Empirical Vs. Analytical Duration
Analytical duration utilizes mathematical models, assuming credit spreads and government bond yields are... Read More
The change in the price of a bond can be summarized as follow:
$$\text{Change in price} = \text{Duration effect} + \text{Convexity effect} $$
$$≈(\text{-AnnModDur}×ΔYield)+(\frac{1}{2}×\text{AnnConvexity}×(ΔYield)^2)$$
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.
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 the 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 par 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.
$$\text{Approx. ModDur}=\frac{98.936070-98.755130}{2×0.0001×98.845543}=9.1527$$
The approximate convexity is 115.3315.
$$\text{Approx. conv}=\frac{98.936070+98.755130-(2×98.845543)}{0.0001^2×98.845543}=115.3315$$
(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 shoots 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- day 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 decrease in the yield-to-maturity?
- 0.0875
- 0.0925
- 0.0953
Solution
The correct answer is C.
\(\%ΔPV^{FULL}≈(-9.00×-0.0100)+(\frac{1}{2}×105.00×(-0.01)^2 )\)
\(≈0.09+0.0053≈0.0953\)
The convexity-adjusted percentage price drop resulting from a 100 bps decrease 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.