CFA Level I Exam · Yield-Based Bond Duration Measures and Properties

Investment Horizon and Macaulay Duration Explained

Updated 7 October 2026 · Fact-checked

When a bond's Macaulay duration equals your investment horizon, price risk and reinvestment risk roughly offset after a one-time parallel yield shift, so the horizon value is close to locked in. The 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

A bond investor who sells before maturity faces two opposite effects when yields change. If yields rise, the bond's price falls (price risk), but coupons can be reinvested at higher rates (reinvestment risk working in your favour). If yields fall, the price rises, but coupons are reinvested at lower rates.

Because the two effects pull in opposite directions, they can cancel. The point where they cancel is set by Macaulay duration, the weighted average time to receive the bond's cash flows, where weights are the present values of each cash flow. If you buy a bond whose Macaulay duration equals your investment horizon, a rise in yields costs you about as much in price as it gains in reinvestment income, and the reverse for a fall.

The duration gap measures how far you are from that match: duration gap = Macaulay duration − investment horizon. A positive gap (duration longer than horizon) means price risk dominates. You lose if yields rise and gain if they fall. A negative gap (duration shorter than horizon) means reinvestment risk dominates. You gain if yields rise and lose if they fall. A zero gap gives the immunized position.

The result holds under conditions the exam expects you to know: a single, instantaneous, parallel shift in yields right after purchase, and a fixed-rate, option-free bond. It is an approximation. In practice Macaulay duration falls more slowly than the remaining horizon as time passes, so the portfolio must be rebalanced to keep the gap near zero. A zero-coupon bond held to maturity has no reinvestment risk, and its Macaulay duration equals its maturity.

Key formulas to remember

Duration gap: Duration gap = Macaulay duration − Investment horizon; Positive: price risk dominates. Negative: reinvestment risk dominates. Zero: the two risks offset.
Immunization condition: Macaulay duration = Investment horizon; Holds approximately, for a one-time parallel yield shift, fixed-rate option-free bonds.
Macaulay duration: MacDur = Σ [t × PV(CFt)] ÷ Bond price; The weighted average time to receive cash flows, weights are present values of cash flows.
Macaulay from modified duration: MacDur = ModDur × (1 + YTM per period); Use YTM per period with the matching number of periods per year, then convert to years.
Zero-coupon bond: MacDur = Maturity; Holding to maturity (horizon = maturity) removes both price risk and reinvestment risk.

How to solve Investment Horizon and Macaulay Duration questions

Use this sequence for any question on horizon matching, duration gap or the two risks.

1Write down the investment horizon and the bond's Macaulay duration, both in years. If you are given modified duration, convert it first: MacDur = ModDur × (1 + YTM).
2Compute the duration gap: Macaulay duration − horizon. Keep the sign.
3Check the conditions: one-time parallel shift in yields right after purchase, fixed-rate bond with no embedded options.
4If the gap is positive, price risk dominates: yields up means the horizon value is lower than expected, yields down means it is higher.
5If the gap is negative, reinvestment risk dominates: yields up means the horizon value is higher than expected, yields down means it is lower.
6If the gap is zero, the two risks offset and the horizon value is approximately protected in either direction.
7Eliminate the two wrong options by direction first (gain or loss), then check any number against your calculation.

Quickest way: Sign of the gap decides the direction

When to use it: Use when the question asks which risk dominates or what happens to the horizon return if yields move.

Compare duration to horizon. Longer duration means price risk wins. Shorter duration means reinvestment risk wins.
Longer duration behaves like a plain bond: yields up, you lose; yields down, you gain.
Shorter duration behaves the opposite way: yields up, you gain; yields down, you lose.
Equal means no clear winner, so the result is protected.
For a calculation, convert modified duration to Macaulay by multiplying by (1 + YTM) before comparing.

Common mistakes in Investment Horizon and Macaulay Duration

Using modified duration as if it were Macaulay duration when comparing to the horizon
Both are called duration, and the question may give only modified duration.
Fix: Multiply modified duration by (1 + YTM per period) first. Then compare with the horizon in years.
Reversing the sign of the duration gap
Students subtract in the order horizon − duration.
Fix: Always compute Macaulay duration − horizon. A positive result means duration exceeds the horizon.
Saying that a negative gap means price risk dominates
Confusing which side is longer when yields change.
Fix: When duration is shorter than the horizon, the price effect is smaller than the reinvestment effect, so reinvestment risk dominates. A negative gap therefore means reinvestment risk dominates.
Thinking that matching removes risk for any yield change at any time
The rule is remembered without its conditions.
Fix: State the conditions: a one-time, instantaneous, parallel shift, and fixed-rate option-free bonds. Later shifts and yield curve twists can still leave risk.
Assuming a matched portfolio stays matched
Students forget that time passing changes duration.
Fix: Macaulay duration of a coupon bond falls by less than the time that passes, so the portfolio must be rebalanced periodically.

Worked examples

Example 1

An investor has a 7-year horizon and buys a fixed-rate, option-free bond with modified duration of 6.5 and YTM of 4% (annual coupons). Yields then rise once, in parallel, right after purchase. Which statement is correct? A. Duration gap is −0.24 and the horizon value is higher than expected. B. Duration gap is 0.24 and the horizon value is higher than expected. C. Duration gap is 0.24 and the horizon value is lower than expected.

Show the solution

Convert to Macaulay duration: 6.5 × (1 + 0.04) = 6.76 years.
Duration gap = 6.76 − 7 = −0.24.
The gap is negative, so Macaulay duration is shorter than the horizon and reinvestment risk dominates.
Yields rose, so reinvestment gains outweigh the price loss and the horizon value is higher than expected.
Option B has the wrong sign for the gap (+0.24 instead of −0.24), even though its direction is right. Option C has the wrong gap sign and the wrong direction, because it says the horizon value is lower. Only A matches both parts.

Answer: A. The duration gap is −0.24 and the horizon value is higher than expected.

Example 2

A 3-year, 5% annual-coupon bond with par value 100 is priced at a YTM of 5%. An investor has a 3-year horizon. What is the duration gap? A. −0.14 B. 0.14 C. 2.86

Show the solution

Price at 5% = 100, since coupon equals yield. On a TI BA II Plus: N = 3, I/Y = 5, PMT = 5, FV = 100, CPT PV = −100.
PV of cash flows: year 1: 5 ÷ 1.05 = 4.7619. Year 2: 5 ÷ 1.1025 = 4.5351. Year 3: 105 ÷ 1.157625 = 90.7029.
Weighted sum: 1 × 4.7619 + 2 × 4.5351 + 3 × 90.7029 = 4.7619 + 9.0703 + 272.1088 = 285.9410.
Macaulay duration = 285.9410 ÷ 100 = 2.8594 years.
Duration gap = 2.8594 − 3 = −0.1406, about −0.14.
The gap is slightly negative, so reinvestment risk dominates mildly. Option C is the duration itself, not the gap; B has the wrong sign.

Answer: A. The duration gap is about −0.14 years.

Exam tips

Questions are three-option MCQs. Decide the direction (gain or loss) first. That usually eliminates one or two options before you calculate.
Always check the sign of the duration gap. Many wrong options differ only by sign.
If a question gives modified duration, convert it to Macaulay duration before comparing with the horizon.
Remember the conditions: a one-time parallel shift and a fixed-rate option-free bond. A question that changes these conditions is testing the limitation of the rule.
A zero-coupon bond held to maturity has no reinvestment risk, so no gap between its duration and the horizon is needed.

Practice questions from Yield-Based Bond Duration Measures and Properties

Investment Horizon and Macaulay Duration 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: frequently asked questions

What is the duration gap in CFA Level I?

It is the bond's Macaulay duration minus the investor's investment horizon. A positive gap means price risk dominates, and a negative gap means reinvestment risk dominates. A zero gap means the two risks roughly offset.

Why does matching Macaulay duration to the horizon reduce risk?

A yield change moves price and reinvestment income in opposite directions. When duration equals the horizon, the two effects are about equal in size, so the value at the horizon changes little. This assumes a single parallel yield shift right after purchase.

How do I immunize a bond portfolio with Macaulay duration?

Choose bonds so that portfolio Macaulay duration equals your investment horizon. Then rebalance over time, because Macaulay duration does not fall exactly as fast as the remaining horizon. The method is approximate and assumes parallel yield shifts.

Does a zero-coupon bond need duration matching?

If you hold a zero-coupon bond to maturity, there is no reinvestment risk and no price risk at the horizon, because its Macaulay duration equals its maturity. The value at the horizon is the face value.