Second National Bank offers an account that earns $4.91\%$ per year, compounded continuously. If a person invests $\$20,000$ in this account, what will be the value of the account at the end of $12$ years? (Round your answer to the nearest cent.)
When I did this problem I used the formula $20,000(1+0.0491)^{12}$ and got $\$35549.43$ but it was marked incorrect so I was wondering what I was doing wrong?
On
Just for clarification
Simple interest is calculated by multiplying the loan amount (e.g. 1000 monetary units) by the interest rate (e.g. 5%) by the number of payment periods over the life of the loan (e.g. 24 months). The thing to keep in mind here is that the interest rate may be expressed in terms of an annual rate, e.g. 5% per annum, whereas the payment periods might be expressed in months, e.g. 24 months. To ensure your calculation of simple interest is accurate, you need to ensure that both the interest rate and the payment periods are expressed in the same manner, say annually or monthly. For example, using the figures above, our 1000 monetary units loan would be at 5% per annum, and taken out over just 2 years (as opposed to 24 months). So here the calculation would be 1000 x 0.05 x 2 (loan x interest x term) = 100. So, the amount of simple interest that we would pay on this loan over the two year term would be 100 monetary units. Interest rates are seldom calculated using the simple interest rate formula however, and are more likely to be calculated using the compound interest formula.
Compound interest relates to charges the borrower must pay not just on the principal amount borrowed, as in simple interest, but also on any interest outstanding at that point in time. To illustrate the difference between compounding interest and simple interest, consider the following very simplified scenario of a $1000 loan taken out at 10% over 2 years (assuming no payments are made until the end of the loan):
Example of Simple vs Compound Interest
Simple Interest:
First year = $1,000 * 1 year * 10/100 = $100 in interest
Second year = $1,000 * 1 year * 10/100 = $100 in interest
Total Interest = $200
Total of the principal amount plus interest = $1,200
In this scenario, the total amount of interest paid over the life of the loan would be $200
Compound Interest:
First year = $1,000 * 1 year * 10/100 = $100 in interest
Second year = $1,100 * 1 year * 10/100 = $110 in interest
Total Interest = $210
Total of the principal amount plus interest = $1,210
In this scenario, with interest compounded annually, the total amount of interest paid is $210
Now keep in mind that this is a somewhat simplified example, but it should be enough to highlight the workings of compound interest. In reality you’ll be expected to make payments on your loan at regular intervals for the life of the loan, and therefore the effects of paying interest on your interest will be different from the above example. For the most part however, any loans you take out will, or should, have regular fixed payments. At the beginning, these payments will go primarily to paying off the interest on the loan, with a smaller percentage going to paying off the principal. Over time, this ratio changes to more of the principal being paid off with each installment.
Today, with the aid of computers, it's possible to compound interest monthly, daily, and in the limiting case, continuously, meaning that your balance grows by a small amount every instant.
See http://www.moneychimp.com/articles/finworks/continuous_compounding.htm
$C*e^{rt}$
$20,000(e^{.0491*12})= 36,050.92$