**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.

**Answer: Price Increase**

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.

**Answer: Price Decrease**

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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