The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 673 Wednesday, February 4, 2004 1:12 PM

Bond Portfolio Management 673

ios are joint paths of all the relevant variables. A probability number is

attached to each scenario. A path of interest rates is a scenario. If we

consider corporate bonds, a scenario will be formed, for example, by a

joint path of interest rates and credit ratings. How scenarios are gener-

ated will be discussed later in this chapter.

Suppose that scenarios are given. Using an LP program, one can find

the optimal portfolio that (1) matches all the liabilities in each scenario

and (2) minimizes initial costs or maximizes final cash positions subject

to budget constraints. The CFM problem can be reformulated as follows:

Minimize ∑αiPi , subject to the constraints

i ∈ U

s s s s s s s s

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

i ∈ U

b

s

m =^0

ai ≥ 0; i ∈ U

In this formulation, all terms are stochastic and scenario dependent

except the portfolio’s weights. Each scenario imposes a constraint.

Scenario optimization can also be used in a more general context.

One can describe a general objective, for instance expected return or a

utility function, which is scenario-dependent. Scenario-dependent con-

straints can be added. The optimization program maximizes or mini-

mizes the objective function subject to the constraints.

Stochastic Programming

Strategies discussed thus far are static (or myopic) in the sense that deci-

sions are made initially and never changed. As explained in Chapter 7,

stochastic programming (or multistage stochastic optimization) is a

more general, flexible framework in which decisions are made at multi-

ple stages, under uncertainty, and on the basis of past decisions and

information then available. Both immunization and CFM discussed

above can be recast in the framework of stochastic programming.

Indeed, multistage optimization is a general framework that allows one

to formulate most problems in portfolio management, not only for

bonds but also for other asset classes including stocks and derivatives.

Stochastic programming is a computerized numerical methodology to

solve variational problems. A variational principle is a law expressed as the