### Duration and Convexity Effect on the Price Change of a Bond

The change in the price of a bond can be summarized as follow:

$$Change \quad in \quad price = Duration \quad effect + Convexity \quad effect$$

$$≈(-AnnModDur×ΔYield)+(\frac{1}{2}×AnnConvexity×(ΔYield)^2)$$

## Example 1

Suppose the yield-to-maturity is expected to fall by 10 bps tomorrow, from 2.95% to 2.85%. A bond has an annual (modified) duration of 24.500 and annual convexity of 775.0. What is the percentage price gain from this fall in interest rate?

$$\%ΔPV^{FULL}≈(-24.500×-0.0010)+(\frac{1}{2}×775.0×(-0.0010)^2 )$$

$$≈0.0245+0.0004≅0.0249$$

The modified duration alone underestimates the gain to be 2.45%. The convexity adjustment adds 4 basis points.

## Example 2

A pension scheme holds a large position in a 6.5% annual coupon payment government bond that matures on 10th March 2034. The bond’s yield-to-maturity is 6.75% for settlement on 15th May 2019, stated as an effective annual rate. That settlement date is 65 days into the 360-day year using the 30/360 day count convention.

(a) Calculate the full price of the bond per 100 of par value.

The full price of the bond is 98.845543 per 100 per value.

$$PV_0=[\frac{6.5}{1.0675^1} +\frac{6.5}{1.0675^2} +⋯+\frac{100+6.5}{1.0675^15} ]×1.0675^{65/360}=98.845543$$

(b) Calculate the approximate modified duration and approximate convexity using a 1 bp increase and decrease in the yield-to-maturity.

$$PV_+=[\frac{6.5}{1.0676^1} +\frac{6.5}{1.0676^2} +⋯+\frac{100+6.5}{1.0676^15} ]×1.0676^{65/360}=98.755130$$

$$PV_-=[\frac{6.5}{1.0674^1} +\frac{6.5}{1.0674^2} +⋯+\frac{100+6.5}{1.0674^15} ]×1.0674^{65/360}=98.936070$$

The approximate modified duration is 9.1527.

$$Approx. \quad ModDur=\frac{98.936070-98.755130}{2×0.0001×98.845543}=9.1527$$

The approximate convexity is 115.3315.

$$Approx. \quad conv=\frac{98.936070+98.755130-(2×98.845543)}{0.0001^2×98.845543}=115.3315$$

(c) Calculate the estimated convexity-adjusted percentage price change resulting from a 100 bp increase in the yield-to-maturity

The convexity-adjusted percentage price drop resulting from a 100 bp increase in the yield-to-maturity is estimated to be 8.576%. Notably, modified duration alone estimates the percentage drop to be 9.1527%. The convexity adjustment adds 57.67 bps.

(100 bps = 1% = 0.0100)

$$\%ΔPV^{FULL}≈(-9.1527×0.0100)+(\frac{1}{2}×115.3315×(-0.0100)^2 )$$

$$≈-0.091527+0.005767=-0.08576$$

(d) How does the estimated percentage price change compare with the actual change, assuming the yield-to-maturity jumps to 7.75% on that settlement date?

The new full price if the yield-to-maturity goes from 6.75% to 7.75% on 15th May 2019 is 90.344807.

$$PV^{FULL}=[\frac{6.5}{1.0775^1} +\frac{6.5}{1.0775^2} +⋯+\frac{100+6.5}{1.0775^15} ]×1.0775^{65/360}=90.344807$$

$$\%ΔPV^{FULL}≈\frac{90.344807-98.845543}{98.845543}=-0.086000$$

As these calculations show, the actual percentage change in the bond price is –8.6%. The convexity-adjusted estimate is –8.576%, whereas the estimated change using modified duration alone is –9.1527%. As such, it is evident that convexity adjustment is paramount.

## Question

An investment bank holds a considerable position in a 7% annual coupon paying bond. The bond’s yield-to-maturity is 8%. The settlement date is 83 days into the 360-year. The approximate modified duration is 9 years and approximate convexity is 105. What is the estimated convexity-adjusted percentage price change resulting from a 100 bps increase in the yield-to-maturity?

A. 0.0953

B. 0.0875

C. 0.0925

Solution

$$\%ΔPV^{FULL}≈(-9.00×-0.0100)+(\frac{1}{2}×105.00×(-0.01)^2 )$$

$$≈0.09+0.0053≈0.0953$$

The convexity-adjusted percentage price drop resulting from a 100 bps increase in the yield-to-maturity is estimated to be 9.53%. The modified duration alone underestimates the gain to be 9.00%, and the convexity adjustment adds 53.0 bps.

Estimate the percentage price change of a bond for a specified change in yield, given the bond’s approximate duration and convexity

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