e

4.6 Effect of a bond’s maturity, coupon, embedded options, and yield level on its interest rate risk

Bond duration is the basic measure of interest rate risk on a fixed-rate bond.

The duration for a fixed-rate bond is a function of the following input variables.

  • Coupon rate or payment per period
  • Yield-to-maturity per period
  • Time-to-maturity (as of the beginning of the period)
  • Fraction of the period that has gone by
  • Presence and nature of embedded options The coupon payments are made on regularly scheduled dates-for example, on 15 June and 15 December each year on a semiannual coupon paying bond. The last coupon is paid together with the face value on the maturity date. The market discount rate is also called the "required yield" or "required rate of return."
  • Coupon rate relation to Macaulay duration:
  • The coupon rate is inversely related to the Macaulay duration. A lower-coupon bond has a higher duration and more interest rate risk than a higher-coupon bond. The Macaulay duration of a zero-coupon bond is equal to its time-to-maturity.
  • Yield-to-Maturity relation to Macaulay duration:
  • The yield-to-maturity is inversely related to the Macaulay duration. A higher yield-to- maturity reduces the weighted average of the time to receipt of cash flow. With a higher yield-to-maturity, there is more weight on the cash flows received in the near term, and less weight is on the cash flows received in the more-distant future periods if those cash flows are discounted at a higher rate.
  • Time-to-Maturity relation to Macaulay duration:
  • Time-to-maturity is typically directly related to the Macaulay duration. This pattern always holds for bonds trading at par value or at a premium above par. The exception is deep- discount bonds, where the relationship does not hold for a long time-to-maturity.
  • Fraction of the period relation to Macaulay duration:
  • Fraction of the period that has gone by (t/T) is inversely related to the Macaulay duration.

Macaulay duration decreases smoothly as t goes from t = 0 to t = T and then jumps upward after the coupon is paid. The duration of perpetuity (consol bond) is the constant.

  • Bonds with embedded options:
  • Bonds with embedded options (e.g., callable, putable) require the use of effective duration because Macaulay and modified yield duration statistics are not relevant.
  • The yield-to-maturity for callable and putable bonds is not well defined because future cash flows are uncertain.
  • When benchmark yields are high (low), the effective durations of the callable (putable) and non-callable (non-putable) bonds are very similar. There is a large discrepancy in durations for callable (putable) and non-callable (non-putable) bonds when yields are low (high).
  • In summary, the presence of an embedded option reduces the sensitivity of the bond price to changes in the benchmark yield curve (lower duration), assuming no change in credit risk.
  • Effective duration measures are also used for bonds with other embedded options, such as asset-backed securities.

Lesson Wrap-Up

This lesson should leave you able to explain the effect of a bond's maturity, coupon, embedded options, and yield level on its interest rate risk in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain effect of a bond's maturity, coupon, embedded options, and yield level on its interest rate risk in your own words.
  2. State one exam-style risk, valuation, or market implication of the effect of a bond's maturity, coupon, embedded options, and yield level on its interest rate risk.