CFA Level I Exam · Interest Rate Risk and Return

Investment Horizon and Macaulay Duration Matching

Updated 7 October 2026 · Fact-checked

Macaulay duration is the present-value-weighted average time to receive a bond's cash flows. When it equals your investment horizon, price risk and reinvestment risk roughly offset after a one-time parallel yield change. Duration gap is Macaulay duration minus horizon. A positive gap means price risk dominates; a negative gap means reinvestment risk dominates.

Understand Investment Horizon and Macaulay Duration Matching

When you buy a bond and plan to sell it later, two things happen when yields change. The bond's price at the sale date moves, and the coupons you receive in the meantime are reinvested at a new rate. These two effects pull in opposite directions.

If yields rise, the price you can sell at falls, but coupons are reinvested at higher rates. If yields fall, the price rises, but reinvestment income shrinks. Price risk and reinvestment risk therefore offset each other.

Macaulay duration is the weighted average time until you receive the cash flows, with weights equal to each cash flow's share of the bond's price. It is measured in years. A zero-coupon bond has a Macaulay duration equal to its maturity. A coupon bond has a Macaulay duration below its maturity, because some cash comes early.

The key result: if the Macaulay duration of the bond equals your investment horizon, the two risks roughly cancel. This holds for a single, one-time, parallel shift in yields right after purchase. The horizon value of the investment is then close to what you expected at the start.

The duration gap measures the mismatch: Macaulay duration minus investment horizon. If the gap is positive, the bond lasts longer than you need and price risk dominates, so rising yields hurt you. If the gap is negative, reinvestment risk dominates, so falling yields hurt you. Immunization means setting the gap to about zero to lock in a target value at the horizon, for example to fund a known liability. The asset's present value must also be at least the liability's present value.

Key formulas to remember

Macaulay duration: MacDur = Σ [t × PV(CF_t)] ÷ Σ PV(CF_t); t is the time of each cash flow in years. The denominator is the bond's full price (PV of all cash flows).
Modified duration: ModDur = MacDur ÷ (1 + y/m); y is the annual yield and m is the number of periods per year. Use it for the approximate percentage price change.
Approximate price change: %ΔPrice ≈ −ModDur × Δy; Accurate for small yield changes. Ignores convexity.
Duration gap: Duration gap = MacDur − Investment horizon; Positive: price risk dominates. Negative: reinvestment risk dominates. Zero: risks roughly offset.
Immunization rule: MacDur of assets ≈ Investment horizon (liability date); Also requires PV of assets ≥ PV of the liability. Holds for a one-time parallel yield shift; rebalancing is needed over time.
Zero-coupon bond duration: MacDur = Maturity; A zero-coupon bond held to maturity has no reinvestment risk.

How to solve Investment Horizon and Macaulay Duration Matching questions

Use this order for any question on horizon, duration gap or immunization.

1Identify the investment horizon (or liability date) in years.
2Find or compute the bond's Macaulay duration. If cash flows are given, weight each time by its PV and divide by the price.
3Compute the duration gap: Macaulay duration minus horizon.
4Decide the sign. Positive gap: price risk dominates. Negative gap: reinvestment risk dominates. Zero: roughly immunized.
5Read the direction of the yield change. Yields up with a positive gap, or yields down with a negative gap, leaves the horizon value below target. The opposite combinations leave it above target.
6For immunization, also check that the asset PV is at least the liability PV and that the shift is a one-time parallel change.
7Eliminate the two options that reverse the risk logic or ignore the direction of the yield move.

Quickest way: Gap-sign shortcut

When to use it: Use it for conceptual questions asking which risk dominates or what happens to the horizon value.

Compute gap = MacDur − horizon.
Gap positive: you are exposed to rising yields (price risk wins).
Gap negative: you are exposed to falling yields (reinvestment risk wins).
Gap zero: the risks offset, so the horizon value is about unchanged.
If the yield move goes against your exposure, the horizon value falls short. If it goes with it, the value exceeds the target.

Common mistakes in Investment Horizon and Macaulay Duration Matching

Treating maturity as the duration of a coupon bond
Zero-coupon bonds have duration equal to maturity, so students assume it applies to all bonds.
Fix: For a coupon bond, Macaulay duration is below maturity. Only a zero-coupon bond has duration equal to maturity.
Reversing which risk dominates when the gap is positive or negative
Students remember that duration is a measure of price sensitivity and forget the horizon comparison.
Fix: Positive gap means the bond is longer than the horizon, so price risk dominates. Negative gap means reinvestment risk dominates.
Using modified duration to match the horizon
Modified and Macaulay duration are both reported, and they look alike.
Fix: The horizon match uses Macaulay duration, which is in years. Modified duration measures price sensitivity, not time.
Assuming immunization works for any yield change at any time
The rule is remembered without its conditions.
Fix: State the conditions: a one-time parallel shift right after purchase. Over time duration drifts, so the portfolio needs rebalancing.
Forgetting that a bond's duration shrinks as time passes
Students treat the purchase-date duration as fixed.
Fix: Macaulay duration falls as time passes, but so does the remaining horizon. The gap can drift, which is why rebalancing is needed.
Dropping the weights when computing Macaulay duration
Students average the cash-flow times instead of PV-weighting them.
Fix: Multiply each time by the PV of its cash flow, sum, and divide by the total PV (the price).

Worked examples

Example 1

A 3-year bond pays a 5% annual coupon on a par value of 100 and has a yield to maturity of 5%. Its price is 100. The Macaulay duration is closest to: A. 2.72 years, B. 2.86 years, C. 3.00 years.

Show the solution

Discount each cash flow at 5%: PV1 = 5 ÷ 1.05 = 4.7619; PV2 = 5 ÷ 1.1025 = 4.5351; PV3 = 105 ÷ 1.157625 = 90.7029.
Check the sum: 4.7619 + 4.5351 + 90.7029 = 99.9999, which is the price of 100 (rounding). Calculator check: N = 3, I/Y = 5, PMT = 5, FV = 100, CPT PV gives −100.
Weight each time: 1 × 4.7619 = 4.7619; 2 × 4.5351 = 9.0703; 3 × 90.7029 = 272.1088.
Sum the weighted times: 4.7619 + 9.0703 + 272.1088 = 285.9410.
Divide by the price: 285.9410 ÷ 100 = 2.859 years.
Check the logic: it is below the 3-year maturity, as it should be for a coupon bond. That rules out C.

Answer: B. 2.86 years.

Example 2

An investor has a 6-year horizon and buys a bond with a Macaulay duration of 7.5 years. Immediately after purchase, yields rise by a one-time parallel shift and stay there. Which statement is correct? A. The horizon value is likely below the target because price risk dominates. B. The horizon value is likely above the target because reinvestment risk dominates. C. The horizon value is approximately unchanged because the risks offset.

Show the solution

Compute the duration gap: 7.5 − 6 = +1.5 years.
A positive gap means the bond's duration is longer than the horizon, so price risk dominates reinvestment risk.
Yields rise, so the bond's price at the sale date falls by more than the extra reinvestment income makes up.
The horizon value therefore ends below the target.
Eliminate B: reinvestment risk dominates only when the gap is negative. Eliminate C: the risks offset only when the gap is about zero.

Answer: A. The horizon value is likely below the target because price risk dominates.

Exam tips

Calculate the gap first. Its sign decides most conceptual questions in under 30 seconds.
When the options differ only in which risk dominates, the sign of the gap picks the answer. Do not compute anything else.
Remember the conditions of immunization: a one-time parallel yield shift, and asset PV at least equal to liability PV.
For a numerical Macaulay duration, sanity-check the result. It must be below maturity for a coupon bond, and it equals maturity for a zero-coupon bond.
Know the difference: Macaulay duration is a time measure in years, and modified duration is the price-sensitivity measure.

Practice questions from Interest Rate Risk and Return

Investment Horizon and Macaulay Duration Matching in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

FRM Part IMacaulay, Modified and Effective Duration

Investment Horizon and Macaulay Duration Matching: frequently asked questions

Why does reinvestment risk offset price risk when duration equals the horizon?

A yield rise lowers the bond's future sale price but raises the income earned by reinvesting coupons. When Macaulay duration equals the horizon, these two effects are about equal in size. The horizon value stays close to the expected value.

How do I calculate the duration gap?

Subtract the investment horizon from the bond's Macaulay duration. A positive result means price risk dominates. A negative result means reinvestment risk dominates.

Does duration matching fully immunize a bond portfolio?

Not completely. It works for a one-time parallel yield shift right after purchase. Non-parallel shifts and the passage of time can break the match, so the portfolio needs to be rebalanced.

Is Macaulay duration the same as modified duration?

No. Macaulay duration is the weighted average time to cash flows, in years. Modified duration equals Macaulay duration divided by (1 + y/m) and measures the approximate percentage price change for a yield change.