Question

Clarrie has just bought a 14-year Treasury bond paying coupon semi-annually
at j2 = 5% p.a. The bond matures at par.
a. Find Clarrie’s purchase price (per $100 face value, rounded
to 3 decimal places) of this Treasury bond, allowing for a 30% tax on
interest only, to give a yield of j2 = 3.2% p.a. (net). Draw a cash flow
diagram that models this scenario to accompany your answer.
b. Find Clarrie’s purchase price (per $100 face value, rounded
to 3 decimal places) of this Treasury bond, allowing for a 30% tax
on interest only. The tax on interest is paid one year later (e.g., for
the coupon payment at t = 0.5 year, the tax payment will be paid at
t = 1.5 years.), to give a yield of j2 = 3.2% p.a. (net). Draw a cash
flow diagram that models this scenario to accompany your answer.
c. Justify the difference in your answers to parts a. and b.
above.
d. If Clarrie paid $95.268 per $100 face value for the bond, and
was exempt from tax, what yield was associated with his purchase?
Use linear interpolation to find this yield and express your yield as a
j2 rate, to one decimal place.

Its all the question information

Answer 1

a)

FV: face value = 100

coupon rate = 5% pa paid semi-annually

semi-annual coupon = 5%*100/2 = 2.5

c: semi-annual coupon after tax = 2.5*(1-30%) = 1.75

yield = 3.2% pa = 1.6% semi-annual

time = 14years = 28 semi-annual years

P: price of bond

P = (c/y)*(1-1/(1+y)^t)+FV/(1+y)^t

P = (1.75/1.6%)*(1-1/(1+1.6%)^28)+100/(1+1.6%)^28 = 103.36

Year	0.5	1	1.5.........	14
Cash flow	1.75	1.75	1.75	1.75+100 = 101.75

(b)

If tax payments are paid after 1-year

P*: price of bond without considering tax payments

P* = (2.5/1.6%)*(1-1/(1+1.6%)^28)+100/(1+1.6%)^28 = 120.18

Tax paid on coupon = 2.5*30% = 0.75

Present value of tax payments = (1/(1+1.6%)^2)*((0.75/1.6%)*(1-1/(1+1.6%)^28)) = 16.29

Price of bond after tax payments = 120.18-16.29 = 103.89

Year	0.5	1	1.5......	14	14.5
Cash inflow	2.5	2.5	2.5	100+2.5 = 102.5
Cash outflow			0.75	0.75	0.75
Net cash-flow	2.5	2.5	1.75	101.75	-0.75

(c)

The price of bond in part b is 103.89 & the price of bond in part a is 103.36

The price of bond in part b has increased to delay in tax payment, thereby reducing its time value of money

(d)

Let the yield be equal to y (semi-annual)

Price of bond = 95.268

95.268 = (2.5/y)*(1-1/(1+y)^28)+100/(1+y)^28

Solving the above equation, we get y = 2.74%

Yield on bond = 2.74%*2 = 5.48% pa compounded semi-annually