π βDuration tells you how much a bond's price will change when interest rates change.β But there are three main types: Macaulay Duration, Modified Duration, and Effective Duration. This article breaks down each one with simple math and real-world examples.
Duration is a measure of a bond's sensitivity to interest rate changes. It helps investors understand the risk and potential price movement of their fixed income investments. While the core idea is the same, the three types of duration are calculated differently and used for different bond structures.
1. Macaulay Duration
Macaulay Duration is the weighted average time until a bond's cash flows are received. It's measured in years. The formula calculates the present value of each cash flow, weights it by the time until receipt, and sums them up, then divides by the bond's price.
Step 1: Calculate Present Value (PV) of each cash flow.
- Year 1 Coupon: $50 / (1.06) = $47.17
- Year 2 Coupon: $50 / (1.06)2 = $44.50
- Year 3 Coupon + Principal: $1050 / (1.06)3 = $881.68
Step 2: Bond Price = Sum of PVs = $47.17 + $44.50 + $881.68 = $973.35.
Step 3: Weight each PV by its time and sum.
= (1 * $47.17) + (2 * $44.50) + (3 * $881.68) = $47.17 + $89.00 + $2,645.04 = $2,781.21.
Step 4: Macaulay Duration = $2,781.21 / $973.35 = 2.86 years.
There is only one cash flow at year 5: $1000.
Bond Price = $1000 / (1.04)5 = $821.93.
Weighted sum = 5 * $821.93 = $4,109.65.
Macaulay Duration = $4,109.65 / $821.93 = 5 years.
2. Modified Duration
Modified Duration is a direct measure of price sensitivity. It tells you the approximate percentage change in a bond's price for a 1% change in yield. It is derived from Macaulay Duration.
Formula: Modified Duration = Macaulay Duration / (1 + YTM/n)
Where 'n' is the number of compounding periods per year.
Macaulay Duration = 2.86 years.
YTM = 6% (0.06). Coupons are annual, so n=1.
Modified Duration = 2.86 / (1 + 0.06/1) = 2.86 / 1.06 = 2.70.
This means if the YTM increases by 1% (from 6% to 7%), the bond's price will fall by approximately 2.70%.
Modified Duration = 4.5 / (1 + 0.05/2) = 4.5 / 1.025 = 4.39.
A 0.5% rise in yield would cause an estimated price drop of 4.39 * 0.5% = 2.195%.
3. Effective Duration
Effective Duration is used for bonds with embedded options, like callable or putable bonds, where cash flows can change with interest rates. It measures sensitivity by calculating price changes for small parallel shifts in the yield curve.
Formula: Effective Duration = (P- - P+) / (2 * P0 * Ξy)
Where P- is price if yield falls, P+ is price if yield rises, P0 is initial price, and Ξy is the change in yield.
Effective Duration = ($1045 - $995) / (2 * $1020 * 0.005)
= $50 / (2 * $1020 * 0.005) = $50 / $10.20 = 4.90.
Effective Duration = ($988 - $970) / (2 * $980 * 0.0025)
= $18 / (2 * $980 * 0.0025) = $18 / $4.90 = 3.67.
Key Differences Summary
| Measure | What it Measures | Best For | Formula Basis | Units |
|---|---|---|---|---|
| Macaulay Duration | Weighted avg. time to cash flows | Understanding time profile | Present value of cash flows | Years |
| Modified Duration | Price sensitivity for non-callable bonds | Plain vanilla bonds | Macaulay Duration / (1+YTM/n) | Unitless (% change) |
| Effective Duration | Price sensitivity for bonds with options | Callable bonds, MBS, complex structures | (P- - P+) / (2*P0*Ξy) | Unitless (% change) |
β οΈ Common Pitfalls & Limitations
- Modified Duration assumes a linear relationship: It gives a good approximation for small yield changes but becomes inaccurate for large shifts because of convexity.
- Effective Duration requires complex modeling: Calculating P- and P+ often needs sophisticated financial models to estimate how optionality affects cash flows.
- All durations assume a parallel shift: They measure sensitivity to a uniform change in yields across all maturities, which is a simplification of real market movements.
- For zero-coupon bonds: Macaulay Duration = Maturity, and Modified Duration is simply Maturity / (1+YTM).