9.35 Combined Effect of Duration and Convexity on Estimated Price
Δ Price = (Δy) (-D) (Initial Price) + (0.5) (Δy)2 (Convexity) (Initial Price)
Example:
Modified Duration = 10.5
Convexity = 175
Assume a 100 BP (basis point) change in rates (or “yield”)
Initial Price: 90.5 (not related to earlier example with same price)
To Solve:
What are the price changes due to both duration and convexity respectively, and in combination? (The formulae in the box above represent the dollar-price adjustments needed for both the Duration and Convexity effects of the Price-Yield Curve in combination. The Duration formula shows how price is affected in inverse relation to the anticipated movement in yield, while the Convexity adjustment adds to the Duration-based estimated change in price.)
Solution:
| Duration: (.0100) (-10.5) (90.5) | = ± 9.5025 | (Inverse to YTM) |
| Convexity: (0.5) (.01002) (175) (90.5) | = + 0.7919 | (Always good news) |
| Combined Change (if yields decline) | = + 10.2943 | |
| Combined Change (if yields rise) | = – 8.7106 | |
| New Price (if yields decline) | = 90.5 + 10.2943 | = 100.7943 |
| New Price (if yields rise) | = 90.5 – 8.7106 | = 81.7894 |
- Notice that the price movement, in reality, is not symmetrical in both directions. That is because the Price-Yield curve is steeper to the left when rates decrease, than to the right when rates increase. As you can see from these numbers, prices move up more than they move down due to Convexity.
- Convexity always adds to price. In contrast, Duration is inversely related to yield, thus the Duration-based estimated price decreases when yields go up, and vice versa.
- After all this, the combined price estimate will still yield a small error. The true price is along the actual Price-Yield curve.
