Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

90 Optimizing Optimization

It is clear that these two versions are equivalent and that we can optimize
either. We choose to maximize:

VGain ()1 λLoss

It is useful to define an array I t that contains T integers where T is the number
of periods. We assign the value 1 to I t if the portfolio makes a net gain in
period t and 0 to I t if the portfolio makes a net loss in that period. Note that
the values in array I t depend on the portfolio weights for each asset, the return
data, and the target return R

Iif wrtRtii i

n

 (^1) ∑ 1 ()
Iif wrtRtii i
n

(^0) ∑ 1 ()
where
w i  weight of the asset I ;
r i ( t )  return of asset i in period t.
The probability that we exceed the target is given by t ITt
T
()∑ 1 /.^
Gain is given by:
i wrt RIii
n
t t
T
()∑∑  11 (() )
Loss is given by:
i wR rt I Iii
n
t
T
()∑∑  11 (())()t
Note that both gain and loss are nonnegative numbers. Our utility V appears
to be a linear expression in portfolio weights (apart from the dependence of I
on w ).
With special care, we can maximize V with respect to w and linear con-
straints on w using an iterative sequential linear programming approach. For
the calculations in this section, we just impose nonnegative upper and lower
bounds on each weight and constrain their sum to be 1.

References

Andersen , E. D. , Roos , C. , & Terlaky , T. ( 2003 ). On implementing a primal-dual inte-
rior point method for conic quadratic optimization. Mathematical Programming ,
95 ( 2 ) , 249 – 277.
Artzner , P. , Delbaen , F. , Eber , J. M. , & Heath , D. ( 1999 ). Coherent measures of risk.
Mathematical Finance , 9 , 203 – 228.