Robust portfolio optimization using second-order cone programming 17
σ maximum portfolio volatility
α * n 1 vector of estimated asset alphas
α (^) p Portfolio return
w (^) max n 1 vector of maximum asset weights in the portfolio
An example is given below where an optimization is first run without any
constraint on the portfolio volatility, but with a constraint on the portfolio
alpha. The optimization is then rerun several times with varying constraints
on the portfolio volatility, and the same constraint on the portfolio alpha. The
universe and benchmark both contain 500 assets. The resulting portfolio vola-
tilities and tracking errors can be seen in Figure 1.11.
The second case, minimizing tracking error against one benchmark, whilst
constraining tracking error against some other benchmark, results in the fol-
lowing SOCP problem when using the SunGard APT risk model:
Minimize[(wb^1 ) BBwb wb( ) ( ) (wb)]
TT T
11 1∑
subject to
αα*wT p
()()()()wbBBwb wb wb
TT T
22222
∑ σ^2
a
ew 1
T
ww max
w^0
where
b 1 n 1 vector of weights for benchmark used in objective function
b 2 n 1 vector of weights for benchmark used in constraint
σ (^) a2 maximum tracking error against second benchmark
An example of this case is given below where an optimization is first run
without any constraint on the tracking error against the internal model portfo-
lio, but with a constraint on the portfolio alpha, minimizing the tracking error
against a market index. The optimization is then rerun several times with vary-
ing constraints on the tracking error against the internal model portfolio, and
the same constraint on the portfolio alpha. The universe and benchmark both
contain 500 assets. The resulting tracking errors against both the market index
and the internal model portfolio can be seen in Figure 1.12.