Question

Compound Interest A bank offers two types of interest-bearing accounts. The first account pays \(6 \%\) interest compounded monthly. The second account pays \(5 \%\) interest compounded continuously. Which account earns more money? Why?

Short answer

Find the future values for both accounts by substituting the known values into the compound and continuous interest formulas. Compare the future values to determine which account yields the most money.

Step-by-step solution

Calculate Compound Interest

First, calculate the compound interest. Assume an initial principle of $1 and one year as time period for simplicity. We know that compound interest is calculated using the formula \(A=P(1+r/n)^{nt}\). Here, the rate of interest r is 6%, compounded monthly, so n=12 and t=1. Interest is to be calculated on $1 (P=1). Substituting these values, we find the total amount after a year.

Calculate Continuous Interest

Next, calculate the continuous interest. This can be accomplished by using the formula \(A=Pe^{rt}\). Here, the rate of interest r is 5%, and t=1. Interest is to be calculated on $1 (P=1). Substituting all the values, we find the total amount after a year.

Compare the Results

Once you've found the total amounts from step 1 and step 2, compare them to find out which one is greater.

Key concepts

Interest Rates

Interest rates are a fundamental concept in understanding how money grows over time, particularly when investing in savings accounts or loans. An interest rate is expressed as a percentage and represents the amount charged by a lender to a borrower for the use of assets. In the context of savings accounts, it's important because it determines how much your investment will grow.

For example, if a bank offers a 6% interest rate compounded monthly, it implies that the interest will be calculated every month on the principal amount, plus any previously accumulated interest. Meanwhile, a 5% interest rate compounded continuously would mean the interest compounds at every instant, theoretically, never stopping.

• A higher interest rate typically yields a greater return.
• The frequency of compounding also influences the final amount.
• More frequent compounding can substantially increase the total amount accrued.

When choosing between different accounts and interest rates, always consider both the rate and the compounding frequency.

Continuous Compounding

Continuous compounding is a unique method of calculating interest that assumes constant compounding across every possible moment. This means your balance grows consistently, almost like a continuous flow of accumulated interest, rather than at specific intervals.

The mathematical formula for continuous compounding is given by the equation \(A = Pe^{rt}\), where:

• \(A\) is the final amount
• \(P\) stands for the initial principal balance
• \(r\) is the annual interest rate in decimal
• \(t\) represents the time in years
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828

With continuous compounding, even with a lower nominal rate, such as 5%, the effects can be quite surprising because of the unending frequency. This form of compounding essentially maximizes the compounding effect. It's particularly beneficial for long-term investments where time allows exponential growth to work its magic.

Monthly Compounding

Monthly compounding is a prevalent method used by banks to calculate interest on various accounts. With monthly compounding, interest is computed each month based on the principal and any accumulated interest from previous months.

This compounding frequency is represented in the compound interest formula \(A = P(1 + \frac{r}{n})^{nt}\), where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount.
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times that interest is compounded per year (12 for monthly).
• \(t\) is the time the money is invested for in years.

Consider an account with a 6% annual interest rate compounded monthly. Every month, the balance is recalculated to include the 1/12th portion of the annual interest rate applied to the sum of the initial deposit and earnings from previous months.
Monthly compounding gives more frequent updates on interest, which can lead to a slightly higher balance over time compared to less frequent compounding intervals. Always compare the effective rate when deciding between different compounding frequencies with similar nominal rates.