Optimal asset allocation under forward performance criteria Oberwolfach, February 2007 Thaleia Zariphopoulou The University of Texas at Austin 1
References Indifference valuation in binomial models (with M. Musiela) Investments and forward utilities (with M. Musiela) Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model (with M. Musiela) Optimal asset allocation under forward exponential criteria (with M. Musiela) Horizon-independent risk measures (with G. Zitkovic) 2
Performance measurement 3
Deterministic environment Preferences u(x, t) : x wealth and t time Monotonicity u x (x, t) > 0 Risk aversion u xx (x, t) < 0 Impatience u t (x, t) < 0 Fisher (1913, 1918), Koopmans (1951), Koopmans-Diamond-Williamson (1964)... 4
Stochastic environment Important ingredients Time evolution concurrent with the one of the investment universe Consistency with up to date information Incorporation of available opportunities and constraints Meaningful optimal performance 5
Performance process U t (x) is an F t -adapted process As a function of x, U is increasing and concave For each self-financing strategy, represented by π, the associated (discounted) wealth X t satisfies E P (U t (X π t ) F s ) U s (X π s ) 0 s t There exists a self-financing strategy, represented by π,forwhich the associated (discounted) wealth Xt π satisfies E P (U t (Xt π ) F s ) = U s (Xs π ) 0 s t 6
Traditional framework A deterministic utility datum u T (x) is assigned at the end of a fixed investment horizon U T (x) =u T (x) Backwards in time generation of optimal utility V t (x) =sup π E P (u T (X π T ) F t; X π t = x) V t (x) =sup π E P (V s (X π s ) F t ; X π t = x) (DPP) V t (x) =E P (V s (Xs π ) F t ; Xt π = x) U t (x) V t (x) 0 t<t The performance process coincides with the traditional value function 7
A deterministic datum u 0 (x) is assigned at the beginning of the trading horizon, t =0 U 0 (x) =u 0 (x) Forward in time generation of optimal performance U s (X π s )=E P (U t (X π t ) F s ) 0 s t Performance can be defined for all trading horizons Performance and allocations take a very intuitive form Difficulties due to the inverse in time nature of the problem Performance criterion is not exogeneously given but is implied/calibrated w.r.t. investment opportunities 8
Motivational examples 9
An incomplete multiperiod binomial example Exponential datum Traded security: S t,t=0, 1,... ξ t+1 = S t+1 S t,ξ t+1 = ξ d t+1,ξu t+1 with 0 <ξ d t+1 < 1 <ξu t+1 Second traded asset is riskless yielding zero interest rate Stochastic factor: Y t,t=0, 1,... η t+1 = Y t+1 Y t,η t+1 = η d t+1,ηu t+1 with η d t <η u t Probability space (Ω, (F t ), P) {S t,y t : t =0, 1,...} : a two-dimensional stochastic process 10
State wealth process: X t, t = s +1,s+2,...,... α i : the number of shares of the traded security held in this portfolio over thetimeperiod[i 1,i] X t = X s + t i=s+1 α i S i Forward exponential performance U s (X α s )=E P (U t (X α t ) F s ) U 0 (x) = e γx, γ > 0 11
A forward performance process U t (x) = e γx if t =0 e γx+ t i=1 h i if t 1 Auxiliary quantities with h i = q i log q i P (A i F i 1 ) +(1 q i)log 1 q i 1 P (A i F i 1 ) A i = {ξ i = ξ u i } and q i = Q (A i F i 1 ) for i =0, 1,.. and Q being the minimal martingale measure 12
Important insights The forward performance process U t (x) = e γx+ t i=1 h i is of the form U t (x) =u(x, A t ) where u(x, t) is the deterministic function u(x, t) = e γx+1 2 t and A t corresponds to a time change depending on the market input A t =2 t h i i=1 13
Important insights (continued) The differential utility input u(x, t) = e γx+1 2 t solves the partial differential equation u t u xx = 1 2 u2 x u(x, 0) = e γx The stochastic market input A t =2 t h i i=1 plays now the role of time. It depends exclusively on the market parameters 14
A continuous-time example Investment opportunities Riskless bond : r =0 Risky security : ds t = σ t S t (λ t dt + dw t ) Datum at t =0: u 0 (x) Wealth process dx t = σ t π t (λ t dt + dw t ) X 0 = x Market input : λ t, A t da t = λ 2 t dt A 0 =0 15
Building the martingale U t (X π t ) Assume that we can construct U t (x) via U t (Xt π )=u(xt π,a t ) U 0 (x) =u(x, 0) = u 0 (x) where u(x, t) is the differential utility input and A t the stochastic market input du t (Xt π)=u x(x t,a t )σ t π t dw t +(u t (Xt π,a t)λ 2 t + u x(xt π,a t)σ t π t λ t + 1 2 u xx(xt π,a t)σt 2π2 t )dt }{{} 0 16
Differential utility input condition u t u xx = 1 2 u2 x u(x, 0) = u 0 (x) The optimal allocations in stock, π t π t = σ 1 t π 0, t = X π t, and in bond, π0, t, u x (X λ t π,a t ) t u xx (X π,a t ) = σ 1 t λ t R t t σt 1 λ t R t R t = r(x π t,a t ) ; r(x, t) = u x(x, t) u xx (x, t) The local risk tolerance r(x, t) and the subordinated risk tolerance process R t emerge as important quantities 16
Forward performance measurement time t 1,informationF t1 asset returns 110 constraints 109.8 market view away from equilibrium benchmark calendar time numeraire subordination 109.6 109.4 109.2 109 1 u(x,t 1 ) 0.8 0.6 0.4 Time 0.2 0 0 0.2 0.4 0.6 Wealth 0.8 1 MI(t 1 ) + u(x, t 1 ) U t1 (x; MI) F t1 π t1 (x; MI) F t1 17
Forward performance measurement time t 2,informationF t2 asset returns constraints market view away from equilibrium benchmark numeraire 110 105 100 95 u(x,t 2 ) calendar time subordination 90 1 0.8 0.6 0.4 Time 0.2 0 0 0.2 0.4 0.6 Wealth 0.8 1 MI(t 2 ) + u(x, t 2 ) U t2 (x; MI) F t2 π t2 (x; MI) F t2 18
Forward performance measurement time t 3,informationF t3 asset returns 110 constraints market view away from equilibrium benchmark calendar time numeraire subordination 105 100 95 90 85 80 75 u(x,t 3 ) 1 0.8 0.6 0.4 0.2 Time 0 0 0.2 0.4 0.6 Wealth 0.8 1 MI(t 3 ) + u(x, t 3 ) U t3 (x; MI) F t3 π t3 (x; MI) F t3 19
Forward performance measurement time t, informationf t u(x,t) asset returns additional market input Time Wealth MI(t) + u(x, t) U t (X t ) F t π t (X t ) F t 20
Forward performance measurement time t 1,informationF t1 u(x,t) asset returns additional market input Time Wealth MI(t 1 ) + u(x, t 1 ) U t1 (X t 1 ) F t1 π t 1 (X t 1 ) F t1 21
Forward performance measurement time t 2,informationF t2 u(x,t) asset returns additional market input Time Wealth MI(t 2 ) + u(x, t 2 ) U t2 (X t 2 ) F t2 π t 2 (X t 2 ) F t2 22
Forward performance measurement time t 3,informationF t3 u(x,t) asset returns additional market input Time Wealth MI(t 3 ) + u(x, t 3 ) U t3 (X t 3 ) F t3 π t 3 (X t 3 ) F t3 23
Construction of a class of forward performance processes 25
Creating the martingale that yields the optimal performance Minimal model assumptions Stochastic optimization problem inverse in time Key idea Stochastic input Market Differential input Individual Maximal performance Optimal allocation 26
Differential input utility surfaces 27
Performance surface A model independent differential constraint on impatience, risk aversion and monotonicity Initial datum u 0 (x) =u(x, 0) Fully non-linear pde u t u xx = 1 2 u2 x u(x, 0) = u 0 (x) 28
Transport equation The u-equation can be alternatively viewed as a transport equation with slope of its characteristics equal to (half of) the risk tolerance r(x, t) = u x(x, t) u xx (x, t) u t + 1 2 r(x, t)u x =0 u(x, 0) = u 0 (x) Characteristic curves: dx(t) dt = 1 2 r(x(t),t) 29
Construction of performance surface u(x, t) using characteristics dx(t) dt = 1 2 r(x(t),t) Performance datum u(x, 0) 30
Construction of characteristics dx(t) dt = 1 2 r(x(t),t) Performance datum u(x, 0) Characteristic curves 31
Propagation of performance datum along characteristics 32
Propagation of performance datum along characteristics 33
Performance surface u(x, t) 34
Two related pdes Fast diffusion equation for risk tolerance r t + 1 2 r2 r xx =0 r(x, 0) = r 0 (x) (FDE) Conductivity : r 2 Porous medium equation for risk aversion γ t = γ(x, t) = 1 r(x, t) ( ) 1 γ xx γ(x, 0) = 1 r 0 (x) (PME) Pressure : r 2 and (PME) exponent: m = 1 35
Difficulties Differential input equation: u t u xx = 1 2 u2 x Inverse problem and fully nonlinear Transport equation: u t + 1 2 r(x, t)u x =0 Shocks, solutions past singularities Fast diffusion equation: r t + 1 2 r2 r xx =0 Inverse problem and backward parabolic, solutions might not exist, locally integrable data might not produce locally bounded slns in finite time Porous medium equation: γ t =( γ 1) xx Majority of results for (PME), γ t =(γ m ) xx, are for m>1, partialresultsfor 1 <m<0 36
A rich class of risk tolerance differential inputs Addititively separable risk tolerance r 2 (x, t; α, β) =m(x; α, β)+n(t; α, β) m(x; α, β) =αx 2 r(x, t; α, β) = Example n(x; α, β) =βe αt αx 2 + βe αt α, β > 0 (Very) special cases r(x, t;0,β)= β u(x, t) = e x β + t 2 r(x, t;1, 0) = x u(x, t) =log x t 2 r(x, t; α, 0) = α x u(x, t) = γ 1xγ e 2(1 γ) t, γ = α 1 γ α 37
Multiplicatively separable risk tolerance r(x, t; α, β) =m(x; α)n(t; β) Example m(x; α) =ϕ(φ 1 (x; α)) n(t; β) = 1 t + β, β > 0 Φ(x; α) = x α ez2 /2 dz ϕ =Φ r(x, t; α, β) =ϕ(φ 1 (x; α)) (Very) special cases m(x; α) =α, n(t; β) =1 u(x, t) = e x α + t 2 m(x; α) =x, n(t; β) =1 u(x, t) =log x 2 t m(x; α) =αx, n(t; β) =1 u(x, t) = 1 γ xγ e γ 2(1 γ) t, γ = α 1 α 37
Summary on differential input Key state variables: wealth and risk tolerance Risk tolerance solves a fast diffusion equation posed inversely in time r t + 1 2 r2 r xx =0 r(x, 0) = u 0 (x) u 0 (x) Transport equation u t + 1 2 r(x, t)u x =0 u(x, 0) = u 0 (x) Forward performance process constructed by compiling differential input and stochastic market input 39
Stochastic market input 40
Investment universe Riskless and risky securities (Ω, F, P) ; W =(W 1,...,W d ) standard Brownian Motion Traded securities 1 i k ds i t = S i t(µ i tdt + σ i t dw t ), S i 0 > 0 db t = r t B t dt, B 0 =1 µ t,r t R, σ i t Rd bounded and F t -measurable stochastic processes Postulate existence of a F t -measurable stochastic process λ t R d satisfying µ t r t 1=σ T t λ t 41
Investment universe Self-financing investment strategies π 0 t,πi t, i =1,...,k Present value of this allocation X t = k πt i i=0 dx t = k i=0 π i t(µ i t r t ) dt + k i=0 π i tσ i t dw t = σ t π t (λ t dt + dw t ) π t =(π 1 t,...,πk t ), µ t r t 1=σ T t λ t 42
Market input processes (σ t,λ t ) and (Y t,z t,a t ) These F t -mble processes do not depend on the investor s differential input They reflect and represent, respectively (λ t,σ t ) : dynamics of traded securites Y t : benchmark numeraire Z t : market view away from market equilibrium feasibility and trading constraints A t : subordination 43
The processes (Y t,z t,a t ) Benchmark and/or numeraire A replicable process Y t satisfying dy t = Y t δ t (λ t dt + dw t ) Y 0 =1 δ t F t, σ t σ + t δ t = δ t σ + t : Moore-Penrose matrix inverse 44
Market input processes Market views, feasibility and trading constraints An exponential martingale Z t satisfying dz t = Z t φ t dw t Z 0 =1, φ t F t Subordination A non-decreasing process A t solving da t = δ t σ t σ + t (λ t + φ t ) 2 dt A 0 =0 45
Forward performance process Optimal asset allocation 46
Forward performance process Stochastic input : (Y t,z t,a t ) Differential input : u(x, t) Time change A t Benchmark Y t u t u xx = 1 2 u2 x u(x, 0) = u 0 (x) Market view Z t U t (x) =u( x Y t,a t )Z t 47
Forward performance process Stochastic market input Differential input λ t,σ t x, r 0 (x) = u 0 (x) u 0 (x) benchmark, views r t + 1 2 r2 r xx =0 (FDE) subordination u t + 1 2 ru x =0 (TE) (Y t,z t,a t ) u(x, t) 7 U t (x) =u( x A t,y t )Z t Model independent construction! 48
What is the optimal allocation? Optimal portfolio processes π t =(π 0 t,π 1 t,...,π k t ) can be directly and explicitly characterized along with the construction of the forward performance! 49
Stochastic input Market The structure of optimal portfolios dx t = σ t π t (λ t dt + dw t ) Differential input Individual (Y t,z t,a t ) wealth x λ t,σ t,δ t,φ t risk tolerance r(x, t) 7 1 πt is a linear combination Y t of (benchmarked) optimal wealth and subordinated (benchmarked) risk tolerance 50
Optimal asset allocation Let X t be the optimal wealth, Y t the benchmark and A t the subordination processes dx t = σ tπ t (λ tdt + dw t ) dy t = Y t δ t (λ t dt + dw t ) da t = σ t σ + t (λ t + φ t ) δ t 2 dt Define r t the subordinated (benchmarked) risk tolerance r t = r ( X t Y t,a t ) Optimal (benchmarked) portfolios 1 Y t π t = σ + t ( (λ t + φ t )r t + δ t ( X t Y t r t )) 51
Stochastic evolution of wealth-risk tolerance 52
A system of SDEs at the optimum X t = X t Y t and r t = r( X t,a t ) d X t = r t (σ t σ + t (λ t + φ t ) δ t ) ((λ t δ t ) dt + dw t ) d r t = r x ( X t,a t )d X t Separability of wealth dynamics in terms of risk tolerance and market input Sensitivity of risk tolerance in terms of its spatial gradient and changes in optimal wealth Utility functional has essentially vanished Universal representation, no Markovian assumptions 53
Wealth-Risk tolerance Optimal wealth-risk tolerance ( X t, r t ) system of SDEs in original market configuration d X t = r t (σ tσ + t (λ t + φ t ) δ t ) ((λ t δ t ) dt + dw t ) d r t = r x ( X t,a t ) d X t change of measure historical benchmarked change of time Levy s theorem 54
Wealth-Risk tolerance Optimal wealth-risk tolerance (x 1 t,x2 t ) system of SDEs in canonical market configuration x 1 t = ( X t Y t ) A ( 1) t ( ) x 2 X t = r t,a t Y t A ( 1) t M t = A t w t = M A ( 1) dx 1 t = x2 t dw t dx 2 t = r x(x 1 t,t)x2 t dw t x 1 0 = x y, x2 0 = r x( x y, 0) 55
Analytic solution of the SDE system dx 1 t = x2 t dw t dx 2 t = r x(x 1 t,t)x2 t dw t Define the budget capacity function h(x, t) via x = h(x,t) x du h(x,t) r(u, t) = γ(u, t)du x x : related to symmetry properties of risk tolerance, reflection point of its spatial derivative and risk aversion front 56
Analytic solutions The budget capacity function h solves the (inverse) heat equation h t + 1 2 h xx 1 2 r x(x,t)h x =0 h(x, 0) = h 0 (x), x = h0 (x) x du r(u, 0) Solution of the SDE system x 1 t = h(z t,t) x 2 t = h z(z t,t) z t = h 1 0 (x) t 0 1 2 r x(x,s)ds + w t Using equivalent measure transformations and time change we recover the original pair of optimal (benchmarked) wealth and (benchmark) risk tolerance 57
Forward performance measurement Market Benchmark, views, constraints Investor Wealth, risk tolerance Market input processes Subordination Fast diffusion eqn Transport eqn Forward evolution > Y t,z t,a t < x, r(x, t), u(x, t) Optimal performance and optimal portfolios measure change time change Wealth-Risk tolerance SDE system Heat eqn Fast diffusion eqn Universal analytic solutions
An example 59
Forward exponential performance Objective: Find an F t adapted process U t (x) such that U 0 (x) = exp ( x y E P (U s (Xs π ) F t ) U t (Xt π) ) E P ( Us ( X π s ) ) ( Ft = Ut X π ) t, s t Solution Differential input u (x, y, z) = exp ( xy ) + z 60
Forward exponential performance (continued) Stochastic market input consists of a pair of Ito processes, (Y,Z), solving, respectively, dy t = Y t δ t (κ t dt + dw t ) and Y 0 = y>0 dz t = η t dt + ξ t dw t Z 0 =0. 61
Dynamic exponential utility (continued) The processes δ t,κ t,η t,ξ t are taken to be bounded and F t progressively measurable. It is, also, assumed that σ t σ + t δ t = δ t and δ t (κ t λ t )=0 The drift η t of the process Z t satisfies 2η t = δt σ t σ + t (λ t + ξ t ) 2 ξt 2 Wlog, the dynamics of the benchmark process Y t can be written as dy t = Y t δ t (λ t dt + dw t ) 62
At the optimum Feedback portfolio control process πt = Y t B t σ t + ( λt + ( Xt Yt 1 1 ) ) δ t + ξ t Optimal wealth process Optimal utility volume dx t = B t 1σ tπ t (λ t dt + dw t ) = ( Y t ( σt σ + t (λ t + ξ t ) δ t ) + X t δ t ) (λt dt + dw t ) du t (X t )=U t (X t ) ( Y 1 t Bt 1 σ t πt + Xt Y 1 ) δ t + ξ t dwt t = U t (X t ) ( σ t σ + t (δ t λ t )+ ( I σ t σ + t ) ξt ) dwt Explicit solutions can be found 63
Explicit solutions Optimal wealth process: X t = E ( t x + t0 Es 1 ( Y s σs σ s + ) (λ s + ξ s ) δ s ((λs δ s ) ds + dw s ) ) E t =exp ( ( t0 δs λ s 1 2 δ s 2) ds + ) t 0 δ s dw s Optimal portfolio process +B t E t ( t 0 E 1 Optimal utility volume U t (X t )=exp π t = B t Y t σ + t (λ t + ξ t δ t )+B t X t σ + t δ t = xe t B t σ + t δ t + B t Y t σ + t (λ t + ξ t δ t ) s Y s ( σs σ + s (λ s + ξ s ) δ s ) ((λs δ s ) ds + dw s ) + t 0 ( x y t 0 1 2 σ s σ + s (δ s λ s )+ ( I σ s σ + s ( σs σ + s (δ s λ s )+ ( I σ s σ + s ) ξs ) dws ) ) σ + t δ t ) ξs 2 ds 64
Case 1: No benchmark and no views δ = ξ =0 Then, Y t = y, fort 0. The forward performance process takes the form U t (x) = exp ( x y + t 0 1 2 ) σ s σ s + λ 2 s ds Note that even in this simple case, the solution is equal to the classical exponential utility only at t =0. The optimal discounted wealth and optimal asset allocation are given, respectively, by and X t = x + t y ( σ s σ + ) s λ s (λs ds + dw s ) 0 π t = yb t σ + t λ t Observe that π is independent of the initial wealth x. 65
Case 1: No benchmark and no views δ = ξ =0 (continued) Optimal performance U t (X t )= exp ( x y t Total amount allocated in the risky assets Amount invested in the riskless asset 0 1 2 σ s σ s + ) λ 2 t s ds σ sσ s + λ s dw s 0 1 π t B t = 1 yσ + t λ t π 0, t = X t 1 yσ + t λ t Such an allocation is rather conservative and is often viewed as an argument against the classical exponential criteria. 66
Case 2: No benchmark and risk neutralization δ =0and λ + ξ =0 Then, E t =1,Y t = y>0 and Z t = t 0 1 2 λ s 2 ds t 0 λ s dw s. Forward exponential performance process ( U t (x) = exp x y 1 2 t 0 λ s 2 ds t 0 λ s dw s ) Optimal discounted wealth X t = x 67
Case 2: No benchmark and risk neutralization δ =0and λ + ξ =0 (continued) Optimal allocations π t =0 and π 0, t = X t = x Optimal exponential performance U t (X t )=U t (x) = exp ( x y 1 2 t 0 λ s 2 ds t 0 λ s dw s ) It is important to notice that, for all trading times, the optimal allocation consists of putting zero into the risky assets and, therefore, investing the entire wealth into the riskless asset. Such a solution seems to capture quite accurately the strategy of a derivatives trader for whom the underlying objective is to hedge as opposed to the asset manager whose objective is to invest. 68
Case 3: Static performance σσ + (δ λ)+ ( I σσ +) ξ =0 Then σ + (δ λ) =0and σσ + ξ = ξ, and Z t = t 0 ξ s dw s Forward exponential performance process ( U t (x) = exp x + Y t t 0 ξ s dw s ) Optimal discounted wealth X t = E t (x + t 0 E 1 s Y s ξ s ((λ s δ s ) ds + dw s ) ) 69
Case 3: Static performance σσ + (δ λ)+ ( I σσ +) ξ =0 (continued) Optimal allocation ( t πt = xe t B t σ t + δ t + Y t B t σ t + ξ t + B t E t Optimal exponential performance U t (Xt )=U 0 (x) = exp 0 E 1 s ( x y Y s ξ s dw s ) ) σ + t δ t Observe that the optimal level of forward performance remains constant across times. 70
Case 4: Following the benchmark δ = λ + ξ with λ + ξ 0 Then δ = σσ + (λ + ξ) and, in turn, Z t = t 0 1 2 ξ s 2 ds + t 0 ξ s dw s Forward exponential performance process Optimal wealth U t (x) = exp ( x Y t t 0 1 2 ξ s 2 ds + t 0 ξ s dw s ) Returns of wealth and of benchmark X t = xe t dx t X t = dy t Y t 71
Case 4: Following the benchmark δ = λ+ξ with λ+ξ 0 (continued) Optimal allocation π t = B t X t σ + t δ t Optimal exponential performance ( U t (Xt )= exp x Y t t 0 1 2 ξ s 2 ds + t 0 ξ s dw s ) Observe that, contrary to what we have observed in traditional (backward) exponential utility problems, the optimal portfolio is a linear functional of the wealth and not independent of it. 72
Case 5: Generating arbitrary portfolio allocations Assume that 1 σ + t (λ t + ξ t )=1.Then 1 π t B t = X t and π 0, t =0 Hence, the optimal allocation π puts zero amount in the riskless asset and invests all wealth in the risky assets, according to the weights specified by the vector σ + (λ + ξ) 73
Case 5: Generating arbitrary portfolio allocations (continued) Note, also, that for an arbitrary vector ν t with 1 σ + t ν t 0, the vector ξ t = 1 1 σ+ t λ t 1 σ t + ν t satisfies the above constraint since 1 σ + t ( ν t λ t + 1 1 σ+ t λ t 1 σ + t ν t ν t ) =1 Can we generate optimal portfolios that allocate arbitrary, but constant, fractions of wealth to the different accounts? The answer is affirmative. Indeed, for p R,set, 1 σ + t (λ t + ξ t )=p Then, the total investment in the risky assets and the allocation in the riskless bond are 1 π t B t = px t and π 0 t B t =(1 p) X t 74
Examples of differential input 75
Risk tolerance r(x, t) = 0.05x 2 +15.5e 0.05t 4.2 4.1 4 3.9 3.8 1 0.8 0.6 Time 0.4 0.2 0 10 5 0 Wealth 5 10 76
Utility surface u(x, t) generated by risk tolerance r(x, t) = 0.05x 2 +15.5e 0.05t 0 2 4 6 8 10 6 4 Time 2 0 0.5 1 1.5 Wealth 2 2.5 Characteristics: dx(t) dt = 1 2 0.05x(t) 2 +15.5e 0.05t 77
Risk tolerance r(x, t) = 10x 2 + e 10t 2.2 2 1.8 1.6 1.4 1.2 1 0.8 1 0.8 0.6 Time 0.4 0.2 0 10 5 0 Wealth 5 10 78
Utility surface u(x, t) generated by risk tolerance r(x, t) = 10x 2 + e 10t 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 1.5 1 0.5 5 10 15 Time 0 0 Wealth Characteristics: dx(t) dt = 1 2 10x(t) 2 + e 10t 79
Risk tolerance r(x, t;0, 1) = 0x 2 +1=1 2 1.5 1 0.5 0 1 0.8 0.6 Time 0.4 0.2 0 10 5 0 Wealth 5 10 80
Utility surface u(x, t) = e x+ t 2 generated by risk tolerance r(x, t) =1 0 20 40 60 80 100 6 4 Time 2 0 0.5 1 1.5 Wealth 2 2.5 Characteristics: dx(t) dt = 1 2 81
Risk tolerance r(x, t;1, 0) = x 2 +0e t = x 6 5 4 3 2 1 0 1 0.8 0.6 Time 0.4 0.2 0 10 5 0 Wealth 5 10 82
Utility surface u(x, t) =logx 2 t, x>0 generated by risk tolerance r(x) =x 4 2 0 2 4 6 6 4 Time 2 0 0 5 10 Wealth 15 20 Characteristics: dx(t) dt = 1 2 x(t) 83
Risk tolerance r(x, t;4, 0) = 4x 2 +0e 4t =2 x 6 5 4 3 2 1 0 1 0.8 0.6 Time 0.4 0.2 0 10 5 0 Wealth 5 10 84
Utility surface u(x, t) =2 xe t 2, x>0 generated by risk tolerance r(x, t) =2x 0.12 0.11 0.1 0.09 0.08 1.5 1 Time 0.5 0 0 5 10 Wealth 15 20 Characteristics: dx(t) dt = x(t) 85
Risk tolerance r(x, t) = ϕ(φ 1 (x;0.5) t +5 0.2 0.15 0.1 0.05 0 1.5 1 Time 0.5 0 5 10 Wealth 15 20 86
Utility surface u(x, t) =Φ(Φ 1 (x;0.5) t +5) generated by risk tolerance r(x, t) = ϕ(φ 1 (x;0.5)) t +5 2.5 2 1.5 1 0.5 1.5 1 Time 0.5 0 5 10 Wealth 15 20 Characteristics: dx(t) dt = ϕ(φ 1 (x(t); 0.5)) t +5 87
Utility function u(x, t 0 ) (fixed time) t 0 =2 5 4.8 4.6 4.4 4.2 Utility 4 3.8 3.6 3.4 3.2 0 2 4 6 8 10 12 14 16 18 20 Wealth 88
Utility function u(x 0,t) (fixed wealth level) x 0 =3.5 3.8 3.6 3.4 3.2 Utility 3 2.8 2.6 2.4 2.2 0.5 1 1.5 2 2.5 3 Time 89
Forward exponential performance Solution U t (x) = exp ( x ) + Z t Yt Idea of the proof: applying Ito calculus and using the structural assumptions on the market input yields du (X) =U (X) ( Y 1 β dw + XY 1 δ dw + ξ dw ) + 1 2 U (X) ( 2Y 1 β λ +2XY 1 δ κ +2η + Y 2 β 2 +2 ( Y 1 XY 2) δ β 2Y 1 ξ β +2XY 1 δ ξ + ( X 2 Y 2 2XY 1) δ 2 + ξ 2) dt = U (X) ( Y 1 β dw + XY 1 δ dw + ξ dw ) + 1 2 U (X) ( Y 1 β ( (λ + ξ)+ ( XY 1 1 ) δ ) 2 +2XY 1 δ (κ λ) +2η + ξ 2 δ (λ + ξ) 2) dt 90
Idea of the proof (continued) and, in turn, du (X) =U (X) ( Y 1 B 1 σπ + XY 1 δ + ξ ) dw + 1 2 U (X) ( Y 1 B 1 σπ σσ + ( (λ + ξ)+ ( XY 1 1 ) δ ) 2 +2XY 1 δ (κ λ) + ( I σσ + ) (λ + ξ) 2 +2η + ξ 2 δ (λ + ξ) 2 ) dt... and, finally, du (X) =U (X) ( Y 1 B 1 σπ + XY 1 δ + ξ ) dw 1 2 U (X) Y 1 B 1 σπ σσ + ( λ + ( XY 1 1 ) δ + ξ ) 2 dt 91