Question

Assume 6-month zero rate is 4.045%. Also use the following table to answer the questions below.

The following table gives the prices of bonds:

Face Value	Time To Maturity	Coupon / Year	Bond Price
100	1 Year	0	97
100	1.5 Year	15	98.5

*Half of the stated coupon is paid every six months ** all rates are continuously compounded

What is the zero rate for 1 year?

1) 3.046%

1) 4.545%

1) 3.455%

1) 5.206%

2. What is the zero rate for 1.5 year?

2) 12.245%

2) 13.522%

2) 16.544%

2) 15.377%

Answer 1

a) As the One year zero coupon bond of face value 100 is priced at 97

Therefore , by the formula for price

Price = present value of all future cashflows  

97 = 100 * e ^-(r*1)

100 = 97 *e ^(r*1)

e^r = 100/97 =1.0309

r = ln (1.0309) = 0.03046 = 3.046% (1st Option )

b) The second bond pays a coupon of 7.5 each at the end of 6 months, 1 year and 1.5 years and 100 as the redemption value at the end of 1.5 years

So , by the bond pricing formula

98.5 = 7.5 *e^{-(0.04045 * 0.5)} + 7.5 * e^-(0.03046*1) + 107.5 * e^-(r*1.5)

98.5 = 7.3598 + 7.2750 + 107.5 *e^-(1.5r)

=> e^-(1.5r) = 83.8751/107.5 = 0.780234

=> -1.5r = ln(0.780234) =-0.24816

So r = .24816/1.5 = 0.16544 = 16.544% (3rd option)