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Effective Convexity

I’ve been having trouble understanding convexity conceptually, but I think I might be starting to understand it.

  • When rates are low (call is in the money), callable bonds have negative convexity because the call option essentially caps the bond’s price and, therefore, the bond is less sensitive to changes in interest rates.
  • Putable bonds exhibit positive convexity at all times, like straight bonds, but, when rates are high (put is in the money), they exhibit “less positive” convexity compared to straight bonds because the put option floors the bond’s price, and, therefore, the putable bond is less sensitive to further changes in rates. When rates are high, a callable bond’s convexity will be positive and similar to the straight bond’s, because the call option is out of the money, and the bond otherwise lacks the downside protection that limits the downside potential of the putable bond, making it sensitive to interest rate changes.

Can someone confirm the above points? It helps me to type it out like this, so I appreciate it.

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That all looks good.

Note that at middling yields a putable bonds have higher (positive) convexity than option-free bonds as the put option comes into the money.

Simplify the complicated side; don't complify the simplicated side.

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Thanks so much, Magician. This was an old thread, but would you be able to confirm what you meant by “extra positive” convexity for putable bonds at rising interest rates? I was under the understanding that putable bonds have “less positive” convexity compared to straight bonds as interest rates rose, as seen through the flatter price/yield curve.

https://www.analystforum.com/forums/cfa-forums/cfa-level-ii-forum/91351867

Thanks again for your time.

I meant there what I wrote above: that as yields rise and the put option comes into the money, the convexity increases above what it would be for an otherwise identical, option-free bond.

Simplify the complicated side; don't complify the simplicated side.

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Still very slightly confused; as yields rise, the price/yield curve for the putable bond flattens, so it appears to me to have less positive convexity compared to the straight bond. Where am I going wrong?.

As yields rise the convexity of a putable bond increases as the option starts to come into the money, then decreases as it moves farther in the money.

Simplify the complicated side; don't complify the simplicated side.

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Again, thank you for your patience. My mental roadblock is: as yields rise, the put option approaches the money. Accordingly, the bond becomes less sensitive to interest rate changes, and, therefore, convexity should decrease.

Unless, does the decrease in the bond’s sensitivity to price not begin until the put is officially in the money?

I’d suggest that you draw a picture of the price vs. yield for a straight bond, and for a bond putable at, say, 100.  If you draw them on the same axes, you’ll see more curvature (more convexity) on the putable bond as the price approaches the put value, then less curvature as it flattens out.

Simplify the complicated side; don't complify the simplicated side.

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Great idea. I now see that the curvature of the putable bond is positive and steeper (“more positive”) than the straight bond’s until the yield hits the coupon rate (roughly). At this point, the put is out of the money, which is consistent with the putable bond’s curve beginning to flatten relative to the straight bond’s. 

How does that sound? 

Sounds good to me.

Simplify the complicated side; don't complify the simplicated side.

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Thanks so much, magician.

My pleasure.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/