4-PrincipCalculus Page 125 Friday, March 12, 2004 12:39 PM
Principles of Calculus 125our bond valuation equation, this means C = $4.5, M = $100, and i =
0.06. Substituting these values into the bond valuation equation, the
price of the bond is $134.6722.
Suppose that we want to know the approximate percentage price
change if the interest rate (i.e., i) increases instantaneously from 6% to
8%. In the bond market, a change in interest rates is referred to in terms
of basis points. One basis point is equal to 0.0001 and therefore 1 per-
centage point is 100 basis points. In our illustration we are looking at
an instantaneous change in interest rates of 200 basis points. We will
use the two terms of the Taylor expansion series to show the approxi-
mate percentage change in the bond’s value for a 200 basis point
increase in interest rates.
We do know what the answer is already. The initial value for this
bond is $134.6722. If the interest rate is 8%, the value of this bond
would be $109.8964. This means that the bond’s value declines by
18.4%. Let’s see how well the Taylor expansion series using only two
terms approximates this change.
The first approximation is the estimate using duration. The duration
for this bond is 10.66 found by using the formula above for duration.
The convexity measure for this bond is 164.11 The change in interest
rates, di, is 200 basis points. Expressed in decimal it is 0.02. The first
term of the Taylor expansion series gives–10.66 × (0.02) = –0.2132 = –21.32%Notice that this approximation overestimates the actual change in
value, which is –18.4% and means that the estimated new value for the
bond is underestimated.
Now we add the second approximation. The second term of the
Taylor series gives¹₂(164.11) × (0.02)^2 = 3.28%The approximate percentage change in the bond’s value found by using
the first term of the Taylor series and the second term of the Taylor series
is –21.32% + 3.28% = –18.0%. The actual percentage change in value is
–18.4%. Thus the two terms of the Taylor series do an excellent job of
approximating the percentage change in value.
Let’s look at what would happen if the change in interest rates is a
decline from 6% to 4%. The exact percentage change in value is +25.04%
(from 134.6722 to 168.3887). Now the change in interest rates di is –0.02.
Notice that the approximate change in value due to duration is the same
except for a change in sign. That is, the approximate change based on the