Negative convexity: explained in simple terms

Question

A callable bond most likely experiences negative convexity when the bond’s yield to maturity is:

less than the bond’s coupon rate.
 	equal to the bond’s coupon rate.
 	greater than the bond’s coupon rate.

A is correct. A callable bond is most likely to experience negative convexity when the bond’s yield to maturity is less than the bond’s coupon rate. When calculating the “convexity effect” for the relationship between bond prices and yields to maturity for a callable bond, the increase in price when the benchmark yield curve is lowered can be smaller than the decrease in price when the benchmark yield curve is raised (in absolute terms). This situation creates negative convexity. When a callable bond moves into the range of negative convexity, it indicates that the call option has more value to the issuer and is more likely to be exercised. When interest rates are low, the value of the call option is much greater because the issuer is more likely to exercise the option to refinance the debt at a lower cost of funds.

Anyone know the “why” to this in simpler terms?

Zee Tan · Answer

When interest rates rise, bond prices fall. Conversely, when interest rates fall, bond prices rise.

But how fast does the price increase/decrease? That's bond duration.

Generally speaking, when interest rates / yields drop, the duration of a bond you hold will increase. The ELI5 way I think about this is because you got a 'good deal' when yields were high, so as yield rates trend to 0, it will send your bond price increasing at a faster rate.

That's positive convexity.

So if you have a bond which duration decreases over time, i.e. your bond price stabilises more as yield rates trend to 0, that's negative convexity.

So why does this happen with a callable bond? Obviously since it's a callable bond, if the bond's coupon rate is too expensive to maintain, the bond issuer will simply exercise the option (recall the bond) to refinance at a lower rate (i.e. reissue bonds at the current, lower rate).

So the price stabilises since it's likely that the issuer will recall the bond.

When does it make sense for an issuer to recall a bond? When the bond's yield-to-maturity (total expected return to maturity) is lower than the coupon rate, i.e. it's more expensive for the issuer continue paying the coupon rate, so they recall the bond and reissue cheaper debt.