Question

A newly issued bond pays its coupons once a year. Its coupon rate is 4.4%, its maturity is 15 years, and its yield to maturity is 7.4%.

a. Find the holding-period return for a one-year investment period if the bond is selling at a yield to maturity of 6.4% by the end of the year. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Holding-period return =17.59

b. If you sell the bond after one year when its yield is 6.4%, what taxes will you owe if the tax rate on interest income is 40% and the tax rate on capital gains income is 30%? The bond is subject to original-issue discount (OID) tax treatment.

	Tax on interest income $21.72 Tax on capital gain Total taxes

c.. What is the after-tax holding-period return on the bond?

d. Find the realized compound yield before taxes for a two-year holding period, assuming that (i) you sell the bond after two years, (ii) the bond yield is 6.4% at the end of the second year, and (iii) the coupon can be reinvested for one year at a 2.4% interest rate.

e.. Use the tax rates in part (b) to compute the after-tax two-year realized compound yield. Remember to take account of OID tax rules.

Answer 1

a. Holding period return = ( Ending Price - Beginning Price + Coupon Earned) / Beginning Price)

Lets find out Beginning Price , Ending price

Let the face value of bond be $1000 * 4

coupon earned = $1000 * 4.4% = $44

Beginning price = 44 * PVAF( 7.4%, 15years) + 1000 PVF(7.4%,15th year)

= 44 * 8.8822 + 1000 * 0.3427

= $733.52

Ending Price = 44 * PVAF( 6.4%, 14years) + 1000 PVF(6.4%,14th year)

= 44 * 9.069 + 1000 * 0.4196

= $818.64

Holding period return = (818.64-733.52+44) / 733.52 = 17.60%

b. Using OID tax rules, the price of the bond under constant yield method is obtained by discounting at an yield 7.4% by taking maturity as 14 years

at YTM = 7.4%, Bond price = 44 * PVAF( 7.4%, 14years) + 1000 PVF(7.4%,14th year)

= 44 * 8.5395 + 1000 * 0.3681

= $743.84

Implicit Interest rate for year 1 = $743.84 - $733.52 = $10.32

Interest income = Coupon earned + Implicit interest

Capital Gain = Actual price - Constant yield price

Tax on Capital Gain = 0.30 (818.64 - 743.84 )

= $22.44

Tax on Interest income = 0.40 ( 44 + 10.32) = $21.73

Total Tax = 22.44 + 21.73 = $44.17

c. After tax Holding rate of return = ( Ending Price - Beginning Price + Coupon Earned- tax) / Beginning Price)

= (818.64 - 733.52 + 44 - 44.17) / 733.52 = 11.58%

d. with 6.4% Ytm and two years holding, Bond price = 44 * PVAF( 6.4%, 13years) + 1000 PVF(6.4%,13th year)

= 44 * 8.6494 + 1000 * 0.4464

= $826.97

Total amount earned on reinvesting coupon amount = $44 + $44 (1+0.024) = $89.06

Total amount earned after two years = $826.97 + $89.06 = $916.03

The realized compound yield (r) =  $916.03 = $733.52 (1 + r )²

r = 11.75%

e.  Using OID tax rules, the price of the bond under constant yield method is obtained by discounting at an yield 7.4% by taking maturity as 13 years

at YTM = 7.4%, Bond price = 44 * PVAF( 7.4%, 13years) + 1000 PVF(7.4%,13th year)

   = 44 * 8.1714 + 1000 * 0.3953

= $754.84

Implicit Interest rate for year 2 = $754.84 - $743.84 = $11

Coupon received in year 1 = $44

Less: Tax on coupon@40% = $17.6

Less: tax on imputed interest =$4.13 ($10.32*0.4)

Cash inflow in year 1 = 44 - 17.6 - 4.13 = $22.27

If you invest year 1 cash flow at an after tax rate of 2.4% * (1-0.4) = 1.44%, it will grow by year 2 to $22.27 * 1.0144 = $22.59

You will sell bond in year 2 @ 6.4% YTM, 13 years for $826.97

Less: Tax on imputed interest@40% on $11 = $4.4

Add: year 2 coupon after tax 44*(1-0.4) = $26.4

Less: capital gain tax on 0.3(826.97-754.84) = $21.64

(Sales price - Constant Yield value)

Add: Cash flow from Y1 (reinvested) = $22.59

Total = 826.97 - 4.4 + 26.4 - 21.64 + 22.59 = $ 849.92

After tax return = $733.52 (1 + r )² = 849.92

r = 7.64%