# Linear programming applied to a financial bond portfolio

#### tiny

Hi everybody,

I've the following linear programming problem with no objective function and only as restrictions only
a set of portfolio indicators (e.g. duration, convexity, sensitivity, yield to maturity).

The problem is to find the correct weights (xi) which satisfies the following constraints:

SUM(xi * di) <= Dl
SUM(xi * ci) <= Cl
SUM(xi * si) <= Sl
SUM(xi * yi) <= Yl
sum(xi) = 1
xi >= 0

where:
xi = i-th asset weight
di = i-th asset duration
ci = i-th asset convexity
si = i-th asset sensitivity
yi = i-th asset yield to maturity
Dl = portfolio duration limit
Cl = portfolio duration limit
Sl = sensitivity portfolio limit
Yl = yield to maturity portfolio limit

Furthemore, Sensitivity limit depend on Duration limit and Convexity limit:
Sl = -Dl*DeltaYl + 0.5*Cl*(DeltaYl)^2, with DeltaYl equal to 0.01% of Yl

I'm concerned about whether consider Yl as a further constraint to be included within the formula of Dl
(e.g. Dl = SUM(xi * di(evaluated at Yl)), di is the i-th Macauly duration in the portfolio).

Could you please tell me if there are any further dependencies to include in the formulation of the problem?

Thanks a lot.