Question

How much should be deposited in an account paying \(7.8 \%\) interest compounded monthly in order to have a balance of \(\$ 21,154.03\) four years from now?

Short answer

To have a balance of \$21,154.03 four years from now in an account that pays 7.8% interest compounded monthly, \$16,858.21 must be deposited.

Step-by-step solution

Understand and Setup the Compound Interest Formula

First, let's comprehend the variables in the compound interest formula \(A = P (1 + \frac{r}{n})^{nt}\). In this problem, we know the following: A (the future value of the investment) which is \$21,154.03, r (annual interest rate in decimal form) which is 0.078, t (time in years the money is invested) which is 4 years and n (number of times that interest is compounded per year) which is 12 times (monthly). We are trying to find P, the initial amount deposited.

Substitute the Known Values

Next, we substitute the known values into the formula and try to solve for the unknown, P. Doing so, we have \(\$21,154.03 = P (1 + \frac{0.078}{12})^{12*4}\).

Solve For P

Now, we isolate P. This can be done by dividing both sides of the equation by \( (1 + \frac{0.078}{12})^{12*4}\). After performing the division, we find the value of P, which will give us the amount to be deposited.

Key concepts

Future Value Calculation

Understanding the future value calculation is essential when planning financial goals like saving for retirement or your children's education. In our example, the future value (\f\(A\f\)) is the amount we want to have in the future - \f\(21,154.03\f\).

The future value calculation involves the compound interest formula, which can be written as \f\(A = P \times (1 + \frac{r}{n})^{nt}\f\). Knowing the desired amount (\f\(A\f\)), the annual interest rate (\f\(r\f\)), the number of times interest is compounded per year (\f\(n\f\)), and the period (\f\(t\f\)) in years, we can work out the initial deposit (\f\(P\f\)) needed to achieve that future sum. In this scenario, we plug the given values into the formula and manipulate it to find \f\(P\f\).

This calculation shows us the power of compound interest and helps establish a clear savings target. It's an invaluable tool for financial planning, ensuring we meet our future financial requirements.

Financial Mathematics

Financial Mathematics encompasses a range of tools and formulas that help individuals and businesses make informed decisions about investments, loans, and savings. The compound interest formula is a cornerstone of this field, reflecting how investments grow over time.

The formula's variables, including the principal amount (\f\(P\f\)), the interest rate (\f\(r\f\)), the compounding frequency (\f\(n\f\)), and the investment period (\f\(t\f\)), must be precisely understood and applied appropriately to calculate either the future value of an investment or the initial sum required to reach a specific goal. In the context of our exercise, understanding these principles allows us to back out the necessary initial deposit that would grow to the desired future value given the specified interest rate and compounding frequency. Financial mathematics enables us to plan for the future by understanding the implications of these variables on our money's growth.

Time Value of Money

The time value of money is a fundamental concept in finance that reflects the idea that money available today is worth more than the same amount in the future due to its potential earning capacity. This core principle underpins the concept of compound interest, where interest is earned on both the initial principle and the accumulated interest from previous periods.

Applying this concept, the formula \f\(A = P \times (1 + \frac{r}{n})^{nt}\f\) gives us the future value of money when the interest is compounded. This teaches us that saving and investing early can dramatically impact the value of our funds down the line. In our textbook problem, we're utilizing the time value of money to determine what amount we'd need to invest today to reach a goal in a specific timeframe. With this knowledge, we can plan our finances to ensure that we harness the full potential of our money over time.