Approximate Convexity
The calculation of convexity is complex. However, an approximation can be used. Like the modified duration approximation, the computation is not too difficult and is very accurate.
The convexity approximation, like the modified duration approximation, uses two prices in its calculation. Prices are computed for increases and decreases in yields. Using the approximation, no actual measure of convexity is required.
Notice that this formula requires the current bond price as well as prices if yields increase or decrease. The inputs are the same as those required for approximate modified duration.
Since $$PV_+$$ is the bond price for an increase in yields, how will the resulting price compare to the current bond price?
Correct!
An increase in yields lowers bond prices. Therefore, the price resulting from $$P^+$$ will be lower than $$P$$.
Incorrect.
An increase in yields lowers bond prices.
Incorrect.
Yield changes always affect prices. An increase in yields lowers bond prices.
For this bond, what would happen to the convexity approximation for a 100 bps change in yields?
To summarize:
[[summary]]
Correct!
A larger change in yields will result in a bigger convexity adjustment. Convexity increases for non-callable bonds as yield changes increase. In fact, the convexity estimate for a 100bp change is 423.581.
Incorrect.
Convexity changes with the magnitude of yield changes.
Incorrect.
Convexity changes with the magnitude of yield changes.
__Approximate convexity__ is computed for an expected change in yields using the following formula:
$$\displaystyle \text{ApproxCon} =\frac{(PV_-) + (PV_+) - [2 \times PV_0]}{(\Delta \text{Yield})^2 \times (PV_0)}$$
where:
$$PV_0$$ is the current bond price, $$PV_-$$ is the bond price for a decrease in yields, $$PV_+$$ is the bond price for an increase in yields, and $$\Delta \text{Yield}$$ is the expected change in yields.
As an example, consider a 4.00%, 30-year non-callable bond that is currently priced at par (100) and has a modified duration of 17.380. An investor wants to estimate the convexity on this bond for a 50 bps change in yields.
If yields decrease by 50 bps to 3.50%, the new bond price is 109.241. If yields increase by 50 bps to 4.50%, the new bond price is 91.813. Therefore, the approximate convexity is equal to:
$$\displaystyle \text{ApproxCon} = \frac{(109.241 + 91.813) - 2 \times 100}{(0.005)^2 \times (100)} = 421.503$$
The result, 421.503, is the approximate convexity and can be used to compute a convexity adjustment.
It will be lower
It will be higher
It will be the same
It will increase
It will decrease
It will stay the same
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