Duration is a weighted average of when the money comes back; modified duration turns it into the percentage the price moves for one point of yield, and convexity fixes the rest.
Modified duration is the Macaulay duration, the present-value-weighted average time to each cash flow, divided by one plus the yield per period. It estimates the percentage fall in price for a one-point rise in yield. An illustrative six-year bond with a 5.45 per cent semiannual coupon bought at par has a Macaulay duration of 5.1976 years and a modified duration of 5.0597: one point of yield takes about 5.06 per cent off the price, and 4.91 per cent once convexity is added, which is the move the bond actually makes.
Worked in full in The Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Excel will return the number in one cell. The point of building it from the cash flows is that the table shows where the sensitivity sits: almost all of it in the final payment, which is why longer bonds and lower coupons carry more of it, and why a high-yield bond's duration is the smaller of its two risks.
| Input | Value |
|---|---|
| Coupon, paid semiannually | 5.45% |
| Maturity | 6 years, 12 half-years |
| Price per 100 of face | 100.00 |
| Yield to maturity (bought at par, so the coupon) | 5.45% |
| Face held | 25,000 |
Macaulay duration (years) = Σ k × PVk ÷ price ÷ 2
Modified duration = Macaulay ÷ (1 + y/2)
Convexity = Σ k(k + 1) × PVk ÷ price ÷ (1 + y/2)2 ÷ 4
Here k counts half-years and PVk is each cash flow discounted at the yield per half-year. In Excel, with settlement and maturity as dates: =MDURATION(DATE(2026,1,1),DATE(2032,1,1),5.45%,5.45%,2) returns 5.0597, and =DURATION(...) with the same arguments returns 5.1976.
Each half-year pays 2.725 per 100 of face and the last pays 102.725. Discount each at 2.725 per cent a half-year and multiply by the period number:
| Half-year k | Cash flow | PV at the yield | k × PV |
|---|---|---|---|
| 1 | 2.725 | 2.6527 | 2.6527 |
| 2 | 2.725 | 2.5823 | 5.1647 |
| 6 | 2.725 | 2.3190 | 13.9143 |
| 11 | 2.725 | 2.0274 | 22.3009 |
| 12 | 102.725 | 74.3982 | 892.7782 |
| All twelve rows | 100.00 | 1,039.5161 |
Divide 1,039.5161 by the price of 100 and the average is 10.3952 half-years, or 5.1976 years: the Macaulay duration. The final row carries 892.7782 of the 1,039.5161, which is the whole story of duration in one cell. Divide by 1.02725 and the modified duration is 5.0597. Run the k(k + 1) column the same way and convexity comes out at 30.44.
| Yield move | Actual new price | Actual change | Duration only | Duration + convexity |
|---|---|---|---|---|
| +0.5 point | 97.51 | -2.49% | -2.53% | -2.49% |
| +1 point | 95.09 | -4.91% | -5.06% | -4.91% |
| -1 point | 105.22 | 5.22% | 5.06% | 5.21% |
| +2 points | 90.46 | -9.54% | -10.12% | -9.51% |
Duration alone overstates the loss when yields rise and understates the gain when they fall, by roughly the same amount, about 0.15 per cent of price at one point. That is convexity: half of 30.44 times the yield move squared. It grows with the square of the move, so at two points the correction is 0.61 per cent and duration alone is off by more than half a point of price.
Turned into money, a holding of 25,000 face has a DV01 of 12.65: each basis point of yield moves its value by that much. A 100 basis point rise is an estimated loss of 1,265 on duration alone and 1,228 actually.
| Maturity | Coupon | Macaulay | Modified | Convexity | Price, +1 point |
|---|---|---|---|---|---|
| 2 years | 4.80% | 1.93 | 1.89 | 4.5 | -1.86% |
| 3 years | 4.95% | 2.82 | 2.76 | 9.2 | -2.71% |
| 4 years | 5.15% | 3.67 | 3.57 | 15.2 | -3.50% |
| 5 years | 5.30% | 4.46 | 4.34 | 22.4 | -4.23% |
| 6 years | 5.45% | 5.20 | 5.06 | 30.4 | -4.91% |
Duration rises with maturity but more slowly, because coupons pull the weighted average forward. Strip the coupons out and the gap closes: a six-year zero-coupon bond has a Macaulay duration of exactly 6.00 years and a modified duration of 5.84 at the same yield. Convexity rises roughly with the square of maturity, so the correction matters most on the long end.
Run the same table on an illustrative CCC bond with an 8.50 per cent coupon, five years to run, bought at 94.00. Its yield to maturity is 10.06 per cent, its Macaulay duration 4.15 years and its modified duration 3.95. A one-point rise in yield costs 3.85 per cent of price.
With an annual default probability of 15 per cent and a 40 per cent recovery, the same bond is expected to lose 9.00 per cent of its value a year to default. That is equivalent to a yield rise of 2.28 points every year, 2.3 times the damage of a one-point rate shock. For low-rated credit, duration measures the risk you can hedge and leaves out the one that decides the return; the arithmetic is in the yield after expected default losses.
Weight each period by the present value it carries, average, convert to years, and divide by one plus the yield per period: 5.0597 for a six-year 5.45 per cent bond at par. Add half the convexity times the move squared for anything larger than a few dozen basis points. Then remember what duration does not see. The Credit Investor follows one portfolio of bonds, loan funds, BDCs and a private credit fund from the yield on the screen to the return actually earned, and the free workbook for this case builds duration and convexity from explicit cash-flow tables for every bond in the ladder.
Macaulay duration is the present-value-weighted average time to each cash flow, in years. Modified duration divides it by one plus the yield per period and measures price sensitivity: the percentage price change for a one-point move in yield. On an illustrative six-year 5.45 per cent bond at par, Macaulay is 5.1976 years and modified 5.0597.
DV01 is the money change in value for a one basis point move in yield: modified duration times market value times one basis point. A holding of 25,000 face in an illustrative bond at par with modified duration 5.0597 has a DV01 of 12.65, so a 100 basis point rise costs about 1,265 before convexity.
Only its rate risk. An illustrative CCC bond at 94.00 yielding 10.06 per cent has modified duration of 3.95, so one point of yield costs about 3.85 per cent of price. Its expected default loss is 9.00 per cent a year, the same as a yield rise of 2.28 points every year. For low-rated credit, default dominates duration.
This article is one calculation from The Credit Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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