Question

Nathan purchased a 6-year Treasury bond with a coupon rate of j2 = 3.50% p.a. and a face value of $100 that matures at par. Coupons can be reinvested at j2 = 3.2% p.a. for the first five and a half years.

a. Calculate Nathan’s purchase price for this bond at a yield rate of j2 = 3.1% p.a. (rounded to three decimal places).

b. Assume that Nathan held this bond to maturity to earn a total realised compound yield of j2 = 3.13% p.a. Based on your result from part a, calculate the reinvestment rate for the last half year. Give your answer in j2 form, rounded to two decimal places.

c. Assume that Nathan held this bond for 3 years and sold it for a yield of j2 = 3.8% p.a. Based on your result from part a, calculate the holding period yield in j2 form, rounded to two decimal places. Include in your answer a cash flow diagram, drawn from Nathan’s perspective, that models the purchase and sale of the bond.

Answer 1

a) Purchase price is the discounted cash flows received in the future.

Year	Cash flows	Reinvested @3.2%	Reinvested @3.2%
1	3.5	3.5 * (1+3.2%)^4.5	4.03
2	3.5	3.5 * (1+3.2%)^3.5	3.91
3	3.5	3.5 * (1+3.2%)^2.5	3.79
4	3.5	3.5 * (1+3.2%)^1.5	3.67
5	3.5	3.5 * (1+3.2%)^0.5	3.56
6	103.5	103.5 *1	103.50

Reinvested Amount at end of 5.5 years = 18.95 discounted at 3.1% yield rate = 16.023

Amount received at end of 6 year ( Maturity value and last coupon payment) = 103.5 discounted at 3.1% yield rate = 86.177

Purchase Price = 16.023 + 86.177 = $102.2

b)

Compounded yield given = 3.13% p.a

which equals to = (Future Value/ Present Value)^(1/n) -1

3.13% = (FV/ 102.2)^(1/6) -1

(1.0313)^6*102.2 = FV

FV= 122.96

Amount received at end of 6 year wo'nt be reinvested thus deducting 103.5

we get = 19.46

Reinvested amount at end of 5.5 years is 18.95

thus Reinvested rate after 5.5 years equals to

FV = PV (1+R)^T

19.46 = 18.95 *(1+R)^0.5

on solving we get R = 5.41%

C)

Yield = 3.8% thus Selling price is = 102.2(1+3.8%)^3

Selling price is thus = $114.3

Holding Period return = Income + (Closing Value - Opening Value) / Opening Value *100

= 3.5*3 + (114.3-102.2) /102.2*100

= (10.5 + 12.1 )/ 102.2 *100

= 22.11%

Cash Flow Diagram

Happy Learning!