Question

Please answer with mathematical formulas, not an excel sheet please

3. John borrows an amount at an annual interest rate of 8%. He repays all interest and principal in a lump sum at the end of ten years John uses the amount borrowed to purchase a 5-year bond with a par value of 1000 with coupons at a nominal rate of 10% payable semiannually, with the first coupon paid at the end of 6-month period from now. The bond is redeemed at par and John's yield rate for the bond is 9% convertible semian- nually As John receives each coupon payment, he immediately puts the money into an account earning nominal rate of 5.8% convertible semiannually At the end of five years (from now), immediately after John receives the final coupon payment, John deposits the accumulated value of the coupons and the redemption amount of the bond into a savings account earning an annual interest rate of 6.5%. At the end of each year from year 6 through 10, John deposits an additional amount of 50 into this savings account Find John's balance at the end of ten years after the loan is repaid. from now

Answer 1

Suppose John borrows an amount X and X is also the price of the bond

The bond pays an amount of 50 every 6 months for 5 years (10 payments) and 1000 at the end of 5 years. The discount rate for a 6 month period is 4.5%.  So

X= 50/1.045 + 50/1.045²+ 50/1.045³+...... 50/1.045¹⁰ + 1000/1.045¹⁰

=50/0.045 * (1- (1/1.045)¹⁰) + 1000/1.045¹⁰

= 395.636 + 643.928 = 1039.564

So, the final amount that John has to pay after 10 years is 1039.564 * 1.08¹⁰ = 2244.34

John reinvests the bond coupons at 5.8% converted semiannually, so for every 6 month,he gets a rate of 2.9% .  After 5 years, the total amount that John will get is given by

50* 1.029⁹ + 50* 1.029⁸ + .... + 50 + 1000 = 50 * (1.029¹⁰-1) / 0.029 +1000 = 1570.561

This amount is invested at a rate of 6.5 % annually for the remaining 5 years , so this amount will turn out to become

1570.561 * (1.065)⁵ = 2151.805 after 10 years

The additional amount of 50 which is deposited at the end of year 6 through 10 will amount to the following  

50* 1.065⁴+ 50* 1.065³+ 50* 1.065²+ 50* 1.065¹+50 = 50/0.065* (1.065⁵-1) = 284.682

In total, John will have in his account 2151.805+284.682 = 2436.487

Out of which , after paying 2244.34 , the balance of 192.147 at the end of 10 years

So, John will have an amount of 192.147 remaining at the end of 10 years