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Explain the investment horizon and the gap in duration relative to it

CFA® Level I · Fixed Income · Interest Rate Risk and Return · FI.4.i

The idea

An investor in a coupon bond faces two offsetting risks. If rates rise, the bond's price falls but coupons are reinvested at higher rates; if rates fall, the reverse. Assuming a single parallel shift in yields soon after purchase, at a horizon equal to the bond's Macaulay duration the two effects cancel and the realised return equals the original yield to maturity, which is the immunisation result. The duration gap is Macaulay duration less the investment horizon: a positive gap leaves the investor exposed to rising rates through price risk, and a negative gap exposes them to falling rates through reinvestment risk.

Formulas

Duration gap

duration gap = MacDur − investment horizon

MacDur
the bond's Macaulay duration, in years
investment horizon
how long the money is to stay invested, in years

At zero the two risks cancel: a rise in yields costs price and repays it in reinvestment income. A positive gap leaves price risk dominant, a negative gap leaves reinvestment risk dominant.

A practice question

From the question bank, with the reason each wrong option is wrong.

What are the two offsetting effects of a rate change on a coupon bond investor?

  1. Credit risk and liquidity riskWhy not: Both real risks, and neither is created by a change in the level of rates. They also do not offset.
  2. Price risk and reinvestment risk — correct
  3. Inflation risk and currency riskWhy not: Also unrelated to the mechanism, and for a domestic bond currency risk does not arise at all.

A rate rise lowers the price and raises reinvestment income. The two move in opposite directions, which is what makes a cancelling horizon exist.

A harder one

An investor's horizon exactly equals the bond's Macaulay duration. What is the result?

  1. The realised return is guaranteed to exceed the yield to maturityWhy not: Nothing is guaranteed, and immunisation aims to *match* the original yield rather than beat it.
  2. The bond cannot defaultWhy not: Immunisation addresses interest rate risk alone. Credit risk is untouched by any duration matching.
  3. Price and reinvestment effects cancel — correct

The realised return is then close to the original yield. This is the immunisation result, and it is the reason Macaulay duration is quoted in years at all.

The hardest tier

An investor holds a bond with Macaulay duration of 8 years against a liability due in 5 years. What is their exposure?

  1. A positive duration gap, leaving them exposed to rising rates through price risk — correct
  2. A negative duration gap, leaving them exposed to falling ratesWhy not: The gap is duration minus horizon, so 8 − 5 is positive. A negative gap is the opposite case, where the horizon is the longer.
  3. No exposure, since the bond matures after the liabilityWhy not: Maturing after the liability is the problem, not the solution — the bond has to be sold at the horizon, at whatever price rates then dictate.

With duration exceeding the horizon, the price effect dominates. A rate rise means selling at a depressed price before the higher reinvestment income has had time to compensate.

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