Question

19. A 20-year, 6.500% annual payment bond settles on a coupon date. The bond's yield to maturity is 9.400%

(b) What is the bond’s approximate modified duration? Use yield changes of +/- 30 bps around the yield to maturity for your calculations

20. Consider the bond from problem (19) above.

(a) Calculate the approximate convexity for the bond.

(b) Calculate the change in the full bond price for a 40 bps change in yield.

I need an answer for 20.  show your calculation using a finance calculator but not excel formulas. Please add 3 decimal places.

Answer 1

19) a)

No of periods = 20 years

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (6.5% / 1) * $1000

Coupon per period = $65

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $65 / (1 + 9.4%)¹ + $65 / (1 + 9.4%)² + ...+ $65 / (1 + 9.4%)²⁰ + $1,000 / (1 + 9.4%)²⁰

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $65 * (1 - (1 + 9.4%)^-20) / (9.4%) + $1,000 / (1 + 9.4%)²⁰

Bond Price = $742.649

Bond price at 30 bps increase in yield

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $65 / (1 + 9.7%)¹ + $65 / (1 + 9.7%)² + ...+ $65 / (1 + 9.7%)²⁰ + $1,000 / (1 + 9.7%)²⁰

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $65 * (1 - (1 + 9.7%)^-20) / (9.7%) + $1,000 / (1 + 9.7%)²⁰

Bond Price at 30 bps increase in yield = $721.893

Bond price at 30 bps decrease in yield

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $65 / (1 + 9.1%)¹ + $65 / (1 + 9.1%)² + ...+ $65 / (1 + 9.1%)²⁰ + $1,000 / (1 + 9.1%)²⁰

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $65 * (1 - (1 + 9.1%)^-20) / (9.1%) + $1,000 / (1 + 9.1% )²⁰

Bond Price at 30 bps decrease in yield = $764.339

Approximate Modified Duration = (Bond Price at 30 bps decrease in yield - Bond Price at 30 bps increase in yield ) / (2 * Bond price * Change in yield)

Approximate Modified Duration = ($764.339 - $721.893) / (2 * $742.649 * 0.3%)

Approximate Modified Duration = 9.526

20) a)

Approximate Convexity = (Bond Price at 30 bps decrease in yield + Bond Price at 30 bps increase in yield - 2 * Bond Price) / (Bond price * (Change in yield)²)

Approximate Convexity = ($764.339 + $721.893 - 2 * $742.649) / ($742.649 * (0.3%)²)

Approximate Convexity = 139.958

b)

Change in the full bond price for a 40 bps increase in yield

Change in the full bond price = (- Modified Duration * Yield change + 0.5 * Convexity * (Yield change)²) * Bond price

Change in the full bond price = (-9.526 * 0.4% + 0.5 * 139.958 * (0.4%)²) * $742.649

Change in the full bond price = -3.698% * $742.649

Change in the full bond price = -$27.466

New Bond price at 40bps increase in yield = Bond price + Change in the full bond price

New Bond price at 40bps increase in yield = $742.649 + (-$27.466)

New Bond price at 40bps increase in yield = $715.183

Change in the full bond price for a 40 bps decrease in yield

Change in the full bond price = (- Modified Duration * Yield change + 0.5 * Convexity * (Yield change)²) * Bond price

Change in the full bond price = (-9.526 * -0.4% + 0.5 * 139.958 * (-0.4%)²) * $742.649

Change in the full bond price = 3.922% * $742.649

Change in the full bond price = $29.129

New Bond price at 40bps decrease in yield = Bond price + Change in the full bond price

New Bond price at 40bps decrease in yield = $742.649 + $29.129

New Bond price at 40bps decrease in yield = $771.778