Question

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

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

(a)   What is the bond’s Macauley Duration (show your work, like you did in problem (16) above.)

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

(c) Calculate the approximate convexity for the bond.

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

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

Illustrating for Time period 1

Discount factor = 1 / (1 + YTM)Time period

Discount factor = 1 / (1 + 9.4%)1

Discount factor = 0.9141

Present value of Cashflow = Discount factor * Cashflow

Present value of Cashflow = 0.9141 * \$65

Present value of Cashflow = \$59.41

Weight = Present value of Cashflow / Total(Present value of Cashflow)

Weight = \$59.41 / \$742.65

Weight = 8%

Weighted average of Time = Weight * Time period

Weighted average of Time = 8.00% * 1

Weighted average of Time = 0.0800

 Time period Yield to Maturity Discount Factor Cashflow Present value of Cashflow Weight Weighted average of Time 1 9.40% 0.9141 \$65 \$59.41 8.00% 0.0800 2 9.40% 0.8355 \$65 \$54.31 7.31% 0.1463 3 9.40% 0.7637 \$65 \$49.64 6.68% 0.2005 4 9.40% 0.6981 \$65 \$45.38 6.11% 0.2444 5 9.40% 0.6381 \$65 \$41.48 5.59% 0.2793 6 9.40% 0.5833 \$65 \$37.91 5.11% 0.3063 7 9.40% 0.5332 \$65 \$34.66 4.67% 0.3267 8 9.40% 0.4874 \$65 \$31.68 4.27% 0.3413 9 9.40% 0.4455 \$65 \$28.96 3.90% 0.3509 10 9.40% 0.4072 \$65 \$26.47 3.56% 0.3564 11 9.40% 0.3722 \$65 \$24.19 3.26% 0.3584 12 9.40% 0.3402 \$65 \$22.12 2.98% 0.3574 13 9.40% 0.3110 \$65 \$20.22 2.72% 0.3539 14 9.40% 0.2843 \$65 \$18.48 2.49% 0.3483 15 9.40% 0.2599 \$65 \$16.89 2.27% 0.3412 16 9.40% 0.2375 \$65 \$15.44 2.08% 0.3326 17 9.40% 0.2171 \$65 \$14.11 1.90% 0.3231 18 9.40% 0.1985 \$65 \$12.90 1.74% 0.3127 19 9.40% 0.1814 \$65 \$11.79 1.59% 0.3017 20 9.40% 0.1658 \$1,065 \$176.61 23.78% 4.7561 Total \$2,300 \$742.65 100.00% 10.4173

Macaulay Duration = 10.4173 years

Using Texas Instruments BA 2 plus calculator

C01 = 65 Press ENTER (65 * 1)
Down Arrow
F01 = 1
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C02 = 130 Press ENTER (65 * 2)
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F02 = 1
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C03 = 195 Press ENTER
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F03 = 1
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C04 = 260 Press ENTER
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F04 = 1
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C05 = 325 Press ENTER
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F05 = 1
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C06 = 390 Press ENTER
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F06 = 1
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C07 = 455 Press ENTER
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F07 = 1
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C08 = 520 Press ENTER
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F08 = 1
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C09 = 585 Press ENTER
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F09 = 1
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C10 = 650 Press ENTER
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F10 = 1
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C11 = 715 Press ENTER
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F11 = 1
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C12 = 780 Press ENTER
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F12 = 1
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C13 = 845 Press ENTER
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F13 = 1
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C14 = 910 Press ENTER
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F14 = 1
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C15 = 975 Press ENTER
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F15 = 1
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C16 = 1040 Press ENTER
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F16 = 1
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C17 = 1105 Press ENTER
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F17 = 1
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C18 = 1170 Press ENTER
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F18 = 1
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C19 = 1235 Press ENTER
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F19 = 1
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C20 = 21300 Press ENTER
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F20 = 1
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PRESS NPV

I = 9.4

Down Arrow

NPV --> CPT NPV = 7736.4125

SET N = 20, PMT = 65, I/Y = 9.4, FV = 1000

CPT ---> PV = -742.6486

Macaulay Duration = NPV / PV

Macaulay Duration = 7736.4125 / 742.6486

Macaulay Duration = 10.4173 years

b)

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

c)

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

d)

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

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