Question

Question: A newly issued bond pays its coupons once a year. Its coupon rate is 5%, its maturity is 20 years, and its yield to maturity is 8%.

a. Find the holding-period return for a one-year investment period if the bond is selling at a yield to maturity of 7% by the end of the year.

b. If you sell the bond after one year when its yield is 7%, 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.

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 7% at the end of the second year, and (iii) the coupon can be reinvested for one year at a 3% 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.

Short answer

Answer

a.19.54%

b. $46.99

c. 12.88%

d. $900.32

e. $829.97

Step-by-step solution

Calculation of the holding period return

Given,n= 20; PMT = 50; FV = 1,000; i= 8

Initial price p₀=705.46

Next year's price, P₁ = 793.29 (n = 19; PMT = 50; FV = 1000; i = 7)

Formulae for HPR = (Income + Value at the end of the holding period - Value at the start of the holding period) / Value at the start of the holding period

HPR = $ 50 + ($ 793.29 - $705.46) / $705.46

= 0.1954

= 19.54%

Calculation of taxes after one year

As per OCD rules, the cost basis and imputed interest under the constant yield method are obtained by discounting bond payments at the original 8% yield to maturity, and simply reducing maturity by one year at a time:

Here,

P₀ = $705.46

P₁ = $711.89 so implicit interest over first year = $711.89 - $705.46 = $6.43

P₂ = $718.84 so implicit interest over second year = $718.84 - $711.89 = $6.95

Coupon received and imputed taxable interest over the first year is treated as ordinary income = 40% x ($50 + $6.43) = $22.57

Capital gain = Actual price at 7% YTM – Constant yield price

= $793.29 - $711.89

= 81.40

Tax on capital gain at 30% = 30% x $81.40 = $24.42

Total taxes = $22.57 + $24.42 = $46.99

Calculation of after tax period holding return 

Formulae for HPR = (Income + Value at the end of the holding period - Value at the start of the holding period) – Total taxes / Value at the start of the holding period

HPR = $ 50 + ($ 793.29 - $705.46) – $46.99 / $705.46

= 0.1288

= 12.88%

Calculation of realized compound yield before taxes

Value of bond after two years = $798.82 where n = 18; i = 7

Total income from the two coupons, including reinvestment income: ($50 x 1.03) + $50 = $101.50

Total funds after two years: $798.82 + $101.50 = $900.32

Therefore, the fund after two years = $900.32.

Þ705.46 (1 + r)² = 900.32

r = 0.1297 = 12.97%

Calculation of after-tax two-year realized compound yield

Coupon received in first year: $50.00

Tax on coupon @ 40% = – 20.00

Tax on imputed interest (0.40 x $6.43) = – 2.57

Net cash flow in first year = $27.43

If you invest the year-1 cash flow at an after-tax rate of: 3% x (1 – 0.40) = 1.8%

Then, by year 2, it will grow to: $ 27.43 x 1.018 = $27.92

You sell the bond in the second year for: $798.82

Tax on imputed interest in second year: – 2.78 [0.40 x $6.95]

Coupon received in second year, net of tax: + 30.00 [$50 x (1 – 0.40)]

Capital gains tax on sales price using constant yield value: – 23.99 [0.30 x ($798.82 – $718.84)]

CF from first year's coupon (reinvested): + 27.92 [from above]

TOTAL $829.97

Therefore, after two years, fund grows to $829.97.