Question

Using semiannual compounding, find the prices of the following bonds:

a. A 10.9%, 15-year bond priced to yield 9.39%.

b. A 5.5 %, 10-year bond priced to yield 7.1 %.

c. An 11.8 %, 20-year bond priced at 10.2 %.

Repeat the problem using annual compounding. Then comment on the differences you found in the prices of the bonds.

Answer 1

Semi annual compounding

a)

Assuming the face value of bond is $1,000.

Coupon = (0.109 * 1000)/2 = 54.5

Number of periods = 15 * 2 = 30

Rate = 9.39% / 2 = 4.695%

Price of bond = Coupon * [1 - 1 / (1 + r)ⁿ] / r + FV / (1 + r)ⁿ

Price of bond = 54.5 * [1 - 1 / (1 + 0.04695)³⁰] / 0.04695 + 1,000 / (1 + 0.04695)³⁰

Price of bond = 54.5 * 15.921683 + 252.476966

Price of bond = $1,120.21

Keys to use in a financial calculator: 2nd I/Y 2, FV 1000, PMT 54.5, I/Y 9.39, N 30, CPT PV

b)

Assuming the face value of bond is $1,000.

Coupon = (0.055 * 1000)/2 = 27.5

Number of periods = 10 * 2 = 20

Rate = 7.1% / 2 = 3.55%

Price of bond = Coupon * [1 - 1 / (1 + r)ⁿ] / r + FV / (1 + r)ⁿ

Price of bond = 27.5 * [1 - 1 / (1 + 0.0355)²⁰] / 0.0355 + 1,000 / (1 + 0.0355)²⁰

Price of bond = 27.5 * 14.148318  + 497.734719

Price of bond = $886.81

Keys to use in a financial calculator: 2nd I/Y 2, FV 1000, PMT 27.5, I/Y 7.1, N 20, CPT PV

c)

Assuming the face value of bond is $1,000.

Coupon = (0.118 * 1000)/2 = 59

Number of periods = 20 * 2 = 40

Rate = 10.2% / 2 = 5.1%

Price of bond = Coupon * [1 - 1 / (1 + r)ⁿ] / r + FV / (1 + r)ⁿ

Price of bond = 59 * [1 - 1 / (1 + 0.051)⁴⁰] / 0.051 + 1,000 / (1 + 0.051)⁴⁰

Price of bond = 59 * 16.926693 + 136.738672

Price of bond = $1,135.41

Keys to use in a financial calculator: 2nd I/Y 2, FV 1000, PMT 59, I/Y 10.2, N 40, CPT PV

Annual compounding

a)

Assuming the face value of bond is $1,000.

Coupon = 0.109 * 1000 = 109

Number of periods = 15

Rate = 9.39%

Price of bond = Coupon * [1 - 1 / (1 + r)ⁿ] / r + FV / (1 + r)ⁿ

Price of bond = 109 * [1 - 1 / (1 + 0.0939)¹⁵] / 0.0939 + 1,000 / (1 + 0.0939)¹⁵

Price of bond = 109 * 7.878413 + 260.217

Price of bond = $1,118.96

Keys to use in a financial calculator: FV 1000, PMT 109, I/Y 9.39, N 15, CPT PV

b)

Assuming the face value of bond is $1,000.

Coupon = 0.055 * 1000 = 55

Number of periods = 10

Rate = 7.1%

Price of bond = Coupon * [1 - 1 / (1 + r)ⁿ] / r + FV / (1 + r)ⁿ

Price of bond = 55 * [1 - 1 / (1 + 0.071)¹⁰] / 0.071 + 1,000 / (1 + 0.071)¹⁰

Price of bond = 55 * 6.99123 + 503.622694

Price of bond = $888.14

Keys to use in a financial calculator: FV 1000, PMT 55, I/Y 7.1, N 10, CPT PV

c)

Assuming the face value of bond is $1,000.

Coupon = 0.118 * 1000 = 118

Number of periods = 20

Rate = 10.2%

Price of bond = Coupon * [1 - 1 / (1 + r)ⁿ] / r + FV / (1 + r)ⁿ

Price of bond = 118 * [1 - 1 / (1 + 0.102)²⁰] / 0.102 + 1,000 / (1 + 0.102)²⁰

Price of bond = 118 * 8.398625 + 143.340234

Price of bond = $1,134.38

Keys to use in a financial calculator: FV 1000, PMT 118, I/Y 10.2, N 20, CPT PV

When the bond is priced at premium i.e when the price is more than face value, price with annual compounding is lower.

When the bond is priced at discount i.e when the price is less than face value, price with annual compounding is higher.