The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 668 Wednesday, February 4, 2004 1:12 PM

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668 The Mathematics of Financial Modeling and Investment Management

can therefore choose an investment strategy such that the change in a port-

folio’s value is offset by changes in the returns earned by the reinvestment

of the cash obtained through coupon payments or the repayment of the

principal of bonds maturing prior to the liability date.

The principle applies in the case of multiple liabilities. To see how

multiple-period immunization works, let’s first demonstrate that—given

a stream of cash flows at fixed dates—there is one instant at which the

value of the stream is insensitive to small parallel shifts in interest rates.

Consider a case where a sum V 0 is initially invested in a portfolio of

risk-free bonds (i.e., bonds with no default risk) that produces a stream

of N deterministic cash flows Ki at fixed dates ti. At each time ti the sum

Ki is reinvested at the risk-free rate. Suppose that there is only one rate r

common to all periods. The following relationship holds:

V 0 =

N

∑Kie

• rti

i = 1

where we have used the formula for the present value in continuous time.

As each intermediate payment is reinvested, the value of the portfo-

lio at any instant t is given by the following expression:

Vt =

N

∑Kie

• rt ( – ti) rt
  = e V 0
  i = 1

Our objective is to determine a time t such that the value Vt at time

t of the portfolio is insensitive to parallel shifts in the interest rates. The

quantity Vt is a function of the interest rate r. The derivative of Vt with

respect to r must be zero so that Vt is insensitive to interest rate changes.

Let’s compute the derivative:

N

∑

dVt (

(

rt

Kitt– i

• ti)
  = )e
  dr i = 1
  N

∑Kitie

• rti

= tV i =^1

t – Vt -----------------------------V -

0

Kie

N –rti

Vt ∑

 

= – ti

i = 1



 V 0



