Effective Convexity
__Effective convexity__, like effective duration, is a way of estimating interest rate risk for unique or complicated fixed-income securities. It is appropriate for measuring convexity of a callable bond.
Actual changes in the bond price are used to measure effective convexity. Because of this, convexity can be measured for bonds with uncertain future prices. Effective convexity is computed as follows:
$$\displaystyle \text{EffCon} = \frac{(PV_-) + (PV_+) - [2 \times PV_0]}{(\Delta \text{Curve})^2 \times (PV_0)}$$
where $$PV_0$$ is the current bond price, $$PV_-$$ is the bond price if yields are shifted down, $$PV_+$$ is the bond price if yields are shifted up, and $$\Delta \text{Curve}$$ is the change in a benchmark yield curve.
At what yield are 6.00% callable and non-callable bonds, that are otherwise the same, priced the same?
Correct!
At 10.00%, both bonds are priced as deep discount bonds and therefore are priced to maturity.
Incorrect.
At 4.00% the callable bond is priced to the call date and the non-callable bond is priced to maturity. That yield is in the negative convexity region.
Incorrect.
At 4.00% the callable bond is priced to the call date and the non-callable bond is priced to maturity. That yield is in the negative convexity region.
Ms. Muni owns a 6.00%, 10-year bond, with semiannual coupons that is callable at par in 8 years. The bond is priced with a yield-to-maturity of 6.00 at 100.
If the yield curve increases by 30 bps, she estimates that the bond will sell for 97.078. However, if the yield curve decreases by 30 bps, the bond will be priced-to-call at an estimated price of 102.551.
Ms. Muni wants to measure convexity for this bond. Using effective convexity, she comes up with the following:
$$\displaystyle \text{EffCon} = \frac{(102.551+97.078) - (2)(100)}{(100)(0.003)^2} = -412.22$$
The result can be used to compute a convexity adjustment.
For what other investment would you also expect to use effective convexity instead of conventional convexity measures?
Incorrect.
US Treasury notes are note callable so conventional convexity measures can be used.
Correct!
For a callable corporate bond, it would be appropriate to use effective convexity because of the call feature.
Incorrect.
For most non-callable bonds, conventional convexity measures can be used.
To summarize:
[[summary]]
The figure below represents the price of two 6.00%, 30-year bonds. One bond is callable in 10 years and the other is non-callable.
The bonds are priced the same for yields equal to or above 6.00%. For yields below 6.00%, the non-callable bond is priced to maturity and the callable bond is priced to the 10-year call date. The region between two pricing lines, where the prices diverge, is called the negative convexity region.
4.00%
5.50%
10.00%
US Treasury Note
Callable corporate bond
Non-callable municipal bond
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