Question

Consider two bonds. The first is a 6% coupon bond with six years to maturity, and a yield to maturity of 4.5% annual rate, compounded semi-annually. The second bond is a 2% coupon bond with six years to maturity and a yield to maturity of 5.0%, annual rate, compounded semi-annually.

1. Calculate the current price per $100 of face value of each bond. (You may use financial calculator to do question 1 and 2, I'm just unsure how to use it.)

2. Given the data for the first two bonds, now consider a third bond: a zero coupon bond with six years to maturity. Calculate the price per $100 of face value of the zero coupon bond. Calculate the yield to maturity for the zero coupon bond. (Express the yield as annual rate, compounded semi-annually).

HINT: Use the Value Additivity principle to answer part c. Create a synthetic zero- coupon bond, that is, a portfolio of the 6% coupon bond and the 2% coupon bond that has the same cash flows as a 6-year, zero coupon bond.

Answer 1

For both the bonds, coupon is compounded semiannualy

a) First bond

Assuming Face value = 100, Semiannual coupon = 6% / 2* 100 =3

Yield to maturity = 4.5%

Years to maturity = 6 , Number of coupon payment period = 6 * 2 = 12

Using excel, Bond current price = $86.3

Period	Coupon	Yield / discount rate	Discount factor	Present value of cash flow
1	3	4.50%	0.957	2.9
2	3	4.50%	0.916	2.7
3	3	4.50%	0.876	2.6
4	3	4.50%	0.839	2.5
5	3	4.50%	0.802	2.4
6	3	4.50%	0.768	2.3
7	3	4.50%	0.735	2.2
8	3	4.50%	0.703	2.1
9	3	4.50%	0.673	2.0
10	3	4.50%	0.644	1.9
11	3	4.50%	0.616	1.8
12	103	4.50%	0.590	60.7
			Bond present value	86.3

Excel formula:

B A C D E 19 Period Coupon Yield/discount rate Discount factor Present value of cash flow - 100 6 % / 2 20 1 =1/(1+C20)^A20 -1/(1+C21)^A21 =1/(1+C22)^A22 -1/(1+C23)^A23 1/(1+C24)^A24 -1/(1+C25)^A25 =1/(1+C26)^A26 -1/(1+C27)^A27 =1/(1+C28)^A28 = 1/ (1+C29)^A29 1/(1+C30)^A30 -1/(1+C31)^A31 -B20 D20 -4.5% 21 A20+1 100 6% / 2 -4.5 % -B21*D21 22 -A21+1 -100*6%/2 -4.5% -B22*D22 -A22+1= 100 * 6% / 2 - B23*D23 -4.5% 23 A23+1-100*6 % / 2 -4.5% =B24*D24 24 -A24+1- 100 * 6% / 2 4.5% -B25*D25 25 26 A25+1 -100*6%/2 -B26 *D26 -4.5% 27 A26+1 100*6% / 2 28 A27+1100*6% / 2 - B27*D27 -4.5 % -4.5% -B28*D28 29A28+1 100*6%/2 4.5 % =B29*D29 - B30*D30 30 A29+1 100*6%/2 4.5% 31 -A30+1 -100*6%/2+ 100 -4.5 % -B31 D31 Bond present value =SUM(E20:E31) 32

b) For second bond coupon is 2% (compounded semmiannually) and Yield to maturity is 5%, years to maturity =6%

Using excel, Bond current price = $64.5

Period	Coupon	Yield / discount rate	Discount factor	Present value of cash flow
1	1	5.00%	0.952	1.0
2	1	5.00%	0.907	0.9
3	1	5.00%	0.864	0.9
4	1	5.00%	0.823	0.8
5	1	5.00%	0.784	0.8
6	1	5.00%	0.746	0.7
7	1	5.00%	0.711	0.7
8	1	5.00%	0.677	0.7
9	1	5.00%	0.645	0.6
10	1	5.00%	0.614	0.6
11	1	5.00%	0.585	0.6
12	101	5.00%	0.557	56.2
			Bond present value	64.5

2) Using addition principal

Price of zero coupon bond = average (value of bond 1, value of bond 2)

= average(86.3,64.5)

Price of zero coupon bond = $75.4

Using rate formula to to calculate yield = RATE(6,0,-75.4,100)

Yield on zero coupon = 4.8%