# Effective Convexities

Effective convexity is the sensitivity of duration to changes in interest rates.

$$\text{Effective convexity} =\cfrac {P_{i-}+P_{i+}-2 \times P_o}{P_0 (\Delta \text{Curve})^2}$$

Both callable and straight bonds experience similar positive convexity when interest rates are high. However, the effective convexity of a callable bond turns negative when the call option is near the money. This is because the upside for a callable bond is much smaller than the downside.

On the other hand, putable and straight bonds have similar positive convexity when interest rates are low. It is worth noting that putable bonds have positive convexity throughout. Besides, the convexity of a putable bond becomes greater than that of a straight bond when the put is near the money.

## Question

Which of the following statements is least accurate: The effective convexity of a callable bond:

1. Becomes negative when the call option is near the money.
2. Is always positive.
3. Is similar to that of a straight when interest rates are high.

#### Solution

The effective convexity for a callable bond is not always positive as it becomes negative when the embedded option is near the money

A is incorrect. The effective convexity of a callable bond turns negative when the call option is near the money because the price response of a callable bond to lower interest rates is capped by the call option. In other words, the price of a callable bond has limited upside potential

C is incorrect. Both callable and straight bonds experience similar positive convexity when interest rates are high.

Reading 30: Valuation and Analysis of Bonds with Embedded Options

LOS 30 (l) Compare effective convexities of callable, putable, and straight bonds.

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