Question

Wan purchased a 7-year Treasury bond with a coupon rate of j₂ = 4.5% p.a. and a face value of $100 that matures at par and is subject to a 30% tax on interest and capital gain. The purchase price was $94.230.

a. Use the approximate bond yield formula to estimate the net yield rate. Give your answer in j₂ form, rounded to 3 decimal places.

b. Use linear interpolation to calculate the net yield rate. Give your answer in j₂ form, rounded to 3 decimal places. Hint: 1.9% per half year and 2% per half year are the lower bound and the upper bound for the net yield rate.

c. Recalculate the bond price if the net yield rate is j₂ = 4.3% p.a. and all tax payments (interest tax payments and capital gain tax payment) are delayed by half year. Round the result to 3 decimal places.

d. Wan decides to hold this bond to maturity. Over the seven years the before-tax reinvestment rates he earned are shown in table 1. Calculate Wan's total realised compound yield rate if he has received a tax exemption and so does not need pay the taxes for this bond. Assume that Wan purchased this bond at a yield rate of j₂ = 4.3% p.a. and the purchase price was $101.20. Give your answer in j₂ form, rounded to 2 decimal places.

Annual reinvestment rates as below:

Year 1 - Year 2	j2 = 4.3% p.a.
Year 3 - Year 7	j2 = 4.7% p.a.

Answer 1

a. Net Yield Rate

Yield Rate = [C + (F - P)/ n] / (F + P)/2

C= Coupon rate

F = Maturity Value

P = Purchase Price

n = Number of years

After tax Coupon Rate = [ 4.5 * ( 1 - 0.30 ) ] = 3.15 %

After tax maturity price = 100 - ( 100 - 94.23 ) * ( 0.30 ) = 98.269

After tax capital gain = 98.269 - 94.23 = 4.039

Yield Rate = [ 3.15 + ( 98.269 - 94.23 ) / 7 ] / (98.269 + 94.230) /2

= [ 3.15 + 0.577 ] / 96.25

= 3.727 / 96.25

= 0.03872 or 3.872 % p.a.

b. Net yield rate by linear interpolation method

Present Value of bond at lower bound rate of 1.9% per half year

PV of bond = Coupon amount * PVIAF (1.9%, 14 half year) + Maturity Price * PVF (1.9%, 14 half year)

= [ ( 3.15 / 2 ) * 12.192 ] + [ 98.269 * 0.768 ]

= [ 19.202 ] + [ 75.471 ]

= 94.673

Present Value of bond at upper bound rate of 2% per half year

PV of bond = Coupon amount * PVIAF (2%, 14 half year) + Face value * PVF (2%, 14 half year)

= [ ( 3.15 / 2 ) * 12.106 ] + [ 98.269 * 0.758 ]

= [ 19.067 ] + [ 74.488 ]

= 93.555

Net yield rate = ( 3.15 / 2 ) + [ 94.673 / ( 94.673 - 93.555) ] * ( 2 - 1.9 )%

= 1.575 + ( 94.673 / 1.118 ) * 0.1%

= 1.575 + 0.084

= 1.659 % per half year

= 3.318 % p.a

c. Value of bond

Let the value of bond be x. Using the formula of net yield we get

4.3 = 4.5 * [( 100 - x ) / 7] / [(100 + x) / 2

thus x = 101.2

Value of bond = 101.2

d. Compound yield rate

Amount to be received at Year 7 from amount reinvested in year 1 = 4.3 * ( 1.043) ^ 6 = 5.54

Amount to be received at Year 7 from amount reinvested in year 2 = 4.3 * ( 1.043) ^ 5 = 5.31

Amount to be received at Year 7 from amount reinvested in year 3 = 4.3 * ( 1.047) ^ 4 = 5.17

Amount to be received at Year 7 from amount reinvested in year 4 = 4.3 * ( 1.047) ^ 3 = 4.94

Amount to be received at Year 7 from amount reinvested in year 5 = 4.3 * ( 1.047) ^ 2 = 4.71

Amount to be received at Year 7 from amount reinvested in year 6 = 4.3 * ( 1.047) ^ 1 = 4.5

Amount to be received at Year 7 = 100 (redemption value of bond) + 4.3 = 104.3

Total receipt at year 7 = 5.54 + 5.31 + 5.17 + 4.94 + 4.71 + 4.5 + 104.3 ( redemption value )

= 134.47

Compound yield rate = ( Total receipt at year 7 / Face Value of bond ) ^ ( 1 / period of bond ) - 1

= (134.47 / 100 ) ^ 1/7 - 1

= 4.32 % p.a