Duration and Convexity
You probably remember by now that duration is a linear estimation of a bond's price change, and convexity adds an adjustment for greater accuracy of that estimation. That's a good start, but a few other details are also worth keeping in mind.
First, duration can mean many things. __Macaulay duration__ is the weighted average of cash flows, and then __modified duration__ is the Macaulay duration divided by 1 plus cash flow yield. So it's a little smaller, but it gives a good linear estimate of percentage price change for a 1% yield curve change. Do you think that price change would have to be based on the dirty price or the clean price?
Yes!
There's no choice there: modified duration estimates an actual price change, which means accrued interest included. Remember that this is the full price, or dirty price. Same thing.
No.
Using the clean price with modified duration as a price change estimate would certainly not improve the accuracy of the estimation. Remember that the difference between these two prices is accrued interest.
Incorrect.
It certainly would. At nearly every point in time, these two prices are different by the amount of any accrued interest.
Of course, that's if the cash flows are known. If there are other things that complicate things, like embedded options, then modified duration won't work. A callable bond, for example, will need effective duration instead for a good price change estimate. What moneyness would make effective duration close to modified duration?
No.
That's when things get most interesting, actually. Effective duration would definitely be needed here. But it's not a big deal if the call option is far out of the money.
Exactly!
If market rates are high enough such that the call is far out of the money, then the bond price will act more like a noncallable bond.
All durations are still just first-order estimates. Those work pretty well for small, parallel yield curve shifts, but then there's convexity.
Similar to duration, convexity can be calculated with known cash flows, and effective convexity can be calculated when there are options to consider. It's usually positive, and if so, what can you say about a bond price estimate following a possible yield curve shift made from just using duration?
Absolutely!
Duration tends to undervalue bonds with its price estimation alone, no matter the direction of the yield curve change. The convexity adjustment is usually positive to fix this undervaluation.
No.
Consider that convexity is most often a positive adjustment.
Zero-coupon bonds offer a few convexity tricks that might be good to remember. First, convexity is proportional to the square of duration. So that means that a 10-year zero has twice the duration of a five-year zero, but four times the convexity, approximately. Compared to that same five-year bond, what would be your estimate of the convexity of a 20-year bond?
No.
Yes!
That's four times the duration, so 16 times the convexity. But still, the convexity of zero-coupon bonds is lower than the convexity of coupon bonds, when you match up durations. For example, a five-year zero-coupon bond would have less convexity than a coupon bond with a duration of five. More dispersion of cash flows, more convexity. Again assuming a positive convexity measure, is that generally good or bad for investors when yield curve changes are expected?
No, it doesn't.
Remember that convexity is usually positive, which means a positive adjustment no matter the direction of the yield curve shift.
Yes.
No, it's usually good.
A positive price adjustment is nice in most scenarios. This means that the "gift" of convexity is met with lower yields in many cases. When markets start really moving, convexity can be nice to have in many portfolios.
To summarize:
[[summary]]
Dirty price
Clean price
It wouldn't matter
At the money
Far out of the money
It's too low
It's too high
It depends on the direction
8 times as big
16 times as big
Bad
Good
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