Actuarial Notes

Duration and Convexity for Pension Liabilities

By Martin McCaulay · Society of Actuaries · 2013

Study guide: Exam RET-201

A short article by Martin McCaulay in the September 2013 (Issue 81) Pension Section News of the Society of Actuaries. It presents duration and convexity as a risk metric for pension plan sponsors: the formula for effective duration can be used to estimate the value of pension liabilities at different interest rates, and a convexity adjustment reflects that the liability increase for a rate decrease is greater than the decrease for a rate increase.

On the syllabus

  • Exam RET-201 — Outcomes 2a-2d and 3c; the whole article.

Estimating Changes in Liabilities

  • Pension liabilities have an inverse relationship to interest rates, and the change can be estimated using the duration of the liabilities.
  • For every 100 basis point change in interest rates, the liability changes by duration divided by 100 in the opposite direction.
  • The typical pension plan has a duration of about 15.

Macaulay Duration

  • The original formula for duration, developed in 1938 by Frederick Robertson Macaulay, measures a bond's weighted average cash flows.

Modified Duration

  • Modified Duration measures the sensitivity of a bond's price to interest rate movements and is related to Macaulay Duration with annual coupons.

Effective Duration

  • Effective Duration, used to price bonds with options, approximates the slope of a bond's value as a function of interest rate movements.
  • Pension liability duration is measured using the formula for Effective Duration, substituting the liabilities for the bond's value.

Duration Example

Convexity Example

Illustrative Examples of Pension Duration with the Convexity Adjustment

  • Making the convexity adjustment results in a lower duration for rate increases and a greater duration for rate decreases.
  • The liability change should be based on a compounded change rather than a simple change.

Convexity

  • The traditional formula for pension duration does not consider convexity.
  • Duration with a convexity adjustment can provide a better estimate of the change in liability when there is significant volatility.

Duration with Convexity Adjustment Example

Negative Convexity

Key Rate Duration

  • Key rate duration considers the sensitivity of a liability's movement to different parts of the yield curve and requires building a yield curve.

Summary

Sources