Sep 25

Duration and Convexity

by David Harper, CFA, FRM, CIPM

FRM |

Learning Outcome

  • LO 23.5: Estimate the price change of a security given the DV01, the duration, and the convexity.

The impact of a yield change on bond price

In two previous posts, I illustrated bond duration and bond convexity. This learning outcome combines them to produce a better estimate of bond price sensitivity to yield changes.

The graph below recaps the idea. Start at the green circle on the price/yield curve. Then lower interest rates (i.e., shock the yield down). Duration moves us along the grey line: it is a linear approximation. Then add the convexity adjustment, illustrated by the orange line, to close the gap toward the blue price/yield curve. Convexity is the second derivative approximation. By adding the duration effect to the convexity adjustment, we get pretty close to the actual price/yield line (but not exactly, convexity is an approximation, too). Again, the idea is just to estimate the price increase that is effected by a yield decrease:

durationCurve

 

Now briefly go in the other direction. Increase the yield. Again, duration moves us along the grey line. Then the convexity adjustment (always positive) closes the gap.

durationCurveOtherDirection

 

Price change = duration + convexity adjustment

The bond's percentage price change is given by duration impact plus convexity adjustment:

durationplusconvexity 

 

And the convexity adjustment can include either a straight convexity measure...

convexitywithouttwo

 

...or one-half the convexity measure depending on whether we scaled the measure in the first place.

convexitywithtwo

 

EditGrid Spreadsheet

To demonstrate with an example, here are the same assumptions as before, plugged into the EditGrid spreadsheet below:

  • Par: $100
  • Coupon: 3% (1.5% every six months)
  • Yield: 5%
  • Years to maturity: 10 years
  • Shock @ 50 basis points

And the EditGrid spreadsheet below calculates the total approximate change in bond price (in percentage terms) for a given yield change. This calculation (that blends duration and convexity) is illustrated in green. You do need to input a yield change assumption (e.g., +1%)

EditGrid Spreadsheet by bt/admin.

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