e

4.7 Bond Convexity

Convexity statistic is the curved (convex) line, derived from the function of bond price and the yield-to- maturity (market discount rate).

The convexity statistic for the bond is used to improve the estimate of the percentage price change provided by modified duration alone.

%DeltaPVFull ~ (-AMD x DeltaYield) + *1/2 x Conv x (DeltaYield)2] Example: A 6% annual payment bond matures on 14 February 2022 and is purchased for settlement on 11 April 2014. The YTM is 4%. Calculate the bond's convexity (actual/actual convention):

Period Time to Receipt CF PV of CF t^2+t (t^2+t) x PV of CF 1 0.8466 6 5.80 1.56 9.07 2 1.8466 6 5.58 5.26 29.34 3 2.8466 6 5.37 10.95 58.76 4 3.8466 6 5.16 18.64 96.19 5 4.8466 6 4.96 28.34 140.58 6 5.8466 106 84.28 40.03 3373.63 111.15 3707.57 Conv = 1/(1 + 0.04)^2 x 3707.57/111.15 = 30.84

  • The formula for traditional convexity is Conv=1/[(1+r)]^2x(sum_(t=1)^T [[CF]_t/[(1+r)]^tx(t^2+t) ])/P_0

4.7.1 Approximate Convexity

Like modified duration, convexity can be accurately approximated. The approximate convexity is calculated by the following:

The money convexity of the bond is the annual convexity multiplied by the full price.

4.7.2 Effective Convexity

The effective convexity of a bond is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve.

4.7.3 Determination of Percentage Price Change of a Bond for a Specified Change in

Yield, Given the Bond's Approximate Duration and Convexity Effects of convexity on bonds

  • For the same decrease in yield-to-maturity, the more convex bond appreciates more in price. And for the same increase in yield-to-maturity, the more convex bond depreciates less in price.
  • The conclusion is that the more convex bond outperforms the less convex bond in both bull (rising price) and bear (falling price) markets.
  • Option-free bonds always have positive convexity.
  • The negative convexity is present in callable bonds but not in putable bonds. As the benchmark yield goes down, the slope of the line tangent to the curve for the non- callable bond steepens, which indicates positive convexity. But the slope of the line tangent to the callable bond flattens as the benchmark yield goes down. Technically, it reaches an inflection point, which is when the effective convexity shifts from positive to negative.

Lesson Wrap-Up

This lesson should leave you able to explain the bond convexity in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain approximate convexity in your own words.
  2. Explain effective convexity in your own words.
  3. Explain determination of percentage price change of a bond for a specified change in in your own words.
  4. State one exam-style risk, valuation, or market implication of the bond convexity.