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

Fixed Income makes up 11 to 14% of the CFA Level I exam (CFA Institute), and duration comes in three versions that are easy to mix up. Macaulay duration measures timing. Modified duration measures price sensitivity for a bond with fixed cash flows, and effective duration measures it for a bond with an embedded option. Most duration questions come down to knowing which one the bond in front of you needs.

When I sat for Level I, I underestimated how often the exam asks you to pick the right measure instead of plugging numbers into a formula. The arithmetic is easy. People lose points by choosing the wrong measure.

Macaulay and modified duration

Macaulay duration is the weighted-average time, in years, until you receive a bond's cash flows. Each cash flow is weighted by its present value as a share of the price.

$$D_{Mac} = \sum_{t=1}^{n} t \cdot \frac{PV(CF_t)}{P}$$

$PV(CF_t)$ is the present value of the cash flow at time $t$, and $P$ is the bond's price, the sum of all the discounted cash flows. A zero-coupon bond has one cash flow, so its Macaulay duration equals its time to maturity. A coupon bond's is always shorter than its maturity, because the coupons pull the average forward.

Take a 3-year, 5% annual-coupon bond with a 5% yield. The cash flows are $5, $5 and $105, and the price is $100.

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

Weighted by present value, the bondholder waits about 2.86 years for the cash flows, a little under 3 because the early coupons count.

Modified duration turns that into price sensitivity, the approximate percentage change in price for a 1% change in yield. You divide by one plus the yield per period: for an annual-coupon bond with annual yield $y$, that's $1+y$, and for semiannual coupons it's $1 + y/2$.

$$ModDur = \frac{D_{Mac}}{1+y}$$

$$\frac{\Delta P}{P} \approx -ModDur \cdot \Delta y$$

For the 3-year bond, modified duration is $2.86 / 1.05 = 2.72$, so a 1% (100bp) rise in yield should drop the price by about 2.72%.

Two more measures come straight from modified duration. Money duration (or dollar duration) is $MoneyDur = ModDur \cdot P$; multiply it by the yield change in decimal and you get the approximate dollar change in price, which is what you use to size positions or estimate P&L. The price value of a basis point, $PVBP = ModDur \cdot P \cdot 0.0001$, is the dollar change for a 1bp move, and it's often quoted on Treasury futures and trading desks. Both assume fixed cash flows, same as modified duration.

Effective duration for bonds with embedded options

Once a bond has an embedded option, its cash flows change when yields change. A callable bond is more likely to be called when rates fall, which shortens its life, and a putable bond is more likely to be put when rates rise. Modified duration assumes fixed cash flows, so on these bonds it can be badly wrong.

Effective duration measures the price change directly for a parallel shift in the benchmark yield curve:

$$EffDur = \frac{P_- - P_+}{2 \cdot P_0 \cdot \Delta y}$$

$P_-$ is the price if the curve shifts down by $\Delta y$, $P_+$ is the price if it shifts up by $\Delta y$, $P_0$ is the current price, and $\Delta y$ is the shift in decimal form. On the exam the prices usually come from a binomial interest-rate tree, which reprices the bond in each scenario and lets the option behave the way it would.

Say a 10-year 6% callable bond is priced at $98.50. The tree gives $97.20 if rates rise 25bp and $99.60 if they fall 25bp.

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

The price rose less when rates fell than it dropped when rates rose. That's the call capping the upside, and effective duration picks it up. Modified duration on the same bond ignores the option and gives you a different, misleading number.

Watch for a stem that describes a callable bond and asks for modified duration. The answer it wants is that modified duration is the wrong measure here and you need effective duration. If you plug the numbers into the modified-duration formula, you'll land on a wrong choice.

Duration is a linear approximation, so for bigger yield changes you add convexity:

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

Option-free bonds have positive convexity: the price rises more when yields drop than it falls when they rise by the same amount. Callable bonds can have negative convexity at low yields because the call caps the price, which is another reason they need effective duration and effective convexity. Level I tests the full duration-plus-convexity form, and on a 200bp move, leaving out convexity throws the price estimate off by a lot.

What the exam tests

From my own Level I sitting and from working through hundreds of Fixed Income questions, these come up:

1. Picking the measure. An embedded option means effective. A pure timing question means Macaulay. A fixed-coupon bond with a price-sensitivity question means modified.
2. Converting Macaulay to modified with the $1+y$ divisor, using $y/2$ for semiannual coupons.
3. Approximating the price change with $\Delta P / P \approx -ModDur \cdot \Delta y$, sometimes with the convexity term.
4. Computing effective duration from a binomial tree, given $P_-$, $P_+$, $P_0$, and $\Delta y$.
5. Explaining the differences. Why is effective duration on a callable bond lower than modified duration on the same bond without the call? Because the call truncates the upside price moves.

The mistakes line up with that list. People skip the division by $1+y$ in a hurry, use modified duration on a callable bond, divide by $1+y$ instead of $1 + y/2$ on a semiannual bond, and leave convexity out of a 200bp-shift question.

Decide which measure the bond needs before you reach for a formula. Review the formulas on the CFA Level I formula sheet, then work through the more than 1,800 free CFA Level I practice questions. The Fixed Income topic filter lets you run duration questions back to back, and the solutions explain both the arithmetic and why a particular measure applies.