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Plate 4 of 19 · Finance

Bonds, yields and duration

About 7 minutes to read, then three questions that count toward your mastery.

A bond’s price is its fixed payments discounted at today’s yield. Duration says how far that price moves when yields do.

The idea

A plain bond pays a fixed coupon every year and its face value at maturity. Its price is those payments discounted at the market’s required return, the yield to maturity. Because the payments are fixed, a higher yield means a lower price: prices and yields always move in opposite directions.

If the coupon is above the yield, buyers pay more than face value (a premium) until their overall return falls to the yield; if it is below, the bond trades at a discount; if they are equal, it trades at face value.

Duration measures sensitivity. Macaulay duration is the average time until you are paid, each payment weighted by its present value. Modified duration, Macaulay ÷ (1 + y), turns that into a rule: price changes by about −modified duration × the change in yield. Convexity corrects for the curve: for big moves, the straight-line estimate overstates the fall and understates the rise.

The formula

P = C(1 + y) + … + C + FV(1 + y)N, ΔPP ≈ −Dmod × Δy + ½ × Conv × Δy2

C, FV
the annual coupon and the face value repaid at maturity
y
yield to maturity, per year
Dmac
Macaulay duration: Σ t × PV(payment at t) ÷ P, in years
Dmod
modified duration: Dmac ÷ (1 + y)
Conv
convexity: how much the price–yield curve bends

A worked example

Plate 4

A five-year bond with a $1,000 face value pays a 6% coupon ($60) once a year. The market yield is 5%. What is it worth, and what happens if the yield rises to 6%?

  1. Price: the five coupons are worth $259.8 today and the $1,000 repayment $783.5, so the bond is worth $1,043.3, a premium, because its 6% coupon beats the 5% yield.
  2. Duration: Macaulay duration is 4.478 years; modified duration is 4.478 ÷ 1.05 = 4.265.
  3. Duration’s estimate for a one-point rise in yield: −4.265 × 1% × $1,043.3 = −$44.5.
  4. Convexity (23.44) adds back ½ × 23.44 × 0.01² × $1,043.3 = $1.2, for an estimate of −$43.3.

At a 6% yield the coupon equals the yield, so the bond is worth exactly $1,000.0: a fall of $43.3. Duration alone was close; with convexity it is almost exact.

Try it

Face value $100, one coupon a year.

5.0%
10
5.0%

Price a bond and its sensitivity

$100.00per $100 of face value, at face value
Macaulay duration
8.11 years
Modified duration
7.72
Convexity
75.0
If the yield rises one point
−7.72% by duration; −7.36% exactly

Duration gets you most of the way; convexity explains the small difference from the exact change.

The common mistake

Using Macaulay duration as the price sensitivity

Macaulay duration is a time in years, not a sensitivity. Plugging it in directly gives −4.478 × 1% × $1,043.3 = −$46.7, overstating the fall by $3.4. Divide by (1 + y) first.

Remember

  • Price and yield move in opposite directions. Coupon above the yield means a premium; below, a discount.
  • Price change ≈ −modified duration × yield change. A modified duration of 7 means about −7% for a one-point rise.
  • Longer maturities and lower coupons mean more duration. More convexity helps the holder whichever way yields move sharply.

Check yourself

Three questions from the Learn room’s bank. Your answers update your mastery of this skill, the same as in Practise.

Use it on real numbers

Open Bonds, in Finance › Bonds and options and try the idea on your own numbers.