R, κ(x, x) = 0 for all x ∈ D0. At =  t 0 β(Xu)du +  t 0 γ(Xu ) · dB_u +  0≤u≤t κ(Xu, Xu−) • At = ψ(Xt) − ψ(X0)

• Exponential of additive processes (strictly positive multiplicative functionals).

Valuation functionals

• A valuation functional {Vt : t ≥ 0} is a multiplicative functional such that {Vt_{St : t ≥ 0} is a martingale.}

• Example: result of continuously reinvesting the payouts of an investment

• If V parameterized by (βv, γv, κv) is strictly positive the implied net return evolution is:

dVt Vt− =  βv(Xt−) + |γv(X t−)|2 2 dt+γv(Xt−)dBt+eκv(Xt,Xt−)−1

• The expected net rate of return is: εv = β. v + |γv|

2

2 +



(exp [κv(y, ·)] − 1) η(dy|·).

• V S is the exponential of an additive process parameterized by: β = βv + βs, γ = γv + γs, κ = κv + κs.

Proposition 2. A valuation process parameterized by

(βv, γv, κv) satisfies the pricing restriction: βv + βs = −|γ

v + γs|2

2 −



e[κv(y,·)+κs(y,·)] − 1_η(dy|·). Corollary 1. The required mean rate of return for the risk

exposure (γv, κv) is εv = −βs − γv · γs − |γs|2

2 −



e[κv(y,·)+κs(y,·)] − eκv(y,·) 

• Instantaneous run risk-return frontier (γv, eκv)  εv,

in particular, −γs is the price of Brownian risk. • Self-financing and multiplicative property.

Example • Breeden model – −γs = (a_Xsfϑf, aϑo). – dcs = X_sods +  XsfϑfdB_sf + ϑodB_so.

– a is ratio between risk prices and consumption volatility. – Instantaneous risk-free rate: b + aXso −

a2(Xsfϑf2+ϑ2o)

2

• Kreps-Porteus model (ρ = 0)

– −γ_s = (ap_Xsf_ϑf + (a − 1)p_Xsfw_{fσf, aϑo} + (a − 1)w_oσo)

– Vol of rate of growth of consumption and vol of vol matter. – Even when ϑo = 0, Bo is priced.

∗ Since wo = _b+ξ1 ₀, if a > 1 risk-price increases more the less Xo

means revert and the less the future is discounted.

– Instantaneous risk-free rate:

Habit • Modify power utility as:

E  _∞ 0 e −bt(Ct − Ht)1−a − 1 1 − a • St = e−bt (C_(C0t−H_−Ht₀)₎−a−a • St = S_tB X_X0t , Xt = (Ct) a (Ct−Ht)a, S B t = e−bt C −a t C₀−a • Menzly, Santos and Veronesi

– dXt = ξ(μx − Xt)dt − κ(Xt − λ)dBt, μx > λ, X0 > λ, κ > 0.

– dct = ucdt + ϑodBt

– Risk-price: aϑo + κ(X_Xt−λ)

t > aϑo, but a does no longer

corresponds to risk-aversion (with respect to consumption or wealth gambles).

• Campbell and Cochrane

– Yt = 1_a log Xt

– dYt = ξy(μy − Yt)dt + λ_|ϑ(Y_ot_|)ϑodBt

– λ(0) = 0 (Guarantees 0 ≤ Ht

Ct ≤ 1) – Short rate of interest rt = r∗

– λ(y) = |ϑo| − 

Some notation introduced previously • Xt = Xtc + X

j

t a Markov process in [0, ∞), Ft. • An Ft n-dimensional Brownian motion {Bt} • Xc t = X0 +  t 0 ξ(Xu)du +  t 0 σ(Xu)dBu • A compensator for Xj , η. • (ξ, σ, η).

• A multiplicative functional is Ft measurable and satisfies M0 = 1 and Mt+u = MtMu(θt). – Examples: 1. Mt1 = e[ R_t 0 β(Xs)ds] ∗ Since, ₀t+u β(Xs)ds =  t 0 β(Xs)ds +  u 0 β(Xt+s)ds 2. Mt2 = e[ R_t 0 γ(Xs)dBs] 3. Mt3 = e[ P 0≤s≤t κ(Xs,Xs−)]_{, with κ(x, x) = 0.} 4. M = M1 × M2 × M3, M = (β, γ, κ).

• Mtψ(x) = E[Mtψ(Xt)|X_{0 = x] is a semigroup.}

– M_{0 = I}

– Mt_+u = Mt(Mu)

• St a stochastic discount factor, that is S0 = 1 and, for any

(Ft-measurable) payoff Π_t, E[St_SΠt|Fs]

s is the price at time s of a

claim to the payoff Π_{t at t.}

• V is a valuation functional if V is multiplicative and V S is a martingale

• If V is a valuation functional parameterized by (βv, γv, κv) the martingale restriction gives βv as a function of (γv, κv).

• Restricts expected return v of V .

• Mapping (γv, κv) → v is instantaneous risk-return relationship.

Long run risk

• Develop “long-run” counterpart of risk-return tradeoff • Slope of the term-structure of “risk-prices.”

• Which aspects of an economic model have transient as opposed to permanent effects.

Growth in payouts

• Risk-premia depend on risk exposure and price of that exposure.

• G a growth process: G adapted, G0 = 1 and Gt+u = GtGu(θt) • Parametrization G = e(Ag) Agt =  t 0 β g (Xu)du +  t 0 γ g (Xu) · dBu +  0≤u≤t κg(Xu, Xu−) • M = SG is also multiplicative • Mtψ(x) = E [Mtψ(Xt)|X0 = x] , is the time-zero price of payoff Gψ(Xt).

Objective: Multiplicative decomposition

• Establish decomposition for multiplicative functionals: Mt = exp(ρt) ˆMt  ϕ(X0) ϕ(Xt) where

– ρ is a deterministic growth rate; – ˆ_Mt is a multiplicative martingale;

– ϕ is a strictly positive function of the Markov state;

• If X is stationary, ϕ(X₀)

ϕ(Xt) stationary component, ˆM the

martingale component of M, and ρ its growth rate.

– Not entirely correct because of possible correlation between

Implications of multiplicative decomposition • If ˆM is a martingale for F ∈ Ft ˆ P r(F ) = E[ ˆMt1_F] • X remains Markovian. • E [Mtψ(Xt)|X0 = x] = exp(ρt)φ(x)E_{P r}ˆ  ψ(Xt) φ(Xt)|X0 = x

• exp(−ρt)φ(Xt) as a numeraire. Applicable when the multiplicative process does not define a price.

• If in addition to stationarity, “stochastic stability” holds lim

t→∞EP rˆ [ψ(Xt)|X0 = x] = 

ψ(Xt)dˆς, whenever  _{ψ(Xt)dˆς is well defined, then}

lim t→∞e −ρt_{E[Mtψ(Xt) | X} 0 = x] = lim t→∞ E ˆ Mt  ψ(Xt) φ(Xt) |X0 = x  φ(x) = (  ψ(Xt) φ(Xt)dˆς)φ(x)

– ρ is the (deterministic) growth rate

– All state dependence is given by the eigenfunction φ

– Mapping “risk in M” to ρ is a long-run risk-return frontier – Approximation is “good” for ψ with | ψ_φ(X(X_tt₎)dˆς| < ∞

Long term bonds • S a stochastic discount factor

St = exp(ρt) ˆMtφ(X0 ) φ(Xt) • Prices of long term discount bonds:

exp(−ρt)E (St|X_{0 = x) ≈ cφ(x)}

• Alvarez and Jerman [2005] estimate the volatility of Mˆ_t+1

ˆ

Mt as a

proportion of volatility of S_St+1

t . (around 75-100%.). Transitory

To do

• Strategy to establish decomposition (1)

– Perron-Frobenius

• Uniqueness (2) • Existence

• Stationarity, recurrence • Some applications (3)

Generators

• Associate to each ψ a function χ such that Mtχ(Xt) is the “expected time derivative” of Mtψ(Xt).

• A Borel function ψ is in the domain of the extended

generator A of the multiplicative functional Mt if for a Borel function χ, Nt = Mtψ(Xt) − ψ(X0) −

 t

0 Msχ(Xs)ds is a local martingale with respect to the filtration {Ft _{: t ≥ 0}. The}

extended generator assigns χ to ψ and we write χ = Aψ. • Extended generator associated with Markov Process (e.g.

Revuz and Yor)

• If ψ smooth, Ito’s lemma.

• Example: If M ≡ 1, Xt ∈ R, and X described by (ξ, σ2_),

Aψ = ξψ ₊ 1

2σ2ψ

• X described by (ξ, σ), Mt the exponential of (β, γ) Aψ(x) =

[β(x) + |γ(x)|₂ 2]ψ(x) + [ξ(x) + σ(x)γ(x)]∂ψ_∂x(x) + 1_2trace(σσ ∂_∂x∂x2ψ(x)_ ) • X described by (ξ, σ, η), Mt the exponential of (β, γ, κ)

Aψ(x) = [β(x) + |γ(x)|₂ 2 +  _{(exp [κ(y, x)] − 1) η(dy, x)]ψ(x) +} [ξ(x) + σ(x)γ(x)]∂ψ_∂x(x) + 1_2trace(σσ ∂_∂x∂x2ψ(x)_ ) + _Rn_−{x}[ψ(y) −

Eigenfunctions and martingales

• A Borel function φ is an eigenfunction of the extended generator (with eigenvalue ρ) if Aφ = ρφ.

• Nt = Mtφ(Xt) − φ(X0) − ρ  t

0 Msφ(Xs)ds is a Ft local martingale.

• Set Yt = Mtφ(Xt). Since dNt = dYt − ρYt−dt, integration by parts yields: exp(−ρt)Yt − Y₀ = −  t 0 ρ exp(−ρs)Ys−ds +  t 0 exp(−ρs)dYs =  t 0 exp(−ρs)dNs_. • exp(−ρt)Mtφ(Xt) is a local martingale.

• A principal eigenfunction of the extended generator is an eigenfunction that is strictly positive.

• If φ is a principal eigenfunction, Mt = exp(ρt) ˆMt  φ(X0) φ(Xt) . where ˆ_Mt = exp(−ρt)M_tφφ_(X(Xt) 0) is a multiplicative local martingale.

Eigenfunctions of the semigroup

Proposition 3. If φ is a principal eigenfunction with eigenvalue ρ

for the extended generator of the multiplicative functional M , then for each t ≥ 0, exp(ρt)φ ≥ Mtφ. If, in addition, ˆM is a martingale then, for each t ≥ 0

Mtφ = exp(ρt)φ. (2)

Conversely, if φ is strictly positive, Mtφ is well defined for t ≥ 0, and (2) holds, then ˆ_{M is a martingale.}

•

1 ≥ E[ ˆ_Mt|X_{0 = x] =} exp(−ρt)

φ(x) E[Mtφ(Xt)|X0 = x],

with equality when ˆ_{M is a martingale, that is (2) . Conversely,} using (2) and the multiplicative property of M one obtains,

E[exp(−ρt)Mtφ(Xt)|Fs] = exp(−ρt)MsE[Mt−s(θs)φ(Xt)|Xs] = −ρs)Ms

Markov diffusion example continued • M = exp(A) where: At = ¯βt+  t 0 βfX f s ds+  t 0 βoX o sds+  t 0  XsfγfdB_sf+  t 0 γodB o s, dXtf = ξf(¯xf − X f t )dt +  XtfσfdB f t , dXto = ξo(¯xo − X o t )dt + σodB o t

• X described by (ξ, σ), Mt the exponential of (β, γ); Aψ(x)

= [β(x)+|γ(x)|₂ 2]ψ(x)+[ξ(x)+σ(x)γ(x)]∂ψ_∂x(x)+1_2trace(σσ ∂_∂x∂x2ψ(x)_ ) • Guess an eigenfunction : exp(cfxf + coxo).

ρ = β + β¯ fxf + βoxo + γf2 2 xf + γo2 2 +c_f[ξf(¯_xf − x_{f) + xfγfσf}] + c_o[ξo(¯_xo − x_{o) + γoσo}] + (cf)2_xf σ 2 f 2 + (co) 2 σo2 2

• 0 = _βf + γ 2 f 2 + cf(γfσf − ξf) + (cf) 2 σf2 2 0 = _βo − c_oξo. • cf = (ξf − γfσf) ±  (ξf − γfσf)2 − σ_f2 2βf + γ_f2  (σf)2 (3) provided that (ξf − γfσf)2 − σ_f2  2βf + γ_f2  ≥ 0. • co = βo ξo . (4) • The resulting eigenvalue is:

ρ = ¯β + γ 2 o + cf_{ξ x}¯ + co_{(ξ x}¯ _{+ γ σ} ) + (co)2σ 2 o . (5)

Write

ˆ

Mt = exp(−ρt)Mt

exp(c_fXtf + c_oXto) exp(c_fX0f + c_oX0o),

• Verify: ˆMt = exp( ˆ_At) where, ˆ_At = ₀t _Xsf_(γf + cf_σf)dBsf +  t 0(γo + coσo)dBso − (γf+cfσf)2 2  t 0 Xsfds − (γ o+coσo)2 2 t.

• ˆM is always a martingale. Use ˆM to change probability. • ˜Btf = B f t −  t 0 

Xsf(γf + cfσf)dt and ˜B_to = B_to − (γo + coσo)t form a Browian in the new probability space.

• The resulting (twisted) drifts are:

ξo(¯xo − xo) + σo(γo + coσo). ξf(¯xf − xf) + xfσf(γf + cfσf),

• Only the negative root produces mean reversion and a stationary distribution.

• Suppose that Xo _{solves instead:} dXto = ξo(¯xo − X o t )dt + σodB o t + dZt

where Z is a pure jump process whose jumps have a fixed probability distribution ν on R and arrive with intensity 1xf + 2 with 1 ≥ 0, 2 ≥ 0. Suppose that the additive functional A has an additional jump term modeled using κ(y, x) = ¯. _κ(yo − xo_{) for y = x and}  exp [¯_{κ(z)] dν(z) < ∞.} • Add to the generator A an extra term:

(1xf + 2) 

[φ(xf, xo + z) − φ(xf, xo)] exp [¯κ(z)] dν(z). • φ(x) = exp(cfxf + coxo) the extra term reduces to:

(1xf + 2) exp(cfxf + coxo) 

• co = βo ξo • cf must solve: 0 = βf + γ_f2 2 + cf(γfσf − ξf) + (cf)2 σ 2 f 2 + 1  exp βo ξo z  − 1exp [¯_{κ(z)] dν(z).} • The resulting eigenvalue is ρ = ¯β + γ_o2

2 + cfξfx¯f + co(ξox¯o + γoσo) + (co)2 σ 2 o 2 + 2   exp βo ξo z  − 1exp [¯_{κ(z)] dν(z).}

Uniqueness

• Suppose the multiplicative process M is strictly positive and φ is a principal eigenfunction of the extended generator A for which the associated process { ˆ_{Mt : t ≥ 0} is a martingale.}

• Suppose there exists an invariant probability measure ˆς for X in (Ω, ˆP r).

– ˆA the generator associated with the multiplicative process

ˆ

M . Suppose there exists probability measure ˆς such that 

ˆ

Aψdˆς = 0

for all ψ in the L∞ domain of the generator ˆA.

• Let ˆE and ˆPr denote the expectation operator and the

probability measure associated with ˆ_{M and ˆς. The process ˆM} determines the distorted transition probabilities and ˆ_{ς is the}

• Suppose that for some Δ > 0 the discrete time process ˆXΔj,

j = 1, 2..., satisfies “stochastic stability”.

– lim j→∞ ˆ E ψ(XΔj)|X0 = x  =  ψ(y)dˆς, whenever  _{ψ(y)dˆς is well defined.}

• Then the associated eigenvalue ρ is the smallest eigenvalue

associated with a principal eigenfunction. Furthermore, φ is the unique eigenfunction (up to scale and ˆ_{ς a.s.) associated with ρ.}

• Proof: Consider another principal eigenfunction φ∗ with associated eigenvalue ρ∗. Then:

Mtφ(x) = exp(ρt)φ(x) Mtφ∗(x) ≤ exp(ρ∗t)φ∗(x).

If ˆ_{M is the martingale associated with the eigenvector φ, then} E  ˆ Mt φ ∗_(X t) φ(Xt) |X0 = x  = exp[−ρt]E [Mt_φ∗_(Xt)|X_{0 = x] ≤}

exp[(ρ∗ − ρ)t]φ_φ∗_(x)(x). Since the discrete-time process ˆM satisfies stochastic stability the left-hand side converges to  φ_φ∗_(y)(y)_{dˆς for} t = Δj as the integer j tends to ∞. While the limit could be +∞, it must be strictly positive, implying, that ρ ≤ ρ∗_{. If} ρ∗ = ρ then for each x,



φ∗(y)

φ(y) dˆς ≤

φ∗(x) φ(x) .

Some notation introduced previously • Xt = Xtc + X

j

t a Markov process in [0, ∞), Ft. • An Ft n-dimensional Brownian motion {Bt} • Xc t = X0 +  t 0 ξ(Xu)du +  t 0 σ(Xu)dBu • A compensator for Xj , η. • (ξ, σ, η).

• A multiplicative functional is Ft measurable and satisfies M0 = 1 and Mt+u = MtMu(θt). – Examples: 1. Mt1 = e[ R_t 0 β(Xs)ds] ∗ Since, ₀t+u β(Xs)ds =  t 0 β(Xs)ds +  u 0 β(Xt+s)ds 2. Mt2 = e[ R_t 0 γ(Xs)dBs] 3. Mt3 = e[ P 0≤s≤t κ(Xs,Xs−)]_{, with κ(x, x) = 0.} 4. M = M1 × M2 × M3, M = (β, γ, κ).

• Mtψ(x) = E[Mtψ(Xt)|X0 = x] is a semigroup.

– M_{0 = I}

– M_t+u = M_t(M_u)

• Decompose multiplicative functionals as: Mt = exp(ρt) ˆMt  φ(X0) φ(Xt) • ˆM a multiplicative martingale • ˆP r(F ) = E[ ˆMt1F] • E [Mtψ(Xt)|X_{0 = x] = exp(ρt)φ(x)EP rˆ}  ψ(Xt) φ(Xt)|X0 = x

• If in addition to stationarity, “stochastic stability” holds lim

t→∞EP rˆ [ψ(Xt)|X0 = x] = 

ψ(y)dˆς, whenever  _{ψ(Xt)dˆς is well defined, then}

lim t→∞e −ρt_E[M tψ(Xt)|X0 = x] = lim t→∞E ˆ Mt  ψ(Xt) φ(Xt) |X0 = x  φ(x) = (  ψ(y) φ(y)dˆς)φ(x)

– ρ is the (deterministic) growth rate

– All state dependence is given by the eigenfunction φ

– Mapping “risk in M” to ρ is a long-run risk-return frontier – Approximation is “good” for ψ with | ψ_φ(y)(y)dˆς| < ∞

• Perron-Frobenius, φ a positive eigenfunction and ρ eigenvalue. • ρ smallest eigenvalue associated with a positive eigenfunction

Plan for remaining two lectures • Examples

– Long run risk-return

• Ideas behind proofs • Heterogeneous beliefs.

Example • dXt = −ξXt_{dt + σdWt, ξ = 0.} • M ≡ 1 • ˆM = M, ρ = 0, φ ≡ 1. • φ(x) = eξx2σ2 , ρ = ξ, ˆM_t = e ξ(X2_{t −X0 )}2 σ −ξt = e R_t 0 2ξXsσ dWs− R_t 0 2ξ2X2_s σ2 ds • If ξ > 0, ρ = 0 is smaller.

• If ξ < 0 then ρ = ξ < 0 and Girsanov states that ˆ

Wt = Wt −  t

0 2ξXσ sds, is a Brownian motion on the probability space obtained when we change measure using ˆ_{M .}

• In any case, after the change in measure we obtain dXt = −|ξ|Xtdt + σd ˆWt

• There always exists a stationary density ˆς ∼ e−|ξ|x2σ2 .

• Stochastic stability obtains lim t→∞e −ρt_E[ψ(X t)|X0 = x] = (  ψ(y) φ(y)dˆς)φ(x)

– If ξ > 0, ρ = 0 and approximation is good for ψ such that

| ψ(y)dˆς| < ∞.

– If ξ < 0, ρ = ξ and approximation is good for ψ such that

Example • dXtf = ξf(¯xf − X f t )dt +  XtfσfdB f t , dXto = ξo(¯xo − Xto)dt + σodBto

with ξi > 0, ¯xi > 0 for i = f, o and 2ξfx¯f ≥ σ_f2 where B = (Bf, Bo) is a bivariate standard Brownian motion. • Per-capita consumption dct = X_todt +  XtfϑfdB f t + ϑodB o t where ct = log(Ct) • Interesting case:

– σo > 0, ϑ0 ≥ 0 (positive Bo’s are unambiguously good)

Breeden model

• Representative investor preferences are given by: E  _∞ 0 exp(−bt)Ct 1−a _{− 1} 1 − a for a and b strictly positive.

• St a stochastic discount factor, that is S0 = 1 and, for any

(Ft-measurable) payoff Π_t, E[St_SΠt|Fs]

s is the price at time s of a

claim to the payoff Π_{t at t.}

• In the additive utility case St = e−bt_u(Cu(Ct)

0)

• Ito’s Lemma implies that the stochastic discount factor in the Breeden model in this example is St = exp(Ast) where

Ast = −a  t 0 X o sds − bt − a  t 0  XsfϑfdB_sf − a  t 0 ϑodB o s.

• V is a valuation functional if V is multiplicative and V S is a martingale

– Example: result of continuously reinvesting the payouts of an

investment

• If V is a strictly positive valuation functional parameterized by (βv, γv), the expected net rate of return is:

εv = β. v + |γv|

2

2

• If V is a valuation functional parameterized by (βv, γv) the martingale restriction gives βv as a function of γv.

– βv − aX_to − b = −|γv−(a √

Xsfϑf,aϑo)|2

2

• Restricts expected return v of V .

– εv = aX_to + b + γv · (a  Xsfϑf, aϑo) − |(a √ Xsfϑf,aϑo)|2 2

• Mapping γv → v is instantaneous risk-return relationship. • (aXsfϑf, aϑo) is vector of (instantaneous) Brownian

risk-prices.

Long rung risk-return • M = V

– V the cumulated results of a self-financing stategy. – Suppose V is parameterized by (γv, κv)

– βv results from martingale restriction.

– ρ(γv, κv) gives the equilibrium long run rate of return as a function of the risk-structure of V

• Dt = Gtψ(Xt), G a growth process. • M = GS

– Negative of eigenvalue is the rate of decay in market value of a

cash flow paid in the far future.

– If γg parameterizes the sensitivity of G to the Brownian motions, long-run price of risk for exposure to the

• G multiplicative, G = exp(Ag t) • Agt = δt +  t 0  Xsfγ_fgdB_sf +  t 0 γ g odB o s −  t 0 Xsf(γ g f)2 + (γ g o)2 2 ds. • γg f parameterizes B f

s risk of cash flow, γog parameterizes Bo risk. • δ is the rate of growth of dividends.

• St = exp(As_t) where Ast = ¯βst+  t 0 β s fXsfds+  t 0 β s oXsods+  t 0  Xsfγ_fsdB_sf+  t 0 γ s odBso

– Breeden: ¯_βs = −b, β_fs _{= 0, βo}s = −a, γ_sf = −aϑf_{, γs}o = −aϑo_.

– Kreps-Porteus • A = As + Ag is given by At = ¯βt+  t 0 βfX f s ds+  t 0 βoX o sds+  t 0  XsfγfdB_sf+  t 0 γodB o s • ¯β = δ − (γog)2 2 + ¯βs, βf = − (γ_fg)2 2 + βfs, βo = βos, γf = γ g f + γ s f and γo = γ_og + γ_os.

• Eigenfunction φ = exp(cfxf + coxo). ρ = β + β¯ fxf + βoxo + γf2 2 xf + γo2 2 +cf_[ξf(¯_xf − xf_{) + xfγfσf}] + co_[ξo(¯_xo − xo_{) + γoσo}] + (cf)2_xf σ 2 f 2 + (co) 2 σo2 2

• cf and co chosen such that the process X is stationary when we

change probability of F ∈ Ft to E[ ˆMt1F] ≡ E[e−ρtMtφ(Xt)].

– unique – c_o = β_ξo o – cf = (ξf−γfσf)− q (ξf−γfσf)2−σ_f2(2βf+γ_f2) (σf)2 – cf = cf_(γfg)

• Long-run risk price for the exposure to the Bo _{risk is:} − dρ dγog = −γ_os − β s o ξo σ o – Breeden: aϑ_o + _ξa oσo.

– Difference from local risk price, _ξa oσo

∗ increases with risk aversion

∗ increases with the volatility of the rate of growth of consumption

∗ decreases with the strength of mean reversion of the rate of growth of consumption.

∗ No difference if rate of growth of consumption is constant.

• Long run risk price for exposure to Bf _{depends on γ}g

f and is state independent (stationarity).

– Local risk price is independent of γ_fg and is proportional to 

Sochastic Stability • Assume ˆM is a martingale

• M is positive with probability one. • ˆAψ = A(φψ)_φ − ρψ

Assumption 1. There exists a probability measure ˆ_{ς such that}

 ˆ

Aψdˆς = 0

for all ψ in the L∞ domain of the generator ˆA.

– This guarantees that ˆ_{ς is a stationary distribution for X}

under ˆ_{P r.}

– ˆ_{E and ˆ}Pr denote the expectation operator and the

probability measure associated with ˆ_{M and ˆς. The process} ˆ

• Let ˆΔ > 0 and consider the discrete time Markov process obtained by sampling the process at ˆ_{Δj for j = 0, 1, ....} skeleton.

Assumption 2. There exists a ˆ_{Δ > 0 such that the discretely}

sampled process {X_{Δjˆ : j = 0, 1, ...} is irreducible. That is,} for any Borel set Λ of the state space D₀ with ˆ_{ς(Λ) > 0,}

ˆ E ⎡ ⎣∞ j=0 1_{Xˆ Δj∈Λ}|X0 = x ⎤ ⎦ > 0 for all x ∈ D0.

Assumption 3. The process X is Harris recurrent under

the measure ˆ_{P r. That is, for any Borel set Λ of the state space} D0 with positive ˆς measure,

ˆ P r  _∞ 0 1_{Xt∈Λ} = ∞|X_{0 = x}  = 1 for all x ∈ D0.

– Among other things, this assumption guarantees that the

stationary distribution ˆ_{ς is unique.}

• These assumptions guarantee the stochastic stability of skeletons and as a consequence, for any ψ for which  (|ψ(y)|/φ(y))dˆς < +∞ lim j→∞ exp(−ρΔj)E[MΔjψ(XΔj)|X0 = x] = φ(x)  ψ(y) φ(y)dˆς

Proposition 4. Suppose that ˆ_{M satisfies Assumptions 1-3, and let}

Δ > 0.

a. For any ψ for which (|ψ|/φ)dˆς < ∞ lim

j→∞ exp(−ρΔj)MΔjψ = φ 

ψ φ dˆς for almost all (ˆ_{ς) x.}

b. For any ψ for which ψ/φ is bounded, lim t→∞ exp(−ρt)Mtψ = φ  ψ φ dˆς for x ∈ D0.

Proof: Note that

exp(−ρt)Mt_{ψ(x) = ˆ}Mt ψ φ  φ(x).

recurrence plus irreducibility of some skeleton chain implies that, lim t→∞ _0≤ψ≤φsup   ˆMt ψ φ  −  ψ φ dˆς   = 0,

which proves (b). Consider any sample interval Δ > 0. Then lim j→∞_0≤ψ≤φsup   ˆMΔj ψ φ  −  ψ φdˆς   = 0. From Proposition 6.3 of Nummelin, the sampled process

{XΔj : j = 0, 1, ...} is aperiodic and Harris recurrent with

stationary density ˆ_{ς. Hence if}  ψ_φ(x) dˆς(x) < ∞, lim j→∞ ˆ MΔj ψ φ  =  ψ φdˆς

for almost all (ˆ_{ς) x, which proves (a). (See for example, Theorem} 5.2 of Meyn and Tweedie 1992)

• If X is a Feller process, these assumptions can be verifyed by using Lyapunov function arguments for A.

– A continuous function V is called norm-like if the set

{x : V (x) ≤ r} is precompact for each r > 0.

– A sufficient condition for the existence of a

stationary distribution and for Harris recurrence is that there exists a norm-like function V for which

A(φV )

φ − ρV = ˆAV ≤ −1 outside a compact subset of the state space.

Existence

• Nummelin (1984), Kontoyiannis and Meyn (2003, 2005)

– R-Recurrence of kernels

• Liapunov functions

• Construction guarantees that φ is an eigenfunction of semigroup M.

Modelling Heterogeneous beliefs

Overconfidence and Speculative Bubbles (Jose Scheinkman and Wei Xiong, J.P.E., December 03)

Equilibrium Portfolio Strategies in the Presence of Sentiment Risk and Excess Volatility (B. Dumas, A. Kurshev and R. Uppal,

Some motivating observations

• Difficulty of explaining prices with fundamentals

• Bubbles frequently accompanied by increased trading activity

– Trading volume (internet stocks: 6% of capital, 20% of

trading, Average of spin offs reported in Lamont and Thaler (2001) 38% daily.)

Some earlier literature

• A static models with heterogeneous beliefs and short-sale constraints Miller (1977)

• With short-sales constraints, pessimistic investors stay out of the market. Sufficient heterogeneity + short-sales constraints ⇒ asset prices are higher than in the absence of short-sale constraints.

• Harrison and Kreps (1978): Heterogeneous beliefs and short sale constraints induces speculative behavior.

Overconfidence as source of heterogeneous beliefs • Overconfidence = tendency of people to overestimate the

precision of their knowledge.

• Psychology studies suggest that people are overconfident.

– Alpert and Raiffa - The 98% confidence interval cover only

60% of realizations.

– Overconfidemce more pronounced if questions more difficult. – Illusion of knowledge: Disagreement more polarized when

given arguments that serve both sides. (Lord, Ross and Lepper,1979)

Consequences of overconfidence in a dynamic model of pricing and trading.

• Elements of the model

• Individuals receive signals on the value of an asset.

• Different groups of individuals have common information but excess confidence on distinct signals.

• This creates two phenomena:

– divergence of opinions.

– buyer knows than in the future others may value the asset

to such an extent that a trade will occur.(Option) • How do bubbles appear and vanish?

• Henry Blodget quoted by Michael Lewis ”We all have the same information, and we’re just making different conclusions about what the future will hold.”

• Michael Lewis quoted by Michael Lewis ”I had been to a

Merrill Lynch conference (them again!) that featured Exodus Communications, and the story Henry Blodget and a few other people told was so good that I figured that even if Exodus

Communications didn’t wind up being a big success, enough people would believe in the thing to drive the stock price even higher and allow me to get out with a quick profit.”

Outline • Two groups of Agents A and B. • Model as linear as possible

• Filtering.

• Buyer acquires option to sell asset. • Option is American.

• Value that buyer is willing to pay today depends on prices he forecasts for the future.

• Equilibrium.

• Solve for infinite horizon model, but can accommodate finite horizon securities.

Payoffs and information • Cumulative dividend process Dt:

dDt = ftdt + σDdZ_tD dft = −λ(ft − ¯f )dt + σfdZ_tf • Two extra signals:

dsAt = ftdt + σsdZtA dsBt = ftdt + σsdZtB

• Group A agents believe: dsAt = ftdt + σsφdZ f t + σs  1 − φ2_dZtA Group B agents believe :

dsBt = ftdt + σsφdZ f

t + σs 

1 − φ2_dZtB

• 0 ≤ φ ≤ 1, agents views on correlations of innovations is public information.

Filtering

• Agents use signals and D to forecast f. • Stationary variance, γ ≡ [ q (λ + φσf/σs)2 + (1 − φ2)(2σ_f2/σs2 + σ_f2/σ_D2 ) −(λ + φσf/σs)]/( 1 σ_D2 + 2 σs2 ) γ decreases with φ

• d ˆfA = −λ( ˆ_fA − ¯_{f )dt +} φσsσf + γ σs2 (ds A _{− ˆ} fAdt) + γ σs2(ds B _{− ˆ} fAdt) + γ σ_D2 (dD − ˆf A dt) d ˆfB = −λ( ˆ_fB − ¯_{f )dt +} γ σs2 (ds A _{− ˆ} fBdt) +φσsσf + γ σs2 (ds B _{− ˆ} fBdt) + γ σ_D2 (dD − ˆf B dt) • “Beliefs”

Difference in beliefs • To A investors: gA _{= ˆf}B − ˆ_fA_,

dgA = −ρgA_{dt + σgdWg}A • Difference in beliefs is a state variable.

Trading • Cost c per trade.

• All agents are risk neutral. • Fixed rate of interest r.

• Finite supply, no short sales and large number of agents of each type guarantee that buyers pay reservation price.

pot = max τ≥0 E o t  t+τ t e−r(s−t)[ ¯_{f + e}−λ(s−t)( ˆ_fto − ¯_{f )]ds} + e−rτ_(pt¯o_+τ − c)  .

• Guess: Demand price of current owner is: pot = p o ( ˆ_fto_{, gt}o) = ¯ f r + ˆ fto − ¯f r + λ + q(g o t ). (6) with q > 0 and q > 0. • q(gto) = sup τ≥0 Eo_t  gto+τ r + λ + q(g ¯o t+τ) − c  e−rτ .

The option problem • q(x) ≥ x r + λ + q(−x) − c (7) 1 2σ 2 gq − ρxq − rq ≤ 0, (8) with equality in (8) if (7) strict,

• Guess: Stop if {x : x ≥ k∗}, k∗ ≥ 0. • 1 2σ 2 gH − ρxH − rH = 0. (9)

• Solutions H to (9) with H > 0 and H > 0 in (−∞, 0) are of the form β1h, h convex, limx→−∞ h(x) = 0, limx→−∞ h(x) = 0.

– An “explicit”formula for h. – Kummer functions.

• q(x) = ⎧ ⎨ ⎩ β1h(x), for x < k∗ x r+λ + β1h(−x) − c, for x ≥ k∗. (10) • “Smooth pasting” ⇒ β1 = 1 [h(k∗) + h(−k∗_{)](r + λ)}, (11) [k∗ − c(r + λ)][h(k∗) + h(−k∗)] +h(−k∗) − h(k∗) = 0 (12)

• There exists unique k∗ _{= k}∗_{(c) that solves (12). k}∗_{(0) = 0 and} k∗(c) > c(r + λ) if c > 0.

– Hold stock even though others value it sufficiently more to

pay for transaction costs.

– Variation of model with one group “rational”

• q defined by (10) is an equilibrium option value function. • The optimal policy consists of exercising immediately if

go ≥ k∗, otherwise wait until first time in which go ≥ k∗.

• Bubble: If x ∈ (−∞, k∗), q(x) the difference between owner’s demand price and his fundamental valuation.

b(c) = _[h_(k∗_(c))+hh(−k_(−k∗(c))∗_(c))](r+λ)

Properties of equilibria with trading costs • Trading volume

– If c = 0, k∗ = 0 and E[τ (−k∗, k∗)] = 0.

– E[τ (−k∗, k∗)] varies continuously with c.

– ∂E[τ(−k_∂c ∗,k∗) = ∞ at c = 0.

• Bubble

– ∂b_∂c(c) is finite.

– Bubble increases with overconfidence φ, volatility of

Greenspan on the possibility of a real estate bubble “While stock market turnover is more than 100% annually, the turnover of home ownership is less than 10 per cent annually -scarcely tinder for speculative conflagration.” (quoted in Financial Times of April 22, 2002).

Price and turnover

• Risk-neutrality allows a separate model for each asset. • Turnover and size of bubble are equilibrium values. • No causality.

• Cochrane: Cross-sectional regression of market value / book value on share turnover for stocks in NASDAQ 96-00.

Risk • Introduce risk averse consumers

• Aggregate consumption is assumed to be a diffusion. • One equity with dividends = aggregate consumption. • Groups not symmetric.

• No transaction costs • Allow short sales

Payoffs and information • Aggregate dividend process Dt:

dDt Dt = f tdt + σDdZ_tD, σD > 0 • dft = −λ(ft − ¯f )dt + σfdZ_tf, λ > 0, σf > 0. • An extra signal: dst = −μst_{dt + σsdZt}s_. • (ZD_{, Z}f_{, Z}s_{) a three dimensional Brownian.}

• Group A agents believe:

dst = −μstdt + σsφdZ_tf + σs 

1 − φ2_dZts • Group B agents are rational

• 0 ≤ φ ≤ 1, agents views on correlations of innovations is public information.

• Filtering implies that stationary variance of agent’s forecasts are: γA = γA(φ), with γA decreasing with φ, γB = γA(0).

• d ˆftA = −λ( ˆf A t − ¯f )dt + γA σD2 (dD D − ˆf A dt) + φσf σs (ds + μs) d ˆftB = −λ( ˆftB − ¯f )dt + γB σ_D2 ( dD D − ˆf B dt) • _σ1_D (dD_D − ˆ_fB_{) is a Brownian innovation for B.}

• _σ1_D (dD_D − ˆ_fA_{) is a Brownian innovation for A.} • Difference in beliefs: ˆg = ˆfB − ˆfA

• dZA

• η change in measure from B’s to A’s beliefs then by Girsanov: dηt ηt = − gˆt σD dZ B t • dˆgt = −(λ + γ A σ_D2 )ˆgt + γB−γA σD dZ B t − φσfdZts

• State variables (D, s, ˆfB, ˆg, η). Two dimensional Brownian (ZB, Zs).

Equilibrium

• Agents in each group C ∈ {A, B} have utility function U (c) = E₀C ₀∞ e−bt c_αα dt and an initial share of rights to output (equity) wC.

• Recall that St is a stochastic discount factor if the date zero price of a payoff Π_{t at t is E0[St}Π_t].

• Arrow’s procedure (Kreps)

– Find stochastic discount factor St (under B’s probabilities) such that when consumers maximize expected utility

subject to a single budget constraint: E₀B[  _∞ 0 Stctdt] ≤ w C E₀B[  _∞ 0 StDt]dt,

Supply = Demand.

– Show that this equilibrium can be decentralized with

long-lived securities.

– Two Brownians require 3 securities, e.g. riskless bond,

• St = e−bt_(cB(cBt )α−1

0 )α−1

• Since group A shares same utility function and ηt transforms Bs beliefs into A’s beliefs, St = _ηηt₀ e

−bt_(cA t )α−1 (cA 0 )α−1 • cB t + cAt = Dt • cB t = θ(ηt)Dt, cAt = [1 − θ(ηt)]Dt. • θ(η) = [ 1 νB ] 1 1−α [_νηAt ] 1 1−α _{+ [} 1 νB ] 1 1−α ,

νC the Lagrande multiplier associated with the budget constraint of group C.

• log St = −b − (α − 1)[log(θ(ηt)) − log(θ(η0)) + log(Dt) − log(D0)]

• Ito’s lemma guarantees that d(log St) = βSdt + γSdZtB

• St only depends on ZS through ˆg.

• Conditional on state, risk-premia are independent of overconfidence

coefficient φ.

• Local risk-price of signal (Zs) risk is zero.

• As t → ∞ positive Martingale ηt concentrates mass at zero.

• θ(η) → 1.

– Calibrations in Dumas et al. show that nonetheless the share of

Long-Run Risk

Lectures at Coll`ege de France Jos´e A. Scheinkman Department of Economics

Princeton University March 2008

In financial economics risk-return tradeoffs show how expected rates of re-turn and consequently asset prices are altered in response to changes in the exposure to the underlying shocks that impinge in the economy. In these lectures we will:

(i) Present some of the recent literature that is concerned with the effect of long run risk on returns and prices.

(ii) Develop an analytical structure that reveals the long-run risk-return relationship in nonlinear continuous time Markov environments. This is done by studying a principal eigenvalue problem for a conveniently chosen family of valuation operators.

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