Question

The 5.3 percent bond of Dan's Auto Parts has a face value of $1,000, a maturity of 12 years, semiannual interest payments, and a yield to maturity of 6.12 percent. What is the current market price of the bond? $945.08 5959.33 $931.01 $1,072.13 $912.40

Answer 1

Current market price of the bond: $931.01

Period	Cash flow	Cash flow ($)	Present value factor @ 3.06%	Discounted cash flow
1	Coupon	26.50	0.97030856	25.71
2	Coupon	26.50	0.9414987	24.95
3	Coupon	26.50	0.91354424	24.21
4	Coupon	26.50	0.8864198	23.49
5	Coupon	26.50	0.86010072	22.79
6	Coupon	26.50	0.83456309	22.12
7	Coupon	26.50	0.8097837	21.46
8	Coupon	26.50	0.78574006	20.82
9	Coupon	26.50	0.7624103	20.20
10	Coupon	26.50	0.73977324	19.60
11	Coupon	26.50	0.71780831	19.02
12	Coupon	26.50	0.69649554	18.46
13	Coupon	26.50	0.67581559	17.91
14	Coupon	26.50	0.65574965	17.38
15	Coupon	26.50	0.6362795	16.86
16	Coupon	26.50	0.61738744	16.36
17	Coupon	26.50	0.59905632	15.87
18	Coupon	26.50	0.58126947	15.40
19	Coupon	26.50	0.56401074	14.95
20	Coupon	26.50	0.54726445	14.50
21	Coupon	26.50	0.53101538	14.07
22	Coupon	26.50	0.51524877	13.65
23	Coupon	26.50	0.49995029	13.25
24	Coupon	26.50	0.48510604	12.86
24	Redemption	1,000.00	0.48510604	485.11
				931.01

• The price of a bond is the present value of all future cash flows from it discounted at a required rate of
  return. This required rate of return is the YTM.
• YTM is the yield the bond-holder earns provided he holds the bond till maturity.
• The cash flows from a bond are periodic coupon payments and the redemption of face value on maturity.
• In this question, cash flows are: semi-annual coupon payments & redemption value on maturity.
  • Coupon is always paid on par value: $1,000 × 2.65% = $26.5.
  • Annual coupon rate = 5.3%. Since coupon payment is semiannual, rate per period of 6 months = 5.3% ÷ 2 = 2.65%. (Conventionally, coupon rates for bonds are always expressed in per annum basis)
• When Cash flows are non-annual convert the entire problem data into periods.
• Thus, there are 24 periods (12 years × 2) (note: 2 periods per year, since coupon payments are semi-annual)
• YTM for each period = 6.12% ÷ 2 = 3.06%

Instead of doing a lengthy calculation, this can be easily solved by finding the present value of annuity. (annuity being semi-annual coupon payments)

Solution Par value semi annual coupon rate 5.3%/2 semi annual yield to maturity 6.12%/2 Number of payments 12*2 1000 2.65% 3.06% 24 PV of bond PV of interest PV of maturity PV of interest Semi annual interest 1000*2.65% x PVIFA(3.06%,24)** PV of interest 26.50 16.82660 445.90 PV of maturity value Par value 1,000.00 x PVIF(3.06%,24)** PV of maturity value 0.48511 485.11 PV of bond/Price of bond 931.01 (1-(1.0306A-24)/0.0306 (1-(1+r)A-n)/r *PVIFA(3.06%,24) formula 16.82660 PVIF(3.06%,24) formula 1/(1r) An | 1/(1.0306 24) 0.48511

PVAF = (1-(1.0306)^-24)÷0.0306 = 16.8266

Now present value of semi annual coupon payments = $26.50 × 16.8266 = 445.9049

Present value of maturity value = $1,000 × PVF (24 periods, 3.06%) = $1,000 × 0.48511 = $485.11

Present value of all cash flows from the bond = $445.9049 + $485.11 = $931.01

Answer 2

Solution Par value semi annual coupon rate 5.3%/2 semi annual yield to maturity 6.12%/2 Number of payments 12*2 1000 2.65% 3.06% 24 PV of bond PV of interest PV of maturity PV of interest Semi annual interest 1000*2.65% x PVIFA(3.06%,24)** PV of interest 26.50 16.82660 445.90 PV of maturity value Par value 1,000.00 x PVIF(3.06%,24)** PV of maturity value 0.48511 485.11 PV of bond/Price of bond 931.01 (1-(1.0306A-24)/0.0306 (1-(1+r)A-n)/r *PVIFA(3.06%,24) formula 16.82660 PVIF(3.06%,24) formula 1/(1r) An | 1/(1.0306 24) 0.48511