Optimal dynamic asset allocation in a non–Gaussian world 277Theorem 1.The optimal value of the ODAA Problem is equal to [18]
p∗=J 0 (x 0 ),where J 0 (x)is given by the last step of the following algorithm,
JN(x)=IN(x),Jk(x)= sup
uk∈Uk∫
k+ 1Jk+ 1 (z)pf(x,uk,wk+ 1 )(z)dz,k=N− 1 ,N− 2 ,..., 1 , 0.(6)
The algorithm proceeds as follows. Suppose that the time horizon of our investment
isN. First, the optimisation algorithm (6) is solved fork=N−1. This optimisation
problem can be automatically solved by using a wealth of optimisation packages
in many computer programs, for example, MATLAB, Mathematica and Maple. The
solution to (6) provides the optimal strategyuˆk(x)to be applied to the investment
when the value of the portfolio isxat timek. Once the optimization problem (6) is
solved fork=N−1, functionJN− 1 (x)is also known. Hence, on the basis of the
knowledge of functionJN− 1 (x), one can proceed one step backwards and solve the
optimisation problem (6) at stepj=N−2. This algorithmic optimisation terminates
whenj=0. The outcome of this algorithm is precisely the optimal control strategy
that solves (3), as formally stated in Theorem 1.
3 Case study: a total return portfolio in the US market
In this section we apply the proposed methodology to the synthesis of a total return
product. The investment’s universe consists of 3 asset classes: the money market, the
US bond market and the US equity market. Details on the indices used in the analysis
are reported below:
Label Asset Index
C Money market US Generic T-bills 3 months
B US bond JP Morgan US Government Bond All Maturity
E US equity S&P500Time series are in local currency and weekly based from January 1st 1988 to December
28th 2007. The total return product consists of a 2-year trade. The investor objective is
to beat a target return of 7% (annualised value) at maturity; his budget risk corresponds
to 7% (ex ante) monthly Value at Risk at 99% (VaR99m) confidence level.^2
The portfolio allocation will be synthesized applying the results presented in the
previous section. We first consider an ODAA problem with a quarter rebalancing
(N=8).
(^2) This budget risk corresponds to an ex ante (annual) volatility of 10.42%.