This document summarizes a research paper on modeling long-run optimal portfolios and risk premia. It includes: 1) An outline describing the goal of developing a tractable framework for portfolio choice and derivatives pricing using a model with stochastic investment opportunities across several assets. 2) A section on the main result regarding long-run portfolios and risk premia, and implications like static fund separation and horizon effects. 3) An overview of the solution method which involves differential equations to identify candidate solutions, finite-horizon bounds to characterize performance, and conditions for long-run optimality.

1. Model Long Run Applications Large Deviations Appendix
Portfolios and Risk Premia for the Long Run
Paolo Guasoni
(joint work with Scott Robertson)
Boston University
Department of Mathematics and Statistics

2. Model Long Run Applications Large Deviations Appendix
Outline
Goal:
A Tractable Framework for
Portfolio Choice and Derivatives Pricing.
Model:
Stochastic Investment Opportunities with Several Assets.
Main Result:
Long Run Portfolios and Risk Premia.
Implications:
Static Fund Separation. Horizon effect.
Large Deviations:
Connection with Donsker-Varadhan results.

3. Model Long Run Applications Large Deviations Appendix
Asset Prices and State Variables
Market with risk-free rate and n risky assets.
Investment opportunities driven by k state variables.
dSti
=r (Yt )dt + dRti 1≤i ≤n
Sti
n
σij (Yt )dZtj
dRti 1≤i ≤n
=µi (Yt )dt +
j=1
k
aij (Yt )dWtj
dYti 1≤i ≤k
=bi (Yt )dt +
j=1
d Zi, Wj 1 ≤ i ≤ n, 1 ≤ j ≤ k
=ρij (Yt )dt
t
Z , W Brownian Motions.
Σ(y ) = (σσ )(y ), Υ(y ) = (σρa )(y ), A(y ) = (aa )(y ).

4. Model Long Run Applications Large Deviations Appendix
(In)Completeness
Υ Σ−1 Υ: covariance of hedgeable state schocks:
Measures degree of market completeness.
A = Υ Σ−1 Υ: complete market.
State variables perfectly hedgeable, hence replicable.
Υ = 0: fully incomplete market.
State shocks orthogonal to returns.
Otherwise state variable partially hedgeable.
One state: Υ Σ−1 Υ/a2 = ρ ρ.
Equivalent to R 2 of regression of state shocks on returns.

5. Model Long Run Applications Large Deviations Appendix
Payoffs
Find optimal portfolio today for a long horizon T .
πt = (πt1 , . . . , πtn ) proportions of wealth in risky assets.
Portfolio value Xtπ evolves according to:
dXtπ
=rdt + πt dRt
Xtπ
xp
Maximize expected power utility U(x) = p, p < 1, p = 0.
Power utility motivated by Turnpike Theorems.

6. Model Long Run Applications Large Deviations Appendix
Stochastic Discount Factors
All stochastic discount factors of the form:
· ·
1
η
(µ Σ−1 + η Υ Σ−1 )σdZ +
E−
MT = η adW
0
ST 0 0 T
for some adapted integrable process (ηt )t≥0 .
η: risk premia of state-variable shocks.
Assumption: there exists a stochastic discount factor.

7. Model Long Run Applications Large Deviations Appendix
Duality Bound
p
If X , M ≥ 0 satisfy E [XM] ≤ 1:
q= p−1 .
1 1 1−p
E [X p ] ≤ E [M q ]
p p
Analougue of Hansen-Jagannathan bound for power utility.
Any payoffs lives below any stochastic discount factor.
If a payoff and a stochastic discount factor live together,
the payoff is maximal, and the discount factor is minimal.

8. Model Long Run Applications Large Deviations Appendix
Certainty Equivalent
Any pair or policies (ˆ , η ) satisfies in a finite horizon T :
πˆ
1y 1y 1−p
η
ˆ
EP (XT )p ≤ uT (y ) ≤ EP (MT )q
π
ˆ
(1)
p p
Certainty Equivalent loss lT measures performance:
increase in risk-free rate that covers utility loss:
1y
E (elT T XT )p = uT (y )
π
ˆ
(2)
pP
(2) in (1) yields upper bound:
1 1 1
1−p
y y
η
ˆ
log EP (MT )q log EP (XT )p
π
ˆ
lT ≤ − (3)
p T T

9. Model Long Run Applications Large Deviations Appendix
Long Run Optimality
Definition
A pair (ˆ , η ) ∈ C 1 (E, Rn ) × C 1 (E, Rk ) is Long-Run Optimal if:
πˆ
1 1 1−p
y y η
ˆ
log EP (XT )p = lim log EP (MT )q
π
ˆ
lim
T →∞ T T →∞ T
Certainty equivalent loss vanishes for long horizons.
Alternative interpretation:
an agent with sufficiently long horizon prefers the long-run
optimal portfolio to the optimal finite horizon portfolio, if the
long-run portfolio has slightly lower fees.
π and η depend on state alone (no t).
ˆ ˆ

10. Model Long Run Applications Large Deviations Appendix
Solution Method
Differential equation delivers candidate solutions.
Finite-horizon bounds.
Characterize performance of candidate.
Most relevant for applications.
Long-run optimality.
Guarantees that candidate prevails in the long run.
Theoretical justification.

11. Model Long Run Applications Large Deviations Appendix
Finite-Horizon Bounds: Assumptions
Theorem
1
i) δ = constant;
1−qρ ρ
ii) for some λ ∈ R, the linear ODE:
A¨ ˙ 1 pr − q µ Σ−1 µ − λ φ = 0
φ + b − qΥ Σ−1 µ φ +
2 2
δ
admits a strictly positive solution φ ∈ C 2 (E, R);
iii) both models:
˙

ˆ
 dRt = 1−p µ + δΥ φ + σd Zt
1
dRt =µdt + σdZt φ
˙
dYt =bdt + adWt ˆt
 dY = b−qΥ Σ−1 µdt + A φ dt +ad W
t φ
ˆ
well-posed under equivalent probabilities P and P.

12. Model Long Run Applications Large Deviations Appendix
Finite-Horizon Bounds
Theorem
The portfolio π and the risk premia η defined by:
ˆ ˆ
˙ ˙
1 φ φ
Σ−1
π=
ˆ µ + Υδ η=δ
ˆ
1−p φ φ
satisfy the equalities:
y y
EP (XT )p = eλT φ(y )δ E ˆ φ(YT )−δ
π
ˆ
P
1−p
1−p δ
− 1−p
y y
η
ˆ
EP (MT )q = eλT φ(y )δ E ˆ φ(YT )
P
π and η as candidate long-run solutions.
ˆ ˆ
Long Run and transitory components
(Hansen and Scheinkman, 2006).

13. Model Long Run Applications Large Deviations Appendix
Long-Run Optimality
Theorem
In addition to the previous assumptions, suppose that:
ˆ
i) for some ¯ > 0, the random variables (Yt ) ¯ are P-tight;
t t≥t
ii) supy ∈E F (y ) < ∞, where F ∈ C(E, R) defined as:
 ˙
 pr − λ − q µ Σ−1 µ + 1 δ(δ − 1) φ2 A φ−δ p<0
φ2
2 2
δ
˙2 − 1−p
ρ) φ2 A
 pr − λ − 1 qµ Σ−1 µ − 1 qδ 2 (1 − ρ 0<p<1
φ
2 2 φ
Then the pair (ˆ , η ) is long-run optimal.
πˆ

14. Model Long Run Applications Large Deviations Appendix
Pricing measure
Long-run version of q-optimal measure:

˜
 dRt =σd Zt
˙
˜
 dYt = b − Υ Σ−1 µ + A − Υ Σ−1 Υ δ φ dt + ad Wt
φ
Indifference pricing, for a long horizon.
Changes drift for Monte Carlo simulation.
First term: original drift.
Second term: effect of correlation.
Third term: effect of preferences.

15. Model Long Run Applications Large Deviations Appendix
Extreme Cases
Υ Σ−1 Υ = A: Complete Market.
Pricing trivial: η = 0.
˙
Portfolio nontrivial: π = 1−p Σ−1 µ + Υδ φ
1
φ
Perfect intertemporal hedging. No unhedgeable risk.
Υ = 0: Fully Incomplete.
1 −1
Portfolio trivial: π = 1−p Σ µ.
˙
φ
Pricing nontrivial: η = δφ
Intertemporal hedging infeasible within asset span.
Unhedgeable risk premia reflect latent hedging demand.

16. Model Long Run Applications Large Deviations Appendix
The Myopic Probability
ˆ
P is neither the physical probability P, nor risk neutral.
xp
An investor with utility and a long horizon behaves
p
ˆ
under P like a logarithmic investor under P.
ˆ
P as myopic probability.

17. Model Long Run Applications Large Deviations Appendix
Several Assets with Predictability
Several assets, one state:

dRt = (σν0 + bσν1 Yt ) dt + σdZt


dYt = − bYt dt + dWt


 d R, Y t =σρadt



r (Yt ) =r0
σ matrix, ν0 , ν1 vectors, b scalar.
Expected Returns Predictable.
Nests Kim and Omberg (1996) and Wachter (2002).
Think of Y as the dividend yield.

18. Model Long Run Applications Large Deviations Appendix
Solution
Long-run portfolios and risk premia linear in the state:
1
Σ−1 (µ(y ) + v0 σρ − v1 y σρ)
π(y ) =
1−p
η(y ) =v0 − v1 y
Utility growth rate:
q 1 1
ν0 ν0 + δ −1 v0 − qv0 ρ ν0 − v1
2
λ = pr0 −
2 2 2
v0 and v1 constants.
Depend on preferences and price dynamics.
√
Θ − 1 + qρ ν1
v1 = δb
√
v0 = qδρ ν0 − qν1 ν0 + qδρ ν0 1 + qρ ν1 /Θ
2
+ δ −1 qν1 ν1
Θ = 1 + qρ ν1

19. Model Long Run Applications Large Deviations Appendix
Static Fund Separation
One state variable: dynamic separation into three funds
(2 + 1): risk-free, myopic, and hedging portfolios.
Drawback: dynamic weights unknown.
Long-run portfolios:
 
1  −1 −1 −1 
Σ µ(y ) +v0 Σ Υ −v1 y Σ Υ
π(y ) =
1−p
3rd fund 4th fund
myopic
Static separation into four “funds”
(preference-free portfolios).
Last two funds span hedging demand.
Weights constant over time. Explicit formulas.

20. Model Long Run Applications Large Deviations Appendix
Long-Run Optimality
Set ν1 = −κρ, for some κ > 0.
Sensitivities proportional to correlations.
Still nests Kim and Omberg (1996) and Wachter (2002).
Long-run optimality holds if:
1 1
qρ ρ < +
4 2κ
Always satisfied if at least one of the following holds:
Sensitivity κ sufficiently low.
Market sufficiently incomplete.
Risk aversion sufficiently low.
Long-run optimality holds: at long horizons
Lifestyle (risk-based) superior to lifecycle (age-based).

21. Model Long Run Applications Large Deviations Appendix
Calibration
Parameters as in Barberis (2000) and Wachter (2002).
One risk asset: equity index.
One state variable: dividend yield.
ρ = −0.935, r = 0.14%, σ = 4.36%, ν0 = 0.0788,
κ = 0.8944, b = 0.0226.
Long-run optimality holds for risk-aversion less than 13.4.

22. Model Long Run Applications Large Deviations Appendix
Long Run vs. Myopic. Risk-aversion 2
1.0
0.8
0.6
0.4
0.2
0 5 10 15 20 25 30

23. Model Long Run Applications Large Deviations Appendix
Long Run vs. Myopic. Risk-aversion 5
3.5
3.0
2.5
2.0
1.5
1.0
0.5
5 10 15 20 25 30

24. Model Long Run Applications Large Deviations Appendix
Conclusion
Long-run asymptotics:
tractable framework for portfolio choice and asset pricing.
Long-run policies widely available in closed form.
Finite-horizon solutions rarely explicit.
Long-run portfolios for investing.
Long-run risk-premia for pricing.
Long-run optimality?
Then horizon does not matter if it is long.
Horizon matters if: (1) high risk aversion (2) nearly
complete markets (3) risk premia sensitive to state shocks.

25. Model Long Run Applications Large Deviations Appendix
Donsker-Varadhan Asymptotics
Y ergodic Feller diffusion with generator L on domain D.
M1 (E) set of Borel probabilities on E.
For V ∈ C(E, R), and under joint conditions on Y and V ,
Donsker and Varadhan (1975,1976,1983) show:
T
1
Vdµ − I(µ)
lim log E exp V (Yt )dt sup
=
T →∞ T 0 E
µ∈M1 (E)
rate function defined as I(µ) = − infu>0,u∈D E Lu dµ
u
For a one-dimensional diffusion with generator
1
Lu = 2 a2 u + bu and invariant density m(y ) the function I
admits the simpler expression:
dµ
˙
a(y )2 ψ(y )2 m(y )dy if dm = ψ 2
E
I(µ) =
∞ otherwise

26. Model Long Run Applications Large Deviations Appendix
Large Deviations Approach
For any strategy π, find explicit utility rate.
Obtain set of lower bounds ot utility growth rate.
Maximize rate lower bound over π.
For any risk premium η, find explicit dual growth rate.
Obtain set of upper bounds ot utility growth rate.
Minimize rate upper bound over η.
Two variational problems coincide.

27. Model Long Run Applications Large Deviations Appendix
Primal Side Calculation
Terminal utility:
·
T
1
(XT )p = exp
π
pr + pπ µ + p(p − 1)π Σπ dt E pπ σdZt
2
0 0
Expected utility under change of measure:
T
pr + pπ µ + 1 p(p − 1)π Σπ dt
E (XT )p = EPπ exp
π
2
0
Donsker-Varadhan asymptotics:
1
log E [(XT )p ] =
π
lim
T
T →∞
2
˙
ψ
pr + pπ µ + 1 p(p − 1)π Σπ − A
ψ 2 mπ
sup 2 2 ψ
ψ 2 mπ =1 E
ψ: E

28. Model Long Run Applications Large Deviations Appendix
More Calculations
Change of variable: ψ 2 mπ = φ2 m0 .
2
˙ ˙
φ
+pπ σ (1 − qρρ ) ν + a φ ρ
1
2
E φ m0 pr − 2 a φ − qρ ν φ
+ 1 p(p − 1)π σ 1 − qρρ σ π
2
Quadratic function of π. Maximizer is candidate optimal:
˙
µ + δΥ φ
1 −1
π=
ˆ 1−p Σ φ
Maximum is the Long Run Risk Return tradeoff:
˙
δ φ2
1
pr − qµ Σ−1 µ − A 2 φ2 m0
sup
2 2φ
φ2 m0 =1 E
φ: E

29. Model Long Run Applications Large Deviations Appendix
Long Run Risk-Return Tradeoff
Suppose there is an invariant density m for the process:
dYt = (b − qΥ Σ−1 µ)dt + adWt
ODE is Euler-Lagrange equation for:
˙2
φ
pr − 2 qµ Σ−1 µ − 1 δA
1
φ2 dm
max 2 φ
φ2 m=1 E
E
If pr − 1 qµ Σ−1 µ constant, maximum achieved at φ ≡ 1.
2
Both portfolios and risk-premia are trivial.
Covers p → 0 (logarithmic utility) or r , µ Σ−1 µ constant.

30. Model Long Run Applications Large Deviations Appendix
Related Topics
Utility Maximization in Incomplete Markets: Karatzas,
Lehoczky, Shreve, Xu (1991), Kramkov and
Schachermayer (1999).
q-optimal measure: Hobson (2004), Henderson (2005),
Goll and Rüschendorf (2001).
Risk Sensitive Control: Bielecki, Pliska (1999), Fleming
and Sheu (2000), Nagai and Peng (2002), Sekine (2006).
Long Term Investment: Pham (2003), Föllmer and
Schachermayer (2007).
Asymptotic Criteria: Dumas and Luciano (1991),
Grossman and Zhou (1993), Cvitanic and Karatzas (1994).
Explicit Solutions: Kim and Omberg (1996), Wachter
(2002), Brendle (2006), Liu (2007).

31. Model Long Run Applications Large Deviations Appendix
Well Posedness
Law of process (R, W ) determined by:
Risk-free rate r (y ), drifts µ(y ) and b(y ).
Covariances Σ(y ) = (σσ )(y ), Υ(y ) = (σρa )(y ),
A(y ) = (aa )(y ).
Assumption
(Ω, F) = C([0, ∞), Rn+k ) with Borel σ-algebra.
E ⊂ Rk open connected set.
b ∈ C 1 (E, Rk ), µ ∈ C 1 (E, Rn ),
A ∈ C 2 (E, Rk ×k ), Σ ∈ C 2 (E, Rn×n ), Υ ∈ C 2 (E, Rn×k ).
A(y ), Σ(y ) nonsingular for all y ∈ E.
For all y ∈ E, there exists a unique probability P y on (Ω, F)
such that (R, Y ) satisfies system (R0 , Y0 ) = (0, y ).
Solution global and internal: P y (Yt ∈ E for all t ≥ 0) = 1.

32. Model Long Run Applications Large Deviations Appendix
Implicit Solution
Value function:
1y
u(y , T ) = max EP [(XT )p ]
π
πp
Implicit solution (Merton 1971):
uy
1
Σ−1 µ + Σ−1 Υ
π=
1−p u
Dynamic (k + 2)-fund separation
risk-free asset
myopic portfolio Σ−1 µ
intertemporal hedging portfolio Σ−1 Υ
Myopic weight constant (passive strategy)
Hedging weight time-varying (active strategy).

33. Model Long Run Applications Large Deviations Appendix
Example
One-asset, one-state model:
dRt =Yt dt + dZt
dYt = − λYt dt + dWt
Closed-form solution for p < 0:
√ y2
λT sinh(αλT )
−q
αe 2
1 √
u(y , T ) = e 2λ sinh(αλT )+α cosh(αλT )
p sinh(αλT )+α cosh(αλT )
1 + q/λ2 .
where y = Y0 , q = p/(p − 1) and α =

34. Model Long Run Applications Large Deviations Appendix
Long Run Limit
Formulas simplify dramatically as T → ∞.
Value function:
λ
u(T ) ∼ e 2 (1−α)T +o(T )
Intertemporal hedging independent of t, and linear in the
state:
uy q
=− y
u 1+α
Mean-reverting drift asymptotically equivalent to constant
drift µ = λ (α − 1):
q
dRt = µdt + dZt

35. Model Long Run Applications Large Deviations Appendix
General Case
Quasilinear PDE in (v , λ)
1 1
v A − qΥ Σ−1 Υ
tr AD 2 v + v
2 2
1
b − qΥ Σ−1 µ + pr − qµ Σ−1 µ = λ
v
+
2
Policies:
1
Σ−1 (µ + Υ v ) v
π=
ˆ η=
ˆ
1−p
Finite-horizon bounds:
y y
EP (XT )p = eλT +v (y ) E ˆ e−v (YT )
π
ˆ
P
1−p
1−p 1
ˆ − 1−p v (YT )
y y
EP (MT )q
h
= eλT +v (y ) E ˆ e
P
Quadratic solution v (y ) = y Ay + By + C for many models.
Includes linear diffusion: r , µ, b affine in y , Σ, A, Υ constant.

36. Model Long Run Applications Large Deviations Appendix
Long Run Optimality
Proposition
Long-run optimality holds if δ < 4.
If δ ≥ 4, long-run optimality fails. In particular:
1
i) if δ > 4, there exists a finite T such that p E[(XT )p ] = −∞.
π
ˆ
ii) if δ = 4 and ξ = 0, the certainty equivalent loss converges
γ
to − 2p ;
iii) if δ = 4 and ξ = 0, the certainty equivalent loss diverges to
∞.
Long-run optimality fails for nearly complete market, and
highly risk-averse investor.
Departure from optimality near T becomes intolerable.

37. Model Long Run Applications Large Deviations Appendix
Finite Horizons as Long-Horizon Expansions
Assume finite horizon solutions solve HJB equations
(Duffie, Fleming, Soner, Zariphopoulou 1997, Pham 2002).
Linear PDE via power transformation (Zariphopoulou
2001).
Separation of variables. Expand value function as a series
of eigenvectors with respect to invariant density.
∞
λn
T
uT (y ) = e φn (y )m(y )
δ
n=1
uy
In the long run, only first eigenvector survives in u.