The market interest rate for Christian Charities is 8% on January 1, 2008. On that day, Christian Charities issued the following bonds.
A. $500,000 7-year 7% bond
B. $300,000 10-year 9% bond
For both bonds, interest is paid semiannually on June 30 and December 31 each year up until maturity. Compounding is done semiannually.
To calculate the cash flows for each bond, we need to determine the interest payments and the principal repayments at each payment interval.
Bond A: $500,000 7-year 7% bond
Interest rate = 7%
Maturity = 7 years
Since interest is paid semiannually, we will have 14 payment periods (7 years * 2).
To calculate the semiannual interest payment:
Step 1: Calculate semiannual interest rate
Semiannual interest rate = Annual interest rate / 2
Semiannual interest rate = 7% / 2 = 3.5%
Step 2: Calculate semiannual interest payment
Semiannual interest payment = Semiannual interest rate * Principal
Semiannual interest payment = 3.5% * $500,000 = $17,500
Principal repayment:
Since this is a bond with a specified maturity, the principal is repaid in full at the end of the term.
Bond B: $300,000 10-year 9% bond
Interest rate = 9%
Maturity = 10 years
Since interest is paid semiannually, we will have 20 payment periods (10 years * 2).
To calculate the semiannual interest payment:
Step 1: Calculate semiannual interest rate
Semiannual interest rate = Annual interest rate / 2
Semiannual interest rate = 9% / 2 = 4.5%
Step 2: Calculate semiannual interest payment
Semiannual interest payment = Semiannual interest rate * Principal
Semiannual interest payment = 4.5% * $300,000 = $13,500
Principal repayment:
Since this is a bond with a specified maturity, the principal is repaid in full at the end of the term.
Now you have the cash flows for both bonds.
To calculate the present value of each bond, we need to use the formula for the present value of a bond:
PV = C * (1 - (1 + r)^(-n)) / r + F / (1 + r)^n
Where:
PV = present value of the bond
C = coupon payment (interest payment)
r = market interest rate per period
n = number of periods
F = face value (principal amount)
For bond A:
Face value (F) = $500,000
Coupon payment (C) = 7% of $500,000 = $35,000 per year (paid semiannually, so it is $17,500 every 6 months)
Market interest rate (r) = 8% per year (compounded semiannually)
Number of periods (n) = 7 years (compounded semiannually, so it is 14 periods)
Using these values, we can calculate the present value (PV) of bond A:
PV = $17,500 * (1 - (1 + 0.08/2)^(-14)) / (0.08/2) + $500,000 / (1 + 0.08/2)^14
Simplifying the equation, we get:
PV = $17,500 * (1 - 0.579741) / 0.04 + $500,000 / 1.490418
PV ≈ $366,798 + $213,358
PV ≈ $580,156
The present value of bond A is approximately $580,156.
Now, let's calculate the present value of bond B:
Face value (F) = $300,000
Coupon payment (C) = 9% of $300,000 = $27,000 per year (paid semiannually, so it is $13,500 every 6 months)
Number of periods (n) = 10 years (compounded semiannually, so it is 20 periods)
Using these values, the equation for calculating the present value of bond B is:
PV = $13,500 * (1 - (1 + 0.08/2)^(-20)) / (0.08/2) + $300,000 / (1 + 0.08/2)^20
Simplifying the equation, we get:
PV = $13,500 * (1 - 0.389886) / 0.04 + $300,000 / 1.485946
PV ≈ $247,288 + $201,742
PV ≈ $449,030
The present value of bond B is approximately $449,030.