Bond Prices and yields Important Question Solution

Bond Analysis: Holding Period Return and Tax Implications

Analysis of a 20-year bond with 5% coupon rate and 8% yield to maturity

Problem Statement

Given Values:

• Coupon rate (i) = 5%
• Maturity period (n) = 20 years
• Yield to maturity (y) = 8%
• Face value = ₹1,000

Questions:

a. Find the holding period return for a 1-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 1 year, what tax do you owe if the tax rate on interest income is 40% and the tax rate on capital gain income is 30%?

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

d. Find the realized compound yield before taxes for a 2-year holding period, assuming that you sell the bond after 2 years, the bond yield is 7% at the end of the second year, and the coupon can be reinvested for 1 year at 3% interest rate.

e. Find the after-tax realized compound yield for the 2-year holding period.

Solution

Part (a): Holding Period Return

Concept: The holding period return is the total return earned from holding a bond for a specific period, including both price appreciation and coupon income.

Holding period return formula:

\[ \text{HPR} = \frac{(P_1 - P_0) + C}{P_0} \]

Where \(P_0\) is the initial price, \(P_1\) is the price after 1 year, and \(C\) is the coupon payment.

First, calculate the initial bond price (\(P_0\)) at 8% YTM:

\[ P_0 = C \times \text{PVIFA}_{8\%,20} + M \times \text{PVIF}_{8\%,20} \] \[ P_0 = 50 \times 9.8181 + 1000 \times 0.2145 = 705.405 \]

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Next, calculate the price after 1 year (\(P_1\)) at 7% YTM (with 19 years remaining):

\[ P_1 = C \times \text{PVIFA}_{7\%,19} + M \times \text{PVIF}_{7\%,19} \] \[ P_1 = 50 \times 10.3356 + 1000 \times 0.2765 = 793.28 \]

Now compute the holding period return:

\[ \text{HPR} = \frac{(793.28 - 705.405) + 50}{705.405} = \frac{137.875}{705.405} = 0.1955 \]

Holding Period Return = 19.55%

Part (b): Tax Owed

Concept: Under the constant yield method, we need to calculate imputed interest (accretion of bond discount) which is taxed as ordinary income along with the coupon payment. Capital gains are taxed separately.

Calculate the constant yield price after 1 year at original YTM (8%):

\[ P_{1,\text{const}} = C \times \text{PVIFA}_{8\%,19} + M \times \text{PVIF}_{8\%,19} \] \[ P_{1,\text{const}} = 50 \times 9.6036 + 1000 \times 0.2317 = 711.88 \]

Imputed taxable interest:

\[ \text{Imputed Interest} = P_{1,\text{const}} - P_0 = 711.88 - 705.405 = 6.475 \]

Tax on ordinary income (coupon + imputed interest at 40%):

\[ \text{Tax} = (50 + 6.475) \times 0.40 = 22.59 \]

Capital gain calculation:

\[ \text{Capital Gain} = P_1 - P_{1,\text{const}} = 793.28 - 711.88 = 81.4 \]

Tax on capital gain (at 30%):

\[ \text{Tax} = 81.4 \times 0.30 = 24.42 \]

Total tax owed:

\[ \text{Total Tax} = 22.59 + 24.42 = 47.01 \]

Total Tax Owed = ₹47.01

Part (c): After-Tax Holding Period Return

After-tax HPR formula:

\[ \text{After-tax HPR} = \frac{(P_1 - P_0) + C - \text{Tax}}{P_0} \]

\[ \text{After-tax HPR} = \frac{(793.28 - 705.405) + 50 - 47.01}{705.405} \] \[ = \frac{90.865}{705.405} = 0.1288 \]

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After-Tax Holding Period Return = 12.88%

Part (d): Realized Compound Yield Before Taxes (2-Year Holding)

Concept: Realized compound yield calculates the equivalent annual return considering reinvestment of coupons and the sale price of the bond.

Calculate bond price after 2 years (\(P_2\)) at YTM = 7% (18 years remaining):

\[ P_2 = C \times \text{PVIFA}_{7\%,18} + M \times \text{PVIF}_{7\%,18} \] \[ P_2 = 50 \times 10.0591 + 1000 \times 0.2959 = 798.86 \]

Compute future value of coupons reinvested at 3%:

• Coupon at year 1: \(50 \times (1.03) = 51.50\)
• Coupon at year 2: \(50\)
• Sale proceeds at year 2: \(798.86\)

\[ \text{FV} = 51.50 + 50 + 798.86 = 900.36 \]

Solve for realized yield (\(R\)):

\[ P_0 (1 + R)^2 = \text{FV} \] \[ 705.405 (1 + R)^2 = 900.36 \] \[ (1 + R)^2 = \frac{900.36}{705.405} = 1.2764 \] \[ 1 + R = \sqrt{1.2764} = 1.1298 \] \[ R = 1.1298 - 1 = 0.1298 \]

Realized Compound Yield Before Taxes = 12.98%

Part (e): After-Tax Realized Compound Yield (2-Year Holding)

After-tax coupons:

• Year 1: \(50 \times (1 - 0.40) = 30\)
• Year 2: \(50 \times (1 - 0.40) = 30\)

Reinvest year 1 coupon at 3%:

\[ 30 \times (1.03) = 30.9 \]

After-tax sale proceeds:

\[ \text{Capital Gain} = P_2 - P_0 = 798.86 - 705.405 = 93.455 \] \[ \text{Tax on Capital Gain} = 93.455 \times 0.30 = 28.0365 \] \[ \text{After-tax Proceeds} = 798.86 - 28.0365 = 770.8235 \]

Total future value after taxes:

\[ \text{FV} = 30.9 + 30 + 770.8235 = 831.7235 \]

Solve for after-tax realized yield (\(R\)):

\[ 705.405 (1 + R)^2 = 831.7235 \] \[ (1 + R)^2 = \frac{831.7235}{705.405} = 1.17907 \] \[ 1 + R = \sqrt{1.17907} = 1.0859 \] \[ R = 1.0859 - 1 = 0.0859 \]

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After-Tax Realized Compound Yield = 8.59%