Question

You borrow \(2000 at 9% annual interest compounded monthly for 2 years. The monthly payment is \)91.37. a. Find the balance after the fifth payment. b. Find the amount of the last payment.

Short answer

a. The balance after the fifth payment is the initial loan plus the total interest over five months minus the total payment made. b. The amount of the last payment will be the remaining balance after making all payments except the last one.

Step-by-step solution

Calculate Interest for Each Month

The interest rate is compounded monthly. The monthly interest rate is thus 9% / 12 = 0.75%. This is calculated in decimal form as 0.0075. Calculate the interest for each month over the first five months using the formula for simple interest, \(PRT\), where \(P = \$2000\), \(R = 0.0075\), and \(T = 1/12\) per month. The total interest over the first five months is the sum of these figures.

Calculate Balance After Fifth Payment

To calculate the balance after the fifth payment, subtract the total payment made, which is \(5 x \$91.37\), from the initial loan of \$2000 plus the total interest over five months obtained in Step 1.

Calculate Interest for Remaining Months

In order to find the amount of the last payment in Step b, calculate the total interest for the remaining months until the second last payment. Use the formula for simple interest with \(P\) being the loan balance after the fifth payment, \(R = 0.0075\) and \(T = 1/12\) for each month.

Calculate Last Payment

The remaining balance after all payments except the last one is obtained by adding the total interest for the remaining months to the balance calculated in Step 2 and then subtracting the total of the payments made, which is \((number of months until second last payment) x \$91.37\). The last payment must clear this balance.

Key concepts

Compound Interest

Compound interest is a powerful concept in finance that reflects the reality that interest is earned on interest. This means that over time, the amount of money grows at an accelerating rate, as the interest you earn each period is added to the original amount, creating a new, larger sum on which to earn interest for the next period.

For example, if you borrow \(2,000 at a 9% annual interest rate compounded monthly, the monthly interest rate would be 9% divided by 12, equating to 0.75% per month. Then you would calculate the interest each month based not only on the original \)2,000 but also on any interest that has been added in prior months. Over numerous periods, this effect can lead to substantial growth in the amount outstanding, which is why borrowers should be very mindful of the compound interest terms of their loans.

Simple Interest Formula

The simple interest formula is a quick way to calculate the interest paid or earned over a certain amount of time. The formula is represented as \( I = PRT \) where

• \( P \) is the principal amount (the initial sum of money),
• \( R \) is the annual interest rate (in decimal),
• and \( T \) is the time the money is invested or borrowed for, in years.

When a loan is issued with interest compounding monthly, like in our example, where the interest compounds monthly, you'll often calculate interest on a monthly basis, with \( T \) being \(1/12\) as used in the provided exercise solution. Even though the term 'simple interest' suggests it isn't compounding, it is a useful stepping stone in monthly compounding calculations to ascertain the interest due at each installment.

Loan Amortization

Loan amortization is the process of spreading out a loan into a series of fixed payments. The loan is paid off at the end of the payment period. Particularly with an amortizing loan, every payment includes both a portion that goes to reducing the principal amount borrowed and a portion that covers the accrued interest.

Under the standard amortization formula, early payments will consist largely of interest rather than reducing the principal. As the balance decreases, the interest portion shrinks, and more of each payment goes towards the principal. The amortization concept is essential in managing long-term loans, ensuring the balance is paid down over time. This is directly relevant in calculating the balance after the fifth payment and determining the last payment of the loan.

Monthly Payments

Monthly payments are regular payments made to repay a loan. These payments are calculated to ensure that the loan is fully repaid, including both principal and interest, by the end of the loan term. In practical terms, when you borrow money, you agree to pay it back in these installments.

The size of each monthly payment depends on the loan amount, the interest rate, the length of the loan, and how the interest is compounded. To keep things affordable, loans are often structured to be paid back over several years. As we see in the exercise, a monthly payment of \(91.37 is set to pay back the \)2,000 loan at a 9% annual interest, compounded monthly across a two-year period. Each of these payments slowly reduces the principal and the accumulated interest until the loan is paid off.