Duration: the average wait, and the rate sensitivity
Macaulay duration is how long, on average, you wait for your money. Modified duration is how many percent the price moves for a one-point move in yield. They are the same idea, because the cash flows far out are the ones most exposed to the rate.
- Compute Macaulay and modified duration for a zero and for a coupon bond
- Estimate a price change from duration alone, in percent and in dollars
- Find the yield move that costs a stated amount
- Rank bonds by duration without calculating
The intuition
A three-year zero pays once, at year three, so the average wait for your money is exactly three years. A three-year 5% coupon bond pays some money back at years one and two, so its average wait, each payment weighted by its share of the price, is shorter: about 2.86 years. That weighted average wait is Macaulay duration, and it is in years.
The same number, divided by one plus the yield, tells you how sensitive the price is to rates: the three-year coupon bond has a modified duration of about 2.75, so a one-point rise in yield knocks roughly 2.75% off the price. The link is not a coincidence. A payment far in the future is discounted many times over, so it is the most sensitive to the discount rate; a bond whose money comes back late is a bond whose price swings most. Zeros have the longest duration for their maturity, and higher coupons pull duration down.
Macaulay = Σ t × PV(CFₜ) ÷ price, in years. Modified = Macaulay ÷ (1 + y). A zero's Macaulay duration is its maturity. % price change ≈ −modified duration × Δy. Dollar change ≈ that × market value, and the market value × modified duration × 0.0001 is the DV01: what one basis point costs.
Why it works
- The conventions here: annual coupons, annual compounding, price per 100, whole years (two to five, so the weighted sum can be done by hand). Macaulay in years; modified = Macaulay ÷ (1 + y). Price changes are first-order, −modified duration × Δy × price, with no convexity. Where a duration and price are printed, the dollar answers are computed from the printed figures.
- Why the average wait is the sensitivity. Differentiate the price with respect to the yield: each term brings down its t and one more (1 + y). Divide by the price and you have −Macaulay ÷ (1 + y). The weights that make the average wait are the weights that make the sensitivity.
- Coupons shorten it. Any coupon brings cash forward and pulls the weighted average below maturity. The higher the coupon, the further below. A zero, with nothing early, sits exactly at maturity.
- Yield shortens it too. A higher yield discounts the far payments more heavily, so they weigh less in the average and the duration falls.
- From percent to dollars. The percentage change applies to the market value, face × price ÷ 100. One basis point is Δy = 0.0001, and the dollar cost of one basis point is the DV01, the number the next chapter is built on.
- Run it backwards. A dollar loss divided by the DV01 is the yield rise that causes it. The true move needed is slightly larger, because each extra basis point costs a little less than the one before: convexity again.
| Weighted sum: 1 × 4.81 + 2 × 4.62 + 3 × 93.34 | 294.09 |
| Macaulay duration: 294.09 ÷ 102.78 | 2.86 years |
| Modified duration: 2.86 ÷ 1.04 | 2.75 |
| A three-year zero: Macaulay 3 years, modified 3 ÷ 1.04 | 2.88 |
| DV01 on $10M face: $10M × 102.78 ÷ 100 × 2.75 × 0.0001 | about $2,826 a basis point |
| Yield rise to lose $250,000: 250,000 ÷ 2,826 | about 88 bp |
The formulas
The average wait for your money, each payment weighted by its share of the price.
The rate sensitivity; a zero waits its whole maturity.
The straight-line estimate.
What one basis point costs in dollars.
The backwards reading.
Worked example
Discount each cash flow, weight it by its year, add, divide by the price; then divide by one plus the yield. The follow-up asks why the answer is below the maturity.
See it move
Same bond. Change the coupon, the yield, the maturity and the face you hold.
- Raise the coupon. Both of the coupon bond's durations fall and the price rises; the zero's durations do not move. The DV01 rises, because the market value rises more than the duration falls.
- Raise the yield. Every duration falls, including the zero's modified duration, and the price and the DV01 fall with them.
- Lengthen the maturity. Every duration rises, and the gap between the zero and the coupon bond widens.
- Hold more face. Only the DV01 moves, in proportion; durations are per unit of price.
Run it backwards
You hold a stated face at a printed price and modified duration. How many basis points must yields rise for the position to lose a stated amount?
Work out what one basis point costs: market value × modified duration × 0.0001. Divide the loss by it. That is the DV01 in use before it has a name.
The true move needed is slightly larger. Convexity means each extra basis point costs a little less than the one before, so it takes a bit more than the straight-line estimate to lose the full amount. The gap only matters for big moves.
Traps
Say it in the interview
“What is duration, and what is the difference between Macaulay and modified?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Macaulay = weighted average wait, in years. A zero's is its maturity.
- Modified = Macaulay ÷ (1 + y). % price change ≈ −modified × Δy.
- Higher coupon, higher yield, shorter maturity: each lowers duration.
- DV01 = market value × modified duration × 0.0001; loss ÷ DV01 is the yield move that causes it.