Question

How much is \(\$ 1\) million to be delivered 20 years in the future worth today if the interest rate is 20 percent?

Short answer

The present value is approximately $26,055.88.

Step-by-step solution

Understand the Present Value Concept

Present value (PV) is the current worth of a future sum of money given a specified rate of return. We use a discount rate to determine how much the future value is worth today.

Identify the Components of the Formula

For this problem, we have: - Future Value (FV) = $1,000,000 - Interest Rate (r) = 20% or 0.2 - Number of Periods (n) = 20 years We will use these components in the present value formula.

Apply the Present Value Formula

The present value formula is given by:\[ PV = \frac{FV}{{(1 + r)^n}} \] Substitute the known values into the formula to calculate the present value.

Calculate the Present Value

Plug the values into the formula:\[ PV = \frac{1000000}{{(1 + 0.2)^{20}}} \] Calculate \((1 + 0.2)^{20}\) first.

Solve the Exponential Part

Calculate \((1.2)^{20}\) which equals approximately 38.3376.

Final Calculation of Present Value

Use this to find the present value:\[ PV = \frac{1000000}{38.3376} \approx 26055.88 \] The present value is roughly $26,055.88.

Key concepts

Future Value

The future value refers to the amount of money an investment will grow to over a specified period at a given interest rate. Think of the future value as the potential value a current sum of money can reach if invested wisely. In the context of the exercise, the future value is the $1 million that will be received 20 years from now. It's important to remember that future value calculations help us determine how much a present investment can be worth in the long term. This concept is essential when planning for financial goals, such as retirement or purchasing a home in the future.

Interest Rate

The interest rate is a key component in calculating both the present and future values of money. It represents the percentage increase in the value of an investment over time. In our problem, the interest rate is 20%, which is significant because it determines how quickly money grows. The higher the interest rate, the greater the future value of an investment. However, for present value calculations, a higher interest rate means a lower present value, as we discount future sums more aggressively. Consider:

• Interest as a way of compensating for the delay in receiving money.
• How this affects investment decisions and cash flow planning.

Discount Rate

The discount rate is used in the present value formula to determine how much a future sum of money is worth today. It reflects both the time value of money and inflation factors. The discount rate is essentially the interest rate, as seen in the formula we used in the exercise. It shows the rate at which money decreases in value over time due to opportunity cost and potential inflation. With this concept, you can understand how much you need to save today to reach a specific financial goal later. In our example, the discount rate of 20% means we expect high returns, which shrink the present value of the future $1 million significantly.

Time Value of Money

The time value of money is a fundamental financial concept that states a sum of money has greater potential value now than the same sum in the future. This principle highlights why receiving money sooner is more beneficial, as it can be invested to earn interest or returns. Understanding the time value of money helps in assessing the present value of future cash inflows or outflows. For instance, in our problem, receiving $1 million today is more advantageous because you can invest this sum now to potentially grow it beyond its future value. This concept is crucial when making financial choices, such as comparing investment projects or planning long-term goals. It encourages maximizing financial benefits by taking into account both present and future funding opportunities.