The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 665 Wednesday, February 4, 2004 1:12 PM

Bond Portfolio Management 665

Minimize ∑αiPi , subject to the constraints

i ∈ U

∑αiKit, ≥ Lt

i ∈ U

αi ≥ 0

The last constraint specifies that short selling is not permitted.

The above formulation of the CFM as an optimization problem is too

crude as it takes into account only the fact that it is practically impossible

to create exactly the required cash flows. In fact, in this formulation at

each date there will be an excess of cash not used to satisfy the liability

due at that date. If borrowing and reinvesting are allowed, as is normally

the case, excess cash can be reinvested and used at the next date while

small cash shortcomings can be covered with borrowing.

Suppose, therefore, that it is possible to borrow in each period an

amount bt at the rate βt and reinvest an amount rt at the rate ρt. Suppose

that these rates are the same for all periods. At each period we will require

that the positive cash flow exactly matches liabilities. Therefore coupon

payments of that period plus the amount reinvested in the previous period

augmented by the interest earned on this amount plus the reinvestment of

that period will be equal to the liabilities of the same period, plus the repay-

ment of borrowing in the previous period plus the eventual new borrowing

of the period. The optimization problem can be formulated as follows:

Minimize ∑αiPi , subject to the constraints

i ∈ U

∑αiKit, + (^1 + ρt)rt – 1 + bt = Lt + (^1 + βt)bt – 1 + rt

i ∈ U

bm = 0

αi ≥ 0; i ∈ U

The CFM problem formulated in this way is a linear programming (LP)

problem.^11 Problems of this type can be routinely solved on desk-top

computers using standard off-the-shelf software.

(^11) The mathematical programming techniques described in this chapter are discussed
in Chapter 7.