APÉNDICES
J. Demostración
III. Propósito: Comprender la esencia e importancia de los enfoques transversales
debt at time T, EX*(T), instead of using the parameters γ or µ. Th is pro-vides a more clever interpretation of the results. Th e substitution of γ may be done f rom equality (5.18). Taking i nto account (5.15) a nd (5.18), t he investment at instant t is
1 ( )d
where z = EX*(T). Th is shows the existing relation between the desired expected levels of debt at time T and the optimal composition of the port-folio at every instant of time t.
Analogously, (5.14) a nd (5.18) a llow us to rewrite t he optimal rate of contribution at instant t as
⎛ −∫ ⎞
By (5.27), the optimal fund satisfi es
T ( )d
We have a nalyzed t he ma nagement of a pens ion-funding process of a n aggregated D B pens ion p lan wh ere t he ben efi ts a re st ochastic a nd t he riskless market interest rate is nonconstant. Th e objective is to determine the contribution rate and investment strategies, maximizing the expected terminal fund, and at the same time minimizing both the contribution and t he solvency r isk. Th e problem is formulated as a m odifi ed mean–
variance optimization problem and has been solved by means of dynamic programming techniques.
We fi nd that there is a linear relationship between the optimal supplemen-tary cost a nd the vector of effi cient investment strategies, with a correction term due to the random behavior of benefi ts. Th e mean–variance effi cient frontier is of a quadratic type, that is to say, the terminal solvency risk has a parabolic dependence on the expected terminal unfunded actuarial liability.
APPENDIX A
Proof of Proposition 5.1: By Assumption 5.1, the conditional expec-tation is
thus, recalling the defi nition of AL and ψAL, we get
Analogously, NC(t) = ψNC(t)P(t).
Now, by means of an integration by parts, and the defi nition of ξAL, we have
which is (5.1). Finally, we deduce the stochastic diff erential equation that the actuarial liability satisfi es. Notice that dψAL(t) = ξAL(t)dt. Th us, using Assumption 5.1
= ψ
Proof of Proposition 5.2: See proof of Proposition 5.2 in Josa-Fombellida and Rincón-Zapatero (2008b).
Proof of Th eo rem 5.1: In order to prove this result, we use the dynamic programming approach, see Fleming and Soner (1993). Consider the value function of the control problem (5.2), (5.10), (5.13),
( Λ ∈)
It is well known that Vˆ is the solution of the Hamilton–Jacobi–Bellman equation:
Note that in (5.20) we have used (5.10) and the stochastic diff erential equa-tion of AL as a funcequa-tion of the Brownian moequa-tions { }wi in=0, obtained from (5.2), that is
= κ + ξ ψ
If there exists a smooth solution V of this equation, strictly convex with respect to X, t hen t he minimizer va lues of t he supplementary cost a nd investments are given by
Aft er substitution of these values in (5.20), we get Vˆ to satisfy
+ − − θ θ + κ + ξ ψ + η
with the fi nal condition (5.21). We try a quadratic solution of the form
= β + β + β + β
so that, from (5.22), the optimal controls must be
− ⎛ −β β ⎞ ⎛ β ⎞
Th e following ordinary diff erential equations are obtained for the above coeffi cients appearing in previous identities:
β = − + θ θ β + β βX
(
r T)
X X XX, βX( )T = − γ2 , (5.24)β = − + θ θ β + βXX
(
2r T)
XX 2XX, βXX( ) 1,T = (5 .25)βX,AL= − − κ + ξ ψ + η+ θ θ β
(
r AL/ AL T)
X,AL+ β βXX X,AL, βX,AL( ) 0.T = (5.26) We a ssume t hat E quation 5.25, of R icatti t ype, ha s a u nique solution βXX(t) = f(t). From Arnold (1974 p. 139), we obtain the solution to (5.24):−∫
β ( )= − γ2 e ( )d ( ).
T t
r s s
Xt f t
Substituting in (5.26) is given by
βX,AL= − − κ + ξ ψ + η+ θ θ + β
(
r AL/ AL T f)
X,AL, βX,AL( ) 0,T =that is to say, βX,AL = 0. Inserting these expressions into (5.23), we obtain (5.14) and (5.15), respectively.
Proof of Th eo rem 5.2: Under the optimal feedback control (5.14), (5.15), the stochastic diff erential equation for process X, (5.10), is
−
−
⎛ ∫ ⎞
⎜ ⎟
=⎜⎜⎝ − θ θ − + θ θ + γ ⎟⎟⎠
⎛ ∫ ⎞
⎜ ⎟
−η − + θ ⎜⎜⎝γ − ⎟⎟⎠
T T ( )d
T T ( )d
0
d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e d
( ) 1 AL( )d ( ) ( ) e ( ) d ( ),
( ) ( )
T t
T t
r s s
r s s
X t r t t t f t X t t t f t t
t q q t w t t X t w t
(5.27)
with X(0) = X0. Applying Ito’s formula to X2 we obtain
with X2(0)=X . Taking expectations on both previous stochastic diff er-02 ential equations, we get functions m1(t) = EX(t) and m2(t) = EX2 (t) to satisfy the linear ordinary diff erential equations
−∫
Following Arnold (1974, p. 139)
T
which, aft er some calculations, and by (5.16), is
−∫ −∫
ACKNOWLEDGMENTS
Th e aut hor g ratefully a cknowledges fi nancial su pport f rom t he Reg ional Government of C astilla y L eón (Spain) u nder project VA004B08 a nd t he Spanish Ministerio de Ciencia e Innovación under project ECO2008-02358.
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