Work out three duration measures for a bond: dollar duration, modified duration and Macaulay duration.
The idea
Duration measures a bond's price sensitivity to yield. Macaulay duration is the present-value-weighted average time to the bond's cash flows, in years. Each time t is weighted by the present value of its cash flow. Macaulay duration = Σ t × PV(CF(t)) ÷ Price. For a zero-coupon bond it equals the maturity, and a coupon bond's is shorter than its maturity. Modified duration adjusts Macaulay duration for compounding. Modified duration = Macaulay duration ÷ (1 + y/m). Here y is the annual yield, m is the compounding frequency and D is the modified duration. For a small yield change Δy, Percentage price change = −D × Δy. Dollar duration puts the sensitivity in money terms. Dollar duration = Modified duration × Price. Dollar duration is the approximate price change per unit of yield, as a positive number. For a one-basis-point rise in yield, the approximate loss is DV01 = Dollar duration × 0.0001. A semiannual bond's duration computed in half-year periods must be divided by 2 to be in years. The usual errors are using Macaulay where modified is needed and mixing percentage points with decimals.
Formulas
Macaulay duration
Macaulay duration = Σ t × PV(CF(t)) ÷ P
- t
- the time of each cash flow, in years
- PV(CF(t))
- the present value of the cash flow at time t
- P
- the bond price
Equals maturity for a zero-coupon bond. If computed in half-year periods, divide by 2 to state it in years.
Modified duration from Macaulay duration
D = DMac ÷ (1 + y ÷ m)
- DMac
- Macaulay duration, in years
- y
- the annual yield
- m
- the compounding frequency of the yield per year
Modified, not Macaulay, duration gives the percentage price change: about −D × Δy for a small yield change.
Dollar duration
dollar duration = D × P
- D
- modified duration
- P
- the bond price or position value
The approximate price change per unit of yield, stated as a positive number.
DV01 from dollar duration
DV01 = dollar duration × 0.0001
- dollar duration
- modified duration × price
The approximate loss for a one-basis-point rise in yield. Mixing percentage points with decimals is the usual error.
A practice question
A 7-year zero-coupon bond and a 7-year bond paying a 6% annual coupon are compared. Which statement about Macaulay duration is correct?
- Both bonds have a Macaulay duration of 7 years, because duration equals maturity for any bondWhy not: Only a zero-coupon bond has a Macaulay duration equal to its maturity; coupons pull the weighted-average time earlier.
- The zero-coupon bond has a Macaulay duration of exactly 7 years and the coupon bond has a shorter one - correct
- The coupon bond has the longer Macaulay duration, because it pays more total cashWhy not: Total cash does not lengthen duration; coupons arrive before maturity, so they shorten the weighted-average time to payment.
- The zero-coupon bond has a Macaulay duration of 3.5 years, half its maturityWhy not: Halves the maturity with no justification; a single cash flow at year 7 has a weighted-average time of 7.
Macaulay duration is the present-value-weighted average time to the bond's cash flows. A zero-coupon bond has one cash flow at maturity, so its duration equals its maturity. Coupons receive some weight at earlier dates, so a coupon bond's duration is below its maturity.
A harder one
A bond portfolio has market value USD 40 million, a Macaulay duration of 6.5 years and a yield of 5% per year with semiannual compounding. The change in portfolio value for a one-basis-point rise in yield (the DV01) is closest to:
- a fall of USD 26,000Why not: Uses Macaulay duration directly instead of converting it to modified duration.
- a fall of USD 2,536,585Why not: Applies a one percentage point change in yield, 0.01, instead of one basis point, 0.0001.
- a rise of USD 25,366Why not: Gets the direction wrong: bond prices fall when yields rise.
- a fall of USD 25,366 - correct
Modified duration = 6.5/1.025 = 6.3415. Dollar duration = 6.3415 x 40 million. DV01 = dollar duration x 0.0001 = 25,366 (a loss).
The hardest tier
A 2-year bond with face value 100 pays a 6% coupon semiannually. The yield is 8% per year with semiannual compounding. The Macaulay duration, expressed in years, is closest to:
- 3.82 yearsWhy not: Leaves the answer in half-year periods and does not convert to years.
- 1.84 yearsWhy not: Divides by (1 + y/2) and so reports the modified duration instead of Macaulay.
- 1.91 years - correct
- 2.00 yearsWhy not: Reports the maturity, which equals the Macaulay duration only for a zero-coupon bond.
Work in half-year periods with 4% per period: price = 96.37, duration = 3.8250 periods. Divide by 2 to get 1.9125 years.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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