Since it is market convention to quote bond prices in terms of yield-to-maturity, it is more insightful to examine the behaviour of the yield-to-maturity in the case of a very short and very long time to maturity. Since a zero coupon bond is a pure discount instrument, its price is written as:
where R ( r, t, T ) is the yield-to-maturity. Equating (2.27) and (2.39) we derive the yield-to-maturity in terms of A ( t, T ) and B ( t, T )as:
As t ’ T , R ( r, t, T ) ’ r , [21] since as the bond approaches maturity, it converges to an instrument with instantaneous maturity. Now consider the yield-to- maturity as T ’ ˆ . This may be viewed as the yield on a perpetual bond:
Consider ln A ( t, T )where A ( t, T ) is given by (2.38):
Now, since
we have:
Also from (2.36):
and hence the yield on a perpetual bond is [22] :
Hence, for bonds with increasing maturity, the yield approaches a limit independent of current rate of interest, but proportional to the mean reversion level [ 45 ].
As for the Vasicek model, the CIR term structure can assume various shapes according to the level of the current interest rate, r ( t ). See Figure 2.1 below. For , the long-term yield, the term structure is uniformly increasing while for the term structure is uniformly decreasing . For values of r ( t ) lying between these two extremes the term structure is humped.
[21] e x can be approximated by its power series expansion as:
and hence [ 45 ]
Therefore as t ’ T we have:
and also
for small T ˆ’ t . Hence from (2.27), P ( r, t, T ) = e ˆ’ r ( T ˆ’ t ) .
[22] Here make use of the following: