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I have recently come across a particularly difficult problem. I was hoping some someone can provide a solution for the problem.
Suppose that, for an effective monthly interest rate r, which is positive, a loan requires equal repayments of dollar A each month for n months, with the first repayment due in exactly one month. Here, n > 1 is an integer. Let us call
it Option 1. For this option, a timeline shows that time 0 is the time when the borrower obtains the loan and that time 1, time 2, time 3, . . . , time n are the n occasions when equal repayments of dollar A each time are to be made by the borrower.. For the same loan, the lender offers the borrower an alternative repayment option at the same effective monthly interest rate r. Let us call it Option 2. For this option, the borrower is to repay a lump sum equal to n times the original monthly amount dollar A, at the mid-point of the timeline for the original monthly repayments. That is, the borrower is to repay the amount of dollar nA, only once, at time (n + 1)/2 on the timeline and there are no other repayments. The task here is establish analytically whether Option 1 or Option 2 is more beneficial to the borrower, based on time value concepts. [Hint: For the task here, we can start with a closed-form
expression of q (weighted average of arrival times for the n coupons) in terms of r, for which dq/dr < 0.
Whereby q = 1 + 1/r − n [(1 + r)^n-1]^-1
Then,
letting
x = 1/1+r
,
we can express q in terms of x instead of r. The sign of dq/dx can be inferred from the sign of dq/dr, which has already been established. The expression of dq/dx has an interesting analytical feature that is relevant for the task here.
Looking forward too see solution to this problem. Thanks
Suppose that, for an effective monthly interest rate r, which is positive, a loan requires equal repayments of dollar A each month for n months, with the first repayment due in exactly one month. Here, n > 1 is an integer. Let us call
it Option 1. For this option, a timeline shows that time 0 is the time when the borrower obtains the loan and that time 1, time 2, time 3, . . . , time n are the n occasions when equal repayments of dollar A each time are to be made by the borrower.. For the same loan, the lender offers the borrower an alternative repayment option at the same effective monthly interest rate r. Let us call it Option 2. For this option, the borrower is to repay a lump sum equal to n times the original monthly amount dollar A, at the mid-point of the timeline for the original monthly repayments. That is, the borrower is to repay the amount of dollar nA, only once, at time (n + 1)/2 on the timeline and there are no other repayments. The task here is establish analytically whether Option 1 or Option 2 is more beneficial to the borrower, based on time value concepts. [Hint: For the task here, we can start with a closed-form
expression of q (weighted average of arrival times for the n coupons) in terms of r, for which dq/dr < 0.
Whereby q = 1 + 1/r − n [(1 + r)^n-1]^-1
Then,
letting
x = 1/1+r
,
we can express q in terms of x instead of r. The sign of dq/dx can be inferred from the sign of dq/dr, which has already been established. The expression of dq/dx has an interesting analytical feature that is relevant for the task here.
Looking forward too see solution to this problem. Thanks