Interest Rate Risk Dynamics

As a final application, think of an investor who purchases a bond with no intention to hold it to maturity. Then the investor is exposed to risk because the price of the bond may fall and the yield will rise between the time of purchase and the time of sale. The question is how much will the price change by? Duration will not help here because the bond is not held to maturity; rather it is liquidated at some intermediate point . To understand the risk, consider a $1 par bond (that is, , and and thus, ).

Clearly, . Now, consider what happens if is allowed to change to, say, , instantaneously.

which, upon solving for P, gives us:

Collecting terms and simplifying,

We began with (the par bond) and end up with a situation in which c and are no longer equal. Since originally at par, then

and the change in price must be this quantity minus , or

Now, to complicate things, let's generalize this so that r changes to at time (k periods from now). That is, has changed to over a period of time equal to k periods. Then the n period bond is now a period bond. If this is the case, then the change in the price of the bond is, in general:

Go to the companion website for more details (see Interest Rate Risk under Chapter 2 Examples).