Bonds (Duration & Convexity)

[Sunil Natarajan]

Hi David,
I have a small doubt.
Holding yield constant , the bond with lower coupon would have higher duration and greater convexity.
But I read somewhere that by keeping both yield and duration constant, the bond with lower coupon has lower convexity.Convexity is a measure of dispersion of cash flows.Could you please explain the same.
If possible could you upload an excel working for this if possible.
Regards,
Sunil

 

[David Harper CFA FRM]

Hi Sunil,

A way to think about convexity (courtesy of the experts at fixedincomerisk.com) is: as (Macualay) duration is the weighted average time to cash flows of a bond, convexity is the weighted average maturity-squared of a bond, where the weights are the present value of the bond's cash flows. In this way, a zero-coupon bond has high convexity; e.g., a 5-year zero coupon bond, with only the one cash flow at 5 years, has convexity of 5^2 = 25.

Now with higher coupons, the weights are being dispersed to interim maturities, and the maturity-squared must come down; i.e., high coupon implies lower convexity (in this case, add coupons, and convexity will be < 25). Just like, in Tuckman, the barbell has higher convexity than the bullet portfolio because the squaring increases the impact of long-term cash flows.

I don't have an XLS - it is good idea for next season....

Thanks, David

 

[mwu888]

Hi Dave,

Do you have a table (similar to the one you did for stock options showing the impact of the five variables on stock options on call and put option premiums) for bonds? Specifically, how would a bond's duration and convexity be impacted by change in: 1) interest rate, 2) maturity, 3) coupon, and 4) yield spread (or upgrade in credit rating).

I understand that higher interest rates would cause duration to lengthen. The same is true for maturity. What is more difficult to see is that the high coupon bonds have low duration relative to low coupon bonds. Finally, I cannot understand how bonds that receive credit upgrades which will cause yields to compress would have higher duration??? How can this be? Since duration is a measure of risk?