HullOFOD8eSolutionsCh04(3)

4e?0?040405?0?5?4e?0?051293?1The solution to this equation isbelow

Maturity (yrs) 0.5 1.0 1.5 2.0 Zero Rate (%) 4.0405 5.1293 5.4429 5.8085 ?0?054429?1?5?R?2?4e?104e?104

R?0?058085. These results are shown in the table

Forward Rate (%) 4.0405 6.2181 6.0700 6.9054 Par Yield (s.a.%) 4.0816 5.1813 5.4986 5.8620 Par yield (c.c %) 4.0405 5.1154 5.4244 5.7778

b) The continuously compounded forward rates calculated using equation (4.5) are shown in the third column of the table

c) The par yield, expressed with semiannual compounding, can be calculated from the formula in Section 4.4. It is shown in the fourth column of the table. In the fifth column of the table it is converted to continuous compounding

d) The price of the bond is

?3?5eThe yield on the bond, y

3?5e?y?0?53?5e?0?040405?0?5?0?051293?1?3?5e?0?054429?1?5?103?5e?y?2?0?0?058085?2?102?13

satisfies

?y?1?0?3?5eThe solution to this equation is

?3?5e?103?5e?102?13

y?0?057723. The bond yield is therefore 5.7723%.

?y?1?5

Problem 4.33.

Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year zero-coupon bond with a face value of $5,000. The current yield on all bonds is 10% per annum.

(a) Show that both portfolios have the same duration.

(b) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same.

(c) What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields?

a) The duration of Portfolio A is

1?2000e?0?1?1?10?6000e?6000e?0?1?102000e?0?1?1?0?1?10?5?95

Since this is also the duration of Portfolio B, the two portfolios do have the same duration.

b) The value of Portfolio A is

2000e2000e?0?1?6000e?6000e?0?1?10?4016?95 ?3993?18

When yields increase by 10 basis points its value becomes

?0?101?0?101?10The percentage decrease in value is

23?77?1004016?95?0?59%

The value of Portfolio B is

5000e5000e?0?1?5?95?2757?81 ?2741?45 ?0?59%

When yields increase by 10 basis points its value becomes

?0?101?5?95The percentage decrease in value is

16?36?1002757?81The percentage changes in the values of the two portfolios for a 10 basis point increase in yields are therefore the same.

c) When yields increase by 5% the value of Portfolio A becomes

2000e?0?15?6000e?0?15?5?95?0?15?10?3060?20

and the value of Portfolio B becomes

5000e956?754016?95709?662757?81?2048?15

The percentage reductions in the values of the two portfolios are:

PortfolioA?PortfolioB??100?23?82

?100?25?73Since the percentage decline in value of Portfolio A is less than that of Portfolio B, Portfolio A has a greater convexity.

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