Macaulay vs Modified vs Effective Duration: A CFA Level I Guide

Macaulay vs Modified vs Effective Duration

Fixed Income makes up 11-14% of the CFA Level I exam (CFA Institute). Of that, duration and convexity are the most heavily tested concepts, and the distinction between Macaulay, modified, and effective duration is the single most common trap candidates fall into.

When I sat for CFA Level I, I underestimated how often the exam asks you to choose the right duration measure rather than just plug numbers into a formula. The arithmetic is easy. The conceptual question (which duration applies to this bond) is what separates a pass from a fail.

This guide walks through each measure, the formulas, and the situations where each one is correct.

Macaulay Duration: Weighted-Average Time

Macaulay duration is the weighted-average time, in years, until a bondholder receives the bond's cash flows. The weights are the present values of each cash flow as a fraction of the bond's price.

\[ D{Mac} = \sum{t=1}^{n} t \cdot \frac{PV(CF_t)}{P} \]

Where \(PV(CF_t)\) is the present value of the cash flow at time \(t\), and \(P\) is the bond's price (the sum of all discounted cash flows).

For a zero-coupon bond, Macaulay duration equals the bond's time to maturity, because there is only one cash flow. For a coupon bond, Macaulay duration is always less than time to maturity, because earlier coupon payments pull the average forward.

A Quick Example

A 3-year, 5% annual-coupon bond with a yield of 5% has cash flows of $5, $5, and $105. At a 5% yield the price is $100. Macaulay duration is:

\[ D_{Mac} = \frac{1 \cdot 4.76 + 2 \cdot 4.54 + 3 \cdot 90.70}{100} \approx 2.86 \text{ years} \]

A bondholder waits, on a present-value-weighted basis, about 2.86 years for the cash flows. Less than 3 because the early coupons matter.

Modified Duration: Price Sensitivity to Yield

Modified duration converts Macaulay duration into a measure of price sensitivity. It tells you the approximate percentage change in a bond's price for a 1% change in yield.

\[ ModDur = \frac{D_{Mac}}{1+y} \]

Where \(y\) is the yield per period (so for an annual-coupon bond with annual yield \(y\), divide by \(1+y\); for semiannual coupons, divide by \(1 + y/2\)).

The price-change approximation is:

\[ \frac{\Delta P}{P} \approx -ModDur \cdot \Delta y \]

For the 3-year bond above with Macaulay duration of 2.86, modified duration is \(2.86 / 1.05 = 2.72\). A 1% (100bp) increase in yield is expected to drop the price by about 2.72%.

Effective Duration: For Bonds with Embedded Options

When a bond has embedded options, the cash flows themselves change as yields change. A callable bond is more likely to be called when rates fall, shortening the bond's effective life. A putable bond is more likely to be put when rates rise. Modified duration cannot capture this because it assumes cash flows are fixed.

Effective duration sidesteps the problem by directly measuring the average price change for a parallel shift in the benchmark yield curve:

\[ EffDur = \frac{P- - P+}{2 \cdot P_0 \cdot \Delta y} \]

Where:

• \(P_-\) is the price if the yield curve shifts down by \(\Delta y\)
• \(P_+\) is the price if the yield curve shifts up by \(\Delta y\)
• \(P_0\) is the current price
• \(\Delta y\) is the size of the parallel shift (in decimal form)

The inputs come from a valuation model (often a binomial interest-rate tree on the exam), which reprices the bond under each scenario while letting the embedded option behave correctly.

When Each Applies

| Bond type | Use this duration |
|---|---|
| Treasury, fixed-coupon corporate, zero-coupon | Modified duration |
| Callable corporate or municipal | Effective duration |
| Putable bond | Effective duration |
| Mortgage-backed security (prepayment option) | Effective duration |
| Convertible bond | Effective duration |
| Floating-rate note | Effective duration (very short) |

A Practical Example

A 10-year 6% callable bond is currently priced at $98.50. Using a binomial tree, the price falls to $97.20 if rates rise 25bp and rises to $99.60 if rates fall 25bp. Effective duration is:

\[ EffDur = \frac{99.60 - 97.20}{2 \cdot 98.50 \cdot 0.0025} = \frac{2.40}{0.4925} \approx 4.87 \]

Notice the price did not rise as much when rates fell as it dropped when rates rose. That asymmetry is the call option at work, and effective duration captures it correctly. Modified duration on the same bond would give a different (and misleading) answer because it ignores the option.

Money Duration and Price Value of a Basis Point

The CFA curriculum also tests two related measures:

• Money duration (or dollar duration): \(MoneyDur = ModDur \cdot P\). Measures the dollar change in price for a 1% yield change. Useful when sizing positions or estimating P&L impact.
• Price value of a basis point (PVBP): \(PVBP = ModDur \cdot P \cdot 0.0001\). The dollar change in price for a 1bp yield move. Often quoted on Treasury futures and in trading desks.

Both are derived from modified duration, so the same caveat applies: they assume cash flows are fixed.

Convexity: The Companion Concept

Duration is a linear approximation. For larger yield changes, you also need convexity to capture the curvature of the price-yield relationship.

\[ \frac{\Delta P}{P} \approx -ModDur \cdot \Delta y + \frac{1}{2} \cdot Convexity \cdot (\Delta y)^2 \]

For option-free bonds, convexity is positive: prices rise more for a yield drop than they fall for an equal-size yield rise. Callable bonds can have negative convexity at low yields, because the call cap limits the price upside. This is another reason effective duration (and effective convexity) is needed for option-embedded bonds.

What the Exam Tests

From my own Level I sitting and from working through hundreds of Fixed Income questions, here is what shows up:

1. Identify the right duration measure. Given a bond description, choose Macaulay, modified, or effective. Embedded option means effective. Pure timing question means Macaulay. Standard fixed-coupon bond, price-sensitivity question, means modified.
2. Compute modified duration from Macaulay. Easy plug-and-chug with the \(1+y\) divisor. Watch for semiannual coupons (use \(y/2\)).
3. Approximate price change. Apply \(\Delta P / P \approx -ModDur \cdot \Delta y\). Sometimes paired with convexity.
4. Compute effective duration from a binomial tree. You will be given \(P-\), \(P+\), \(P_0\), and \(\Delta y\). Plug into the formula.
5. Reason about the differences. Why is effective duration on a callable bond lower than modified duration on the equivalent option-free bond? Because the call truncates upside price moves.

Common Mistakes

• Forgetting to divide by \(1+y\). Macaulay and modified are not the same number. Many candidates write the formula correctly but skip the conversion in a hurry.
• Using modified on callable bonds. The single most common Fixed Income error.
• Mismatching the period. For semiannual coupons, the yield in the modifier is \(y/2\), not \(y\). Get this wrong and your duration is off by a factor of 2.
• Ignoring convexity in large-shift questions. A 200bp shift question without the convexity term will give the wrong percentage answer.

Practice This Topic

Review the duration formulas on the CFA Level I formula sheet, then drill the concepts on real exam-style questions.

FreeFellow has over 1,050 free CFA Level I practice questions, including a dedicated Fixed Income topic filter so you can practice duration questions back-to-back. The solutions explain not just the arithmetic but also why a particular duration measure applies.

If you can identify the right duration measure in 15 seconds (before reaching for the formula) you will save several minutes across the exam, time you can spend on the harder Ethics and FSA questions.

Start with the free Fixed Income practice set, and pair it with the formula sheet for fast review.