Interest Rates and Rates of Return on Bonds

Core Concepts of Investment Returns and Holding Periods

• Primary Investment Objective: When undertaking an investment, the central focus is measuring what is earned over a specific timeframe, referred to as the holding period.

• Definition of Holding Period: The given period of time over which an investor owns and holds an asset or security.

• Components of a Bond's Return: For an investor who purchases a coupon bond and holds it for a holding period of one year, the total return consists of two distinct components:

  • The coupon payment received during the year.

  • The change in the market price of the bond over the year, which produces either a capital gain (if the price rises) or a capital loss (if the price falls).

• Rate of Return ($R$): The return expressed as a percentage of the initial investment amount.

• Core Learning Objective: Explain the fundamental differences between interest rates and rates of return.

Rate of Return Calculations and Worked Scenarios

• Baseline Investment Parameters (Ford Bond Example):

  • Initial purchase price: $\$1{,}000$

  • Face value (par value): $\$1{,}000$

  • Coupon rate: $5\%$

  • Annual coupon payment: \1{,}000 \times 0.05 = \$50$

• Scenario A: Bond Price Increase (Capital Gain):

  • At the end of the one-year holding period, the market price of the bond increases from $\$1{,}000$ to $\$1{,}169.84$.

  • Capital gain calculation:     $\text{Capital Gain} = \$1{,}169.84 - \$1{,}000 = \$169.84$

  • Total cash flow return:     $\text{Coupon Payment} + \text{Capital Gain} = \$50 + \$169.84 = \$219.84$

  • Rate of return ($R$) calculation:     $R = \frac{\text{Coupon} + \text{Capital gain}}{\text{Purchase price}} = \frac{\$50 + \$169.84}{\$1{,}000} = 0.220\text{, or } 22.0\%$

  • In this scenario, the investor earns a positive rate of return equal to $22.0\%$.

• Scenario B: Bond Price Decrease (Capital Loss):

  • At the end of the one-year holding period, the market price of the bond declines from $\$1{,}000$ to $\$864.09$.

  • Capital loss calculation:     $\text{Capital Loss} = \$864.09 - \$1{,}000 = -\$135.91$

  • Total cash flow return:     $\text{Coupon Payment} + \text{Capital Change} = \$50 - \$135.91 = -\$85.91$

  • Rate of return ($R$) calculation:     $R = \frac{\$50 - \$135.91}{\$1{,}000} = -0.086\text{, or } -8.6\%$

  • Although the investor received the positive $\$50$ coupon payment, the capital loss of $\$135.91$ exceeded the coupon cash flow, resulting in an overall negative rate of return of $-8.6\%$.

General Equation for the Rate of Return on a Bond

• Extension to a General Formula: The rate of return on any coupon bond over a one-year holding period can be decomposed into two distinct percentage rates:

  • Current yield

  • Rate of capital gain (or loss)

• Current Yield Component:

  • Defined as the coupon payment divided by the beginning/initial price of the bond:     $\text{Current Yield} = \frac{\text{Coupon}}{\text{Initial price}}$

• Rate of Capital Gain Component:

  • Defined as the dollar change in bond price over the holding period divided by the beginning/initial price of the bond:     $\text{Rate of Capital Gain} = \frac{\text{Change in price}}{\text{Initial price}}$

• Decomposed Rate of Return Equations:   $\text{Rate of return} = \text{Current yield} + \text{rate of capital gain}$   $R = \frac{\text{Coupon}}{\text{Initial price}} + \frac{\text{Change in price}}{\text{Initial price}}$

Critical Principles Governing Rates of Return

• Price Benchmark for Current Yield: When calculating the rate of return for a holding period, the bond's price at the beginning of the year (initial purchase price) must be used as the denominator to determine the current yield component.

• Realized versus Unrealized Capital Gains and Losses:

  • An investor incurs a capital gain or loss regardless of whether the bond is sold at the conclusion of the holding period.

  • Realized capital gain/loss: Occurs if the bondholder actually sells the bond at the end of the period.

  • Unrealized capital gain/loss: Occurs if the bondholder continues to hold the bond at the end of the period without selling.

  • In both realized and unrealized situations, the market value of the investment has changed; therefore, price appreciation or depreciation must be incorporated into the rate of return calculation.

• Limitations of Current Yield and Yield to Maturity:

  • For a coupon bond, neither the current yield nor the yield to maturity (YTM) serves as an accurate indicator of the actual rate of return earned during a specific holding period.

  • Current yield and yield to maturity fail to account for the impact of future capital gains or capital losses that arise from market price fluctuations during the holding period.