Effective Convexity
The figure shown represents the price of a bond and current yield:
Duration provides a linear approximation of a bond's price change when yields shift, but the approximation gets worse the larger the shift is. Bond prices exhibit convexity, as you can see here. What does this mean for price approximations using only duration?
Well, no. That would suggest that duration estimates are always above the convexity curve, which isn't the case.
That's right.
In either direction, the positive convexity most bonds have means that the linear approximation leaves you with a price that's too low.
No. The error in pricing is in the same direction either way.
Now that's the case for _most_ bonds. For straight bonds, there's positive convexity. For putable bonds, there's positive convexity. But as yields fall, eventually those callable bonds are in the money, and price stops increasing. Now if convexity is positive, but then the price hits a ceiling because of the exercise price of the call option, what do you think will happen to convexity?
Yes!
Convexity drops to zero and actually goes negative past a certain point. This is why __effective convexity__ is an important measure for bonds with embedded options.
$$\displaystyle \mbox{Effective Convexity } = \frac{(PV_{-}) + (PV_{+}) - [2 \times (PV_0)]}{(\Delta \mbox{Curve})^2 \times (PV_0)}$$
Here, _PV_0 is the current bond price, _PV_- is the full bond price if the yield curve shifts down, _PV_+ is the full bond price if the yield curve shifts up, and ∆ Curve is the size of the yield curve shift.
Not exactly.
Consider the convexity curve in the graph hitting a price ceiling as you move to the left.
No.
It's decelerating toward zero, not accelerating.
A callable bond might already be up at 100.85, for example, where the bond can be called at 101. Perhaps a 30 bps increase would drop the price to 100.68, where a 30 bps decrease would cause the price to rise to 100.99. Then the effective convexity could be calculated as
$$\displaystyle \frac{(PV_{-}) + (PV_{+}) - [2 \times (PV_0)]}{ (\Delta \mbox{Curve})^2 \times (PV_0)} $$
$$\displaystyle = \frac{100.99 + 100.68 - (2 \times 100.85)}{(0.003)^2 \times 100.85} = -33.052388 $$.
How do you interpret this result in terms of this bond's price when yields fall?
No, actually.
If yields fall, the price will rise. That's shown by the _PV_- value in the example.
Absolutely.
The duration measure estimates a price change, and this convexity measure tells you that it's on a negative convexity region, so the price has to be adjusted downward to account for the bond being so close to its call price.
The final intuition is consistent with all of this: If yields fall, putable bonds will do a lot better than similar callable bonds due to the "ceiling" call price. If yields rise, then callable bonds will be riskier than similar putable bonds, as putable bonds provide the "floor" put price to investors.
No, it wouldn't.
That would be the case only if the effective convexity measure were positive.
To summarize:
[[summary]]
They're overstated
They're understated
They're overstated or understated, depending on the direction of the yield shift
It will drop to zero and stop
It will accelerate to large values
It will reach zero and continue to go negative
Price will fall
Price will rise less than duration suggests
Price will rise more than duration suggests
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