Question

Assume the current Treasury yield curve shows that the spot rates for six​ months, one​ year, and one and a half years are 1 %1%​, 1.1 %1.1%​, and 1.3 %1.3%​, all quoted as semiannually compounded APRs. What is the price of a ​$1 comma 0001,000 ​par, 4.25 %4.25% coupon bond maturing in one and a half years​ (the next coupon is exactly six months from​ now)? The price of this bond is ​$nothing.

Answer 1

Answer:

present value of bond = $1042.96

Explanation:

given data

spot rates for six​ months = 1%

spot rates for one and = 1.1%​

spot rates for one and half years = 1.3%​

price = $1000

coupon bond = 4.25%

time = 6 month

solution

we get here first price on bond paid that is

coupon paid = $1000 × 4.25 × 0.5   = $21.25

we get here present value of 6 month and 1 year and 1 and half  year

present value  =   $\frac{coupon\ payment }{(1+\frac{spot \ rate}{2})^t}$     ..............1

present value of 6 month = $\frac{21.25}{(1+\frac{0.1}{2})^1}$    = 20.23

present value of 1 year = $\frac{21.25}{(1+\frac{0.011}{2})^2}$   = 21.01  

present value of 1 year and half year = $\frac{21.25}{(1+\frac{0.013}{2})^2}$   =  20.97

and

now we get present value of par value in 1 and half year

present value of par value in 1 and half year = $\frac{par\ value}{(1+\frac{spot rate}{2})^3}$  

present value of par value in 1 and half year = $\frac{1000}{(1+\frac{0.013}{2})^3}$

present value of par value in 1 and half year = 980.75

so

present value of bond will be as

present value of bond = 20.23 + 21.01 + 20.97 + 980.75

present value of bond = $1042.96