Mathematical and Statistical Methods for Actuarial Sciences and Finance

Tracking error with minimum guarantee constraints 19

qikt=( 1 +rikt)

[

qif(kt)+aif(kt)−vif(kt)

]

i= 1 ,...,n 1 (9)

bjkt=( 1 +rjkt)

[

bjf(kt)+ajf(kt)−vjf(kt)

]

j= 1 ,...,n 2 (10)

ckt=( 1 +rckt)

[

cf(kt)−

∑n^1

i= 1

(κ+)aif(kt)+

∑n^1

i= 1

(κ−)vif(kt)

+

∑n^2

j= 1

(κ+)ajf(kt)+

∑n^2

j= 1

(κ−)vjf(kt)+

∑n^2

j= 1

gktbjf(kt)

]

(11)

aikt≥ 0 vikt≥ 0 i= 1 ,...,n 1 (12)

ajkt≥ 0 vjkt≥ 0 j= 1 ,...,n 2 (13)

qikt≥ 0 i= 1 ,...,n 1 (14)

bjkt≥ 0 j= 1 ,...,n 2 (15)

θk+tθk−t= 0 (16)

θk+t≥ 0 θk−t≥ 0 (17)

γk−t≥ 0 (18)

ckt≥ 0 (19)

qi 0 = ̄qi i= 1 ,...,n 1 (20)

bj 0 =b ̄j j= 1 ,...,n 2 (21)

c 0 = ̄c (22)

kt=Kt− 1 + 1 ,...,Kt

t= 1 ,...,T

where equation (8) represents the portfolio composition in nodekt; equations (9)–
(11) describe the dynamics of the amounts of stocks, bonds and cash in the portfolio
moving from the ancestor nodef(kt), at timet−1, to the descendent nodeskt, at time
t, withK 0 =0. In equation (11), withgktwe denote the inflows from the bonds in the
portfolio. Equation (16) represents the complementarity conditions which prevent
positive and negative deviations from being different from zero at the same time.
Equations (20)–(22) give the initial endowments for stocks, bonds and cash.
We need to specify the value of the benchmark and the value of the minimum
guarantee at each time and for eachnode. The stochastic benchmarkyktand the prices
of the risky assets in the portfolio must be simulated according to given stochastic
processes in order to build the corresponding scenario trees. Other dynamics for the
minimum guaranteed level of wealth can be designed. In particular, we can discuss
a time-varying rate or returnρtalong the planning horizon, or we can include the
accruedbonuses as in [8].
A second approach to tackle the problem of the minimum return guarantee is
to introduce probabilistic constraints in the dynamic optimisation problem. Denot-
ing withθthe desired confidence level, we can formulate the shortfall constraints
both on the level of wealth at an intermediate timetand on the terminal wealth as