194 Optimizing Optimization
and since γ θ 1 and ( u v ) 0 at ( u , v ) from Equation (8.89), the mini-
mum point is established.
We note that the condition γ θ 1 corresponds to the condition given
in Proposition 1 of Alexander and Bapista (2001). By substituting Equation
(8.69) back into Equation (8.73), we can recover the mean – variance portfolio
that is the minimum value at risk portfolio as described by their Equation (10).
8.5 Simulation methodology
We consider simulation of portfolios ω of length ω 2 ∗ subject to symmetric lin-
ear constraints, i.e., ∑ωi 1 and a ω (^) i b. We assume ω 1 is distributed
as uniform [ a,b ] and let ω 1 ∗ be the sampled value. Then, ω 2 is also sampled
from a uniform [ a,b ] distribution with sampled value ω 2 ∗. The procedure is
repeated sequentially as long as j ωj
m
∑ 1 ∗()Nma−.^ If a value ωm∗ 1
is chosen such that j ωj
m
∑ ∗Nm a
1 ()^1
1
, we set ωm∗ 1 so that
(^) j ωj
m
∑ ∗ Nm a
1 ()^11
1
. This is always feasible. Because the sequential
sampling tends to make the weights selected early in the process larger, for any
feasible ω we consider all N! permutations of ω *. From the symmetric con-
straints, these will also be feasible portfolios. With N 8, N! 40,320, so it
may be feasible, but at N 12, N! 479,001,600, which may no longer be a
feasible approach. We have to rely on random sampling in the first instance.
If we have a history of N stock returns for T periods, then for any vector
of portfolio weights, we can calculate the portfolio returns for the T periods.
These values can then be used to compute an estimate of the expected return
μ (^) i , and the risk measure φ (^) i , which can be mapped into points on a diagram as
in Figure 8.1. If we believe the data to be elliptically generated, then, following
Proposition 8.2, we can save the weights of the set of minimum risk portfolios.
We amend an algorithm suggested by Bensalah (2000). If the risk – return
value of individual portfolios can be computed, then an intuitive procedure is
to draw the surface of many possible portfolios in a risk – return framework
and then identify the optimal portfolio in a mean minimum risk sense. In the
case of no short selling, an algorithm for approximating any frontier portfolios
of N assets each with a history of T returns can be described as follows:
Step 1: Define the number of portfolios to be simulated as M.
Step 2: Randomize the order of the assets in the portfolio.
Step 3: Randomly generate the weight of the first asset
ω 1 U [0,1] from a Uniform distribution, ω 2 U [0,1 ω 1 ],
ω 3 U [0,1 ω 1 ω 2 ], ω (^) N 1 ω 1 ω 2 ... ω (^) ( (^) N (^) 1)
Step 4: Generate a history of T returns for this portfolio, compute
the average return and risk measure.