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, ...
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