Long Term Risk : An Operator Approach

Long Term Risk: An operator approach

• Long run risk return tradeoff • Motivation

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

∗ Market microstructure, transaction costs... may make it

hard to evaluate these models using short run data.

∗ Behavioral biases

– How risk averse agents value the risks in permanent shocks – Long run risk-return frontier

– Complementary to work using short run data (Bansal-Yaron...)

– Hansen, Heaton and Li

Sustainable development

• Risks for which markets that may reveal information are absent • Risks for which markets are present

– Technological vs. economic risk.

– Correlation with other factors that determine “utility.” – Catastrophe insurance, pricing of CAT bonds.

Stochastic discount factor

• Xt a Markov process, Ft the associated (completed) filtration.

• A Stochastic Discount Factor S is a strictly positive adapted

process such that if s ≤ t

E [StΠt|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

• Garman (1984), Rogers (1998) • S0 = I and St+u = StSu

• dXt = ϑ(¯x − Xt) + σdBt

• Per-capita consumption

dct = Xtdt + γdBt. where ct = log(Ct)

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

• The implied stochastic discount factor is St = exp(As_t) where

Ast = −a  t

0 Xsds − bt − a  t

• θt the shift operator:

(θtX)u = Xt+u.

• Since Su only depends on the history of the Markov process X between dates 0 and u, 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).

Generalizations

• G a growth process

• G adapted, G0 = 1 and Gt+u = GtGu(θt)

• M = SG is also multiplicative •

Mtψ(x) = E [Mtψ(Xt)|X₀ = x] , is the time-zero price of payoff D0Gψ(Xt).

• Valuation functional V

– V S a martingale –

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 stationary and martingale components.

Implications of multiplicative decomposition • If ˆM is a martingale for F ∈ Ft ˆ P r(F ) = E[ ˆMt1F] • X remains Markovian. • E [Mtψ(Xt)|X₀ = x] = exp(ρt)φ(x) ˆE  ψ(Xt) φ(Xt)|X0 = x 

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

• If X stationary under ˆP r then in fact ρ is the asymptotic

• If in addition to stationarity, recurrence holds lim t→∞E [ψ(Xˆ t)|X0 = x] = ˆE [ψ(Xt)] . lim t→∞e −ρt_E[M tψ(Xt)|X₀ = x] = lim t→∞E  ˆ Mt  ψ(Xt) φ(Xt)  |X0 = x  φ(x) = _Eˆ  ψ(Xt) φ(Xt)  φ(x)

– ρ is the (deterministic) growth rate

Long term bonds

• S a stochastic discount factor

St = exp(ρt) ˆMtφ(X0)

φ(Xt)

• Prices of long term discount bonds:

exp(−ρt)E (S_t|X_{0 = x) ≈ ˆE}  1 φ(Xt)  φ(x)

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

M_t as a proportion of volatility of S_St+1

t . (around 75-100%.). Transitory component has low estimated conditional volatility.

Remainder of lecture

• Strategy to establish decomposition

– Perron-Frobenius

• Long-run dominance • Uniqueness

• Existence • Example

Restrictions on Markov Process

• {Xt : t ≥ 0} be a continuous time Markov process on a state space D0. The sample paths of {Xt : t ≥ 0} are continuous from the right and with left limits and we will sometimes also assume that this process is stationary and ergodic. Let F_t be completion of the sigma algebra generated by {X_u : 0 ≤ u ≤ t}.

• Semimartingale • X = Xc

+ Xj

• Xj _{with a finite number of jumps in any finite interval and} compensator η[dy|x]dt.

• An Ft n-dimensional Brownian motion {Bt}

• Σ = σσ • Xtc = X0 +  t 0 ξ(Xu)du +  t 0 σ(Xu)dBu • (ξ, Σ, η)

Example • X = (Xv , Xm) • dXtv = ϑv(¯xv − Xtv) +  XtvσvdBtv, dXtm = ϑm(¯xm − Xtm) + σmdBtm

with ϑi > 0, ¯xi > 0 for i = v, m and 2κvx¯v ≥ |σv|2 and

B =

⎡ ⎣ Bv

Bm

⎤

⎦ is a bivariate standard Brownian motion.

• The parameter restrictions guarantee that there is a stationary

distribution associated with X with support contained in R+ × R.

Parameterizing multiplicative functionals

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

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

Mt+u = Mu(θt)Mt. Here θ is the shift operator.

• 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.

• (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(X_u)|2_{du < ∞ 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, X_u−) • At = ψ(Xt) − ψ(X₀)

• Exponential of additive processes (strictly positive

multiplicative functionals).

– Parameterized by the additive process (β, γ, κ)

Example continued : SDF - Breeden model • Per-capita consumption dct = X_tmdt +  XtvγvdBtv + γmdBtm. where ct = log(Ct)

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

• The implied stochastic discount factor is St = exp(As_t) where

Ast = −a  t 0 X m s ds − bt − a  t 0  XsvγvdBsv − a  t 0 γmdB m s .

• local risk prices

SDF: Kreps-Porteous model

• Preferences satisfy recursion (a > 1 and b > 0):

lim ↓0

E (Wt+ − Wt|Ft)

 = Wt [b(a − 1)ct + b log Wt]

where −W_t is the continuation value for the consumption plan.

• Bansal and Yarom (2004) • Guess and verify:

Wt = exp 

(1 − a)(w_fXtf + wo_Xto _{+ ct} + ¯w) 

• St = exp(As_t) × exp(Aw_t ) • Ast = −  t 0 X m s ds − bt −  t 0  XsvϑvdBsv −  t 0 ϑmdB m s . • Awt = (1 − a)  t 0  Xsv(ϑv + wvσv)dBsv +(1 − a)  t 0 (ϑm + wm_σm)dBsm −(1 − a)2 2  t 0 X v s |ϑv + wv_σv|2 2 ds − (1 − a)2 2 t

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 M_t if for a Borel function χ, Nt = Mtψ(Xt) − ψ(X₀) − ₀t Msχ(Xs)ds is a local martingale with respect to the filtration {F_{t : t ≥ 0}. The}

extended generator assigns χ to ψ and we write χ = Aψ.

• If ψ smooth and M ≡ 1 apply Ito’s lemma.

• X described by (ξ, σσ, 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) − ρ₀t Msφ(Xs)ds is a Ft local martingale.

• Set Yt = Mtφ(Xt). Since dNt = dYt − ρY_t−, integration by parts yields: exp(−ρt)Y_t − Y₀ = −  t 0 ρ exp(−ρs)Ys−ds +  t 0 exp(−ρs)dY_s =  t 0 exp(−ρs)dN_s.

• 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_tφ(Xφ(Xt) 0) is a multiplicative local martingale.

• Will discuss assumptions under which ˆM is actually a

Example: Risk-return frontier

• Cash flow process D0Gtψ(Xt)

• Gt = exp(Ag_t ) where Agt = δt+  t 0  XsvvdBsv+  t 0 mdB m s −  t 0 Xsv|v|2 + |m|2 2 ds – G = exp(δt) ˆG, ˆG a martingale.

– v parameterizes Bv risk of cash flow, m parameterizes

Bm risk • M = SG = exp(A) = exp(As + Ag) At = (δ − b)t +  t 0  Xsv(−aγv + v)dBsv +  t 0 (−aγm + m)dB_sm −  t 0  Xsv|v|2 + |m|2 2 + aX m s  ds

• Compute A (Ito’s)

• Guess an eigenfunction of the form: exp(cvxv + cmxm).

• Real solutions if H = [ϑv + (aγv − v)σv]2 − |σv|2[|aγv|2 − |v|2] ≥ 0 cm = − a ϑm cv = ϑv + (aγv − v)σv ± √ H |σv|2

Only the minus sign will lead to stationarity of X under the distorted probability distribution. Can check directly that ˆ_{M is} a martingale and recurrence holds.

• The eigenvalue is

λ = δ − b + |aγm₂ |2 − aγmm + ϑvx¯vcv + ϑmx¯mcm

• −λ is the decay rate in value of the cash flow over time. • Risk adjusted asymptotic interest rate

−λ + δ = b − |aγm₂ |2 + aγmm − ϑvx¯vcv − ϑmx¯mcm

− (−aγm + m)σmcm − (cm)2 |σm|

2

2 .

• Risk-return frontier: mapping

(v, m) → −λ + δ

• The long run risk prices to the cash flow risk exposure to the Bm risk is:

aγm + a

ϑmσm

Long run dominance

Assumption 1. The multiplicative functional M is strictly

positive with probability one.

Assumption 2. There exists a probability measure ˆ_{ς such that} 

ˆ

Aψdˆς = 0

for all ψ in the L∞ domain of the generator ˆA of ˆ_{M .}

Assumption 3. 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 ∈ D .

Assumption 4. 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 ˆ_{ς 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.}

• Sufficient conditions for Assumptions 1-4 using Liapunov

functions (Meyn-Tweedie).

– Restrictions on coefficients of M and X – X Feller

– A function V is called norm-like if {x : V (x) ≤ r} is precompact for each r > 0.

– Example: A sufficient condition for the existence of stationary distribution (Assumption 2) and for Harris

recurrence (Assumption 4) is that there exists a norm-like function V for which

A(φV )

φ − ρV = ˆAV ≤ −1

Proposition 1. Suppose that ˆ_{M is a martingale and satisfies}

Assumptions 1 - 4, 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.

Uniqueness

• Under the above assumptions there exists at most one principal

eigenfunction for which ˆ_{M is a martingale and X is stationary} and recurrent under ˆ_{P r.}

• Proof: Consider two such principal eigenvectors, φ and φ∗, and

let ρ ≥ ρ∗ be the corresponding eigenvalues. Since ˆ_{M is a} martingale, exp(ρt)φ(x) = exp(ρt)φ(x)E[ ˆMt|X₀ = x] =

E[Mtφ(Xt)|X₀ = x], and since ˆM∗ is a martingale exp(ρ∗t)φ∗(x) = E[Mtφ∗(Xt)|X₀ = x]. Thus

E  ˆ Mt φ ∗_(Xt) φ(Xt) |X0 = x  = exp[(ρ∗ − ρ)t]φ ∗_(x) φ(x) .

Since the discrete time sampled Markov process associated with ˆ_{M is stationary, aperiodic and Harris recurrent the} left-hand side converges to:

ˆ E  φ∗(Xt) φ(Xt)  .

for t = Δj as j → ∞. While the limit could be +∞, it must be strictly positive, implying, since ρ ≥ ρ∗, that ρ∗ = ρ and

ˆ E  φ∗(Xt) φ(Xt)  = φ ∗_(x) φ(x) .

Existence

• Nummelin (1984), Kontoyiannis and Meyn (2003, 2005) • Recurrence of kernels