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# Portfolios and Risk Premia for the Long Run

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### Portfolios and Risk Premia for the Long Run

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.