General Electric issued 1000 debenture bonds 3 years ago with a face value of $5000 each and a bond interest rate of 8% per year payable semiannually. The bonds have a maturity date of 20 years from the date they were issued. If the interest rate in the market place is 10% per year, compounded semiannually, what is the present worth of one bond to an investor who wishes to purchase it today?
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4190.35
To calculate the present worth of the bond, we need to discount the future cash flows generated by the bond to their present value.
The cash flows from the bond include the semi-annual coupon payments and the face value received at maturity.
Step 1: Calculate the present value of the coupon payments.
The bond has a coupon rate of 8%, which is the annual percentage of the face value that is paid as interest. Since the coupons are paid semiannually, the coupon rate per semiannual period is 8%/2 = 4%.
The bond has a maturity period of 20 years, which means there will be 40 semiannual periods (20 years * 2 periods/year).
The semiannual coupon payment is 4% * $5000 = $200.
To calculate the present value of the coupon payments, we need to discount each semiannual payment to its present value using the market interest rate of 10% compounded semiannually. The bond pays interest semiannually, so the compounding period is also semiannual.
We can use the formula for the present value of an ordinary annuity:
PV = C * (1 - (1 + r)^(-n)) / r
Where:
PV = Present value of the annuity
C = Cash flow per period (coupon payment)
r = Discount rate per period
n = Number of periods
Using the given values:
C = $200
r = 10%/2 = 5% (semiannual rate)
n = 40
PV = $200 * (1 - (1 + 5%)^(-40)) / 5% = $4,579.37
Step 2: Calculate the present value of the face value received at maturity.
The face value of the bond is $5000, which will be received at the end of the 20-year period.
To calculate the present value of the face value, we can use the formula for the future value (FV) of a single sum:
PV = FV / (1 + r)^n
Where:
PV = Present value
FV = Future value (face value)
r = Discount rate per period
n = Number of periods
Using the given values:
FV = $5000
r = 10%/2 = 5% (semiannual rate)
n = 40
PV = $5000 / (1 + 5%)^40 = $676.30
Step 3: Calculate the present worth of the bond.
The present worth of the bond is the present value of the coupon payments plus the present value of the face value.
Present Worth = Present Value of Coupon Payments + Present Value of Face Value
= $4,579.37 + $676.30
= $5,255.67
Therefore, the present worth of one bond to an investor who wishes to purchase it today is approximately $5,255.67.
To find the present worth of the bond, we need to calculate the discounted value of all future cash flows associated with the bond. Here's how you can calculate it:
Step 1: Find the number of semiannual periods until the bond matures.
Since the bonds were issued 3 years ago and have a maturity period of 20 years, there are 40 semiannual periods (20 years * 2) left until maturity.
Step 2: Calculate the semiannual interest rate using the market interest rate.
The market interest rate is given as 10% per year, compounded semiannually. To calculate the semiannual interest rate, divide it by 2: 10% / 2 = 5% per semiannual period.
Step 3: Find the semiannual coupon payment.
The bond has a face value of $5000 and an annual interest rate of 8% payable semiannually. To find the semiannual coupon payment, divide the annual interest rate by 2: 8% / 2 = 4% per semiannual period. Multiply the face value by the semiannual interest rate to find the semiannual coupon payment: $5000 * 4% = $200 per semiannual period.
Step 4: Calculate the present value of the future cash flows.
To calculate the present value, we need to discount each future cash flow to its present value using the semiannual interest rate. We can use the formula for the present value of an annuity:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
PV = Present value
C = Cash flow per period
r = Interest rate per period
n = Number of periods
In this case, C = $200, r = 5%, and n = 40 (semiannual periods until maturity).
Plug in these values into the formula:
PV = $200 * [1 - (1 + 5%)^(-40)] / 5%
Calculating this equation will give you the present value of the bond to an investor who wishes to purchase it today.