278 Gianni Pola
Ta b le 1 .Probabilistic model assumptions
CB E
return (ann) 3.24% 5.46% 10.62%
vol (ann) 0% 4.45% 14.77%
skewness 0 − 0. 46 − 0. 34
kurtosis 3 4.25 5.51corr to C 1 0 0
corr to B 0 1 0.0342
corr to E 0 0.0342 1The first step consists in building up a probabilistic model that describes the
asset classes’ return dynamics. Risk figures and expected returns^3 are reported in
Table 1. Asset classes present significant deviations to the Gaussian nature (Jarque–
Bera test; 99% confidence level): bond and equity markets are leptokurtic and negative
skewed. We assume stationarity in the dynamics of the returns distribution. This
market scenario has been modelled with a 2-states Mixture of Multivariate Gaussian
Models (MMGM), as detailed in the Appendix. The proposed MMGM modelling
exactlyfits up to the fourth-order the asset-classes’ performance and risk figures, and
up to the second order the correlation pattern.
The investment requirements are translated into the model as follows. The opti-
misation criterion consists in maximising the probabilityP(x 8 > 1. 072 ),x 8 being
the portfolio value at the end of the second year. The target setss formalisation is
given below:
0 ={ 1 },k=[0,+∞), ∀k= 1 , 2 ,..., 7 , 8 =[1. 072 ,+∞). (7)More precisely, the optimisation problem consists in determining the (optimal) dy-
namic allocation gridsuk(k= 0 , 1 ,...,7) in order to maximise the joint probability
P(x 1 ∈ 1 ,...,x 8 ∈ 8 )subjected to the Value-at-Risk budget constraint. By apply-
ing Theorem 1 we obtain the optimal control strategy that is illustrated in Figure 1.
(Budget and long-only constraints have been included in the optimisation process.)
The allocation at the beginning of the investment (see Figure 1, upper-left panel)
is 46% Bond and 54% Equity market. After the first quarter, the fund manager re-
vises the portfolio allocation (see Figure 1, upper-right panel). Abscissas report the
portfolio valuex 1 at timek =1. For each portfolio realisationx 1 ,themapgives
the corresponding portfolio allocation. As the portfolio strategy delivers higher and
higher performance in the first quarter, the optimal rebalancing requires a reduction
in the risky exposure. Ifx 1 reaches a value around 1.0832, a 100% cash allocation
guarantees the target objective will be reached at maturity. Conversely, a portfolio
(^3) In the present work we do not face the problem of returns and risk-figures forecasting.
Volatility, skewness, kurtosis and the correlation pattern have been estimated by taking the
historical average. Expected returns have been derived by assuming a constant Sharpe ratio
(0.50), and a cash level given by the US Generic T-bills 3 months in December 31st 2007.