## Monday, November 21, 2011

### Convexity

CFA Level 1 - Fixed Income Investments

Convexity helps to approximate the change in price that is not explained by duration. If you go back to the third property of a bond's price volatility you will see that when there is a large change in rates, the duration measure can be way off because of the convex nature of the yield curve.

To calculate convexity the formula is:

Formula 14.16

 Total Price change = (-duration * change in yield * 100) + (C * change in yield squared * 100)

To find the C in the equation, use this equation that has the same notation as duration:

C = V3 +V2 - 2(V1) / 2V1(change in yield) squared

Estimate a Bond's Price Given Duration, Convexity and Change in Yield
This is done by simply adding the convexity adjustment and the percentage price change due to duration equations to achieve an estimate that is closer than just a duration measure.

Formula 14.17

 Convexity adjustment to the percentage price change= C* change in yield squared * 100

Example: Total Price Change
Using the Stone & Co. bonds that had duration of 5.5, let's add a convexity of 93 and an increase of 150 bps in yield.

Total Price Change = (-5.5 * .0150 * 100) + (93 * .0150 squared * 100)
= -8.25 + 2.0925
= 6.157 So if rates increase by 150 bps, the price will decrease by 6.157%

Now let's look at a decrease of 150 bps in yield.

Total Price Change = (-5.5 * -.0150* 100) + (93 * -.0150 squared * 100)
= 8.25 + 2.0925
= 10.34

So if rates decrease by 150 bps, the price will increase by 10.34 %

Again, if you refer to the properties of price volatility, you can see that as rates decrease, the price increase will be greater than the decrease in price when rates rise.

Modified Convexity vs. Effective Convexity
With modified convexity the cash flows do not change due to a change in interest rates.

Effective Convexity, on the other hand, assumes that cash flow does change due to a change in interest rates.

When bonds have options, it is best to use effective convexity just like you should use effective duration. For option-free bonds, either convexity measure will be a positive value, whereas when it comes to bonds with options, the effective convexity could be negative even if the modified convexity is positive.

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