Question

Nathan purchased a 5-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 four and a half years.
a. [2 marks] Calculate Nathan’s purchase price for this bond at a yield
rate of j2 = 3.1% p.a. (rounded to three decimal places).
b. [4 marks] 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. [3 marks] Assume that Nathan held this bond for 2 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.
d. [2 marks] Without any further calculations, explain how the holding
period yield will change if the sale yield is lower than j2 = 3.8% p.a.
e. [4 marks] Assume that this bond is subject to a 30% tax on interest and
capital gains. Recalculate the price Nathan paid (in part a) if the net
yield rate is j2 = 3% p.a. and all tax payments (interest tax payments
and capital gains tax payment) are delayed by one year from when
taxable cash flows occur. Round your result to three decimal places.
Accompany your answer with a cash flow diagram, from Nathan’s perspective,
that models this scenario.

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%)^3.5	3.91
2	3.5	3.5 * (1+3.2%)^2.5	3.79
3	3.5	3.5 * (1+3.2%)^1.5	3.67
4	3.5	3.5 * (1+3.2%)^0.5	3.56
5	103.5	103.5 *1	103.50

Reinvested Amount at end of 4.5 years = 14.93 discounted at 3.1% yield rate = 13

Amount received at end of 5 year ( Maturity value and last coupon payment) = 103.5 discounted at 3.1% yield rate = 88.85

Purchase Price = 13 + 88.85 = $101.85

b)

Compounded yield given = 3.13% p.a

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

3.13% = (FV/ 101.85)^(1/5) -1

(1.0313)^5*101.85 = FV

FV= 118.82

Amount received at end of 5 year wo'nt be reinvested thus deducting 103.5

we get = 15.32

Reinvested amount at end of 4.5 years is 14.93

thus Reinvested rate after 4.5 years equals to

FV = PV (1+R)^T

15.32 = 14.93 *(1+R)^0.5

on solving we get R = 1.3% (Approax)

C)

Yield = 3.8% thus Selling price is = 101.85(1+3.8%)^2

Selling price is thus = $109.74

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

= 3.5*2 + (109.74-101.85) /101.85*100

= (7 + 7.89 )/ 101.85 *100

= 14.62%

Cash Flow Diagram

d.

If the Sale yield is lower than 3.8%, Holding period return would also be lower.

e.

Yield = [Intt (1- tax rate) + (Maturity Value- (maturity Value -Price)*Tax rate - Price) divided by Term to maturity ] divided by Average of Maturity Value and Initial Price

3% = 3.5%(1-30%) + (100- (100- Price)*30% - Price)/ 5 divided by (100+ price)/2

3% = 2.45% +(100 -(100-P)*0.3)/5 divided by (100+P)/2

1.5+ 0.015P = 0.0245 + (100- 30+0.3P)/5

1.5 +0.015P = 0.0245 + 14 + 0.06P

12.5245 = 0.045P

p = 278.32

Happy Learning.