[PDF] - Optimal Consumption and Investment with Habit Formation and PDF Document

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Optimal Consumption and Investment with Habit Formation and Hyperbolic discounting Mihail Zervos Department of Mathematics London School of Economics Joint work with Alonso P´ erez-Kakabadse and Dimitris Melas

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counting

the SOP (Many references!...)

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The Standard Optimisation Problem (SOP)

ket driven by a standard one-dimensional Brownian motion W.

(Π, C) satisfies the SDE dXt = rXt dt − Ct dt + σΠtθ dt + σΠt dWt, (1) where r ≥ 0 is the short-term interest rate, σ is the volatility of the risky asset, and θ is the market relative risk.

process X given by (1) is well-defined and strictly positive.

E T

U1(s, Cs) ds + U2(XT) | Xt = x

utility functions.

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A Model with Hyperbolic Discounting

v(t, T, x) = sup

E T

q(s)U1(Cs) ds + U2(XT) | Xt = x

subject to dXt = rXt dt − Ct dt + σΠtθ dt + σΠt dWt, (4) where U1(c) = cp p and U2(x) = ζxp, (5) for some p ∈ ]0, 1[ and ζ > 0, and q(t) = (1 + βt)−γ/β, (6) for some constants β, γ > 0.

arises from the hyperbolic discounting function q given by (24). Experimental evidence suggests that economic agents may have rel- atively high discounting rates over short horizons and relatively low discounting rates over long horizons: − ˙ q(t) q(t) = γ 1 + βt. (7)

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wt(t, T, x) + sup

1 2σ2π2wxx(t, T, x) +

p cp

(8) with boundary condition w(T, T, x) = ζxp. (9) In light of the standard theory regarding the SOP, the value function v of our optimal control problem satisfies the PDE vt(t, T, x) − 1 2θ2 v2

vxx(t, T, x) + rxvx(t, T, x) + 1 − p p q1/(1−p)(t)v−p/(1−p)

(t, T, x) = 0, (10) with boundary condition v(T, T, x) = ζxp. (11)

v(t, T, x) = f(t, T)xp. (12)

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ft(t, T) = −ξf(t, T) − h(t)f −p/(1−p)(t, T), (13) f(T, T) = ζ. (14) where ξ = p

2(1 − p) + r

(15) and h(t) = (1 − p)p−p/(1−p)q1/(1−p)(t). (16) The solution of this ODE is given by f(t, T) = e−ξt

T

e− ξ

1−p . (17)

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v(t, T, x) = f(t, T)xp, (18) the optimal investment strategy is given by Π∗

θ σ(1 − p)X∗

(19) and the optimal consumption rate is given by C∗

pf(t, T) 1/(1−p) X∗

(20)

they depend on both t and T, not just on the time to “maturity” T − t. Non-stationary models are indeed used in finance. For instance, the Hull and White interest rate model results in non-stationary discount bond prices. The non-stationarity of this investment model, which arises by the introduction of hyperbolic discounting, may be appropriate for indi- vidual investors, but may not be so for institutional investors such as pension funds.

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Investment Decisions with Habit Formation and Hyperbolic Discounting

v(t, T, x, a) = sup

E T

q(s)U1(Cs − As) ds + U2(XT) | Xt = x, At = a

(21) subject to dXt = rXt dt − Ct dt + σΠtθ dt + σΠt dWt, dAt = −δAt dt + Ct dt, (22) where U1(c − a) = (c − a)p p and U2(x) = ζxp, (23) for some p ∈ ]0, 1[ and ζ > 0, and q(t) = (1 + βt)−γ/β, (24) for some constants β, γ > 0.

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wt(t, T, x, a) + sup

1 2σ2π2wxx(t, T, x, a) +

+ (c − δa)wa(t, T, x, a) + q(t) p (c − a)p

(25) with boundary condition w(T, T, x, a) = ζxp. (26) Incorporating the first order conditions σ2πwxx(t, T, x, a) + σθwx(t, T, x, a) = 0, (27) and wa(t, T, x, a) − wx(t, T, x, a) + q(t)(c − a)p−1 = 0, (28) that arise from the choice of π and c, we obtain wt(t, T, x, a) − 1 2θ2 w2

wxx(t, T, x, a) + (rx − a)wx(t, T, x, a) + (1 − δ)awa(t, T, x, a) + q(t)(1 − p) p (wx − wa)(t, T, x, a) q(t) p/(p−1) = 0, (29) with boundary condition w(T, T, x, a) = ζxp. (30)

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w(t, T, x, a) = f(t, T)

p. (31)

ft(t, T) − p

2(p − 1) + kt(t, T)a + rx − a + (1 − δ)k(t, T)a x + k(t, T)a

+ q(t)(1 − p) p p[1 − k(t, T)]f(t, T) q(t) p/(p−1) = 0, (32) f(T, T) = ζ and k(T, T) = 0. (33) To eliminate a from (32), we set kt(t, T)a + rx − a + (1 − δ)k(t, T)a = r[x + k(t, T)a]. (34) This expression and (32) imply that ft(t, T) − p

2(p − 1) + r

+ q(t)(1 − p) p p[1 − k(t, T)]f(t, T) q(t) p/(p−1) = 0. (35) Also, (34) implies that kt(t, T) = −(1 − δ − r)k(t, T) + 1. (36)

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(36) with boundary conditions (33) is given by k(t, T) = 1 − e(1−δ−r)(T−t) 1 − δ − r (37) and f(t, T) = e−ξt

+ p− 1

T

e− ξ

1 − δ − r p

(1 + βs)

1−p . (38)

the value function and the optimal strategy. Again, these are non- stationary.

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Portfolio Optimisation with Habit Formation as a Special Case of the SOP

dAt = −δAt dt + Ct dt. (39) Given ˜ C > 0, the process C defined by Ct = e−(δ−1)t

Ct + t e−(δ−1)s ˜ Cs ds

and the corresponding process A defined by (39) satisfy C −A = ˜ C.

mance criterion ˜ Jt,T,x(Π, ˜ C) = E T

˜ U1(s, ˜ Cs) ds + U2( ˜ XT) | ˜ Xt = x

(41)

C), subject to d ˜ Xt = r ˜ Xt dt − ˜ Ct dt + σΠtθ dt + σΠt dWt. (42)

habit formation is to maximise the performance criterion Jt,T,x,a(Π, C) = E T

U1(s, Cs − As) ds + U2(XT) | Xt = x, At = a

(43) subject to dXt = rXt dt − Ct dt + σΠtθ dt + σΠt dWt, dAt = −δAt dt + Ct dt. (44)

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d(Xt + k(t, T)At) = r(Xt + k(t, T)At) dt + [kt(t, T) − (δ + r)k(t, T)]At dt − [1 − k(t, T)]Ct dt + σΠtθ dt + σΠt dWt. (45) If we let k satisfy kt(t, T) − (δ + r)k(t, T) = 1 − k(t, T), (46) and we define ˜ Xt = Xt + k(t, T)At and ˜ Ct = [1 − k(t, T)](Ct − At), (47) then (...) d ˜ Xt = r ˜ Xt dt − ˜ Ct dt + σΠtθ dt + σΠt dWt. (48) Furthermore, if k(T, T) = 0, (49) then ˜ XT = XT, (50) and Jt,T,x,a(Π, C) = ˜ Jt,T,x+k(t,T)a(Π, ˜ C), (51) provided that ˜ U1(s, z) = U1