Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Find the present value of amount \(A\) invested at rate \(r\) for \(t\) years, compounded \(n\) times per vear.\(A=\$ 50,000, r=7 \%, t=10\) years, \(n=12\)

Step-by-Step Solution

Answer

The present value of the investment is approximately \$25,518.37.

Step 1: Interpret the given values

Here, \(A=\$50,000\) is the future value of the investment, \(r=7\%\) is the annual interest rate, \(n=12\) is the number of times interest is compounded per year, and \(t=10\) years is the time duration of the investment.

Step 2: Convert the percentage to a decimal

The annual interest rate is given as 7%. To use this in our formula, we need to convert it into decimal form by dividing by 100. So, \(r = 7/100 = 0.07\).

Step 3: Plug the values into the formula

Now, we plug all the given values into the formula to calculate present value. The formula is \[PV = \$50,000 / (1+0.07/12)^{(12*10)}\].

Step 4: Solve the equation

After performing the operations in the equation, the present value (PV) is calculated to be approximately \$25,518.37.

Key Concepts

Time Value of Money

The concept of the time value of money is central to finance and the foundation for understanding how money grows over time. Simply put, it's the idea that a certain amount of money today is worth more than the same amount of money in the future. This is because the money you have right now can be invested and earn a return, making it grow over time.

For example, if you have \(1,000 today and can earn a 5% annual return on your investment, in one year you would have \)1,050. On the other hand, if someone promises to give you \(1,000 a year from now, it would be worth less than \)1,000 today because you've lost the opportunity to earn that year's worth of interest. Understanding this concept is crucial for making smart financial decisions regarding investments, loans, and savings.

Compound Interest

Compound interest is what makes invested money grow at a faster rate than simple interest. It is interest calculated on the initial principal as well as the accumulated interest from previous periods on a deposit or loan. This means that each time interest is calculated and added to the account, the larger balance then earns more interest the next period, creating a snowball effect.

In our exercise example, interest is compounded 12 times per year, or monthly. The formula to find the final amount including compound interest is: \[A = P(1 + \frac{r}{n})^{nt}\]where P is the principal amount (original sum), r is the annual interest rate in decimal, n is the number of times interest is compounded per year, and t is the time the money is invested for in years.

One important note is that the more frequently interest is compounded, the more total interest will be paid or earned. Therefore, compounding has a significant impact on the growth of investments.

Present Value Formula

The present value formula is used to determine the current worth of a future amount of money or stream of cash flows given a specified rate of return. Future cash flows are discounted at the discount rate, and the higher the discount rate, the lower the present value of the future cash flows.

Based on the compound interest principle, the formula for calculating present value (PV) is:
\[PV = \frac{FV}{(1+\frac{r}{n})^{nt}}\]where FV is the future value or amount of money you're aiming to have in the future, r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years you expect to invest your money.

In our exercise, to find out how much you'd need to invest today to reach a goal of \(50,000 in 10 years at a 7% interest rate compounded monthly, you'd use the present value formula as shown in the steps to arrive at the answer of roughly \)25,518.37. This calculation tells us that if you invest about \(25,518.37 today with the given conditions, you would have \)50,000 in 10 years.