Mathematical and Statistical Methods for Actuarial Sciences and Finance

18 D. Barro and E. Canestrelli

(v 1 kt,...,vnkt)denoting the value of each asset purchased and sold at timetin node
kt, while we denote withκ+andκ−the proportional transaction costs for purchases
and sales.
Different choices of tracking error measures are possible and different trade-offs
between the goals on the minimum guarantee side and on the enhanced tracking error
side, for the risky benchmark, are possible, too. In this contribution we do not tackle
the issue of comparing different choices for tracking error measures and trade-offs
in the goals with respect to the risk attitude of the investor. Among different possible
models, we propose the absolute downside deviation as a measure of tracking error
between the managed portfolio and the minimum guarantee benchmark, while we
consider only the upside deviations between the portfolio and the risky benchmark

φkt(ykt,xkt)=[ykt−xkt]+=θk+t; (3)

ψkt(ykt,zt)=[ykt−zt]−=γk−t, (4)

where [ykt−xkt]+=max[ykt−xkt,0] and [ykt−zt]−=−min[ykt−zt,0]. The
minimum guarantee can be assumed constant over the entire planning horizon or it can
follow a deterministic dynamics, i.e, it is not scenario dependent. Following [14] we
assume that there is an annual guaranteed rate of return denoted withρ. If the initial
wealth isW 0 =

∑n+ 1
i= 1 xi^0 , then the value of the guarantee at the end of the planning
horizon isWT=W 0 ( 1 +ρ)T. At each intermediate date the value of the guarantee
is given byzt=eδ(t,T−t)(T−t)W 0 ( 1 +ρ)T,whereeδ(t,T−t)(T−t)is a discounting factor,
i.e., the price at timetof a zcb which pays 1 at terminal timeT.
The objective function becomes a weighted trade-off between negative deviations
from the minimum guarantee and positive deviations from the risky benchmark. Given
the choice for the minimum guarantee, the objective function penalises the negative
deviations from the risky benchmark only when these deviations are such that the
portfolio values are below the minimum guarantee and penalises them for the amounts
which are below the minimum guarantee. Thus, the choice of the relative weights for
the two goals is crucial in the determination of the level of risk of the portfolio strategy.
The obtained dynamic tracking error problem in its arborescent form is

max

qkt,bkt,ckt

∑T

t= 1

⎡

⎣αt

∑Kt

kt=Kt− 1 + 1

θk+t−βt

∑Kt

kt=Kt− 1 + 1

γk−t

⎤

⎦ (5)

θk+t−θk−t=ykt−xkt (6)

−γk−t≤ykt−zt (7)

ykt=ckt+

∑n^1

i= 1

qikt+

∑n^2

j= 1

bjkt (8)