Question

36. Estimate the convexity for each of the following three bonds, all of which trade at a yield to maturity of 8 percent and have face values of $1.000. A 7-year, zero-coupon bond. A 7-year, 10 percent annual coupon bond A 10-year, 10 percent annual coupon bond that has a duration value of 6.994 years (i.e., approximately 7 years) Rank the bonds in terms of convexity and express the convexity relationship between zeros and coupon bonds in terms of maturity and duration equivalencies

Answer 1

Duration measures linear relationship of bond price and yield whereas convexity is second order derivative of duration which measures non linear relationship of bond price and yield.

Convexity is calculated as follows = (Pi + Pj - 2P0) / 2*P0*(change in yield)^2

Pi refers to bond price when interest rate decreases by 10bps

Pj refers to bond price when interest rate increases by 10 bps

P0 refers to bond price

Change in yield refers to 10bps

Lets calculate each component for all 3 bonds and calculate convexity based on that

1. 7 year zero coupon bond

Here, based on TVM calculator - i/y = 8%, FV = 1000, PMT = 0, N = 7

Calculate PV or P0 = 583.4

Lets increase, i/y to 8.1%, Pj = 579.7

Lets decrease i/y to 7.9%, Pi = 587.2

Substituting values in convexity formula = 587.2 + 579.7 - (2* 583) / (2*583.4*0.001^2) = 0.9/0.001167 = 771.3

2. 7 year 10% annual coupon bond

Here, based on TVM calculator - i/y = 8%, FV = 1000, PMT = 100 (10%*1000), N = 7

Calculate PV or P0 = 1104.1

Lets increase, i/y to 8.1%, Pj = 1098.5

Lets decrease i/y to 7.9%, Pi = 1109.7

Substituting values in convexity formula = 1109.7 + 1098.5 - (2* 1104.1) / (2*1104.1*0.001^2) = 0

3. 10 year 10 percent annual coupon bond

Here, based on TVM calculator - i/y = 8%, FV = 1000, PMT = 0, N = 10

Calculate PV or P0 = 1134.20

Lets increase, i/y to 8.1%, Pj = 1126.91

Lets decrease i/y to 7.9%, Pi = 1141.54

Substituting values in convexity formula = 1141.54 + 1126.91 - (2* 1134.2) / (2*1134.2*0.001^2) = 0.05/0.00226 = 22.04

Conclusion

As we can see zero coupon bond will have highest convexity followed by 10 year coupon bond and last 7 year coupon bond.

For zero coupon bond, duration is equivalent to maturity of bond and hence change in yield of 10 bps will have drastic effect on convexity of this bond.

For coupon bonds, duration will be lesser compared to zero coupon bond as we will have intermediate coupons to recover the investment value hence majority of convexity relationship is driven by maturity. As maturity increase, interest rate uncertainty increases and hence duration and convexity will increase

This is what supported by our analysis mentioned above, 10 year coupon bond will be more sensitive to interest rates than 7 year coupon bond and hence will have higher duration and convexity.