Question

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.

Answer 1

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