Question

Bond	Coupon Rate	Maturity Year	Par Value
1	7.5%	2032	1000
2	8.25%	2029	1000
3	6.0%	2023	1000

a.) Assuming that bonds pay annual coupon, estimate the market value of each bond at a discount rate of 7.4%

b.) Assuming that bonds pay annual coupon, what will happen to the price of each bond if market rates suddenly decrease from 7.4% to 6.2%? Which of the three bonds will have the greatest percentage change in price?

c.) Assuming that bonds pay annual coupon, what will happen to the price of each bond if market interest rates suddenly increase from 7.4% to 8.6%? Which of the three bonds will have the greatest percentage change in price?

d.) Assuming that bonds pay annual coupon. Also, the bonds are currently trading in the market at $973.63, $932.37, and $1,075.58 respectively. What is the yield-to-maturity of each bond?

e.) Assuming that bonds pay annual coupon. Suppose Rhea purchased bond 1 today at a price of $973.63, but would like to sell the bond in 7 years at which time similar bonds yield 6.753%. At what price can Rhea expect to sell the bonds? If she sells the bond in 7 years at the price computed, what would be her realized rate of return over her holding period?

f.) Repeat analysis in parts a)-e) assuming bonds pay semi-annual coupon.

Answer 1

Price of the bond = Present value of coupons + present value of cash flow at maturity

a)Given discount rate 7.4%

• Bond1: Coupon rate = 7.5%; Maturity year = 2032 (2032 - 2019 = 13 years to maturity)
• Coupon amount = 7.5% * 1000 = 75 p.a (paid annually)
• P1 = 75/(1+0.074) + 75/(1+0.074)^2 + ...+ 1075/(1+0.074)^13 = 1,008.17

Similar calculations for Bond2, Bond3 as below:

• Bond2: Coupon rate = 8.25%; Maturity year = 2029 (2029 - 2019 = 10 years to maturity)
• Coupon amount = 8.25% * 1000 = 82.5 p.a (paid annually)
• P2 = 82.5/(1+0.074) + 82.5/(1+0.074)^2 + ...+ 1082.50/(1+0.074)^10 = 1,058.61

Bond3:

• Bond3: Coupon rate = 6%; Maturity year = 2023(2023 - 2019 = 4 years to maturity)
• Coupon amount = 6% * 1000 = 60 p.a (paid annually)
• P3 = 60/(1+0.074) + 60/(1+0.074)^2 + ...+ 1060/(1+0.074)^4  = 953.00

b) If the discount rate changes from 7.40% to 6.20% then bond prices change as below:

• P1 = 75/(1+0.062) + 75/(1+0.062)^2 + ...+ 1075/(1+0.062)^13 = 1,113.75
• P2 = 82.5/(1+0.062) + 82.5/(1+0.062)^2 + ...+ 1082.50/(1+0.062)^10 = 1,149.46
• P3 = 60/(1+0.062) + 60/(1+0.062)^2 + ...+ 1060/(1+0.062)^4  = 993.10
• Percentage change in the price
  • Bond1: (1113.75/1008.17)-1 = 10.47%
  • Bond2: (1149.46/1058.61)-1 = 8.58%
  • Bond3: (993.10/953) - 1=4.21%
  • So Bond1 had the greatest change in the price

c) If the discount rate changes from 7.40% to 8.60% then bond prices change as below:

• P1 = 75/(1+0.086) + 75/(1+0.086)^2 + ...+ 1075/(1+0.086)^13 = 915.86
• P2 = 82.5/(1+0.086) + 82.5/(1+0.086)^2 + ...+ 1082.50/(1+0.086)^10 = 977.14
• P3 = 60/(1+0.086) + 60/(1+0.086)^2 + ...+ 1060/(1+0.086)^4  = 915.02
• % change in Price:
  • Bond1: (915.86/1008.17)-1 = -9.16%
  • Bond2: (977.14/1058.61)-1= -7.70%
  • Bond3: (915.02/953)-1 = -3.99%
  • Bond1 has the greatest change in the price.

d) Yield to maturity of the bond with the prices

• 973.63 = 75/(1+x) + 75/(1+x)^2 + ...+ 1075/(1+x)^13
  • Solving for x, we get YTM for Bond1 = x = 7.83%
• 932.37 = 82.5/(1+y) + 82.5/(1+y)^2 + ...+ 1082.50/(1+y)^10
  • Solving for y, YTM of Bond2 = y = 9.319%
• 1,075.58 = 60/(1+z) + 60/(1+z)^2 + ...+ 1060/(1+z)^4
  • Solving for z, YTM of Bond3 = z = 3.92%

e)Holding period return for Rhea

Sell bond1 after 7 years. Price of Bond1 after 7 years is the present value of cash flows till 7 years as below:

Price of Bond1 after 7 years = 75/(1+0.06753) + 75/(1+0.06753)^2 + ...+ 1075/(1+0.06753)^7 = 1,040.61

• Purchase price = 973.63
• Selling price = 1,040.61
• Holding period return = (1,040.61/973.63) - 1 = 1.068791 - 1 = 0.068791 = 6.88%

Analysis with semi-annual coupons:

a)Given discount rate 7.4%

• Bond1: Coupon rate = 7.5%; Coupon amount = 7.5% * 1000 = 75 p.a. So Coupon = 75/2 = 37.50(paid semi-annually)
• number of compounding periods = 13 * 2 = 26
• YTM semi-annually = 7.40%/2 = 3.70%
• P1 = 37.5/(1+0.037) + 37.5/(1+0.037)^2 + ...+ 1037.5/(1+0.037)^26 = 1,008.26

Similar calculations for Bond2, Bond3 as below:

• Bond2: Coupon rate = 8.25%; Coupon amount = 8.25% * 1000 = 82.5 p.a (paid annually) So semi-annual coupon = 82.5/2 = 41.25
• number of compounding periods = 10 * 2 = 20
• YTM semi-annually = 7.40%/2 = 3.70%
• P2 = 41.25/(1+0.037) + 41.25/(1+0.037)^2 + ...+ 1041.25/(1+0.037)^20 = 1,059.32

Bond3:

• Bond3: Coupon rate = 6%; Coupon amount = 6% * 1000 = 60 p.a (paid annually) So half-yearly coupon = 60/2 = 30
• number of compounding periods = 4 * 2 = 8
• YTM semi-annually = 7.40%/2 = 3.70%
• P3 = 30/(1+0.037) + 30/(1+0.037)^2 + ...+ 1030/(1+0.037)^8 = 952.28

b) If the discount rate changes from 7.40% to 6.20% then bond prices change as below:

Semi-annual yield = 6.20%/2 = 3.10%; number of compounding periods = 2 * maturity; Semi-annual coupon = annual coupon/2

• P1 = 37.5/(1+0.031) + 37.5/(1+0.031)^2 + ...+ 1037.5/(1+0.031)^26 = 1,114.87
• P2 = 41.25/(1+0.031) + 41.25/(1+0.031)^2 + ...+ 1041.25/(1+0.031)^20 = 1,151.09
• P3 = 30/(1+0.031) + 30/(1+0.031)^2 + ...+ 1030/(1+0.031)^8 = 993.01
• Percentage change in the price
  • Bond1: (1,114.87/1008.26)-1 = 10.57%
  • Bond2: (1,151.09/1059.32)-1 = 8.66%
  • Bond3: (993.01/952.28) - 1=4.28%
  • So Bond1 had the greatest change in the price

c) If the discount rate changes from 7.40% to 8.60% then bond prices change as below:

Semi-annual yield = 8.60%/2 = 4.30%; number of compounding periods = 2 * maturity; Semi-annual coupon = annual coupon/2

• P1 = 37.5/(1+0.043) + 37.5/(1+0.043)^2 + ...+ 1037.5/(1+0.043)^26 = 914.90
• P2 = 41.25/(1+0.043) + 41.25/(1+0.043)^2 + ...+ 1041.25/(1+0.043)^20 = 976.84
• P3 = 30/(1+0.043) + 30/(1+0.043)^2 + ...+ 1030/(1+0.043)^8 = 913.55
• Percentage change in the price
  • Bond1: (914.90/1008.26)-1 = -9.26%
  • Bond2: (976.84/1059.32)-1 = -7.79%
  • Bond3: (913.55/952.28) - 1= -4.07%
  • So Bond1 had the greatest change in the price

d) Yield to maturity of the bond with the prices

• 973.63 = 37.5/(1+x) + 37.5/(1+x)^2 + ...+ 1037.5/(1+x)^26
  • Solving for x, we get semi-annual YTM for Bond1 = x = 3.9135%;
  • So annual YTM = 3.9135% * 2 = 7.827%
• 932.37 = 41.25/(1+y) + 41.25/(1+y)^2 + ...+ 1041.25/(1+y)^20
  • Solving for y, half-yearly YTM of Bond2 = y = 4.653%
  • So annual YTM = 4.653% * 2 = 9.306%
• 1,075.58 = 30/(1+z) + 30/(1+z)^2 + ...+ 1030/(1+z)^8
  • Solving for z, half-yearly YTM of Bond3 = z = 1.97%
  • So annual YTM = 1.97% * 2 = 3.94%

e)Holding period return for Rhea

Sell bond1 after 7 years. Price of Bond1 after 7 years is the present value of cash flows till 7 years as below:

Price of Bond1 after 7 years, calculated with values:

• compounded semi-annually, with 7 * 2 = 14 compounding periods
• half-yearly coupon = 75/2 = 37.5
• semi-annual discount rate = 6.753% / 2 = 3.3765% =0.033765

Price = 37.5/(1+0.033765) + 37.5/(1+0.033765)^2 + ...+ 1037.5/(1+0.033765)^14 = 1,041.13

• Purchase price = 973.63
• Selling price = 1,041.13
• Holding period return = (1,041.13/973.63) - 1 = 1.069326 - 1 = 0.069326 = 6.93%