The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 669 Wednesday, February 4, 2004 1:12 PM

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Bond Portfolio Management 669

From this expression it is clear that the derivative

dVt

dr

is zero at a time horizon equal to the portfolio duration. In fact, the

quantity

N 

Kie

• rti

∑ ti----------------

i = 1  V 0 

is the portfolio’s duration expressed in continuous time.

Therefore, if the term structure of interest rates is flat, we can match

a given liability with a portfolio whose duration is equal to the time of

the liability and whose present value is equal to the present value of the

liability. This portfolio will be insensitive to small parallel shifts of the

term structure of interest rates.

We can now extend and generalize this reasoning. Consider a

stream of liabilities Lt. Our objective is to match this stream of liabili-

ties with a stream of cash flows from some initial investment insensitive

to changes in interest rates. First we want to prove that the present

value of liabilities and of cash flows must match. Consider the frame-

work of CMF with reinvestment but no borrowing:

∑αiKit, + (^1 + ρt)rt – 1 = Lt + rt

i ∈ U

∑αiKit, – Lt ≥^0

i ∈ U

ai ≥ 0; i ∈ U

We can recursively write the following relationships:

∑αiKi, 1 – Lt = r 1

i ∈ U

∑αiKi, 2 + (^1 + ρ 2 )∑ αiKi, 1 = (^1 + ρ 2 )L 1 + L 2 + r 2

i ∈ U i ∈ U