CFA Program Level I
Macaulay vs modified duration: CFA Level I explained
Macaulay duration is the weighted-average time until a bond's cash flows arrive, with each cash flow weighted by its share of the bond's present value. Modified duration is Macaulay duration divided by (1 + yield per period), and it estimates the percentage price change for a change in yield. Use Macaulay for time, modified for price sensitivity.
Macaulay duration in three steps
Step 1: discount each coupon and principal payment at the bond's yield. The present values add up to the price.
Step 2: multiply each present value by the number of years until it is paid, and add the results.
Step 3: divide the step 2 total by the price. The answer is in years.
A zero-coupon bond has a duration equal to its maturity, because it has one cash flow. A coupon bond has a duration shorter than its maturity. For the same maturity, a higher coupon gives a shorter duration.
Modified duration and the price estimate
Modified duration = Macaulay duration / (1 + i/c), where i is the annual yield and c is the number of coupon payments a year. For an annual-pay bond that is simply Macaulay / (1 + yield).
Estimated % price change = -modified duration x change in yield (in decimals). The minus sign is the inverse price-yield relationship: yields up, prices down.
Example from the OTS handbook glossary: a semiannual bond with a Macaulay duration of 4 years and a 10% yield has a modified duration of 4 / (1 + 0.10/2) = 3.8. A 100 basis point move in rates changes its value by about 3.8%.
Worked example: 3-year 6% annual-coupon bond at a 5% yield
Cash flows are 6, 6 and 106 per 100 of par. Present values at 5%: 6 / 1.05 = 5.7143; 6 / 1.05^2 = 5.4422; 106 / 1.05^3 = 91.5668. Price = 102.7232.
Time-weighted: 1 x 5.7143 + 2 x 5.4422 + 3 x 91.5668 = 291.2990. Macaulay duration = 291.2990 / 102.7232 = 2.84 years.
Modified duration = 2.8358 / 1.05 = 2.70. If the yield rises 50 basis points, the estimate is -2.70 x 0.0050 = -1.35%.
Repricing the bond at 5.5% gives an actual change of -1.34%. The small gap is convexity.
Convexity: the second-order correction
The price-yield relationship is curved. A fall in yield raises the price by more than an equal rise in yield lowers it, so duration alone is accurate only for small yield changes.
Second-order estimate: % price change = -modified duration x change in yield + 1/2 x convexity x (change in yield)^2.
Example: modified duration 8, convexity 90, yield change 100 basis points. Duration alone gives -8% for a rise and +8% for a fall. With convexity: -8 x 0.01 + 0.5 x 90 x 0.0001 = -7.55% for a rise, and +8.45% for a fall. With positive convexity, the convexity term adds to the price in both directions.
Where the exam catches people
- Believing modified duration is measured in years like Macaulay duration: it is a sensitivity, the approximate % price change for a 1 percentage point change in yield.
- Believing you divide by (1 + annual yield) for a semiannual or quarterly bond: divide by (1 + yield per period), for example 1 + 0.08/4 for a quarterly-pay bond yielding 8%.
- Believing a coupon bond's Macaulay duration equals its maturity: only a zero-coupon bond's does; coupons paid earlier pull a coupon bond's duration below maturity.
- Believing duration gives the exact price change for a large yield move: for an option-free bond, duration overstates the loss when yields rise and understates the gain when they fall.
- Forgetting the sign: a yield fall is a negative change, so -modified duration x (negative) gives a price gain.