A B Coupon 8% 9% YTM 8% 8% Term (yrs) 2 5 Par 100.00 100.00 Price 100.00 104.055
ID: 2489133 • Letter: A
Question
A
B
Coupon
8%
9%
YTM
8%
8%
Term (yrs)
2
5
Par
100.00
100.00
Price
100.00
104.055
2. [8pts] For the bond B in question 1,
Calculate the actual, exact price change for a 100bp increase in interest rates;
Using duration only, estimate the price of the bonds for a 100bp increase in rates;
Using duration and convexity, estimate the price change of the bonds for a 100bp increase in rates;
Comment on the accuracy of the approximations in (b) and (c) and explain why one is better than the other;
Without an actual calculation, indicate (“explain”) whether the duration of the bond would be higher or lower if the YTM were 10%, rather than 8%. Hint: recall that the price-yield curve is convex for risk- and option-free bonds.
A
B
Coupon
8%
9%
YTM
8%
8%
Term (yrs)
2
5
Par
100.00
100.00
Price
100.00
104.055
Explanation / Answer
(a) Calculate the actual price of the bonds B for a 100-basis-point increase in interest rates.
To price the bonds, recognize we have 8% yields. If we increase the bond yields by 100 basis points, then the new yield is 9%. At a YTM of 9% the prices of bonds B are:
Assuming the interest is paid semi annually. The formula to calculate the price is given below
+
Where
C = coupon amount paid 9% on $100 is $9, semi – annually it is $4.5
R = interest rate 4.5% semi annually
N = period which is 10 years
M= par value $100
PB=+ = $100.00
(b) Using duration, estimate the price of the bonds for a 100-basis-point increase in interest rates.
The first step is to compute modified duration: MD =
Where ,
C = coupon amount paid 9% on $100 is $9, semi – annually it is $4.5
R = interest rate 4.5% semi annually
N = period which is 10 years
M= par value $100
Y= YTM is 8% annual and semiannual is 4%
MDB = = 7.9888
These are modified durations based on semiannual cash flows. To “annualize” them we divide each by 2:
Modified duration for Bond B is 3.9944.
Now we estimate the percentage price change of bonds B assuming a 100 bp yield increase (i.e., dy = 0.01).
For bond B: = –modified duration (dy) = –3.9944(0.01) = –0.039944
This means that duration estimates that bonds B will decline by 3.9944%, respectively, when interest rates increase by 100 bps.
Therefore, the new bond price predicted by duration is:
Bond B: 104.055(1 – 0.039944) = $99.899.
(c) Using both duration and convexity measure, estimate the price of the bonds for a 100-basis-point increase in interest rates.
Now we need to calculate convexity for B.
We have: convexity measure (half years) = =
.
Where
C = coupon amount paid 9% on $100 is $9, semi – annually it is $4.5
R = interest rate 4.5% semi annually
N = period which is 10 years
M= par value $100
Y= YTM is 8% annual and semiannual is 4%
P = price of the bond $104.055
For Bonds B, we get:
CMB = = 79.0547
To “annualize” these convexity measures, we divide by 4:
Convexity measure for Bond B is 19.7637.
Now we estimate the percentage price change using both modified duration and convexity, assuming a 100 bp yield increase (i.e., dy = 0.01).
= –modified duration(dy) + convexity measure(dy)2
For bond B: = –3.9944(0.01) + (19.7637)(0.01)2 = –0.03896
This means that duration and convexity together predict that bonds B will decline by 3.896%, respectively, when interest rates increase by 100 bps.
Therefore, the new bond prices estimated are:
Bond B: 104.055(1 – 0.03896) = $100.0014.
(d) Comment on the accuracy of your results in parts b and c, and state why one approximation is closer to the actual price than the other.
The following table summarizes the results:
Bond B
Actual new price
100.00
Price predicted by duration
99.899
Price predicted by duration and convexity
100.001
It is clear that the new price estimated by duration and convexity together is closer to the actual new price estimated by duration alone. This occurs because duration is based on a linear approximation of a relationship that is inherently convex. Thus, providing an adjustment” for convexity improves the estimate.
However, even with the convexity adjustment, the new price is still an estimate. We can see this when we look at the Taylor Series expansion of the percentage price change in a bond as a function of changes in y:
.
The first term on the RHS of the above equation is modified duration and the second term is the convexity measure. Notice the third term. It contains higher order derivatives (order ³ 3). The number of derivatives in this term is infinite (although each diminishes in importance). The more terms on the RHS we include in our estimate of bond price changes, the better our estimate will be. Since we use only the first two terms means our result will not be exact.
Bond B
Actual new price
100.00
Price predicted by duration
99.899
Price predicted by duration and convexity
100.001
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