13. Write down a model for installment savings, savings that is increased by mak

Question

13. Write down a model for installment savings, savings that is increased by making installment

deposits. Call this model (5). Solve (5) as an IVP with initial balance (Ao)

Model If we deposit money in a savings account, take out a car loan or maintain a balance on a credit card, then we face the implications of compound interest. Consider using a credit, card. A person obtains the card, usually through a bank or credit union, and agrees to reimburse the credit company each time the card is used. As the person uses the card, a balance accumulates in his or her account. At the end of the month, if the balance is not completely paid, then the credit company applies interest to the account. At the end of the following month interest is applied again to any remaining balance and so on. Usually, the credit company expects a monthly payment, however, in order to derive a model, let us suppose that no payment is expected and interest rates never change, Let A(t) = account balance (in dollars) r = annual interest, rate, note 0

Explanation / Answer

When the loan is first taken out, no payments have been made, so the loan balance is the same as the loan amount:

What happens at the end of the first period? Interest has accrued. The interest rate per period is i, and the balance is B_0, so the accrued interest is i times B_0, which equals iA; this gets added to the loan balance. On the other hand, the payment P is subtracted. Therefore:

What happens to the loan balance at the end of the second period? We add in the interest on the previous balance, which is i times B_1, and subtract the next payment P:

This process repeats at the end of each period: add in the accrued interest and subtract the payment. I won

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