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Controlling of Pension Fund Investment by Using Bellman's Optimality Principle
IFAC
Copyright © IFAC Control Systems Design, Bratislava, Slovak Republic, 2003
\ J Publications www.elsevier.comlIocate/ifac
CONTROLLING OF PENSION FUND INVESTMENT BY USING BELLMAN'S OPTIMALITY PRINCIPLE Miroslav Simandl * Marek Lesek **
* [email protected], University of West Bohemia, Pilsen
** [email protected]. University of West Bohemia,
Pilsen
Abstract: This paper analyzes the financial risk in a contribution defined pension fund in the Czech republic. The Bellman's optimality principle is used to derive the best allocation of a pension fund asset in the two-asset world. The principal results concern the suitability of the optimal pension fund strategy and the large variability of the level of achievement in pension fund asset in the case of variable rates of assets return. Copyright © 20031FAC Keywords: Pension fund, Mathematical modeling, Bellman's optimal principle
1. INTRODUCTION
fund. The optimal allocation decision in two-asset world was derived. The net replacement ratio was used as level of reference asset. The asset of contribution defined pension fund is valorized by exponential function.
The present work analyzes the financial risk in contribution defined pension fund. The financial risks of contribution defined pension fund are analyzed for investing in a two-level system of distributing investment. In this work, the best way of investing between low and high-risk investment is suggested. For computing the optimal investment strategy for multiplicative rates of return dynamical programming is used.
This paper is focused on deriving optimal strategy for allocation decision in a two-asset world for multiplication valorized contribution defined pension fund in Czech republic.
In the work (Simandl, 1998), the Mathematical Model of Pension Fund in the Czech Republic was firstly established. This basic model was extended and adapted by other author for a better description of reality. We will use the modification which was described in work (Simandl and Lesek, 2002). The main change, that was done, allows the model to invest assets to two groups of capital - high risk (stocks etc.) and low risk (bonds etc.).
The use of dynamic programming is not new in the pension fund area. (Sung and Haberman, 1994) use Bellman's optimality principle to • minimize simultaneously the contribution risk and the solvency risk in a defined benefit pension scheme and derive the optimal contribution rate. (Cairns, 1997) in a continuous time framework and (Owadally, 1998) a discrete time framework apply this principle to a defined benefit pension scheme in order to derive the optimal contribution rate and the optimal allocation decision in a twoasset world.
The paper has the following structure. Section 2 is denoted to describe a mathematical model of a contribution defined pension fund which was designed in work (Simandl and Lesek, 2002). The contents of the section 3 is focused on a problem of dynamic programming. In the first part it is
(Vigna and Haberman, 2001) use Bellman's optimality principle in contribution defined pension
419
defined problem of dynamic programming for the best allocation of pension fund asset in two-asset world and in the second part the Bellman's optimality principle is used for solving the problem of dynamical programming. In section 4 it is given a illustration example of applying the Bellman's optimal principle to the best allocation of pension fund asset and the brief summary is done in section 5.
is number of the clients which have contributed to the other pension fund and save some money. The last equation, (5), describes a dynamical evolution of the size of the pension fund assets, where up(t) is a size of the amount which clients bring to the pension fund from other pension fund in which they have saved. We assume that annual rates of returns v(t) (for low-risk investment) and A(t) (for high-risk investment) from asset (IS (t)) are normally distributed random variables.
2. THE MATHEMATICAL MODEL OF PENSION FUND IN THE CZECH REPUBLIC
v(t) ~ N(v, wi)
(Simandl and Lesek, 2002) describe the design of pension fund mathematical model in two-asset world in state-space form. The state x(t) is described by set of stochastic non-linear equations.
a
A(t) ~ NeX, w~)
where
wi
~ A, ~ w~ and it is also assumed that the random variables (v(t) and A(t)) and pension fund asset (xs(t)) are independent
v
XI(t + 1) = h(XI(t) + WI(t))
(1)
X2(t + 1) = X2(t) + W2(t)
(2)
+ 1) = X3(t) + W3(t)
(3)
cov(xs(t), v(g)) = 0 V 9
~
t
(4)
cov(xs(t), A(g)) = 0 V 9
~
t
X3(t
X4(t + 1) =
Xl
(t)X4(t) + Uk(t)
COV(V(t),A(g)) = 0 V t,g
xs(t + 1) = [(1 - s(t))v(t) + S(t)A(t)].
(7)
The fifth equation of mathematical model contains the investment strategy and the Bellman's optimal principle will be applied to it. The states X2(t), X3(t) and X4(t) are measurable then it is possible to arrange the equation for fifth state (xs(t)) to following form:
,[(X3(t) - X2(t))X4(t) + + xs(t) + up(t)]
(6)
(5)
In following part, equations of pension fund mathematical model will be described. Equation (1) describes an evolution of the portion of clients who stay in the pension fund at time t + 1 (except new incoming) to number of clients at time t, where h (... ) represents saturation function that holds value of expression Xl (t)+w(t) in define interval (0,1) and is valid for all t, that h(XI(t) + WI(t)) = 0, if XI(t) + UIJ(t) < 0 and at the same time fl(XI(t) +WI(t)) = 1, ifxI(t) + WI(t) > 1 and WI(t) is a random variable with mean III and variance ai.
Xs(t + 1) = [xs(t) + c(t)]. .[(1 - s(t))v(t) + S(t)A(t)]
(8)
where
The output equation is designed in virtue of fact, that a part of the states (X2(t), X3(t), X4(t) and Xs (t)) is directly measurable. Hence, the output equation is in linear form.
Equation (2) describes a development of an average value of benefit pay-off per client in the pension fund (this amount already include money that clients carry out to other pension funds if they are transferring), where W2(t) is a random variable with mean 1L2 and variance a~.
y(t)
= C.x(t)
(9)
where C will be in form:
c=
The next equation, (3), represents a evolution of an average value of contribution per client including a contribution which the State supports the supplementary pension secure, where W3(t) is a random variable with mean 1L3 and variance a~.
0100001 001000 [ 000 1 0 0 o0 0 0 1 0
(10)
and x( t) is the state vector.
Equation (4) represents a dynamical development of number of the pension fund clients, where Uk (t) = Uk, + Uk2 is number of clients that join the pension fund at time t and Uk, is number of the clients which have not already contributed and Uk 2
3. PROBLEM OF DYNAMIC PROGRAMMING We start with the documents (Sung and Haberman, 1994) and (Owadally, 1998), where the
420
"costs" function was defined for benefit defined pension fund, and (Vigna and Haberman, 2001) where is defined "costs" function for contribution defined pension fund in form
The objective of mathematical programming is to find investment strategy 7rtl which would get minimal average value of future cost G(t).
(ll)
3.1 Bellman's optimal principle
where 8 j > 0 is weight coefficient and X5(t) is expected asset of a pension fund, which the fund should own at time t. The amount of expected asset will be counted on voted rate of return, which is defined as arithmetical average of chosen rates of return for investment to high-risk (~) and low-risk (v) asset and it is defined by equation 1
Bellman's optimal principle, which is given in (Bellman and Kabala, 1965), has for above noticed problem the Bellman's equation in form
J(I(t)) =
-
n;~nE
[t,
,C(S)II(t)] =
X5(t) = (X5(t - 1) + c(t - 1))"2(17 + A).
= min [C(t) + ,E[J(I(t + 1)II(t))]]
The set of expected assets X5 (t) is known in advance and it is fixed at each discrete time moment {X5(t)}t=I,2, ... ,N and X5(0) = X5(0)
We notice that {v(t)} and {A(t)} are independent, X5(t) is Marcovian process and it is valid for conditional density of a probability that
P[X5(t + 1)II(t)]
The border condition from "costs" function at time N is
C(N) = 80 (X5(N) - x5(N))2
(15)
8(t)
=
P[X5(t + 1)ly(t)]
and also
(12) P[X5(t + 1), X5(t
For pension fund managers, it is more important to reach final target than particular targets. This principle is applied in weight coefficients 8 1 and 80 , where condition for weight coefficients is 81 ::; 80 , Summary of future cost at time t is defined by equation, which count all future values of the criteria functions multiplied by a discount factor b):
+ 2), ... , x5(N)II(t)]
=
= P[X5(t + 1),x5(t + 2), ... ,x5(N)ly(t)] It follows from the previous equation that conditional densities of probability are in form
p(G(t)II(t)) = p(G(t)ly(t))
(16)
and
N
G(t) =
L ,b-tC(b)
(13)
J(X5(t), t)
b=t
, y(t),
J(I( t)) = min E[G( t) II( t)],
(14)
+ S(t)2(~2 + w~) +
t = 0, 1, ... ,N
+ 2s(t)(1 - s(t))V~l
where 7rt represents the set of future investment strategies
=
(17)
+ 1)ly(t)] = (X5(t) + c(t)) (18) [(1 - s(t))v + s(t)~l E[x~(t + 1)ly(t)] = (X5(t) + C(t))2 (19) [(1 - S(t))2p;2 + w~) +
7rt
=
+ ,E[J(X5(t + 1), t + 1)1
E[X5(t
, s(t - I)}
Criteria function at time t can be defined as
7rt
8(t)
On the basis of the asset equation (8) and the fact, that {v(t)}, P(t)} and X5(t) are independent (7), it is possible to derive the conditioned mean E[X5(t + 1)ly(t)], the conditioned mean of second power E[x~(t+l)ly(t)] and the conditioned variation var[x5(t + 1)ly(t)]. It easy to verify that
For solving the problem of optimal investment distribution, it is necessary to define set I(t) as set of all available information at time t:
s(O), s(I),
min[C(t)
Iy(t))]], t=O,I, ... ,tN-l
where , is the inter-temporal discount factor, which can be seen as a "psychological" discount rate or as a risk discount rate (Cairns, 1997).
I(t) = {y(O), y(I),
=
var[x5(t + 1)ly(t)] = (X5(t) + c(t)f [(1 - S(t))2w~
{{S(tb)}tb=t,t+l....,N-j : 0::; S(tb) ::; I} = {{s(t), s(t + 1), ... , s(t(N - I))} :
(20)
+ s2(t)w~1
Without loss of generality it is possible to assume that 80 is multiple of 8 j • If it is assumed that Bj = 1 then it is possible to define Bo = B, where
0::; S(tb) ::; I}
421
() ?: 1. The dynamic programming problem is
M(t) and N(t) and unknown parameter is value of optimal distribution of invested asset s(t)
transformed to:
J(X5(t), t) = min[(x5(t) - X5(t))2 +
E[J(X5(t + 1), t + 1)ly(t)] = L(t)S2(t) +
(21)
s(t)
+M(t)s(t) + N(t) = 'ljJ(s(t)) (27)
+ ,E[J(X5(t + 1), t + 1)ly(t)]] =
with boundary condition t = N: where
L(t) = P(t + l)(x5(t) + C(t))2
where x5(N) is expected amount of pension fund asset from time t = 0 when the cost c( t) and the amount of profit from high-risk investment (with mean "X) and amount of profit from lowrisk investment (with mean 17) to time t = N. We notice, that the realized pension fund target (x5(N)), where half of asset is invested to highrisk capital and second half is invest to low-risk capital, is in form:
(172 + W~ + ).2 + W~
+
L
("X)k c(N - k) 17;
217"X)
(28)
M(t) = 2P(t + 1)(x5(t) + c(t))2(17"X _17 N(t)
=
2
-
wi) -
- 2Q(t + 1)(x5(t) + c(t))("X -17) P(t + 1)(x5(t) + c(t))2(17 2 + wi) -
(29)
- 2Q(t + 1)(X5(t) + c(t))17 + R(t + lX30) From equation (27) it is possible to retrieve unique global extreme 'ljJ*(t) and it is assumed that it is valid L(t) > 0 for Vt, then the extreme is unique global minimum
17 +"X) N x5(N) = (X5(0) + c(O)) ( -2+ N-l
-
'ljJ(s*(t)) = Z*(t) (23)
and
k=l
3.2 Solution of the dynamic programming problem We assume that it is possible to verify a hypothesis. This hypothesis expects to find and solve the minimal of equation (21) as LQ problem. We expect the solution in form:
2 Z*(t) = N(t) _ M (t) 4L(t)
(32)
+ Q'(t)X5(t) + R'(t)
Z*(t) = P'(t)x~(t) where
P'(t)
Q(N) = ()x5(N)
=
HP(t + 1)
(34)
Q'(t) = 2Hc(t)P(t + 1) + 21\Q(t + 1) (35) 2 R'(t) = Hc (t)P(t + 1) + 2h'c(t)Q(t + 1) _
(25)
Q(t + 1 ("X _ 17)2 DP(t + 1)
(36)
where
R(N) = ()x~(N) Assuming that the hypothesis is satisfied for t + 1. We rewrite the equation (24) to this time moment.
= 1/2 + w~ + "X + w~ - 21/"X H = 1/2 + w~ - ~ (I/"X - 1/2 - w~) 2
D
D
J(X5(t + 1), t + 1) = P(t + l)x~(t + 1) - 2Q(t + l)x5(t
+ R(t + 1)
(33)
(24)
For boundary condition (22) at time N the values of coefficient (P( t), Q( t) a R( t)) are acceptable for solving expectations (24)
P(N) = ()
(31)
Substitution (28), (29) and (30) to (32) we get
J(X5(t), t) = P(t)x~(t) - 2Q(t)X5(t) +
+ R(t)
*( ) __ M(t) s t 2L(t)
K
+ 1) + (26)
= -1 (-' lI/I D
-2 II
-
(' W 2) 1 /I
-
-) II
-
-II
(37)
(38) (39)
Equation (17) becomes now
It has to be shown that the hypothesis is satisfied for time t.
J(X5(t), t)
Using (19), (20) and (26) it is not difficult to verify that conditioned mean at time t + 1 is equal to second degree polynomial with coefficients L(t),
= min[C(t) + s(t)
+ ,E[J(X5(t + 1), t + 1)ly(t)]] =
= min[(x5(t) - X5(t))2 + s(t)
422
+ ,E[J(X5(t + 1), t + 1)ly(t)]] = = (X5(t) - X5(t))2 + ,Z*(t) =
For simplification, we made assumption, that client will not transfer any finances from the other pension funds, so it is assumed that X5(0) = 0 and up(t) = 0 for '
= P(t)x;(t) - 2Q(t)X5(t) + R(t) (40) where
P(t) = 1 + ,P'(t)
(41)
Q(t) = X5(t) - 0.5,Q'(t)
(42)
R(t) = x;(t)
+ ,R'(t)
The amount of contributions (costs=O) is constant and it is equal to c = 420 Kc, where the client contributes 300 Kc and amount 120 Kc is paid as state contribution for supporting supplementary pension insurance.
(43)
The rates of return assume the following values. The mean for low-risk investment (v) is 4% and for high-risk (>:) it is 6%. The standard deviation for low-risk investment (Wl) is 2.5% and for highrisk (W2) it is 20%.
where P'(t), Q'(t) and R'(t) are given by (34) and (35) respectively by (36) above. The hypothesis which, finds solution of equation (24) as second degree polynomial, was proved.
In the figure (1), behaviour of optimal investment strategy s(t) is shown. This simulation scenario is very close to the current situation in the Czech republic, where it is possible to invest with a low rate of return in a low-risk asset. On the other hand, if pension fund management wants to get higher rate of returns, they have to invest in a very risky asset.
3.3 Optimal investment strategy It is now possible to determine the optimal investment strategy (s*(t)) by substituting (28) and (29) to (31). This leads to
*
W
Q(t+l)V
s (t) = """'P-'--(t-+-l--:-)-:-(X-5(,-',t)'---+--:-c)-D
D
(44)
In this case, the optimal investment strategy has a strictly descending characteristic. The differences between a real asset x5(t)and a target asset X5(t) depends on time of service. In all cases the real asset is higher then the target asset.
where the sequences {P(t) }t=l,... ,N, {Q(t)}t=l, ... ,N and {R(t)}t=l, ... ,N are given recursively by
P(t) = 1 + ,HP(t + 1)
(45)
Q(t) = X5(t) -,c(t)HP(t
+ 1) -
-,KQ(t + 1) R(t) = x;(t) + ,c2(t)H P(t + 1) +
(46) 5. CONCLUSION
+ 2,c(t)KQ(t + 1) + R(t + 1) _ Q(t+l) (\_17)2 DP(t + 1)
This paper was focused on utilization of the mathematical model of the contribution defined pension fund in the Czech republic. The structure of mathematical model has been already given in (Simandl and Lesek, 2002). The method of the best allocation of a pension fund pension fund asset in two-asset world was introduced.
(47)
where the boundary conditions P(N), Q(N) and R(N) are given by (25), D, H and K are given by (37), (38) and (39) above and V and W are given by
V=>:-v W = v>: - 172
Analysis of behaviour of the pension fund investment for distribution of capital to low-risk investment and high-risk investment was performed. It is possible to expect that pension fund managers know the values for each type of rates of return. And we proved that on basis, we can automatically allocate capital between high-risk and lowrisk investment.
(48) -
w~
(49)
4. ILLUSTRATION EXAMPLE In the illustration example, it is assumed that we have a pension fund with only one client who saves a constant amount during whole period and finishes saving in strictly defined term.
In the work (Vigna and Haberman, 2001), it is mentioned that managers in the Great Britain firstly invested in high-risk investments and at the end of service period, they moved capital to low-risk investments. Simulation results show the same strategies in how to best allocate in a twoasset world are similar to the empirical behaviour that pension fund managers in actual pension fund employ.
The optimal distribution of asset depends on many parameters. Firstly, it is necessary to decide on length of time invested period. In this illustration, the time period for managing asset was set to 10, 20, 30 and 40 years.
423
O.
0.3
O.
025 00
00
02
04
015
O'
0' 02
02
0.05
0 0
r",.
o
'0
°OL--.-<:::.---:o:----:--J'o
o L---- - 15 - - : '20 o 10
°O!---,...----',:-O---:,:-.--:'20
rme
Tome
Tome
2~'~,o'=:::==:::===::::;-'---'---'-~-~----"-I Real asset Target asset
IS
3000 2000
0.'
'000
----=---7---:---:-----:'.0
°oL=~-~---:---'--:-. r",.
o0J.-,=c----''------'----:-~'O:--~'2:-----,'-:---:':'O--:':'.----:!.20 Tme
(h) Investment strategies and fund asset for 20 years investing period
(a) Investment strategies and fund asSl't for 10 ypars investing period
,
,
Mean cA optimal nvtlSlmenl dlltribuborl strategy
Val\8nCe of optimal .,YUtmenl dlstnbubofl slfmeg)'
Mean 01 opllmaJ Invutmenl d15tnbubon strlilegy
04
00
03
00
00
02
00
04
0'
04
Vanat1Cll!l of optllNlJ lnVetltmem dlStnbutlon a1ra1eqy
O. 04 03 02
0.20L---::'0---::20,.----":
02'L-----------J
°oL---'O....L.:....--:':20,.----:'JO
o
T,me
T'me
0' 10
20
30
°O~---:'-:':0--:20~----:30:::------J40
40
Tome
Tome
7~"~0·=='=====~-~--~----,.--~-J 2.
Real asset Target asset
Real asset Target asset
IS
O. °OL~=~---'"::O--~1S:---,.--20=----::2S;---~JO
O'OL~===----,~o
-----',=-.---:20:----::2.:----:':JO,.----::3.:---J40 T'me
Time
(d) Investment strategies and fund asset for 40 years investing period
(c) Investment strate!!;ies and fund ass('t for 30 years investing period
Fig. 1. Optimal strategies and fund asset for different investing period 6. ACKNOWLEDGMENT
Vigna. E. and S. Haberman (2001). Optimal investment strategy for defined cotrihution pension schemes. Insumnce: A/ath.cmatic and ECOTwmics. Simandl. t\1. (1998). Nel1'rh matematic/,·:tjch. modelll. pro nal'rh nekteryjch para1ll.et711. prnzijnt71.O fondll.. Zi1padoeeska univerzita. Plzefl. (ill Czech). Simandl. t\1. and t\I. Lesek (2001). Design of stfllCtrure and state estimation of pension fund model. In: Nostmdarnns. 4th Intc7lIar'ional Confercnce: on Pr'ediction awl Nonlinear DylUL7IIics. TOlllas Bata Uni\"Crsity in Zlin. Simalldl. t\1. and t\I. Lesek (2002). Rozloienf ill1'est'ic penzijm7w fondu S l'yuiitlm BcllmanOI'(l p1"i1lcipll opti1ll.ahty. Zapadoceska univerzita. Plzefl. (in Czech).
The work was supported h:v the Grant Agency of the Czech Republic. project GA CR 102/01/0021 and by the Ministry of Education. Youth and Sports of Czech Repuhlic t\lSt\1 2352 00004. REFERENCES Bellman. R. and R. Kabala (1965). Dynamic proqmmminq and modern control theory. Academic press. New York. Cairns. A..1. (1997). A cornparition of optimal and stochatic control strategies for continuoustime pension fund models. AFIR'97. O\\·adall:v. t\1..1. (1998). The d:vnamics and control of pension funding. PhD thesis. The Cit:v University. London. Sung. .I.H. and S. Haherman (1994). Dynamic approaches to pension funding. Ins'llmnce: Mathematic and Economics.
424
Copyright © IFAC Control Systems Design, Bratislava, Slovak Republic, 2003
\ J Publications www.elsevier.comlIocate/ifac
CONTROLLING OF PENSION FUND INVESTMENT BY USING BELLMAN'S OPTIMALITY PRINCIPLE Miroslav Simandl * Marek Lesek **
* [email protected], University of West Bohemia, Pilsen
** [email protected]. University of West Bohemia,
Pilsen
Abstract: This paper analyzes the financial risk in a contribution defined pension fund in the Czech republic. The Bellman's optimality principle is used to derive the best allocation of a pension fund asset in the two-asset world. The principal results concern the suitability of the optimal pension fund strategy and the large variability of the level of achievement in pension fund asset in the case of variable rates of assets return. Copyright © 20031FAC Keywords: Pension fund, Mathematical modeling, Bellman's optimal principle
1. INTRODUCTION
fund. The optimal allocation decision in two-asset world was derived. The net replacement ratio was used as level of reference asset. The asset of contribution defined pension fund is valorized by exponential function.
The present work analyzes the financial risk in contribution defined pension fund. The financial risks of contribution defined pension fund are analyzed for investing in a two-level system of distributing investment. In this work, the best way of investing between low and high-risk investment is suggested. For computing the optimal investment strategy for multiplicative rates of return dynamical programming is used.
This paper is focused on deriving optimal strategy for allocation decision in a two-asset world for multiplication valorized contribution defined pension fund in Czech republic.
In the work (Simandl, 1998), the Mathematical Model of Pension Fund in the Czech Republic was firstly established. This basic model was extended and adapted by other author for a better description of reality. We will use the modification which was described in work (Simandl and Lesek, 2002). The main change, that was done, allows the model to invest assets to two groups of capital - high risk (stocks etc.) and low risk (bonds etc.).
The use of dynamic programming is not new in the pension fund area. (Sung and Haberman, 1994) use Bellman's optimality principle to • minimize simultaneously the contribution risk and the solvency risk in a defined benefit pension scheme and derive the optimal contribution rate. (Cairns, 1997) in a continuous time framework and (Owadally, 1998) a discrete time framework apply this principle to a defined benefit pension scheme in order to derive the optimal contribution rate and the optimal allocation decision in a twoasset world.
The paper has the following structure. Section 2 is denoted to describe a mathematical model of a contribution defined pension fund which was designed in work (Simandl and Lesek, 2002). The contents of the section 3 is focused on a problem of dynamic programming. In the first part it is
(Vigna and Haberman, 2001) use Bellman's optimality principle in contribution defined pension
419
defined problem of dynamic programming for the best allocation of pension fund asset in two-asset world and in the second part the Bellman's optimality principle is used for solving the problem of dynamical programming. In section 4 it is given a illustration example of applying the Bellman's optimal principle to the best allocation of pension fund asset and the brief summary is done in section 5.
is number of the clients which have contributed to the other pension fund and save some money. The last equation, (5), describes a dynamical evolution of the size of the pension fund assets, where up(t) is a size of the amount which clients bring to the pension fund from other pension fund in which they have saved. We assume that annual rates of returns v(t) (for low-risk investment) and A(t) (for high-risk investment) from asset (IS (t)) are normally distributed random variables.
2. THE MATHEMATICAL MODEL OF PENSION FUND IN THE CZECH REPUBLIC
v(t) ~ N(v, wi)
(Simandl and Lesek, 2002) describe the design of pension fund mathematical model in two-asset world in state-space form. The state x(t) is described by set of stochastic non-linear equations.
a
A(t) ~ NeX, w~)
where
wi
~ A, ~ w~ and it is also assumed that the random variables (v(t) and A(t)) and pension fund asset (xs(t)) are independent
v
XI(t + 1) = h(XI(t) + WI(t))
(1)
X2(t + 1) = X2(t) + W2(t)
(2)
+ 1) = X3(t) + W3(t)
(3)
cov(xs(t), v(g)) = 0 V 9
~
t
(4)
cov(xs(t), A(g)) = 0 V 9
~
t
X3(t
X4(t + 1) =
Xl
(t)X4(t) + Uk(t)
COV(V(t),A(g)) = 0 V t,g
xs(t + 1) = [(1 - s(t))v(t) + S(t)A(t)].
(7)
The fifth equation of mathematical model contains the investment strategy and the Bellman's optimal principle will be applied to it. The states X2(t), X3(t) and X4(t) are measurable then it is possible to arrange the equation for fifth state (xs(t)) to following form:
,[(X3(t) - X2(t))X4(t) + + xs(t) + up(t)]
(6)
(5)
In following part, equations of pension fund mathematical model will be described. Equation (1) describes an evolution of the portion of clients who stay in the pension fund at time t + 1 (except new incoming) to number of clients at time t, where h (... ) represents saturation function that holds value of expression Xl (t)+w(t) in define interval (0,1) and is valid for all t, that h(XI(t) + WI(t)) = 0, if XI(t) + UIJ(t) < 0 and at the same time fl(XI(t) +WI(t)) = 1, ifxI(t) + WI(t) > 1 and WI(t) is a random variable with mean III and variance ai.
Xs(t + 1) = [xs(t) + c(t)]. .[(1 - s(t))v(t) + S(t)A(t)]
(8)
where
The output equation is designed in virtue of fact, that a part of the states (X2(t), X3(t), X4(t) and Xs (t)) is directly measurable. Hence, the output equation is in linear form.
Equation (2) describes a development of an average value of benefit pay-off per client in the pension fund (this amount already include money that clients carry out to other pension funds if they are transferring), where W2(t) is a random variable with mean 1L2 and variance a~.
y(t)
= C.x(t)
(9)
where C will be in form:
c=
The next equation, (3), represents a evolution of an average value of contribution per client including a contribution which the State supports the supplementary pension secure, where W3(t) is a random variable with mean 1L3 and variance a~.
0100001 001000 [ 000 1 0 0 o0 0 0 1 0
(10)
and x( t) is the state vector.
Equation (4) represents a dynamical development of number of the pension fund clients, where Uk (t) = Uk, + Uk2 is number of clients that join the pension fund at time t and Uk, is number of the clients which have not already contributed and Uk 2
3. PROBLEM OF DYNAMIC PROGRAMMING We start with the documents (Sung and Haberman, 1994) and (Owadally, 1998), where the
420
"costs" function was defined for benefit defined pension fund, and (Vigna and Haberman, 2001) where is defined "costs" function for contribution defined pension fund in form
The objective of mathematical programming is to find investment strategy 7rtl which would get minimal average value of future cost G(t).
(ll)
3.1 Bellman's optimal principle
where 8 j > 0 is weight coefficient and X5(t) is expected asset of a pension fund, which the fund should own at time t. The amount of expected asset will be counted on voted rate of return, which is defined as arithmetical average of chosen rates of return for investment to high-risk (~) and low-risk (v) asset and it is defined by equation 1
Bellman's optimal principle, which is given in (Bellman and Kabala, 1965), has for above noticed problem the Bellman's equation in form
J(I(t)) =
-
n;~nE
[t,
,C(S)II(t)] =
X5(t) = (X5(t - 1) + c(t - 1))"2(17 + A).
= min [C(t) + ,E[J(I(t + 1)II(t))]]
The set of expected assets X5 (t) is known in advance and it is fixed at each discrete time moment {X5(t)}t=I,2, ... ,N and X5(0) = X5(0)
We notice that {v(t)} and {A(t)} are independent, X5(t) is Marcovian process and it is valid for conditional density of a probability that
P[X5(t + 1)II(t)]
The border condition from "costs" function at time N is
C(N) = 80 (X5(N) - x5(N))2
(15)
8(t)
=
P[X5(t + 1)ly(t)]
and also
(12) P[X5(t + 1), X5(t
For pension fund managers, it is more important to reach final target than particular targets. This principle is applied in weight coefficients 8 1 and 80 , where condition for weight coefficients is 81 ::; 80 , Summary of future cost at time t is defined by equation, which count all future values of the criteria functions multiplied by a discount factor b):
+ 2), ... , x5(N)II(t)]
=
= P[X5(t + 1),x5(t + 2), ... ,x5(N)ly(t)] It follows from the previous equation that conditional densities of probability are in form
p(G(t)II(t)) = p(G(t)ly(t))
(16)
and
N
G(t) =
L ,b-tC(b)
(13)
J(X5(t), t)
b=t
, y(t),
J(I( t)) = min E[G( t) II( t)],
(14)
+ S(t)2(~2 + w~) +
t = 0, 1, ... ,N
+ 2s(t)(1 - s(t))V~l
where 7rt represents the set of future investment strategies
=
(17)
+ 1)ly(t)] = (X5(t) + c(t)) (18) [(1 - s(t))v + s(t)~l E[x~(t + 1)ly(t)] = (X5(t) + C(t))2 (19) [(1 - S(t))2p;2 + w~) +
7rt
=
+ ,E[J(X5(t + 1), t + 1)1
E[X5(t
, s(t - I)}
Criteria function at time t can be defined as
7rt
8(t)
On the basis of the asset equation (8) and the fact, that {v(t)}, P(t)} and X5(t) are independent (7), it is possible to derive the conditioned mean E[X5(t + 1)ly(t)], the conditioned mean of second power E[x~(t+l)ly(t)] and the conditioned variation var[x5(t + 1)ly(t)]. It easy to verify that
For solving the problem of optimal investment distribution, it is necessary to define set I(t) as set of all available information at time t:
s(O), s(I),
min[C(t)
Iy(t))]], t=O,I, ... ,tN-l
where , is the inter-temporal discount factor, which can be seen as a "psychological" discount rate or as a risk discount rate (Cairns, 1997).
I(t) = {y(O), y(I),
=
var[x5(t + 1)ly(t)] = (X5(t) + c(t)f [(1 - S(t))2w~
{{S(tb)}tb=t,t+l....,N-j : 0::; S(tb) ::; I} = {{s(t), s(t + 1), ... , s(t(N - I))} :
(20)
+ s2(t)w~1
Without loss of generality it is possible to assume that 80 is multiple of 8 j • If it is assumed that Bj = 1 then it is possible to define Bo = B, where
0::; S(tb) ::; I}
421
() ?: 1. The dynamic programming problem is
M(t) and N(t) and unknown parameter is value of optimal distribution of invested asset s(t)
transformed to:
J(X5(t), t) = min[(x5(t) - X5(t))2 +
E[J(X5(t + 1), t + 1)ly(t)] = L(t)S2(t) +
(21)
s(t)
+M(t)s(t) + N(t) = 'ljJ(s(t)) (27)
+ ,E[J(X5(t + 1), t + 1)ly(t)]] =
with boundary condition t = N: where
L(t) = P(t + l)(x5(t) + C(t))2
where x5(N) is expected amount of pension fund asset from time t = 0 when the cost c( t) and the amount of profit from high-risk investment (with mean "X) and amount of profit from lowrisk investment (with mean 17) to time t = N. We notice, that the realized pension fund target (x5(N)), where half of asset is invested to highrisk capital and second half is invest to low-risk capital, is in form:
(172 + W~ + ).2 + W~
+
L
("X)k c(N - k) 17;
217"X)
(28)
M(t) = 2P(t + 1)(x5(t) + c(t))2(17"X _17 N(t)
=
2
-
wi) -
- 2Q(t + 1)(x5(t) + c(t))("X -17) P(t + 1)(x5(t) + c(t))2(17 2 + wi) -
(29)
- 2Q(t + 1)(X5(t) + c(t))17 + R(t + lX30) From equation (27) it is possible to retrieve unique global extreme 'ljJ*(t) and it is assumed that it is valid L(t) > 0 for Vt, then the extreme is unique global minimum
17 +"X) N x5(N) = (X5(0) + c(O)) ( -2+ N-l
-
'ljJ(s*(t)) = Z*(t) (23)
and
k=l
3.2 Solution of the dynamic programming problem We assume that it is possible to verify a hypothesis. This hypothesis expects to find and solve the minimal of equation (21) as LQ problem. We expect the solution in form:
2 Z*(t) = N(t) _ M (t) 4L(t)
(32)
+ Q'(t)X5(t) + R'(t)
Z*(t) = P'(t)x~(t) where
P'(t)
Q(N) = ()x5(N)
=
HP(t + 1)
(34)
Q'(t) = 2Hc(t)P(t + 1) + 21\Q(t + 1) (35) 2 R'(t) = Hc (t)P(t + 1) + 2h'c(t)Q(t + 1) _
(25)
Q(t + 1 ("X _ 17)2 DP(t + 1)
(36)
where
R(N) = ()x~(N) Assuming that the hypothesis is satisfied for t + 1. We rewrite the equation (24) to this time moment.
= 1/2 + w~ + "X + w~ - 21/"X H = 1/2 + w~ - ~ (I/"X - 1/2 - w~) 2
D
D
J(X5(t + 1), t + 1) = P(t + l)x~(t + 1) - 2Q(t + l)x5(t
+ R(t + 1)
(33)
(24)
For boundary condition (22) at time N the values of coefficient (P( t), Q( t) a R( t)) are acceptable for solving expectations (24)
P(N) = ()
(31)
Substitution (28), (29) and (30) to (32) we get
J(X5(t), t) = P(t)x~(t) - 2Q(t)X5(t) +
+ R(t)
*( ) __ M(t) s t 2L(t)
K
+ 1) + (26)
= -1 (-' lI/I D
-2 II
-
(' W 2) 1 /I
-
-) II
-
-II
(37)
(38) (39)
Equation (17) becomes now
It has to be shown that the hypothesis is satisfied for time t.
J(X5(t), t)
Using (19), (20) and (26) it is not difficult to verify that conditioned mean at time t + 1 is equal to second degree polynomial with coefficients L(t),
= min[C(t) + s(t)
+ ,E[J(X5(t + 1), t + 1)ly(t)]] =
= min[(x5(t) - X5(t))2 + s(t)
422
+ ,E[J(X5(t + 1), t + 1)ly(t)]] = = (X5(t) - X5(t))2 + ,Z*(t) =
For simplification, we made assumption, that client will not transfer any finances from the other pension funds, so it is assumed that X5(0) = 0 and up(t) = 0 for '
= P(t)x;(t) - 2Q(t)X5(t) + R(t) (40) where
P(t) = 1 + ,P'(t)
(41)
Q(t) = X5(t) - 0.5,Q'(t)
(42)
R(t) = x;(t)
+ ,R'(t)
The amount of contributions (costs=O) is constant and it is equal to c = 420 Kc, where the client contributes 300 Kc and amount 120 Kc is paid as state contribution for supporting supplementary pension insurance.
(43)
The rates of return assume the following values. The mean for low-risk investment (v) is 4% and for high-risk (>:) it is 6%. The standard deviation for low-risk investment (Wl) is 2.5% and for highrisk (W2) it is 20%.
where P'(t), Q'(t) and R'(t) are given by (34) and (35) respectively by (36) above. The hypothesis which, finds solution of equation (24) as second degree polynomial, was proved.
In the figure (1), behaviour of optimal investment strategy s(t) is shown. This simulation scenario is very close to the current situation in the Czech republic, where it is possible to invest with a low rate of return in a low-risk asset. On the other hand, if pension fund management wants to get higher rate of returns, they have to invest in a very risky asset.
3.3 Optimal investment strategy It is now possible to determine the optimal investment strategy (s*(t)) by substituting (28) and (29) to (31). This leads to
*
W
Q(t+l)V
s (t) = """'P-'--(t-+-l--:-)-:-(X-5(,-',t)'---+--:-c)-D
D
(44)
In this case, the optimal investment strategy has a strictly descending characteristic. The differences between a real asset x5(t)and a target asset X5(t) depends on time of service. In all cases the real asset is higher then the target asset.
where the sequences {P(t) }t=l,... ,N, {Q(t)}t=l, ... ,N and {R(t)}t=l, ... ,N are given recursively by
P(t) = 1 + ,HP(t + 1)
(45)
Q(t) = X5(t) -,c(t)HP(t
+ 1) -
-,KQ(t + 1) R(t) = x;(t) + ,c2(t)H P(t + 1) +
(46) 5. CONCLUSION
+ 2,c(t)KQ(t + 1) + R(t + 1) _ Q(t+l) (\_17)2 DP(t + 1)
This paper was focused on utilization of the mathematical model of the contribution defined pension fund in the Czech republic. The structure of mathematical model has been already given in (Simandl and Lesek, 2002). The method of the best allocation of a pension fund pension fund asset in two-asset world was introduced.
(47)
where the boundary conditions P(N), Q(N) and R(N) are given by (25), D, H and K are given by (37), (38) and (39) above and V and W are given by
V=>:-v W = v>: - 172
Analysis of behaviour of the pension fund investment for distribution of capital to low-risk investment and high-risk investment was performed. It is possible to expect that pension fund managers know the values for each type of rates of return. And we proved that on basis, we can automatically allocate capital between high-risk and lowrisk investment.
(48) -
w~
(49)
4. ILLUSTRATION EXAMPLE In the illustration example, it is assumed that we have a pension fund with only one client who saves a constant amount during whole period and finishes saving in strictly defined term.
In the work (Vigna and Haberman, 2001), it is mentioned that managers in the Great Britain firstly invested in high-risk investments and at the end of service period, they moved capital to low-risk investments. Simulation results show the same strategies in how to best allocate in a twoasset world are similar to the empirical behaviour that pension fund managers in actual pension fund employ.
The optimal distribution of asset depends on many parameters. Firstly, it is necessary to decide on length of time invested period. In this illustration, the time period for managing asset was set to 10, 20, 30 and 40 years.
423
O.
0.3
O.
025 00
00
02
04
015
O'
0' 02
02
0.05
0 0
r",.
o
'0
°OL--.-<:::.---:o:----:--J'o
o L---- - 15 - - : '20 o 10
°O!---,...----',:-O---:,:-.--:'20
rme
Tome
Tome
2~'~,o'=:::==:::===::::;-'---'---'-~-~----"-I Real asset Target asset
IS
3000 2000
0.'
'000
----=---7---:---:-----:'.0
°oL=~-~---:---'--:-. r",.
o0J.-,=c----''------'----:-~'O:--~'2:-----,'-:---:':'O--:':'.----:!.20 Tme
(h) Investment strategies and fund asset for 20 years investing period
(a) Investment strategies and fund asSl't for 10 ypars investing period
,
,
Mean cA optimal nvtlSlmenl dlltribuborl strategy
Val\8nCe of optimal .,YUtmenl dlstnbubofl slfmeg)'
Mean 01 opllmaJ Invutmenl d15tnbubon strlilegy
04
00
03
00
00
02
00
04
0'
04
Vanat1Cll!l of optllNlJ lnVetltmem dlStnbutlon a1ra1eqy
O. 04 03 02
0.20L---::'0---::20,.----":
02'L-----------J
°oL---'O....L.:....--:':20,.----:'JO
o
T,me
T'me
0' 10
20
30
°O~---:'-:':0--:20~----:30:::------J40
40
Tome
Tome
7~"~0·=='=====~-~--~----,.--~-J 2.
Real asset Target asset
Real asset Target asset
IS
O. °OL~=~---'"::O--~1S:---,.--20=----::2S;---~JO
O'OL~===----,~o
-----',=-.---:20:----::2.:----:':JO,.----::3.:---J40 T'me
Time
(d) Investment strategies and fund asset for 40 years investing period
(c) Investment strate!!;ies and fund ass('t for 30 years investing period
Fig. 1. Optimal strategies and fund asset for different investing period 6. ACKNOWLEDGMENT
Vigna. E. and S. Haberman (2001). Optimal investment strategy for defined cotrihution pension schemes. Insumnce: A/ath.cmatic and ECOTwmics. Simandl. t\1. (1998). Nel1'rh matematic/,·:tjch. modelll. pro nal'rh nekteryjch para1ll.et711. prnzijnt71.O fondll.. Zi1padoeeska univerzita. Plzefl. (ill Czech). Simandl. t\1. and t\I. Lesek (2001). Design of stfllCtrure and state estimation of pension fund model. In: Nostmdarnns. 4th Intc7lIar'ional Confercnce: on Pr'ediction awl Nonlinear DylUL7IIics. TOlllas Bata Uni\"Crsity in Zlin. Simalldl. t\1. and t\I. Lesek (2002). Rozloienf ill1'est'ic penzijm7w fondu S l'yuiitlm BcllmanOI'(l p1"i1lcipll opti1ll.ahty. Zapadoceska univerzita. Plzefl. (in Czech).
The work was supported h:v the Grant Agency of the Czech Republic. project GA CR 102/01/0021 and by the Ministry of Education. Youth and Sports of Czech Repuhlic t\lSt\1 2352 00004. REFERENCES Bellman. R. and R. Kabala (1965). Dynamic proqmmminq and modern control theory. Academic press. New York. Cairns. A..1. (1997). A cornparition of optimal and stochatic control strategies for continuoustime pension fund models. AFIR'97. O\\·adall:v. t\1..1. (1998). The d:vnamics and control of pension funding. PhD thesis. The Cit:v University. London. Sung. .I.H. and S. Haherman (1994). Dynamic approaches to pension funding. Ins'llmnce: Mathematic and Economics.
424