Question

What amount must be invested at \(4 \%\) interest compounded daily to have \(\$ 15,000\) in 3 years?

Short answer

The required principal is approximately \( \$13,306.21.

Step-by-step solution

Understand the Compound Interest Formula

The compound interest formula is \[ A = P \times \bigg(1 + \frac{r}{n}\bigg)^{n \times t} \] where:- \( A \) is the future value of the investment/loan, including interest.- \( P \) is the principal investment amount (the initial deposit or loan amount).- \( r \) is the annual interest rate (decimal).- \( n \) is the number of times that interest is compounded per year.- \( t \) is the number of years the money is invested or borrowed for.

Identify the Given Values

From the problem statement, we have:- \( A = 15,000 \)- The annual interest rate \( r = 4\% = 0.04 \)- The investment period \( t = 3 \) years- Since interest is compounded daily, \( n = 365 \).

Solve for the Principal \( P \)

We need to rearrange the formula to solve for \( P \):\[ P = \frac{A}{\bigg(1 + \frac{r}{n}\bigg)^{n \times t}} \]Substituting the known values:\[ P = \frac{15,000}{\bigg(1 + \frac{0.04}{365}\bigg)^{365 \times 3}} \]

Calculate the Compound Interest Factor

First calculate the compound interest factor:\[ \bigg(1 + \frac{0.04}{365}\bigg)^{365 \times 3} \]\( \frac{0.04}{365} \rightarrow 0.04 / 365 = 0.00010958904 \)\[ 1 + 0.00010958904 = 1.00010958904 \]\[ 1.00010958904^{1095} \approx 1.12749685 \]

Calculate the Principal \( P \)

Now, substitute the compound interest factor back into the rearranged formula:\[ P = \frac{15,000}{1.12749685} \approx 13,306.21 \]

Key concepts

compound interest formula

To understand how money grows in an investment or loan, we need to grasp the compound interest formula. This formula considers interest on both the original principal and the accumulated interest from previous periods. The formula is:\[ A = P \times \bigg(1 + \frac{r}{n}\bigg)^{n \times t} \]Here's a breakdown:

• A: Future value of the investment or loan, including interest.
• P: Principal amount (initial deposit or loan).
• r: Annual interest rate in decimal form (for example, 4% becomes 0.04).
• n: Number of times interest is compounded per year.
• t: Number of years the money is invested or borrowed.

Understanding this formula is crucial because it not only helps calculate the future value of your investment but also shows how different variables like the interest rate and compounding frequency impact growth.

investment calculations

When dealing with investments, it's important to input accurate values into the compound interest formula for precise results. Let's break down the given problem. We want to know how much to invest now to get \(15,000 in 3 years with a 4% interest rate compounded daily. Here's what we know:

• A: \)15,000 (future value).
• r: 4%, which we convert to the decimal 0.04.
• t: 3 years.
• n: 365 times a year, because the interest is compounded daily.

With the given values, our formula setup is almost complete. By rearranging the compound interest formula to solve for the principal \( P \), we help ensure we are making well-informed investment decisions.

principal amount determination

Finding out how much money you need to invest today to reach a specific goal requires determining the principal amount. Here’s how you do it from our problem:First, rearrange the compound interest formula to solve for \( P \):\[ P = \frac{A}{\bigg(1 + \frac{r}{n}\bigg)^{n \times t}} \]Substitute the given values:\[ P = \frac{15,000}{\bigg(1 + \frac{0.04}{365}\bigg)^{365 \times 3}} \]Then, calculate the base growth factor inside the parentheses:\[ 1 + \frac{0.04}{365} = 1.00010958904 \]Now raise this number to the power of the total compounding periods \( (365 \times 3) = 1095 \):\[ 1.00010958904^{1095} \rightarrow 1.12749685 \]Finally, use this to find \( P \):\[ P = \frac{15,000}{1.12749685} \rightarrow 13,306.21 \]This calculation shows how much you need to invest initially, which is $13,306.21.

daily compounding interest

Daily compounding has a powerful effect on how much your initial investment grows. Compounding means you earn interest on your interest. When it happens every day, even small amounts of interest can add up significantly.

Interest is calculated and added to your principal every day, so the principal grows slightly each day.

Each subsequent interest calculation is based on this slightly larger principal.

For a 4% annual interest rate compounded daily, each day's rate is:\[ \frac{0.04}{365} = 0.00010958904 \]While this seems tiny, because it's applied daily, it leads to a noticeable increase over time. Compounding 365 times within a year can yield a higher effective interest rate than less frequent compounding methods.By understanding daily compounding interest, you can see how even small changes in how often interest is compounded can greatly impact your investment’s outcome over time.