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Solutions to Homework 3

FM 5021 Mathematical Theory Applied to Finance

4.1 A bank quotes you an interest rate of 14% per annum with quarterlycompounding. What is the equivalent rate with (a) continuous compoundingand (b) annual compounding?(a) The rate with continuous compounding is

4 ln

1 +

0.14

4

= 0.1376,

i.e. 13.76% per annum.

(b) The rate with annual compounding is1 +

0.14

4

4 1 = 0.1475,

i.e. 14.75% per annum.

4.3 The 6-month and 1-year zero rates are both 10% per annum. For abond that has a life of 18 months and pays a coupon of 8% per annum (withsemiannual payments and one having just been made), the yield is 10.4% perannum. What is the bonds price? What is the 18-month zero rate? All rates

are quoted with semiannual compounding.Suppose that the bond has a face value of $100. There would be a cash flow of $4 after6 months, $4 after 1 year, and $104 (coupon plus priciple) after 18 months. The price ofthe bond is obtained by discounting these cash flows at 10 .4% per annum. The price is,therefore,

4

1 + 0.1042

+4

1 + 0.1042

2

+104

(1 + 0.1042

)3=

4

1.052+

4

1.0522+

104

1.0523= $96.74.

If the 18-month zero rate is R, we must have

41 + 0.1

2

+ 41 + 0.1

2

2

+ 1041 + R

2

3

= 96.74,

which gives R = 0.1042, i.e. R = 10.42%.

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4.5 Suppose that zero interest rates with continuous compounding are asfollows:

Maturity (months) Rate (% per annum)

3 8.06 8.29 8.4

12 8.515 8.618 8.7

Calculate forward interest rates for the second, third, fourth, fifth, and sixthquarters.Using formula (4.5) on page 85, we obtain

RF for the second quarter = 8.2(0.5) 8.0(0.25)

0.5 0.25= 8.4%

RF for the third quarter =8.4(0.75) 8.2(0.5)

0.75 0.5= 8.8%

RF for the fourth quarter =8.5(1) 8.4(0.75)

1 0.75= 8.8%

RF for the fifth quarter =8.6(1.25) 8.5(1)

1.25 1= 9.0%

RF for the sixth quarter =8.7(1.5) 8.6(1.25)

1.5 1.25= 9.2%

4.7 The term structure of interest rates is upward-sloping. Put the followingin order of magnitude:(a) The 5-year zero rate(b) The yield on a 5-year coupon-bearing bond(c) The forward rate corresponding to a period between 5 and 5.25 years in thefutureWhat is the answer to this question when the term structure of interest ratesis downward-sloping?

When the term structure of interest rates is upward-sloping, (b) < (a) < (c). When theterm structure of interest rates is downward-sloping, (c) < (a) < (b).

4.12 A 3-year bond provides a coupon of 8% semiannually and has a cashprice of 104. What is the bonds yield?The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond yieldis the value ofy that solves

4e0.5y + 4e1.0y + 4e1.5y + 4e2.0y + 4e2.5y + 104e3.0y = 104

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Solving this nonlinear equation numerically (using Newtons method, for example), youshould obtain that y = 0.06407, i.e. 6.407%.

4.22 A 5-year bond with a yield of 11% (continuous compounded) pays an8% coupon at the end of each year.(a) What is the bonds price?(b) What is the bonds duration?(c) Use the duration to calculate the effect on the bonds price of a 0.2% decreasein its yield.(d) Recalculate the bonds price on the basis of a 10.8% per annum yield andverify that the result is in agreement with your answer to (c).(a) The bonds price is

8e0.11 + 8e0.112 + 8e0.113 + 8e0.114 + 108e0.115 = $86.80

(b) The bonds duration is

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86.80[8e0.11 + 2 8e0.112 + 3 8e0.113 + 4 8e0.114 + 5 108e0.115] = 4.256 years

(c)B = BDy = 86.80 4.256 (0.002) = 0.74.

Therefore, the bonds price would increase from $86.80 to $87.54.(d) With a 10.8% yield the bonds price is

8e0.108 + 8e0.1082 + 8e0.1083 + 8e0.1084 + 108e0.1085 = $87.54,

which agrees with our answer for (c).

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