Macaulay and Modified Duration: When Shortcuts Are Safe on CFA Questions
Duration questions become hard when candidates treat every number as a calculator target. The CFA exam is usually testing whether you know which duration measure fits the prompt, what the measure means, and when an approximation is good enough.
The safe workflow is concept first, shortcut second. A calculator can help with bond cash flows, but it cannot tell you whether the answer should be Macaulay duration, modified duration, effective duration, or a first-order price-change estimate. That classification has to happen before the arithmetic.
The Core Map
Macaulay Duration Is A Weighted Time Measure
Macaulay duration is the present-value-weighted average time to receive a bond's promised cash flows. Its unit is years because it is literally a timing measure.
Original example:
Seabrook Water Authority issues a 3-year annual coupon bond with a 5 percent coupon and yield of 4 percent. The promised cash flows are:
- Year 1: 5
- Year 2: 5
- Year 3: 105
To compute Macaulay duration manually, discount each cash flow, divide each present value by total bond price, and multiply each cash-flow weight by the year received.
The final result will be below 3 years because some present value arrives before maturity through coupons. If the bond were a zero-coupon bond, all value would arrive at maturity and Macaulay duration would equal maturity.
Modified Duration Translates Timing Into Price Sensitivity
Modified duration converts the timing measure into an approximate percentage price sensitivity to a yield change:
modified duration = Macaulay duration / (1 + yield per period)
If a bond's Macaulay duration is 4.60 and the yield per period is 6 percent, modified duration is:
4.60 / 1.06 = 4.34
The interpretation is local and approximate: for a small yield increase of 1 percent, the bond price falls by about 4.34 percent. For a 25 basis point increase:
approximate percentage price change = -4.34 x 0.0025 = -1.085 percent
Effective Duration Is For Changing Cash Flows
Modified duration assumes the expected cash flows are fixed. If the bond has embedded options or expected cash flows can change when rates move, effective duration is usually the better measure. That is a classification issue, not a calculator issue.
When A Calculator Shortcut Is Safe
Safe Shortcut 1: Verifying Bond Price Or Cash-Flow Present Values
Using a financial calculator to find a bond price can be reasonable if the question gives clean coupon, yield, maturity, and par information. But the calculator price alone is not Macaulay duration. Macaulay duration still requires the timing weights of the cash flows.
If a shortcut gives you a duration value, ask two questions before trusting it:
- Does the output represent Macaulay duration or modified duration?
- Does the input yield match the correct compounding period?
Many wrong answers come from using an annual yield in a semiannual setup or reading a modified-duration output as Macaulay duration.
Safe Shortcut 2: Converting Macaulay To Modified Duration
If the prompt gives Macaulay duration directly, the conversion to modified duration is usually quick:
modified duration = Macaulay duration / (1 + yield per period)
For semiannual coupon bonds, use the semiannual yield in the denominator. If the stated annual yield is 8 percent with semiannual compounding, the period yield is 4 percent. Do not divide by 1.08 when the period is six months.
Safe Shortcut 3: Price Change From Modified Duration
For small yield changes, the first-order estimate is:
percentage price change approximately equals -modified duration x change in yield
Original example:
Northline Rail has a bond with modified duration of 5.7. Its yield increases by 40 basis points.
Convert the yield change to decimal:
40 basis points = 0.0040
Estimate price impact:
-5.7 x 0.0040 = -0.0228, or about -2.28 percent.
This is the exam workhorse approximation. It is fast, but it is not exact for large yield changes because bond prices are curved, not linear.
Where Convexity Fits
Convexity is the second-order correction to the duration estimate. It does not replace duration. Duration gives the first-order slope; convexity improves the estimate when the yield change is large enough for curvature to matter.
If a question only gives modified duration and a small yield change, use the first-order duration estimate. If it also gives convexity and the yield move is meaningful, the prompt is likely inviting the combined duration-plus-convexity estimate.
Worked Example: Manual Concept, Fast Exam Answer
Harbor Glass issues a 4-year annual coupon bond with a 6 percent coupon, a yield of 5 percent, and a Macaulay duration of 3.56 years. The exam asks for the approximate percentage price change if the yield rises by 30 basis points.
Step 1: Identify The Measure Needed
The question asks for price change from a yield move, so modified duration is needed. Macaulay duration is not the final answer.
Step 2: Convert Macaulay To Modified Duration
modified duration = 3.56 / 1.05 = 3.39
Step 3: Convert The Yield Change
30 basis points = 0.0030
Step 4: Estimate Price Change
percentage price change approximately equals -3.39 x 0.0030 = -0.01017
The bond price falls by about 1.02 percent.
Step 5: State The Limitation
The answer is a duration approximation. It ignores convexity unless the prompt provides convexity and asks for the more refined estimate.
Exam Framing
What CFA Questions Tend To Reward
The exam often rewards the candidate who can classify the task:
- Timing question: Macaulay duration.
- Fixed-cash-flow sensitivity question: modified duration.
- Option-sensitive cash-flow question: effective duration.
- Price-change approximation: negative modified duration times yield change.
- Larger yield change with convexity data: duration estimate plus convexity adjustment.
What To Avoid
Do not assume every duration question is a manual cash-flow table. Do not assume every calculator result is the final answer. And do not use convexity as a vague explanation for any duration mistake. Match the method to the data provided.