Question

Balance in an Account You deposit \(\$ 10,000\) in an account with an annual interest rate of \(6.75 \%\) for 12 years. Determine the balance in the account when the interest is compounded (a) daily \((n=365)\), (b) weekly, (c) monthly, and (d) quarterly. How is the balance affected by the type of compounding?

Short answer

After 12 years with a yearly interest rate of 6.75%, the account balance with the interest compounded daily will be approximately \$25,447.48, weekly will be approximately \$25,413.73, monthly will be approximately \$25,367.68, and quarterly will be approximately \$25,320.98. The type of compounding affects the account balance; the more frequently the interest is compounded, the more the account balance increases.

Step-by-step solution

Define Given Values and Formula

In this exercise, the initial amount \(P\) is \$10,000, the annual interest rate \(r\) is 6.75% or 0.0675 in decimal form, and the number of years \(t\) is 12. Now define the formula for compound interest, \(A = P (1 + r/n) ^ {nt}\).

Compounding Daily (n=365)

For daily compounding, \(n = 365\). Substitute these values into the formula: \(A = 10000(1 + 0.0675/365) ^ {365*12} \). Calculate the value and round to the nearest cent.

Compounding Weekly

For weekly compounding, \(n = 52\). Substitute these values into the formula: \(A = 10000(1 + 0.0675/52) ^ {52*12} \). Calculate the value and round to the nearest cent.

Compounding Monthly

For monthly compounding, \(n = 12\). Substitute these values into the formula: \(A = 10000(1 + 0.0675/12) ^ {12*12} \). Calculate the value and round to the nearest cent.

Compounding Quarterly

For quarterly compounding, \(n = 4\). Substitute these values into the formula: \(A = 10000(1 + 0.0675/4) ^ {4*12} \). Calculate the value and round to the nearest cent.

Analyze the Results

Compare the results from steps 2-5 to determine how the type of compounding affects the account balance. The more frequently the interest is compounded, the more money is earned.

Key concepts

Interest Rates

Interest rates are a key part of financial mathematics, used to calculate how much extra money is earned or paid out on a principal amount over time. An interest rate is normally expressed as a percentage, representing a proportion of the principal over a set period. It tells you how much a lender charges a borrower, or how much a saver earns, on deposited money.

• A higher interest rate means you earn more on your savings.
• Interest rates can be annual, but sometimes converted to monthly or even daily rates for more precise calculations.
• Always convert percentages to decimals when using them in mathematical formulas.

In our example, the annual interest rate is 6.75%, which equals 0.0675 when expressed as a decimal. Knowing how to handle these conversions is essential for calculating compound interest accurately.

Compounding Frequency

Compounding frequency refers to how often interest is applied to the principal balance. This can be daily, weekly, monthly, or quarterly, or even yearly, and it greatly affects the amount of interest earned or paid.

• The basic rule: the more frequent the compounding, the larger the amount of interest accrued.
• Common compounding frequencies include annual (once per year), semi-annual (twice a year), quarterly (four times a year), monthly (twelve times a year), weekly (52 times a year), or daily (365 times a year).

In the given exercise, we explored several frequencies:

• Daily compounding (n=365)
• Weekly compounding (n=52)
• Monthly compounding (n=12)
• Quarterly compounding (n=4)

With each step, more frequent compounding results in a higher balance at the end of the 12-year period.

Mathematical Formulas

Mathematical formulas form the backbone of solving compound interest problems. The formula to determine the balance when interest is compounded is:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]This formula calculates the final amount \(A\), where:

• \(P\) is the initial principal (the initial amount of money).
• \(r\) is the annual interest rate (as a decimal).
• \(n\) is the number of times interest is compounded per year.
• \(t\) is the time in years.

The power \(nt\) in the formula illustrates how interest gets "compounded" over time. With these variables, you fill the formula to adapt to any compounding frequency scenario.

Financial Mathematics

Financial mathematics, or quantitative finance, is a mathematical approach to solving financial problems. It blends mathematics, finance theory, and statistics to help us understand and predict financial markets and figures. Understanding compound interest is a part of this discipline, as it influences how investments and savings grow over time.

• Compound interest is a fundamental concept in finance that reveals how quickly money can grow.
• It teaches the importance of both interest rates and compounding frequency.
• Knowing how to calculate compound interest helps individuals make better financial decisions.

In our example, the use of compound interest shows how choosing different compounding frequencies impacts financial outcomes. It underscores the significance of financial mathematics in achieving personal and professional financial goals.