mean-variance dynamic optimality for dc pension schemes€¦ · mean-variance dynamic optimality...

Mean-variance dynamic optimality for DC pension schemes

Mean-variance dynamic optimality for DCpension schemes

Francesco Menoncin, Elena Vigna

EAJ 2018, Leuven, 9-11 September 2018

Francesco Menoncin, Elena Vigna (UniBs, UniTo, CCA) EAJ 2018, Leuven, 9-11 September 2018 1 / 34


Outline

1 Motivation

2 The model

3 The mean-variance optimization problem

4 Approaches to mean-variance problem

5 The dynamically optimal strategy in closed form

6 Simulations

7 Conclusions

8 References



Motivation

Motivation

investment risk in DC pension funds is importantone possible criterium to attack investment risk is mean-variancemean variance portfolio selection is a time-inconsistent problemtypically one adopts precommitment or Nash equilibrium approachwe investigate a third approach: dynamically optimal or naiveapproachthis is novel in DC pension schemes, because there is no paper thatadopts the dynamically optimal naive strategywe compare it with the precommitment strategy



The model

General model (s state variables, n risky assets)The financial market is described by the following variables:

s state variables z (t) (with z (0) = z0 ∈ Rs known) whose values solve thestochastic differential equation (SDE)

dz (t)s×1

= µz (t, z)s×1

dt + Ω (t, z)s×n

dW (t)n×1

; (1)

one riskless asset whose price G (t) solves the (ordinary) differential equation

dG(t) = G(t)r (t, z) dt,

where r (t, z) is the spot instantaneously riskless interest rate;

n risky assets whose prices P (t) (with P (0) = P0 ∈ Rn known) solve thematrix stochastic differential equation

dP (t)n×1

= IPn×n

[µ (t, z)

n×1dt + Σ (t, z)

n×ndW (t)

n×1

], (2)

where IP is the n× n square diagonal matrix that reports the pricesP1,P2, ...,Pn on the diagonal and zero elsewhere.



The model

General model, wealth dynamics

The wealth dynamics are given by the following stochastic differentialequation (SDE)

dX (t) =(X (t) r (t, z) + c (t, z) + w (t)′ (µ (t, z)− r (t, z) 1)

)dt

+ w (t)′Σ (t, z) dW (t) . (3)



The model

Particular case (stochastic salary and interest rate)

Here we assume s = n = 2. There are two state variables: (i) the interest rater (t) which follows a Vasicek process, and (ii) the contribution c (t) whichfollows a GBM. Thus, the vector z (t) = [r(t) c(t)]′ can be represented inthe following way:

[dr (t)dc (t)

]︸︷︷︸

dz(t)

=

[a (b− r (t))

c (t)µc

]︸︷︷︸

µz(t,z)

dt +

[σr 0

c (t)σcr c (t)σcs

]︸︷︷︸

Ω(t,z)

[dWr (t)dWs (t)

]︸︷︷︸

dW(t)

.



The model

Particular case (stochastic salary and interest rate)

On the financial market we have two risky assets: (i) a rolling zero-couponbond BK (t) with maturity K, and (ii) a stock whose price S (t) follows aBGM. The two prices can be represented in the following way[

dBK(t)BK(t)dS(t)S(t)

]︸︷︷︸

I−1P dP(t)

=

[r (t)− g (0,K)σrξr

r (t) + ξrσsr + ξsσs

]︸︷︷︸

µ(t,z)

dt+[−g (0,K)σr 0

σsr σs

]︸︷︷︸

Σ(t,z)

[dWr (t)dWs (t)

]︸︷︷︸

dW(t)

.

The fund follows the SDE (the dependence on z has been omitted):

dX (t) =(X (t) r (t) + c (t) + w (t)′ (µ (t)− r (t) 1)

)dt + w (t)′ Σ (t) dW (t) .

(4)



The model

Black and Scholes case (constant salary and interest rate)

Here we assume s = 0 and n = 1, no state variables, interest rate andcontribution constant and positive: r ≥ 0, c ≥ 0. The Black and Scholesmarket consists of two assets, riskless with price G(t):

dG(t) = rG(t)dt,

and risky asset, whose price S(t) follows a GBM:

dS(t) = µS(t)dt + σS(t)dW(t).

The amount invested in the risky asset at time t is w(t) ∈ R, and the fundgrows according to

dX(t) = (X(t)r + w(t)(µ− r) + c) dt + w(t)σdW(t). (5)



The mean-variance optimization problem


Pension fund member has a wealth x0 at initial time t0 and wants tomaximize expected final wealth at retirement time T , while minimizing itsvariance:

[PMV

t0,x0

]supw∈U

JMV(t0, x0,w) = supw∈UEt0,x0 [X

w(T)]− αVt0,x0 [Xw(T)] ,

(6)where Xw(t) is fund at time t when the control strategy w is appliedEt0,x0(Z) is expectation at time t0 with wealth x0 of ZVt0,x0(Z) is variance at time t0 with wealth x0 of Zα > 0 measures risk aversion




About time inconsistency

The mean-variance problem is time-inconsistent!

Time inconsistency (Strotz (1956)) arises in an intertemporal optimizationproblem when the optimal strategy selected at t is no longer optimal at s > t.

A strategy is time-inconsistent when the individual at future time s > t istempted to deviate from the strategy decided at time t.

What are the sources of time inconsistency?

We use setup introduced by Bjork, Khapko and Murgoci (2017).




Time inconsistent problems

LetdXs = µ(s,Xs, us)ds + σ(s,Xs, us)dWs Xt0 = x0

be the evolution of some state variable and uss∈[t0,T] be a control actionthat the controller can choose at any time s.Optimizer wants to solve problem (Bjork, Khapko and Murgoci, 2017):

supu∈U

Et0,x0

[∫ T

t0U1(s,Xs, us)ds + U2(XT)

]+ U3 [Et0,x0(XT)]

(7)

where U1(·), U2(·) and U3(·) are utility functions.

If U3(·) is NON-LINEAR, then the problem is TIME-INCONSISTENT.




Why mean variance is time-inconsistent

In mean-variance portfolio selection problem (6), U3(·) is non-linear,because of the variance term.

Indeed, in (6) we have

U1(x) = 0 U2(x) = x− αx2 U3(x) = αx2

Therefore, the mean-variance optimization problem is time-inconsistent.




Three approaches

There are 3 approaches to attack a time inconsistent problem (7):1. precommitment⇒ time inconsistent2. dynamic optimality naive⇒ time consistent3. consistent planning⇒ time consistent.

In this paper we focus only on the first and the second.



Approaches to mean-variance problem

Precommitment approach

To solve the mean-variance problem[PMV

t0,x0

]with the precommitment

approach, one fixes the initial point (t0, x0) and finds the control plan w thatmaximizes only JMV(t0, x0,w), i.e., the precommitment strategy:

Definition

Given the mean-variance problem (6), the strategy w that maximizesJMV(t0, x0,w), i.e., the control map

wt0,x0 : [t0,T]× R→ Rn (8)

such thatJMV(t0, x0, wt0,x0) = sup

w∈UJMV(t0, x0,w)

if it exists, is called the precommitment strategy.




Dynamically optimal naive approach

Dynamically optimal naive strategy for the mean-variance problem[PMV

t0,x0

]in 3 steps:Step 1. Assume that for the initial time-wealth point (t0, x0) theprecommitment strategy maximizing JMV(t0, x0,w) is given by:

wt0,x0 : [t0,T]× R→ Rn. (9)

Step 2. Define the new control map

w(t, x) = wt,x(t, x), for (t, x) ∈ [t0,T]× R, (10)

where the right hand side of (10) is obtained by replacing t0 with t and x0with x in the function (9).Step 3. The strategy w : [t0,T]× R→ Rn is called the dynamically optimalnaive strategy.




Intuition behind dynamically optimal naive approach

Strict link between dynamically optimal naive and precommitment:At time t ∈ [t0,T] with wealth x dyn opt naive investor faces problemPMV

t,x and solves it with precommitment approach at time t, as if hisinitial time-wealth point was (t, x).In fact, he plays the first control of the precommitment strategy forPMV

t,x , because he plays ut,x(t, x).Then, the dynamically optimal investor can be seen as the continuousreincarnation of the precommitted investor!This investor is called dynamically optimal by Pedersen and Peskir(2017) and is called naive by Pollak (1968).




Link with target-based approach

Interesting link between both strategies (precommitment and dyn optnaive) and so-called target-based approach:

infw

J(t, x) = infw

Et,x

[(Xw(T)− γ(t, x))

2]

where at time t with wealth x the investor want to hit the target γ(t, x) atfinal time T

precommitment⇒ constant target from t0 to T: γ(t, x) = γ(t0, x0)

dyn opt naive⇒ time-varying stochastic targets: γ(t, x) 6= γ(t0, x0)



The dynamically optimal strategy in closed form

Dynamically optimal strategy for the general model

w (t, x) =1

2αEt[e2Φ(t,T)

]B (t,T)

(Σ′)−1

(ξ + Ω′

∂ ln B (t,T)

∂z (t)− Ω′

∂ lnEt[e2Φ(t,T)

]∂z (t)

)(11)

+

(H (t,T)

∂ ln B (t,T)

∂z (t)− ∂H (t,T)

∂z (t)

)(Σ′)

−1Ω′ for (t, x) ∈ [t0,T]× R.

where H(t,T) is

H (t,T) := X∗ (t) +

∫ T

tEFs

t [c (s, z)] B (t, s) ds.




Dynamically optimal strategy for the particular case

w (t, x) =1

2αEt[e2Φ(t,T)

]B (t,T)

(Σ′)−1

(ξ + Ω′

∂ ln B (t,T)

∂z (t)− Ω′

∂ lnEt[e2Φ(t,T)

]∂z (t)

)

+

(H (t,T)

∂ ln B (t,T)

∂z (t)− ∂H (t,T)

∂z (t)

)(Σ′)

−1Ω′ for (t, x) ∈ [t0,T]× R

=1

2αeEt[Φ(t,T)]+ 3

2 Vt[Φ(t,T)]

[− ξr+g(t,T)σr

g(0,K)σr+ σsr

g(0,K)σrσsξs

1σsξs

].




Dynamically optimal strategy for the Black and Scholescase

w (t, x) =1

2αβ

σe−(r−β2)(T−t) for (t, x) ∈ [t0,T]× R. (12)

whereβ =

µ− rσ

is the Sharpe ratio of the risky asset




General case (naive strategy): Time-varying targets

The time-varying targets for the dynamically optimal naive approach in thegeneral case are

F (t, x, α) =1

2αEt[e2Φ(t,T)

]B (t,T)

2 +x +

∫ Tt EFs

t [c (s, z)] B (t, s) dsB (t,T)

,

in which

Φ (t,T) = −∫ T

tr (u, z) du−1

2

∫ T

tβ (u, z)′ β (u, z) du−

∫ T

tβ (u, z)′ dW (u) ,

B (t,T) = Et

[eΦ(t,T)

].




Particular case (naive strategy): Time-varying targetsIn the particular case

β is constantr (t, z) is Gaussianc (t, z) is a GBM

The first two points imply that Φ (t,T) is Gaussian and so the time-varyingtargets are

F (t, x, α) =1

2αeVt[Φ(t,T)] +

x + ct∫ T

t eµF(s−t)+Et[Φ(t,s)]+ 12 Vt[Φ(t,s)]ds

eEt[Φ(t,T)]+ 12 Vt[Φ(t,T)]

in which

Φ (t,T) = −∫ T

tr (u, z) du− 1

2β2 (T − t)− β′ (W (T)−W (t))




B&S case (naive strategy): Time-varying stochastic targets

The time-varying targets for the dynamically optimal naive approach for theBlack and Scholes case are (constant β, r, c):

γ (t, x, α) =1

2αeβ

2(T−t) + xer(T−t) +cr

(er(T−t) − 1

)= K ×

[xer(T−t) +

cr

(er(T−t) − 1

)](13)

where K > 1. We see that γ(t, x, α)

increases with β, r, c, T , K

decreases with αdecreases with t

increases with x




B&S case (precommitment strategy): Constant target

The constant target for the precommitment approach in the Black andScholes case is

γ(t, x, α) = γ (t0, x0, α) =1

2αeβ

2(T−t0) + x0er(T−t0) +cr

(er(T−t0) − 1

)= K ×

[x0er(T−t0) +

cr

(er(T−t0) − 1

)](14)

where K > 1. We see that γ(t, x, α)

increases with β, r, c, T , K

decreases with αdoes not depend on t and x

depends on t0 and x0: increases with x0 and decreases with t0



Simulations

Simulations assumptionsWe make Monte Carlo simulations in the Black and Scholes case.Assumptions are

x0 = 1t0 = 0T = 20c = 0.1r = 0.03µ = 0.08, σ = 0.15⇒ β = 0.333K = 1.2⇒ α = 5.056weekly discretization, 1000 scenarios

As a consequence the constant target for the precommitment approach is

γ(t, x, α) = γ(0, 1, α) = 1.2× 4.56 = 5.475



Simulations

Precommitment wealth (XtP) vs Naive wealth (XtN)



Simulations




Simulations

Precommitment strategy (ytP) vs Naive strategy (ytN)



Simulations




Simulations

Constant target and time-varying stochastic targets



Conclusions

Conclusions

first comparison between precommitment and dyn opt naive approachesfor mean-variance portf selection in DC pension fundprecommitment is target-based approach with constant target, dyn optnaive is target-based with time-varying targetson average the wealth growth over time is exactly the samestand deviation of wealth lower with precommit than dyn opt naivein worst cases, precommitment wealth worse than dyn opt naiveon average precommitment strategy less risky than dyn opt naiveprecommitment strategy highly more volatile than dyn opt naivein general time-varying targets larger than constant precommitmenttarget



References

References

Bajeux-Besnainou, I., and Portait, R. (1998). Dynamic Asset Allocation in a Mean-Varianceframework, Management Science 44, S79–S95.

Basak, S. and Chabakauri, G. (2010). Dynamic mean-variance asset allocation, Review ofFinancial Studies 23, 2970–3016.

Bjork, T., Khapko, M., and Murgoci, A. (2017). On time-inconsistent stochastic control incontinuous time, Finance and Stochastics, 21, 331–360.

Karnam, C., Ma, J., and Zhang, J. (2016). Dynamic Approaches for Some Time InconsistentProblems, arXiv:1604.03913v1 [math.OC].

Pedersen, J., and Peskir, G. (2016). Optimal mean-variance portfolio selection, Mathematicsand Financial Economics, 11, 137–160.

Richardson, H. R. (1989). A minimum variance result in continuous trading portfoliooptimization, Management Science, 35, 1045–1055.

Strotz, R. H. (1956). Myopia and inconsistency in dynamic utility maximization, The Review ofEconomic Studies 23, 165–180.

Zhou, X.Y., and Li, D. (2000). Continuous time mean variance portfolio selection: a stochasticLQ framework, Applied Mathematics and Optimization, 42, 19–33.


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