GF540 Unit 5 Duration and Convexity Problem Set Example

Where Duration Stops Working: A Unit 5 Duration and Convexity Problem Set on a Twelve-Year Bond

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Purdue University Global

GF540: Investment and Securities Analysis

Unit 5 Duration and Convexity Problem

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[Date]

The bonds and every figure are illustrative; the textbooks and monograph cited are real.

Problem 1: Macaulay and Modified Duration

Given. A corporate bond with a 4.5 percent coupon paid semiannually, a final maturity twelve years out, a market yield of 5.0 percent and twenty-four coupon dates left.

Price. Discounting twenty-three payments of $22.50 and a final $1,022.50 at 2.5 percent per half-year gives about $955.29.

Macaulay duration. Each payment's present value is weighted by the number of periods until it arrives; the weighted sum divided by the price gives about 18.66 periods, or 9.33 years. Macaulay (1938) introduced the measure as an average time to receipt of a bond's cash flows, weighted by value.

Modified duration. Scaling 9.33 down by 1.025, the half-year growth factor, gives about 9.10.

Meaning. For a small change in yield, the bond's price changes by about 9.10 percent for each percentage point, in the opposite direction.

Problem 2: Testing Duration and Adding Convexity

Convexity. Computed from the same cash flows, convexity is about 101.3.

A 100 basis point rise. Duration alone predicts a fall of 9.10 percent. Adding half of convexity times the yield change squared, 0.5 times 101.3 times 0.0001, or 0.51 percent, gives an estimate of a fall of about 8.60 percent. Full repricing at 6.0 percent gives a fall of 8.62 percent.

A 100 basis point fall. Duration predicts a rise of 9.10 percent, the convexity-adjusted estimate is 9.61 and full repricing gives 9.63.

A 200 basis point rise. Duration predicts a fall of 18.21 percent; full repricing gives 16.33, a miss of nearly two points. With convexity the estimate is 16.18.

A 200 basis point fall. Duration predicts a rise of 18.21 percent; full repricing gives 20.41, a miss of about 2.2 points. With convexity the estimate is 20.23.

Effective duration. Repricing at 4.0 and 6.0 percent and dividing the price difference by twice the price times the yield change gives about 9.12, very close to modified duration, as expected for a bond with no embedded options.

Meaning. Duration is a straight-line estimate of a curved relationship. It overstates losses and understates gains, and the error grows with the square of the yield change (Fabozzi, 2016).

Problem 3: A Zero of the Same Maturity

Given. A twelve-year zero-coupon bond at the same 5.0 percent yield.

Macaulay duration. Twelve years, since all the value arrives at maturity.

Modified duration. 12 divided by 1.025, about 11.71.

Meaning. The coupon bond's duration is 9.33 years against the zero's 12, because coupons return part of the value early. Two bonds of equal maturity can differ in rate sensitivity by more than a quarter.

Problem 4: A Barbell and a Bullet

Given. A bullet portfolio holds only a ten-year zero. A barbell holds about 55.6 percent in a two-year zero and 44.4 percent in a twenty-year zero, weights chosen so both portfolios have the same modified duration of about 9.76. All yields are 5.0 percent.

Convexity. The bullet's convexity is about 99.9. The barbell's is about 176.1.

Meaning. For a parallel shift of 200 basis points, the convexity terms add about 2.0 percent to the bullet's estimated return and about 3.5 percent to the barbell's, so the barbell outperforms whichever way rates move.

Equal duration hides unequal convexity: the barbell and the bullet respond identically to a small parallel shift, yet the barbell gains about one and a half points more on a 200 basis point move in either direction, which is exactly what a duration-only comparison would miss.

Putting the Measures to Use

A pension fund that owes a single payment in ten years could hold either portfolio from Problem 4 and be protected against small parallel shifts, because both durations match the liability's. The barbell offers more protection against large moves but requires two trades instead of one and leaves the fund exposed to changes in the curve's shape. In practice, a fund matching a liability would choose the bullet for simplicity unless it had a view on large rate moves. For the twelve-year corporate in Problem 1, a portfolio manager expecting rates to rise a full point would expect a loss close to 8.6 percent, not the 9.1 percent duration alone suggests.

What Duration Leaves Out

Three things. First, convexity, as Problems 2 and 4 show: for large moves the straight-line estimate is wrong, and portfolios with the same duration can behave differently. Second, the shape of yield changes: duration assumes every maturity moves by the same amount, but in practice short and long rates often move differently, and a twist that raises twenty-year yields while lowering two-year yields would hurt the barbell's long leg. Third, the price of convexity: in real markets, investors usually accept a lower yield for more convexity, so the barbell's advantage in this flat-curve example would partly be paid for in income (Bodie et al., 2024).

Conventions

Prices assume settlement on a coupon date. Yields are bond-equivalent. Convexity is in years squared and applied as half of convexity times the squared yield change in decimals.

References

Bodie, Z., Kane, A., & Marcus, A. J. (2024). Investments (13th ed.). McGraw Hill.

Fabozzi, F. J. (2016). Bond markets, analysis, and strategies (9th ed.). Pearson.

Macaulay, F. R. (1938). Some theoretical problems suggested by the movements of interest rates, bond yields and stock prices in the United States since 1856. National Bureau of Economic Research.