A loan of face value $30,000,000 is issued in bonds each face value $300. The bonds will be repaid with a bonus of 20%.
1/5th of the total capital will be repaid each year, the first payment being at the end of the 10th year. Coupons on the outstanding bonds are 7% p.a. paid half yearly [ in arrears]. Find the price at which the bonds should be issued if the loan issue is priced so that an investor who purchased the entire loan would earn 6%pa payable half yearly.
To find the price at which the bonds should be issued, we need to calculate the present value of the future cash flows associated with the bonds and then solve for the issue price.
To calculate the present value, we need to determine the cash flows for each bond. For the loan of face value $30,000,000, each bond has a face value of $300. So, the total number of bonds issued is:
Total number of bonds = Face value of loan / Face value of each bond
Total number of bonds = $30,000,000 / $300 = 100,000
Now, let's calculate the cash flows for each bond. The bonds will be repaid with a bonus of 20% at the end of the loan term. Therefore, the repayment amount for each bond is:
Repayment amount = Face value of each bond + Bonus amount
Repayment amount = $300 + (20% * $300) = $360
The total repayment amount for all bonds is:
Total repayment amount = Total number of bonds * Repayment amount
Total repayment amount = 100,000 * $360 = $36,000,000
Next, we need to determine the coupon payments for each bond. The coupon rate is 7% p.a., paid half-yearly in arrears. Since the coupon payments are made half-yearly, the annual coupon rate needs to be divided by 2, and the number of coupon payments is double the number of years.
Annual coupon rate = 7%
Coupon rate per period = Annual coupon rate / 2 = 0.07 / 2 = 0.035
Total number of coupon payments = Number of years * 2
Total number of coupon payments = 10 * 2 = 20
Now, we can calculate the present value of the cash flows. The present value of each cash flow can be calculated using the formula:
PV = CF / (1 + r)^n
Where PV is the present value, CF is the cash flow, r is the discount rate, and n is the number of periods.
In this case, the discount rate is 6% p.a. payable half-yearly, which needs to be divided by 2 to get the periodic discount rate:
Periodic discount rate = Annual discount rate / 2 = 0.06 / 2 = 0.03
Let's calculate the present value of the bond's cash flows:
Present value of the repayment amount = $36,000,000 / (1 + 0.03)^10 = $25,411,225.51
Present value of the coupon payments:
PV of each coupon payment = ($300 * 0.035) / (1 + 0.03) + ($300 * 0.035) / (1 + 0.03)^2 + ... + ($300 * 0.035) / (1 + 0.03)^20
This is a series of cash flows, so we can use the formula for the present value of an annuity:
PV of annuity = PMT * (1 - (1 + r)^(-n)) / r
Applying this formula, we can calculate the present value of the coupon payments:
PV of coupon payments = ($300 * 0.035) * (1 - (1 + 0.03)^(-20)) / 0.03 = $47,091.49
Finally, the issue price of the bond is the sum of the present value of the repayment amount and the present value of the coupon payments:
Issue price = Present value of repayment amount + Present value of coupon payments
Issue price = $25,411,225.51 + $47,091.49 = $25,458,317.00
Therefore, the bonds should be issued at a price of $25,458,317.00 to meet the requirements of earning a 6% p.a. return for the investor.