Question

The Treasury department issued a 10-year bond on January 1, 2015. The par value is $1,000 and the annual coupon rate is 10%. The bond pays two coupons every year, one at the end of June and one at the end of December. The required annual yield is 8%. An investor bought the bond on March 31, 2015. What is the price that he should pay for the bond

Answer 1

To calculate the price of the bond, we can use the present value formula. The bond has a par value of $1,000 and an annual coupon rate of 10%, paying two coupons every year. By discounting the future cash flows at the required annual yield rate, we find that the investor should pay $834.31 for the bond.

Step-by-step explanation:

To calculate the price that the investor should pay for the bond, we can use the present value formula. The bond has a par value of $1,000 and an annual coupon rate of 10%. Since the bond pays two coupons every year, one at the end of June and one at the end of December, the coupon rate for each coupon payment is 5% ($1,000 * 10% / 2).

The required annual yield is 8%. We can calculate the present value of the bond by discounting the future cash flows (coupon payments and the face value) at the required annual yield rate.

We can apply the present value formula to calculate the price of the bond as follows:

1. Calculate the present value of each individual cash flow using the formula PV = CF / (1 + r)ⁿ, where PV is the present value, CF is the cash flow, r is the required annual yield rate, and n is the number of periods until the cash flow is received.
2. Sum the present values of all the cash flows to get the price of the bond.

Let's calculate the price of the bond step by step:

1. Calculate the present value of each coupon payment:

• First coupon payment: PV = $50 / (1 + 0.08/2)^(2/12) = $48.54
• Second coupon payment: PV = $50 / (1 + 0.08/2)^(2/12) = $48.54

Calculate the present value of the face value:

• PV = $1,000 / (1 + 0.08/2)¹⁰ = $737.23

Sum the present values of all the cash flows:

• The price of the bond is $48.54 + $48.54 + $737.23 = $834.31

Therefore, the investor should pay $834.31 for the bond on March 31, 2015.

Answer 2

$1,135.90

Step-by-step explanation:

The bond price formula is given below, the formula implies that the price of the bond is the present value of future cash flows which are the semiannual coupons and the face value

Bond price=face value/(1+r)^n+semiannual coupon*(1-(1+r)^-n/r

face value=$1,000

r= semiannual yield to maturity=8%*6/12=4%

n=number of semiannual coupons in 10 years=10*2=20

semiannual coupon=face value*coupon rate*6/12=$1,000*10%*6/12=$50

bond price=$1,000/(1+4%)^20+$50*(1-(1+4%)^-20/4%

bond price=$1000/(1.04)^20+$50*(1-(1.04)^-20/0.04

bond price=$1000/2.191123143 +$50*(1-0.456386946 )/0.04

bond price=$1000/2.191123143 +$50*0.543613054/0.04

bond price= $456.38 +$679.52

= $1,135.90