Suppose a family member approaches you to borrow $2,000 for the down payment on an automobile. You have the cash available in a savings account that currently earns 5% annual interest. You and the family member consider the following repayment options:(i) Borrower repays $259 each year over the next ten years(ii) Borrower repays $300 each year over the next five years, plus a lump-sum payment of $895 in the fifth year.(iii) Borrower repays you $2,100 at the end of one year. For each of the options above, show that the present values of each option are approximately equal. Then, relate each of the options above to the four types of bonds, indicating which option is equivalent to which type of bond. Explain why.
What will be an ideal response?
The present values are calculated as follows:
(i) Fixed-payment loan. The borrower repays a loan in fixed annual payments for a pre- determined period of time.
Present value of fixed payment loan =
= Fixed payment/(1 + i) + Fixed payment/(1 + i)2 + + Fixed payment/(1 + i)n
= $259/(1 + 0.05) + $259/(1 + 0.05)2 + + $259/(1 + 0.05)10 = $2,000
(ii) Coupon bond. The borrower repays the loan in fixed annual payments ("coupons") and pays a one-time lump sum payment in the last year of the loan ("face value").
Present value of coupon bond = PCB
PCB = Coupon payment/(1 + i) + Coupon payment /(1 + i)2 + + Coupon payment /(1 + i)n + Face value/(1 + i)n
= $300/(1 + 0.05) + $300/(1 + 0.05)2 + + $300/(1 + 0.05)5 + $895/(1 + 0.05)5
= $2,000
(iii) Zero-coupon bond. The borrower repays the principal plus interest in one annual payment, with no intermediate payments.
Price of zero coupon bond = Payment/(1 + i) = $2,100/(1 + 0.05) = $2,000
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