Module III·Article II·~3 min read
Bond Duration
Fixed Income: Foundation Level
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Duration: a measure of interest rate risk
Duration is a key risk metric for bonds, indicating their price sensitivity to changes in interest rates. It is the "elasticity" of price with respect to rate: if duration = 7, a 1% increase in rates will lead to approximately a 7% drop in price.
Types of Duration
| Type | Definition | Application |
|---|---|---|
| Macaulay Duration | Weighted average time to receive cash flows (in years) | Theoretical calculation, immunization |
| Modified Duration | Macaulay / (1 + YTM/n) | Practical assessment of price sensitivity |
| Effective Duration | Numerical calculation with curve shift | For bonds with embedded options (callable, puttable) |
| Key Rate Duration | Sensitivity to individual points on the curve | Curve exposure management |
| Dollar Duration (DV01) | Change in price in $ with 1 bp shift | Portfolio risk management |
Modified Duration Formula
$ \Delta P/P \approx -\text{Modified Duration} \times \Delta y $
Where $\Delta P$ — change in price, $P$ — price, $\Delta y$ — change in yield.
Practical Example
| Bond | Price | Modified Duration | Rate Increase +1% | New Price |
|---|---|---|---|---|
| 2Y Treasury | 100 | 1.9 | -1.9% | 98.1 |
| 10Y Treasury | 100 | 8.5 | -8.5% | 91.5 |
| 30Y Treasury | 100 | 19.0 | -19.0% | 81.0 |
Conclusion: Long bonds are 10 times more sensitive to rates than short bonds!
Factors Influencing Duration
| Factor | Impact on Duration |
|---|---|
| ↑ Time to Maturity | ↑ Duration |
| ↑ Coupon | ↓ Duration (more cash flow earlier) |
| ↑ YTM | ↓ Duration (discounting is stronger) |
| Zero-coupon bond | Duration = Maturity |
Convexity: Second Order
Duration is a linear approximation. For large rate changes, convexity is needed:
$ \Delta P/P \approx -\text{Duration} \times \Delta y + \frac{1}{2} \times \text{Convexity} \times (\Delta y)^2 $
Convexity Properties
- Positive convexity — price rises faster with rate declines than it falls with increases (beneficial to holder)
- Negative convexity — reverse effect (callable bonds, MBS)
- The higher convexity, the better for the investor (all else being equal)
Portfolio Duration
$ \text{Portfolio Duration} = \sum (w_i \times \text{Duration}_i) $
Example:
| Position | Weight | Duration | Contribution |
|---|---|---|---|
| 2Y Treasuries | 30% | 1.9 | 0.57 |
| 10Y Treasuries | 50% | 8.5 | 4.25 |
| IG Corporate | 20% | 6.0 | 1.20 |
| Portfolio Total | 100% | 6.02 |
Duration Management Strategies
| Strategy | Action | When to Apply |
|---|---|---|
| Shorten Duration | Sell longs, buy shorts | Expecting rate increases |
| Extend Duration | Sell shorts, buy longs | Expecting rate decreases |
| Barbell | Concentrate on shorts and longs | Expecting yield curve changes |
| Bullet | Concentrate in one point of curve | Minimizing reinvestment risk |
| Ladder | Even distribution by maturity terms | Stability, automatic rolling |
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