A new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009

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1 A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1

2 Performance measurement of investment strategies 2

3 Market environment Riskless and risky securities (Ω, F, P) ; W =(W 1,...,W d ) standard Brownian Motion Traded securities 1 i k dst i = St (μ i tdt + σt i ) dw t, S0 i > 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 an F t -measurable stochastic process λ t R d satisfying μ t r t 1=σ T t λ t No assumptions on market completeness 3

4 Market environment Self-financing investment strategies π 0 t, π t =(π 1 t,...,πi t,...,πk t ) Present value of this allocation X t = k πt i i=0 dx t = k i=1 π i tσ i t (λ t dt + dw t ) = σ t π t (λ t dt + dw t ) 4

5 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) No market input to the choice of terminal utility Backwards in time generation of the indirect utility V (x, s; T )=sup π E P (u T (X π T ) F s; X π s = x) V (x, s; T )=sup π E P (V (X π t,t; T ) F s ; X π s = x) (DPP) V (x, s; T )=E P (V (Xt π,t; T ) F s ; Xs π = x) The value function process becomes the intermediate utility for all t [0,T) 5

6 The value function process V (x, s; T ) F s V (x, t; T ) F t 0 s t u T (x) F 0 T For each self-financing strategy, represented by π, the associated wealth X π t satisfies E P (V (X π t,t; T ) F s ) V (X π s,s; T ), 0 s t T There exists a self-financing strategy, represented by π,forwhichthe associated wealth Xt π satisfies E P (V (Xt π,t; T ) F s )=V(Xs π,s; T ), 0 s t T At expiration, V (x, T ; T )=u T (x) F 0 6

7 Study of the value function process Arbitrary environments Duality methods Martingale representation results Markovian environments HJB equation Feedback optimal controls Weak solutions. 7

8 A stochastic PDE for the value function process 8

9 Intuition Assume that, for t [0,T], the value function V (x, t) solves dv (x, t) =b (x, t) dt + a (x, t) dw t where b, a are F t measurable processes. Recall that for an arbitrary admissible portfolio π, the associated wealth process, X π, solves dx π t = σ t π t (λ t dt + dw t ) Applying the Ito-Ventzell formula to V (X π t,t) yields dv (X π t,t) = b (X π t,t) dt + a (X π t,t) dw t +V x (X π t,t) dx π t V xx (X π t,t) d X π t + a x (X π t,t) d W, X π t = ( b (X π t,t)+v x (X π t,t) σ t π t λ t + σ t π t a x (X π t,t)+ 1 2 V xx (X π t,t) σ t π t 2) dt +(a (X π t,t)+v x (X π t,t) σ t π t ) dw t 9

10 Intuition (continued) By the monotonicity and concavity assumptions, the quantity sup (V x (Xt π,t) σ t π t λ t + σ t π t a x (Xt π,t)+ 1 π 2 V xx (Xt π,t) σ t π t 2) is well defined Calculating the optimum π yields π t = σ + t M ( Xt π,t ) = ( V x X π t,t ) ( λ t + a x X π t,t ) ( V xx X π t,t ) Deduce that the above supremum is given by σ t σ t + The drift coefficient b must satisfy ( ( Vx X π t,t ) ( λ t + a x X π t,t )) 2 ( 2V xx X π t,t ) b ( Xt π,t ) = M ( Xt π,t ) 10

11 SPDE for the value function process Market (σ t,σ + t,λ t); volatility a(x, t) F t dv = 1 σσ + A(Vλ+ a) 2 2 A 2 V V (x, T )=u T (x) F 0 ; dt + a dw A = x Feedback optimal portfolio vector πt = π (Xt π, + A(Vλ+ a),t)= σ A 2 (X π, V t t) Choices for the volatility process a? 11

12 A Markovian example r t = r(y t ), μ t = μ(y t ), σ t = σ(y t ) dy t = θ(y t ) dt +Θ T (Y t ) dw t Value function v(x, y, t; T )=sup π E (u T (X π T ) Xπ t = x, Y t = y) HJB equation ( 1 v t + sup π 2 σπ 2 v xx + σπ σσ + (λv x +Θv xy )+ 1 ) 2 ΘT Θ v yy + θ v y = v t 1 2 σσ + (v x λ +Θv xy ) 2 v xx ΘT Θ v yy + θ v y =0 12

13 The SPDE for the value function process V (x, t) =v(x, Y t,t; T ) dv (x, t) =v t dt + v y dy v yy d Y HJB = ( 1 2 σσ + (v x λ +Θv xy ) 2 v xx 1 2 ΘT Θ v yy θ v y ) dt + v y (θdt+θ T dw ) v yy Θ T Θ dt = 1 2 σσ + (V x λ + a x (x, t)) 2 V xx dt + a(x, t) dw The volatility process is uniquely determined: a(x, t) =Θv y (x, Y t,t; T ) 13

14 Going beyond the deterministic terminal utility problem 14

15 Motivation (partial) Terminal utility might be ω-dependent Liability management, indifference valuation u T (x, ω) = e γ(x C T (ω)) ; C T F T Numeraire consistency u T (x, ω) = e γ T (ω)x Need to extend the value function process beyond T Need to manage liabilities of arbitrary maturities How do we formulate investment performance criteria? 15

16 Investment performance process U(x, t) is an F t -adapted process, t 0 The mapping x U(x, t) is increasing and concave For each self-financing strategy, represented by π, the associated (discounted) wealth X π t satisfies E P (U(X π t,t) F s ) U(X π s,s), 0 s t There exists a self-financing strategy, represented by π,forwhich the associated (discounted) wealth Xt π satisfies E P (U(Xt π,t) F s )=U(Xs π,s), 0 s t 16

17 Optimality across times 0 U(x, s) F s 0 U(x, t) F t U(x, t) F t U(x, s) =sup A E(U(X π t,t) F s,x s = x) What is the meaning of this process? Does such a process aways exist? Is it unique? 17

18 Forward performance process Adatumu 0 (x) is assigned at the beginning of the trading horizon, t =0 U(x, 0) = u 0 (x) Forward in time criteria E P (U(X π t,t) F s) U(X π s,s), E P (U(Xt π,t) F s )=U(Xs π,s), 0 s t 0 s t Many difficulties due to inverse in time nature of the problem 18

19 The forward performance SPDE 19

20 The forward performance SPDE Let U (x, t) be an F t measurable process such that the mapping x U (x, t) is increasing and concave. Let also U = U (x, t) be the solution of the stochastic partial differential equation du = 1 2 σσ + A (Uλ+ a) 2 A 2 U dt + a dw where a = a (x, t) is an F t adapted process, while A = x. Then U (x, t) is a forward performance process. The process a may depend on t, x, U, its spatial derivatives etc. 20

21 Optimal portfolios and wealth At the optimum The optimal portfolio vector π is given in the feedback form πt = π (Xt,t)= σ +A (Uλ+ a) A 2 U (X t,t) The optimal wealth process X solves dxt = σσ +A (Uλ+ a) A 2 U (X t,t)(λdt + dw t ) 21

22 Solutions to the forward performance SPDE du = 1 2 σσ + A (Uλ+ a) 2 A 2 U dt + a dw Local differential coefficients a (x, t) =F (x, t, U (x, t),u x (x, t)) Difficulties The equation is fully nonlinear The diffusion coefficient depends, in general, on U x and U xx The equation is not (degenerate) elliptic 22

23 The zero volatility case: a(x, t) 0 23

24 Space-time monotone performance process The forward performance SPDE simplifies to The process du = 1 2 σσ + A (Uλ) A 2 U U (x, t) =u (x, A t ) with A t = 2 dt t 0 σ s σ + s λ s 2 ds with u : R [0, + ) R, increasing and concave with respect to x, and solving is a solution. MZ (2006) Berrier, Rogers and Tehranchi (2007) u t u xx = 1 2 u2 x 24

25 Performance measurement time t 1,informationF t1 risk premium A t1 = t1 0 λ 2 ds u(x,t 1 ) Time Wealth A t1 + u(x, t 1 ) U(x, t 1 )=u(x, A t1 ) F t1 25

26 Performance measurement time t 2,informationF t2 risk premium A t2 = t2 0 λ 2 ds u(x,t 2 ) Time Wealth A t2 + u(x, t 2 ) U(x, t 2 )=u(x, A t2 ) F t2 26

27 Performance measurement time t 3,informationF t3 A t3 = risk premium t3 0 λ 2 ds u(x,t 3 ) Time Wealth A t3 + u(x, t 3 ) U(x, t 3 )=u(x, A t3 ) F t3 27

28 Forward performance measurement time t, informationf t u(x,t) market Time Wealth MI(t) + u(x, t) U(x, t) =u(x, A t ) F t 28

29 Properties of the performance process U (x, t) =u (x, A t ) the deterministic risk preferences u (x, t) are compiled with the stochastic market input A t = t 0 λ 2 ds the evolution of preferences is deterministic the dynamic risk preferences u(x, t) reflect the risk tolerance and the impatience of the investor 29

30 Optimal allocations 30

31 Optimal allocations Let X t be the optimal wealth, and A t the time-rescaling processes dx t = σ tπ t (λ tdt + dw t ) da t = λ t 2 dt Define R t r(x t,a t ) r(x, t) = u x(x, t) u xx (x, t) Optimal portfolios π t = σ + t λ tr t 31

32 A system of SDEs at the optimum dx t = r(x t,a t )λ t (λ t dt + dw t ) dr t = r x (X t,a t )dx t π t = σ + t λ tr t The optimal wealth and portfolios are explicitly constructed if the function r(x, t) is known 32

33 Concave utility inputs and increasing harmonic functions 33

34 Concave utility inputs and increasing harmonic functions There is a one-to-one correspondence between strictly concave solutions u(x, t) to u t = 1 2 u 2 x u xx and strictly increasing solutions to h t h xx =0 34

35 Concave utility inputs and increasing harmonic functions Increasing harmonic function h : R [0, + ) R is represented as h (x, t) = R e yx 1 2 y2t 1 ν (dy) y The associated utility input u : R [0, + ) R is then given by the concave function u (x, t) = 1 2 t e h( 1) (x,s)+ s ( ) x 2h x h ( 1) (x, s),s ds e h( 1) (z,0) dz The support of the measure ν plays a key role in the form of the range of h and, as a result, in the form of the domain and range of u as well as in its asymptotic behavior (Inada conditions) 35

36 Examples 36

37 Measure ν has compact support ν(dy) =δ 0,whereδ 0 is a Dirac measure at 0 Then, and h (x, t) = R e yx 1 2 y2t 1 δ y 0 = x u (x, t) = 1 2 t 0 e x+s 2ds + x 0 e z dz =1 e x+ t 2 37

38 Measure ν has compact support ν (dy) = b 2 (δ a + δ a ), a,b > 0 Then, δ ±a is a Dirac measure at ±a If, a =1,then u (x, t) = 1 2 If a 1,then ( ln u(x, t) = ( a) 1+ 1 a a 1 ( x + e 1 a 2 t h (x, t) = b a e 1 2 a2t sinh (ax) x 2 + b 2 e t) et b 2x ( x x 2 + b 2 e t) t ) 2 β a e at +(1+ ( ax+ ) a )x ax 2 + βe at ( ax+ ax 2 + βe at ) 1+ 1 a 38

39 Measure ν has infinite support ν(dy) = 1 2π e 1 2 y2 dy Then h(x, t) =F ( ) x t +1 F (x) = x 0 ez2 2 dz and u(x, t) =F ( F ( 1) (x) t +1 ) 39

40 Optimal processes and increasing harmonic functions 40

41 Optimal processes and risk tolerance dx t = r(x t,a t)λ t (λ t dt + dw t ) dr t = r x(x t,a t ) dx t Local risk tolerance function and fast diffusion equation r t r2 r xx =0 r(x, t) = u x(x, t) u xx (x, t) 41

42 Local risk tolerance and increasing harmonic functions If h : R [0, + ) R is an increasing harmonic function then r : R [0, + ) R + given by r (x, t) =h x ( h ( 1) (x, t),t ) = R eyh( 1) (x,t) 1 2 y2t ν (dy) is a risk tolerance function solving the FDE 42

43 Optimal portfolio and optimal wealth Let h be an increasing solution of the backward heat equation h t h xx =0 and h ( 1) stands for its spatial inverse Let the market input processes A and M by defined by A t = t 0 λ 2 ds and M t = t 0 λ dw Then the optimal wealth and optimal portfolio processes are given by and X,x t = h ( h ( 1) (x, 0) + A t + M t,a t ) πt ( = h x h ( 1) ( Xt,x ) ),A t,at σ + t λ t 43

44 Complete construction Utility inputs and harmonic functions u t = 1 u 2 x h t u xx 2 h xx =0 Harmonic functions and positive Borel measures h(x, t) ν(dy) Optimal wealth process X,x = h ( h ( 1) (x, 0) + A + M,A ) M = λ dw s, M = A 0 Optimal portfolio process t π,x = h x ( h ( 1) (X,x,A),A ) σ + λ The measure ν emerges as the defining element ν h u How do we choose ν and what does it represent for the investor s risk attitude? 44

45 Inferring investor s preferences 45

46 Calibration of risk preferences to the market Given the desired distributional properties of his/her optimal wealth in a specific market environment, what can we say about the investor s risk preferences? Desired future expected wealth Investor s investment targets Desired distribution References Sharpe (2006) Sharpe-Golstein (2005) 46

47 Distributional properties of the optimal wealth process The case of deterministic market price of risk Using the explicit representation of X,x we can compute the distribution, density, quantile and moments of the optimal wealth process. P ( X,x t y ) = N h( 1) (y, A t ) h ( 1) (x, 0) A t At f X,x t (y) =n h( 1) (y, A t ) h ( 1) (x, 0) A t At 1 r (y, A t ) y p = h ( h ( 1) (x, 0) + A t + A t N ( 1) (p),a t ) EX,x t = h ( h ( 1) (x, 0) + A t, 0 ) 47

48 Target: The mapping x E ( Xt,x ) is linear, for all x>0. Then, there exists a positive constant γ>0 such that the investor s forward performance process is given by and by Moreover, U (x, t) = γ γ 1 xγ 1 γ e 1 2 (γ 1)A t, if γ 1 U t (x) =lnx 1 2 A t, if γ =1 E ( Xt,x ) = xe γa t Calibrating the investor s preferences consists of choosing a time horizon, T, and the level of the mean, mx (m >1).Then, the corresponding γ must solve xe γa T = mx and, thus, is given by γ = ln m A T The investor can calibrate his expected wealth only for a single time horizon. 48

49 Relaxing the linearity assumption The linearity of the mapping x E ( Xt,x ) is a very strong assumption. It only allows for calibration of a single parameter, namely, the slope, and only at a single time horizon. Therefore, if one intends to calibrate the investor s preferences to more refined information, then one needs to accept a more complicated dependence of E ( Xt,x ) on x. Target: Fix x 0 and consider calibration to E ( X,x ) 0 t, for t 0 The investor then chooses an increasing function m (t) (with m (t) > 1) to represent E ( X,x ) 0 t, E ( X,x ) 0 t = m (t), for t 0. What does it say about his preferences? Moreover, can he choose an arbitrary increasing function m (t)? 49

50 Relaxing the linearity assumption For simplicity, assume x 0 = 1 and that ν is a probability measure. h ( 1) (1, 0) = 0 and we deduce that Then, E ( Xt,1 ) = h (At, 0) = 0 e ya t ν (dy) Clearly, the investor may only specify the function m (t),t>0, which can be represented, for some probability measure ν in the form m (t) = 0 e ya t ν (dy) 50

51 Conclusions Space-time monotone investment performance criteria Explicit construction of forward performance process Connection with space-time harmonic functions Explicit construction of the optimal wealth and optimal portfolio processes The trace measure as the defining element of the entire construction Calibration of the trace to the market Inference of dynamic risk preferences 51