Problem 59 - Precalculus

Question

How much money should you put in a savings account now that earns \(5 \%\) a year compounded daily if you want to have \(\$ 32,000\) in 18 years?

Step-by-Step Solution

Answer

Deposit approximately \( \$ 12,876.01 \) today.

Step 1: Understanding the Compound Interest Formula

Compound interest is calculated using the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), 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, and \( t \) is the number of years the money is invested or borrowed for.

Step 2: Identify Known and Unknown Variables

In this problem, we need to find the present value \( P \). The future value \( A \) is given as \( 32000 \), the annual interest rate \( r \) is \( 0.05 \) (5% as a decimal), \( t \) is 18 years, and since the interest is compounded daily, \( n \) is 365.

Step 3: Rearrange Formula to Solve for Present Value \( P \)

Rearrange the formula to solve for \( P \): \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]. This will allow us to determine the amount to deposit now.

Step 4: Substitute Known Values into the Equation

Substitute the known values into the rearranged formula: \[ P = \frac{32000}{\left(1 + \frac{0.05}{365}\right)^{365 \times 18}} \].

Step 5: Calculate the Denominator

Calculate the denominator: \( \left(1 + \frac{0.05}{365}\right)^{365 \times 18} = \left(1 + 0.0001369863\right)^{6570} \approx 2.484652787 \). This represents the compound growth factor over 18 years.

Step 6: Divide to Find the Present Value

Divide the future value by the calculated factor: \( P = \frac{32000}{2.484652787} \approx 12876.01 \).

Step 7: Conclusion

The present value, or the amount you need to deposit today, is approximately \( \$ 12,876.01 \).

Key Concepts

Present Value

The concept of present value helps us understand how much money we should invest today to reach a specific amount in the future. In the context of compound interest, the present value is crucial because it reveals the baseline investment needed when money grows over time.
When you know your target future value, you also need details about:

• Interest rate
• Time frame for the investment
• Frequency of compounding

These factors help calculate the initial amount you need to set aside today (present value) to accomplish your future financial goals. By rearranging the compound interest formula, we can solve for the present value, giving us a clear picture of what needs to be done now to achieve a set target later.
Simply put, if you want a specific amount in the future, knowing the present value tells you how much you need to start with today.

Future Value

The future value is the amount of money you want after a specific period. With compound interest, your investment grows larger each period because earnings are reinvested to earn additional returns. In our case example, the future value is given as $32,000.
Understanding the future value is vital for setting clear financial goals. It helps answer questions like "How much will my investment be worth?" or "What financial target am I aiming for?"
To determine the future value, you utilize the compound interest formula. In our step-by-step solution, we used this value as a starting point, working backwards to find out how much we needed initially in today's dollars.
Ultimately, the future value serves as a beacon, ensuring that financial plans align with desired outcomes by the end of the investment period.

Principal Amount

The principal amount, in terms of investments, is the starting sum of money that you initially deposit or invest. It forms the basis upon which interest is calculated.
Consider it as your initial 'seed' money:

• The bigger the principal, the more substantial the potential returns due to compounding.
• The principal amount is crucial since it influences how quickly and efficiently your money can grow over time.

In the exercise, the principal amount was what we set out to find. By rearranging the compound interest formula, we aimed to determine the present value, which serves as our principal amount that needs to be invested.
The principal amount is the foundation. It lays the groundwork for future wealth accumulation, making it an essential component in any investment plan.