Macaulay, Modified, and Effective Duration

Duration is one of the most tested ideas in CFA® Level I fixed income, and the exam expects you to keep three versions straight. Macaulay, modified, and effective duration answer different questions, and the trap is using the wrong one for a given bond. This note shows what each measure means and when to reach for it.

Key Takeaways: Macaulay Duration vs Modified Duration and Effective Duration

• Macaulay duration measures the timing of cash flows, expressed in years.

• Modified duration estimates the percentage price change for a change in the bond's own YTM.

• Effective duration estimates price sensitivity to a benchmark yield curve shift and suits bonds with uncertain cash flows.

• Modified duration links directly to Macaulay duration for option-free bonds with known cash flows.

• The main trap is using modified duration for a bond with embedded options when effective duration fits better.

What You Need to Know for CFA Level I

• Define each of the three duration measures in one clear sentence.

• Match the right measure to the bond type and the wording of the question.

• Recall that modified duration comes from Macaulay duration for option-free bonds.

• Recognize when uncertain cash flows or embedded options call for effective duration.

• Read whether a question is about a change in the bond's YTM or a shift in the benchmark curve.

Macaulay vs Modified vs Effective Duration: Main Difference

The one-sentence distinction is this: Macaulay duration is about timing, modified duration is about price sensitivity to the bond's own yield, and effective duration is about price sensitivity to the benchmark curve when cash flows may change. Use the table below as your quick map.

What Is Macaulay Duration?

Macaulay duration is the weighted average time to a bond's cash flows, measured in years. Each cash flow is weighted by its present value as a share of the bond's price, so cash flows that arrive sooner and carry more value pull the number down, while later cash flows push it up.

Where:

• = Macaulay duration

• = time period when the cash flow is received

• = total number of cash flow periods

• = cash flow received at time t

• = yield or discount rate per period

• = discount factor for time t

Because it is a time measure, Macaulay duration connects naturally to an investment horizon. A candidate matching a bond to a future funding need cares about when the money actually arrives, and that is exactly what this measure captures.

What Is Modified Duration?

Modified duration estimates the percentage change in a bond's price for a change in its own yield to maturity. It is the first measure most candidates use for a standard option-free bond, and it comes straight from Macaulay duration.

Where:

• = modified duration

• = Macaulay duration

• = annual yield to maturity

• = number of compounding periods per year

• = yield per compounding period

Here r is the yield per period and k is the number of periods per year. For a bond with annual Macaulay duration of 4.5 and an annual yield of 5 percent, modified duration is 4.5 divided by 1.05, which is about 4.29. The price impact of a small yield change then follows from a simple relationship.

The negative sign reflects the inverse link between yield and price. Modified duration is a linear approximation, so it works well for small yield changes and loses accuracy as the change grows.

What Is Effective Duration?

Effective duration estimates how much a bond's price moves when the benchmark yield curve shifts, rather than when the bond's own YTM changes. It is the right measure when future cash flows are not certain, which is the case for bonds with embedded options.

In this formula, is the bond price after a downward curve shift, is the bond price after an upward curve shift, and is the original bond price. is the size of the benchmark yield curve shift expressed in decimal form.

The key difference from modified duration is what changes. Effective duration measures the effect of a benchmark curve shift, not a change in the bond's own yield. This makes Deff useful for callable or putable bonds, where changing interest rates can alter expected cash flows and make a measure based on fixed cash flows less suitable.

Comparison Table

This table lines up the three measures on the points that decide exam questions.

How to Choose the Right Duration Measure in CFA Questions

Read the question wording before you reach for a formula. The clues are usually in plain sight: what is changing, and are the cash flows certain?

• If the question asks about timing or matching a horizon, think Macaulay duration.

• If the bond is option-free and the question changes the bond's own YTM, use modified duration.

• If the question shifts a benchmark yield curve, lean toward effective duration.

• If the bond has an embedded option or uncertain cash flows, use effective duration.

Common Exam Traps

• Using Macaulay duration as a price sensitivity measure without converting to modified duration first.

• Using modified duration for a bond with embedded options, where effective duration is the better fit.

• Confusing a change in the bond's YTM with a shift in the benchmark yield curve.

• Forgetting that duration is an approximation and becomes less accurate for large yield changes.

Practice Question

A portfolio analyst is reviewing two bonds. Bond X is an option-free, fixed-rate corporate bond. Bond Y is a callable bond whose expected cash flows change when interest rates move. Which duration measure is most appropriate for estimating Bond Y's price sensitivity?

1. Macaulay duration, because it measures the timing of cash flows.

2. Modified duration, because it estimates price sensitivity to the bond's YTM.

3. Effective duration, because the bond's cash flows can change when rates move.

FAQs About Macaulay, Modified, and Effective Duration