Interest-Rate Risk and Maturity

Fundamentals of Interest-Rate Risk

  • Definition of Interest-Rate Risk: Interest-rate risk refers to the risk that the price of a financial asset will fluctuate in response to changes in market interest rates.

  • Impact on Bondholders: When market interest rates rise, holders of existing fixed-income bonds suffer a capital loss because their securities pay lower coupon rates relative to newly issued bonds in the prevailing market.

  • Maturity Differential: Bonds are not equally vulnerable to interest-rate risk:

    • Bonds with fewer years to maturity experience significantly less price fluctuation when market interest rates change.

    • Bonds with more years to maturity experience much greater price sensitivity and fluctuation.

  • Economic Mechanism:

    • The longer the duration until a bond matures, the greater the number of years the investor is locked into receiving a coupon rate that sits below current market rates.

    • Because future cash flows will be below market levels for a prolonged period, prospective buyers require a substantial discount, driving down the price they are willing to pay for existing debt securities.

Pricing Mechanics and Rate of Return Calculations

  • Hypothetical Portfolio Scenario:

    • Initial investment: Purchase of a bond at par value at the beginning of the year (t=0t = 0).

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

    • Initial purchase price (P0P_0): $1,000\$1{,}000.

    • Coupon rate: 5%5\%, yielding an annual coupon payment (CC) of $50\$50 ($1,000×0.05\$1{,}000 \times 0.05).

    • Initial current yield: 5%5\%.

    • Interest rate shock: At the end of the first holding year (t=1t = 1), the market yield to maturity on comparable bonds increases to 7%7\%.

  • Pricing a Two-Year Bond at the End of Year One:

    • After one year of ownership, a bond initially issued with a 22-year maturity has exactly 11 year remaining until maturity.

    • The buyer at t=1t = 1 discounts the single remaining annual coupon payment ($50\$50) and the par principal repayment ($1,000\$1{,}000) at the updated market interest rate of 7%7\% (0.070.07):     \text{Price} = \frac{\50}{(1 + 0.07)} + \frac{\1{,}000}{(1 + 0.07)}     Price=$1,0501.07=$981.31\text{Price} = \frac{\$1{,}050}{1.07} = \$981.31

  • Capital Loss Calculation for the Two-Year Bond:

    • Capital gain or loss measures the percentage change between initial purchase price and the price realized at the end of the year:     Rate of Capital Gain or Loss=$981.31−$1,000$1,000×100%=−1.87%\text{Rate of Capital Gain or Loss} = \frac{\$981.31 - \$1{,}000}{\$1{,}000} \times 100\% = -1.87\%

  • Total Rate of Return for the Two-Year Bond:

    • Total rate of return combines the current yield (coupon income) with the rate of capital gain or loss:     Rate of Return=Current Yield+Rate of Capital Gain or Loss\text{Rate of Return} = \text{Current Yield} + \text{Rate of Capital Gain or Loss}     Rate of Return=5.0%+(−1.87%)=3.13%≈3.1%\text{Rate of Return} = 5.0\% + (-1.87\%) = 3.13\% \approx 3.1\%

    • Because the investor sustains a capital loss of −1.87%-1.87\%, the realized rate of return (3.1%3.1\%) falls below the current coupon yield (5%5\%).

Maturity and Interest-Rate Sensitivity Analysis

Table 3.2: The Effect of Maturity on Interest-Rate Risk During the First Year of Owning a Bond When the Interest Rate Rises from 5% to 7%

  • Comparative Data Across Maturities (Table 3.2):

    • 1-Year Maturity:

    • Current yield: 5%5\%

    • Initial price: $1,000\$1{,}000

    • Price at end of year: $1,000.00\$1{,}000.00

    • Rate of capital gain or loss: 0%0\%

    • Rate of return during the year: 5.0%5.0\%

    • Observation: Because the bond matures at the end of the holding year, the investor receives the full $1,000\$1{,}000 face value; changes in prevailing interest rates exert zero impact on capital value.

    • 2-Year Maturity:

    • Current yield: 5%5\%

    • Initial price: $1,000\$1{,}000

    • Price at end of year: $981.31\$981.31

    • Rate of capital gain or loss: −1.87%-1.87\%

    • Rate of return during the year: 3.1%3.1\%

    • 10-Year Maturity:

    • Current yield: 5%5\%

    • Initial price: $1,000\$1{,}000

    • Price at end of year: $869.70\$869.70

    • Rate of capital gain or loss: −13.0%-13.0\%

    • Rate of return during the year: −8.0%-8.0\%

    • Observation: The capital loss (−13.0%-13.0\%) exceeds the coupon yield (5%5\%), turning total return negative.

    • 20-Year Maturity:

    • Current yield: 5%5\%

    • Initial price: $1,000\$1{,}000

    • Price at end of year: $793.29\$793.29

    • Rate of capital gain or loss: −20.7%-20.7\%

    • Rate of return during the year: −15.7%-15.7\%

    • 30-Year Maturity:

    • Current yield: 5%5\%

    • Initial price: $1,000\$1{,}000

    • Price at end of year: $754.45\$754.45

    • Rate of capital gain or loss: −24.6%-24.6\%

    • Rate of return during the year: −19.6%-19.6\%

    • 50-Year Maturity:

    • Current yield: 5%5\%

    • Initial price: $1,000\$1{,}000

    • Price at end of year: $724.66\$724.66

    • Rate of capital gain or loss: −27.5%-27.5\%

    • Rate of return during the year: −22.5%-22.5\%

  • Core Empirical Findings:

    • Inverse Relationship Between Maturity and Total Return: The longer the term to maturity of a bond, the lower (more negative) the investor's rate of return when market interest rates rise.

    • Extreme Sensitivity in Ultra-Long Debt: For a bond with 5050 years to maturity, a 2%2\% increase in market yield (from 5%5\% to 7%7\%) triggers a severe price decline of −27.5%-27.5\% and leaves the bondholder with a first-year rate of return of −22.5%-22.5\%.

    • Immunization at Maturity: When the holding period perfectly matches the bond's maturity (as with the 11-year bond held for 11 year), interest-rate risk is eliminated, and the realized return equals the initial yield.