Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose you put \(\$ 100,000\) in a bank account with \(6 \%\) interest and leave it for one year. How much money will there be in the account if the interest is compounded (a) annually? (b) monthly? (c) daily? (d) hourly?

Step-by-Step Solution

Answer

(a) When compounded annually, there will be \$106,000. (b) When compounded monthly, there will be approximately \$106,182.05. (c) When compounded daily, there will be approximately \$106,183.28. (d) When compounded hourly, there will be approximately \$106,183.70.

Step 1: Calculate Annual Compounding

For annual compounding, interest is added once per year. So, \(n = 1\). Using the compound interest formula: \(A = \$100,000(1 + \frac{0.06}{1})^{1*1} = \$100,000 * 1.06 = \$106,000.\)

Step 2: Calculate Monthly Compounding

For monthly compounding, interest is added twelve times per year. So, \(n = 12\). Using the compound interest formula: \(A = \$100,000(1 + \frac{0.06}{12})^{12*1} ≈ \$106,182.05.\)

Step 3: Calculate Daily Compounding

For daily compounding, interest is added 365 times per year. So, \(n = 365\). Using the compound interest formula: \(A = \$100,000(1 + \frac{0.06}{365})^{365*1} ≈ \$106,183.28.\)

Step 4: Calculate Hourly Compounding

For hourly compounding, interest is added 24*365 times per year. So, \(n = 24*365 = 8760\). Using our formula: \(A = \$100,000(1 + \frac{0.06}{8760})^{8760*1} ≈ \$106,183.70.\)

Key Concepts

Annual Compounding

Annual compounding refers to the process of calculating interest once per year. This means that after one year, the interest earned over that period will be added to the initial principal amount. For example, if you deposit \(\\( 100,000\) into a savings account with an annual interest rate of \(6\%\), the calculation will be straightforward. You'll use the formula: \[A = P \left(1 + \frac{r}{n}\right)^{nt}\]where

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (in this case, \(\\)100,000\)).
• \(r\) is the annual interest rate (expressed as a decimal, \(0.06\)).
• \(n\) is the number of times that interest is compounded per year (1 for annual compounding).
• \(t\) is the time the money is invested or borrowed for, in years (1 in this example).

Plugging in these values, you get \(A = 100,000 \cdot (1 + \frac{0.06}{1})^{1 \times 1} = 106,000\). Therefore, after one year, your savings will grow to \$ 106,000. Annual compounding is the simplest compounding frequency, as it involves just one application of interest per year.

Monthly Compounding

Monthly compounding increases the frequency of how often interest is calculated and added to your principal amount within a year. In this method, you compound your interest twelve times a year, significantly accelerating your earnings accrued from interest. Let's consider the same scenario: \(\$ 100,000\) principal at an \(6\%\) annual rate. For monthly compounding, update the compounding frequency to 12: \[A = 100,000 \left(1 + \frac{0.06}{12}\right)^{12 \cdot 1}\]Calculations are carried out using this updated frequency (= 12), resulting in \(A \approx 106,182.05\). Compared to annual compounding, you'd have slightly more at the end of the year due to the more frequent compounding.

• Higher compounding frequency generally results in higher returns.
• A small interest rate over multiple compounding periods can significantly impact overall returns.

Monthly Compounding is especially beneficial for investments or bank accounts where interest is calculated frequently, ensuring that your savings grow more rapidly.

Daily Compounding

With daily compounding, your interest is calculated and added to your principal every single day. This is more frequent than both annual and monthly compounding, which is advantageous if you aim to maximize your return. The technical formula remains the same, but \(n\) changes to 365 to reflect daily compounding: \[A = 100,000 \left(1 + \frac{0.06}{365}\right)^{365 \cdot 1}\]This results in an accumulated amount of about \(\$ 106,183.28\) after one year.

• Daily compounding takes advantage of small, frequent additions to your balance.
• The difference in returns increases slightly with daily versus monthly frequency.

It is ideal in contexts such as savings accounts or investments that provide daily interest calculations, allowing every day’s interest to build on top of the preceding days.

Interest Calculation

Understanding interest calculation is key to deciding how to grow money effectively in a bank or investment account. The basic compound interest formula is: \[A = P \left(1 + \frac{r}{n}\right)^{nt}\]However, the frequency (\(n\)) plays a critical role in how much interest you ultimately earn. Different frequencies—like annually, monthly, or daily—result in different final amounts for the same rate, exemplifying the compounding effect. Key considerations include:

• Higher frequency of compounding generally provides greater returns.
• More sophisticated calculations might involve calculus for continuous compounding.
• Choosing the right compounding period depends on the policy of your bank or investment and your financial goals.

Knowing how to calculate interest using these methods helps you make informed decisions regarding savings and investment strategies. The strategy for maximizing your wealth involves choosing a compounding option that balances frequency with your financial needs and the policies offered by financial institutions.