9.33 Dollar Convexity
Definition:
Dollar Convexity estimates the additional change in price for a given change in rates, i.e., the “vertical error” in the Duration-based price estimate; this approximation is useful for refining price estimates, given different interest rate environments.
Dollar Price Change Due to Convexity = (.5)(Δy)2 (Dollar Convexity)
Where, Dollar Convexity = Convexity (Initial Price)
Example:
Convexity = 94.36
Initial Price = $84.63
Solution:
Dollar Convexity = (94.36) (84.63) = $7,985.69
Dollar Price Change (up or down) Due to Convexity:
-
-
- 100 b.p. change = (.5) (.01)2 (7,985.69) = $0.399 ≈ + $0.40
- 200 b.p. change = (.5) (.02)2 (7,985.69) = $1.597 ≈ + $1.60
-
Note:
Convexity always adds to price. The Price-Yield Curve must be above the Duration line. That is why there is a plus sign attached to the price change.
Interpretation:
For a 100 and 200 b.p. change, the price change would be $0.40 and $1.60 respectively per $100 of par value – a fourfold difference! Do not infer from this that the price change will be double for a double increase in yield. The vertical error increases with increasing expected yield movements.
Notes:
- Dollar Duration changes in proportion to change in yield – i.e., approximately double the yield, double the price change due to duration (which is linear).
Approximate Dollar Price Change = (-D) (Initial Price) (Yield Change)
- Dollar Convexity changes more so – because of the squaring of yield change, i.e., curvature. The Convexity-based price change will be added to the Duration-based price change.
Dollar Price Change Due to Convexity = (.5) (Dollar Convexity) (Δy)2
