# low coupon issues exhibit less negative convexity

Can someone explain the reasoning behind why low coupons exhibit less negative convexity? I’m not really understanding the logic why there would be greater price appreciation for low coupon bonds when interest rates decline. Thank you in advance!

This example proves it. I guess it has to do with when the coupon is smaller and closer to the yield - there is not so much of an impact on price. Since the coupons are discounted at factors pretty close to 1. As the coupon goes higher - and the rate drops - the number becomes bigger (after discounting).

FV=100, PMT=2, N=10, I/Y=5%

PV=76.83

FV=100, PMT=2, N=10, I/Y=4%

PV=83.77

FV=100, PMT=6, N=10, I/Y=5%

PV=107.72

FV=100, PMT=6, N=10, I/Y=4%

PV=116.22

lower coupon = payments come in slower - so longer duration. hence they exhibit less negative convexity.

cpk123 “I guess it has to do with when the coupon is smaller and closer to the yield - there is not so much of an impact on price.”

Don’t you mean the opposite, it’s when coupons are low that the price appreciation is greater

I guess I misspoke there. a lower coupon bond would have a higher duration - so it would show a higher price rise.

check page 101, reading 24 solution 19.

“low coupon issues exhibite less negative convexity than high coupon issues. there is a greater price appreciation for low coupon issues when rates decline”

as I said - typo on my part…

Zero coupon bonds have highest duration and greatest price appreciation when interest rates fall, because there are no coupons and no reinvestment. All the coupons are componded in up until maturity. More coupon implies more reinvestment risk and when interest rates drop your coupons must be reinvested at lower rates. On the contarry, when interest rates rise the behaviour is opposite, zero coupon bonds loose more than the coupon bonds.

I would think this has quite a bit to do with the prepayment motivation. If you are paying a high interest rate you are more lmotivated to prepay as rates decline than if you are paying a lower interest rate.

But it is also an expectations thing.

If rates are decilining , then well before prepayments rise , the value of the mortgage issues decrease because the market expects prepayments to rise in the future.

The ones that are high coupon will fall faster because the market expects prepayment to rise fastest in those.

nobody read my note above…

so am posting it again.

Low Coupon = High Duration. So Price rises more when the Rate Falls.

while for a high coupon because the duration is lower - the price rise is not as much.

This is a question of change in duration and not necessarily the level of duration

convexity = change in duration as interest rates change.

The lower the coupon the less the duration fluctuates as interest rates change, this is because payments for low coupon bonds do not change much when interest rates change. Therefore low coupons exhibit less negative convexity compaired to high coupons. (they also exhibit less positive convexity).

Bonds with lower coupons are more convex…

I’m also slightly confused why people are using the term “negative convexity” for regular bonds. Vanilla bonds are always positively convex.

aaron - since when did this become a part of the syllabus you are reading, if you are ?

Think of the coupon rate as the trigger point for prepayments - and it is the timing and scale prepayment activity that dictates the negative convexity - payers prepay when the rate they are paying is higher than the market and so therefore market must fall more to hit lower coupon rate.

I am not registered for a CFA exam currently (school got in the way), so I’m just going off of material from my finance classes. I have no idea what the exam actually covers.

This is a duration effect, not a convexity effect. If you have a zero coupon bond and a coupon-paying bond with the same duration, the zero-coupon bond’s higher convexity implies that it will always have a higher price regardless of the direction of interest rates. High duration is risky, but high convexity is super cool.

thanks everyone for responding.

There is a professor’s note on page 62 in SS9 in schweser that says: unless rates are very high (relative to coupon rate), the embedded option in a callable bond has value.

Not really understanding this…

Rate is high compared to coupon. So bond is selling at a discount.

Why would an issuer call this bond at Par?

I think what the note is trying to say is that the option is worth something as long as there is still a reasonable chance of rates falling to the point where it is rational to call the bond. The only time the option wouldn’t have value is when the bond has no chance of being called, and that’s when rates are very high – rates would need to drop a lot before the option would make a difference. Compare it to a call option on a stock: as long as the option is not so far out of the money that it will never be exercised, it still retains time value, even if it has no intrinsic value.

Now that I know that the OP is referring to callable bonds, I understand the concept being discussed. However, I’m not in the right state of mind to explore what effect the coupon rate has on convexity. I can promise that almost everything in the thread so far isn’t relevant to the real answer, though.

Edit: FinNinja and janakisri might be right conceptually.