Question

All calculations are Semi Annual!! ABC Inc. has three different types of bonds outstanding. Bond X carries a 6.25% coupon and has 8 years to maturity, Bond Y is a zero coupon bond also with 8 years to maturity, and Bond Z is a 6.25% coupon bond with 15 years to maturity. All three bonds currently have a YTM =10%. 1.What is today’s market price for Bond X? Bond Y? Bond Z? 2.What is the true annualized yield of Bond X? Of Bond Y? Of Bond Z? (i.e., find their EARs) 3.Suppose interest rates suddenly drop, such that the yield on all three bonds declines by 200 basis points. What is the new price for each of these bonds? What is the percentage change in price of the three bonds? Explain why the percentage change in the price of Bond X is different than that of the other two bonds. 4.Suppose in the previous problem that yields increase by 200 basis points instead. Without performing any calculations, do you expect the percentage price change in the bonds to be of greater or lesser magnitude than what you calculated in problem #3? Why? What principle lies behind your answer? 5. Suppose you decide to sell Bond X 4 years from now for $880.00. What is your HPY? When you first bought the bond 4 years ago, what did you expect would to be its year 4 price?

Answer 1

Following are the data extracted from the question given above to solve the present value of the bonds:-

Formula to be use:-

Coupon = Face value* coupon rate/2 (as the coupon payment is half yearly)

PV (present value)= coupon value/(1+YTM)^year

Current price of Bond = Summation of the PV of the bond value.

Bond X
			year	coupon	PV
Current Price of Bond	$718.54		0.5	$31.25	$29.80
Duration	8	years	1	$31.25	$28.41
Face Value	$1,000		1.5	$31.25	$27.09
Coupon Rate	6.25%	semi annually	2	$31.25	$25.83
Current Yield	10%		2.5	$31.25	$24.62
			3	$31.25	$23.48
			3.5	$31.25	$22.39
			4	$31.25	$21.34
			4.5	$31.25	$20.35
			5	$31.25	$19.40
			5.5	$31.25	$18.50
			6	$31.25	$17.64
			6.5	$31.25	$16.82
			7	$31.25	$16.04
			7.5	$31.25	$15.29
			8	$31.25	$14.58
			8.5	$31.25	$13.90
			9	$31.25	$13.25
			9.5	$31.25	$12.64
			10	$31.25	$12.05
			10.5	$31.25	$11.49
			11	$31.25	$10.95
			11.5	$31.25	$10.44
			12	$31.25	$9.96
			12.5	$31.25	$9.49
			13	$31.25	$9.05
			13.5	$31.25	$8.63
			14	$31.25	$8.23
			14.5	$31.25	$7.85
			15	$31.25	$7.48
			15.5	$31.25	$7.13
			16	$1,031.25	$224.43

Bond Y
Current Price of Bond	$466.51
Duration	8	years
Face Value	$1,000
Coupon Rate	0.00%	semi annually
Current Yeild	10%

In the case of bond Y, there will not be any coupon payment during the course of time an investor holds the bond. Its value will be decided entirely by discounting the bond present value with the YTM.

Bond z
			year	coupon	PV
Current Price of Bond	$660.87		0.5	$31.25	$29.80
Duration	15	years	1	$31.25	$28.41
Face Value	$1,000		1.5	$31.25	$27.09
Coupon Rate	6.25%	semi annually	2	$31.25	$25.83
Current Yeild	10%		2.5	$31.25	$24.62
			3	$31.25	$23.48
			3.5	$31.25	$22.39
			4	$31.25	$21.34
			4.5	$31.25	$20.35
			5	$31.25	$19.40
			5.5	$31.25	$18.50
			6	$31.25	$17.64
			6.5	$31.25	$16.82
			7	$31.25	$16.04
			7.5	$31.25	$15.29
			8	$31.25	$14.58
			8.5	$31.25	$13.90
			9	$31.25	$13.25
			9.5	$31.25	$12.64
			10	$31.25	$12.05
			10.5	$31.25	$11.49
			11	$31.25	$10.95
			11.5	$31.25	$10.44
			12	$31.25	$9.96
			12.5	$31.25	$9.49
			13	$31.25	$9.05
			13.5	$31.25	$8.63
			14	$31.25	$8.23
			14.5	$31.25	$7.85
			15	$31.25	$7.48
			15.5	$31.25	$7.13
			16	$31.25	$6.80
			16.5	$31.25	$6.48
			17	$31.25	$6.18
			17.5	$31.25	$5.89
			18	$31.25	$5.62
			18.5	$31.25	$5.36
			19	$31.25	$5.11
			19.5	$31.25	$4.87
			20	$31.25	$4.65
			20.5	$31.25	$4.43
			21	$31.25	$4.22
			21.5	$31.25	$4.03
			22	$31.25	$3.84
			22.5	$31.25	$3.66
			23	$31.25	$3.49
			23.5	$31.25	$3.33
			24	$31.25	$3.17
			24.5	$31.25	$3.03
			25	$31.25	$2.88
			25.5	$31.25	$2.75
			26	$31.25	$2.62
			26.5	$31.25	$2.50
			27	$31.25	$2.38
			27.5	$31.25	$2.27
			28	$31.25	$2.17
			28.5	$31.25	$2.07
			29	$31.25	$1.97
			29.5	$31.25	$1.88
			30	$1,031.25	$59.10

2. True annualized of Bond X, Bond Y, Bond Z comprises that instead of using coupon for half yearly, we will take it annually. By considering the the question. following are the calculations:-

Bond X
Current Price of Bond	$718.54
Duration	8
Face Value	$1,000
Coupon Rate	6.25%
Current Annualized Yield	11.90%

year	coupon	PV
1	$62.50	$55.85
2	$62.50	$49.91
3	$62.50	$44.61
4	$62.50	$39.86
5	$62.50	$35.62
6	$62.50	$31.83
7	$62.50	$28.45
8	$1,062.50	$432.20

In this case, every calculation will be same but instead of half yearly coupon rate, it will yearly coupon rate.

There will not be any change for the coupon Y bond, every thing will same as usual for the previous question.

Bond Y
Current Price of Bond	$466.51
Duration	8	years
Face Value	$1,000
Coupon Rate	0.00%	semi annually
Current Annualized Yield	10%

Bond Z
Current Price of Bond	$660.87
Duration	15
Face Value	$1,000
Coupon Rate	6.25%
Current Annualized Yield	10.96%

year	coupon	PV
1	$62.50	$56.33
2	$62.50	$50.76
3	$62.50	$45.75
4	$62.50	$41.23
5	$62.50	$37.16
6	$62.50	$33.49
7	$62.50	$30.18
8	$62.50	$27.20
9	$62.50	$24.51
10	$62.50	$22.09
11	$62.50	$19.91
12	$62.50	$17.94
13	$62.50	$16.17
14	$62.50	$14.57
15	$1,062.50	$223.27

3. Yield drop by 200 basis points i.e 2%.

Lets take case by case:-

Bond X
			year	coupon	PV
Current Price of Bond	$855.95		0.5	$31.25	$30.07
Duration	8	years	1	$31.25	$28.94
Face Value	$1,000		1.5	$31.25	$27.84
Coupon Rate	6.25%	semi annually	2	$31.25	$26.79
Current Yield	8%	drop by 2%	2.5	$31.25	$25.78
Percentage Change		19.12%	3	$31.25	$24.81
			3.5	$31.25	$23.87
			4	$31.25	$22.97
			4.5	$31.25	$22.10
			5	$31.25	$21.27
			5.5	$31.25	$20.47
			6	$31.25	$19.69
			6.5	$31.25	$18.95
			7	$31.25	$18.23
			7.5	$31.25	$17.55
			8	$31.25	$16.88
			8.5	$31.25	$16.25
			9	$31.25	$15.63
			9.5	$31.25	$15.04
			10	$31.25	$14.47
			10.5	$31.25	$13.93
			11	$31.25	$13.40
			11.5	$31.25	$12.90
			12	$31.25	$12.41
			12.5	$31.25	$11.94
			13	$31.25	$11.49
			13.5	$31.25	$11.06
			14	$31.25	$10.64
			14.5	$31.25	$10.24
			15	$31.25	$9.85
			15.5	$31.25	$9.48
			16	$1,031.25	$301.01

Bond Y
Current Price of Bond	$540.27
Duration	8	years
Face Value	$1,000
Coupon Rate	0.00%	semi annually
Current Yield	8%

Percentage change in value 15.81%

Bond z
			year	coupon	PV
Current Price of Bond	$816.79		0.5	$31.25	$30.07
Duration	15	years	1	$31.25	$28.94
Face Value	$1,000		1.5	$31.25	$27.84
Coupon Rate	6.25%	semi annually	2	$31.25	$26.79
Current Yield	8%		2.5	$31.25	$25.78
Percentage Change		23.59%	3	$31.25	$24.81
			3.5	$31.25	$23.87
			4	$31.25	$22.97
			4.5	$31.25	$22.10
			5	$31.25	$21.27
			5.5	$31.25	$20.47
			6	$31.25	$19.69
			6.5	$31.25	$18.95
			7	$31.25	$18.23
			7.5	$31.25	$17.55
			8	$31.25	$16.88
			8.5	$31.25	$16.25
			9	$31.25	$15.63
			9.5	$31.25	$15.04
			10	$31.25	$14.47
			10.5	$31.25	$13.93
			11	$31.25	$13.40
			11.5	$31.25	$12.90
			12	$31.25	$12.41
			12.5	$31.25	$11.94
			13	$31.25	$11.49
			13.5	$31.25	$11.06
			14	$31.25	$10.64
			14.5	$31.25	$10.24
			15	$31.25	$9.85
			15.5	$31.25	$9.48
			16	$31.25	$9.12
			16.5	$31.25	$8.78
			17	$31.25	$8.45
			17.5	$31.25	$8.13
			18	$31.25	$7.82
			18.5	$31.25	$7.53
			19	$31.25	$7.24
			19.5	$31.25	$6.97
			20	$31.25	$6.70
			20.5	$31.25	$6.45
			21	$31.25	$6.21
			21.5	$31.25	$5.97
			22	$31.25	$5.75
			22.5	$31.25	$5.53
			23	$31.25	$5.32
			23.5	$31.25	$5.12
			24	$31.25	$4.93
			24.5	$31.25	$4.74
			25	$31.25	$4.56
			25.5	$31.25	$4.39
			26	$31.25	$4.23
			26.5	$31.25	$4.07
			27	$31.25	$3.91
			27.5	$31.25	$3.76
			28	$31.25	$3.62
			28.5	$31.25	$3.49
			29	$31.25	$3.35
			29.5	$31.25	$3.23
			30	$1,031.25	$102.48

The value of a bond depends on the YTM, coupon rate, duration for the bond is being held by the investor.

As the duration of all the three bonds are bond different, so there will be change in the percentage change in the price. As the duration of the bond will increase the price fluctuation will also increase, so this is also one of the reason where issuer gives incentive to long term bond holder to counter the risk of interest rate fluctuation.