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Q-10-44I

Expert-verified
Found in: Page 333

### Essentials Of Investments

Book edition 9th
Author(s) Zvi Bodie, Alex Kane, Alan Marcus, Alan J. Marcus
Pages 748 pages
ISBN 9780078034695

# 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.

a.19.54%

b. $46.99 c. 12.88% d.$900.32

e. $829.97 See the step by step solution ### Step by Step Solution ## Step 1: Calculation of the holding period return Given, n = 20; PMT = 50; FV = 1,000; i = 8 Initial price p0 =705.46 Next year's price, P1 = 793.29 (n = 19; PMT = 50; FV = 1000; i = 7) Formulae for HPR = (Income + Value at the end of the holding period - Value at the start of the holding period) / Value at the start of the holding period HPR =$ 50 + ($793.29 -$705.46) / $705.46 = 0.1954 = 19.54% ## Step 2: 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

## Step 3: 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% ## Step 4: 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% ## Step 5: 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.