Multiple-payments
In many real-world financial instruments, we see a repeating-payment structure. For example, loan repayments are often monthly.
In multiple-payment scenarios, we have a payment plan that repeats the same amount every period. There are two formulae for multiple-payments:
equation 1: F=P(1+i)^n
F is “future value”
P is “present value”
i is the interest rate per period
n is the number per periods
equation 2: P=A( ((1+i)^n-1)/ (i(1+i)^n) )
equation 3: F=A( ((1+i)^n-1)/ (i) )
where all of the variables are the same as in equation 1 and A is the amount paid per period. Important: It’s very important that the compounding period of interest, the number of periods, and the amount paid per period all use the same period.
We can choose between the two equations equations 2 and 3 based on whether the lump sum amount occurs at the before (equation 2) or after (equation 3) all of the periodic payments.
A-Suppose you deposit $100 every month into an account that pays 3% interest per annum, compounded monthly. How much will you have saved if you keep this up for 5 years? (Hint: Since the lump amount of “how much you’ll have” is after the monthly payments, use equation 3).
B-Suppose you have $x in your bank account today. You’ve done the math, and it seems that you can withdraw $100 from your bank account every month for the next 5 years. If your bank pays 3% per annum compounded monthly, how much do you have in your account today?
The total withdrawn amount is 100 × 12 × 5 = 6000. Do you expect x to be greater or less than 6000?
The difference between $6000 and x is the extra you earned due to interest.
C-In order to pay for tuition, you take out a loan of $10,000. If the loan repayment terms are 2%/a compounded monthly, repayable over 10 years, how much must you repay every month?