The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 664 Wednesday, February 4, 2004 1:12 PM

664 The Mathematics of Financial Modeling and Investment Management

based on optimization methodologies. Ultimately, all methodologies can

be cast in the framework of optimization, but duration and convexity play

an important role from the practical as well as conceptual point of view.

We will begin by discussing the cash-flow matching approach in a deter-

ministic context and then successively discuss strategies based on duration

and convexity and lastly a full stochastic programming approach.

Cash Flow Matching

Cash flow matching (CFM), also referred to as a dedicated portfolio strat-

egy, in a deterministic environment is the problem of matching a predeter-

mined set of liabilities with an investment portfolio that produces a

deterministic stream of cash flows.^10 In this context, fluctuations of inter-

est rates, credit risk, and other sources of uncertainty are ignored. There

are, however, conditions where financial decisions have to be made.

Among them we will consider:

■ Reinvestment of excess cash

■ Borrowing against future cash flows to match liabilities

■ Trading constraints such as odd lots

To formulate the model, consider a set of m dates {t 0 ,t 1 ,...,tm} and a

universe U of investable assets U = {1,2,...,n}. Call {Ki,0,...,Ki,m} the

stream of cash flows related to the i-th asset. We will consider only

bonds but most considerations that will be developed apply to broader

classes of assets with positive and negative cash flows. In the case of a

bond with unit price Pi per unit par value 1, with coupon ci,t, and with

maturity k, the cash flows are

{–Pi,ci,1,...,ci,k–1,ci,k + 1,0,...,0}

Let’s call Lt the liability at time t. Liabilities must be met with a

portfolio

∑αiPi

i ∈ U

where αi is the amount of bond i in the portfolio. The CFM problem can

be written, in its simplest form, in the following way:

(^10) For an illustration of cash flow matching applied to pension fund liabilities, see
Frank J. Fabozzi and Peter F. Christensen, “Dedicated Bond Portfolios,” Chapter 45
in Frank J. Fabozzi (ed.), The Handbook of Fixed Income Securities (New York, NY:
McGraw Hill, 2000).