Modified Duration

What it is

Modified duration measures a bond’s price sensitivity to changes in interest rates. It estimates the approximate percentage change in a bond’s price for a 100-basis-point (1 percentage point) change in yield, assuming small changes and a parallel shift in the yield curve. It is derived from Macaulay duration and is widely used for risk management, portfolio construction, and interest-rate exposure analysis.

Key formulae

  • Modified duration:
    Modified Duration = Macaulay Duration / (1 + YTM / n)

where:
Macaulay Duration = weighted average time (in years) to receive the bond’s cash flows
 YTM = yield to maturity (expressed as a decimal)
* n = number of coupon periods per year

  • Macaulay duration (practical form):
    Macaulay Duration = (Σ (PV of CF at t × t)) / Market Price

where PV of CF at t is the present value of the cash flow at time t, and t is time in years.

How to calculate (step by step)

  1. Compute the market price by discounting each coupon and the principal at the bond’s YTM.
  2. Compute the present value of each cash flow and multiply each by its time t (in years).
  3. Sum those PV×t products and divide by the market price to get Macaulay duration (years).
  4. Convert to modified duration by dividing Macaulay duration by (1 + YTM / n).

Modified duration is expressed in years but interpreted as a percentage price change: a modified duration of 3 implies roughly a 3% price change for a 1% change in yields (inverse direction).

Example

A $1,000 face-value bond, 3-year maturity, 10% annual coupon, YTM = 5% (annual):

  1. Market price:
  2. Year 1 coupon PV = 100 / 1.05 = 95.24
  3. Year 2 coupon PV = 100 / 1.05^2 = 90.70
  4. Year 3 coupon + principal PV = 1,100 / 1.05^3 = 950.22

  5. Market price = 95.24 + 90.70 + 950.22 = $1,136.16

  6. Macaulay duration:

  7. (95.24 × 1 + 90.70 × 2 + 950.22 × 3) / 1,136.16 = 2.753 years

  8. Modified duration (n = 1):

  9. 2.753 / (1 + 0.05) = 2.622

Interpretation: For a 1% increase in yields, the bond’s price would fall by about 2.62% (and rise by about 2.62% for a 1% decrease), ignoring convexity.

What modified duration tells you

  • It quantifies interest-rate risk: higher modified duration → greater price volatility for a given change in yields.
  • Useful for:
  • Estimating price impact from interest-rate moves
  • Constructing/immunizing portfolios to a target interest-rate exposure
  • Comparing sensitivity across bonds

Key principles

  • Longer maturity → higher duration (more sensitivity).
  • Higher coupon → lower duration (less sensitivity), because more cash flows arrive earlier.
  • Higher yield → lower duration (reduced sensitivity), since future cash flows are discounted more heavily.

Limitations

  • Linear approximation: accurate for small yield changes; for larger moves use convexity-adjusted estimates.
  • Assumes parallel yield-curve shifts; does not capture non-parallel or shape changes.
  • Bonds with embedded options (callable/putable) require option-adjusted duration measures.

Quick FAQs

  • Difference between Macaulay and modified duration?
  • Macaulay duration is the weighted average time to cash flows (in years). Modified duration converts that into a price-sensitivity measure (approximate % price change per 1% change in yield).
  • Do zero-coupon bonds pay interest?
  • No. Zero-coupon bonds pay no periodic coupons and trade at a discount; their duration equals their time to maturity (and modified duration = maturity / (1 + YTM/n)).

Bottom line

Modified duration is a practical, widely used metric for estimating how bond prices respond to interest-rate changes. It helps investors quantify and manage interest-rate risk, but it is a first-order (linear) approximation and should be used with awareness of its assumptions and limitations.