Examples of Present Value and Future Value Concepts

Explore practical examples of Present Value and Future Value concepts in financial mathematics.
By Jamie

Understanding Present Value and Future Value Concepts

In financial mathematics, the concepts of Present Value (PV) and Future Value (FV) are critical for evaluating investments and understanding the time value of money. Present Value calculates the current worth of a sum that is to be received in the future, whereas Future Value determines how much a current investment will grow over time at a specified interest rate. Below are three practical examples that illustrate these concepts in real-world scenarios.

Example 1: Saving for a Future Purchase

Context

Imagine you want to buy a new car in five years, and you estimate that it will cost $30,000. You decide to invest a lump sum now to reach your goal.

To find out how much you need to invest today, we can use the Present Value formula:

[ PV = \frac{FV}{(1 + r)^n} ]

Where:

  • FV = Future Value ($30,000)
  • r = annual interest rate (let’s assume 5% or 0.05)
  • n = number of years (5)

Calculating this:

[ PV = \frac{30,000}{(1 + 0.05)^5} ]

[ PV = \frac{30,000}{1.27628} \approx 23,508.80 ]

Thus, you would need to save approximately $23,508.80 today to have $30,000 in five years at a 5% interest rate.

Notes

  • If the interest rate were higher, the present value would decrease, meaning you would need to invest less today.
  • Consider different interest rates to see how it affects your savings plan.

Example 2: Investment Growth Over Time

Context

You currently have $10,000 that you plan to invest for 10 years. You want to know how much this investment will grow if it earns an annual interest rate of 7%. This is a classic Future Value calculation.

The Future Value formula is:

[ FV = PV \times (1 + r)^n ]

Where:

  • PV = Present Value ($10,000)
  • r = annual interest rate (7% or 0.07)
  • n = number of years (10)

Calculating this:

[ FV = 10,000 \times (1 + 0.07)^{10} ]

[ FV = 10,000 \times 1.967151 \approx 19,671.51 ]

After 10 years, your investment would grow to approximately $19,671.51.

Notes

  • Increasing the interest rate or time period will significantly enhance the future value.
  • Consider the impact of compounding frequency on your investment returns.

Example 3: Loan Repayment

Context

Suppose you are considering taking out a loan of $5,000 to be paid back in 3 years with an annual interest rate of 8%. You want to determine the total amount you’ll repay in the future.

Using the Future Value formula, we can find out how much you will owe at the end of the loan period:

[ FV = PV \times (1 + r)^n ]

Where:

  • PV = Present Value ($5,000)
  • r = annual interest rate (8% or 0.08)
  • n = number of years (3)

Calculating this:

[ FV = 5,000 \times (1 + 0.08)^{3} ]

[ FV = 5,000 \times 1.259712 \approx 6,298.56 ]

At the end of 3 years, you would repay approximately $6,298.56 on your loan.

Notes

  • The total amount repaid can vary significantly based on the interest rate and loan term.
  • Understanding these calculations can help you make informed borrowing decisions.

These examples of Present Value and Future Value concepts not only illustrate how to apply these formulas but also emphasize the importance of the time value of money in financial decision-making.