Question

Suppose the interest rate is 10 percent. If \(\$ 100\) is in vested at this rate today, how much will it be worth after one year? After two years? After five years? What is the value today of \(\$ 100\) paid one year from now? Paid two years from now? Paid five years from now?

Short answer

Placement of $100 today will be worth $110 after one year, $121 after two years, and $161.05 after five years. The current value of $100 that is to be paid in the future is $90.91 one year from now, $82.64 two years from now, and $62.09 five years from now.

Step-by-step solution

Find the future worth of $100 after one, two and five years respectively

Use the simple interest formula with P as $100, r as 10% (or 0.10), n as 1 (since it is compounded annually), and t as the respective years for each case. So, after one year, the amount \( A = 100 * (1 + 0.10/1)^1 = \$110 \). Similarly, after two years \( A = 100 * (1 + 0.10/1)^2 = \$121 \) and after five years \( A = 100 * (1 + 0.10/1)^5 = \$161.05 \)

Find the present value of $100 paid after one, two and five years

Using the Present Value formula, \( PV = FV / (1 + r)^t \), calculate the present value for each of the specified years. For a payment after one year, the present value is \( PV = 100 / (1 + 0.10)^1 = \$90.91 \). Similarly, for two years \( PV = 100 / (1 + 0.10)^2 = \$82.64 \) and for five years \( PV = 100 / (1 + 0.10)^5 = \$62.09 \)

Key concepts

Understanding Interest Rates

Interest rates represent the cost of borrowing money or the return on investment for deposited funds. They're typically expressed as a percentage and can vary based on economic conditions.
When you hear that an interest rate is 10%, it means that you will earn 10% of the principal amount as additional money over a given period. In terms of investment, this could mean earning extra money on top of the amount you initially put in.
Interest rates can be:

• Simple interest - calculated only on the principal amount.
• Compound interest - calculated on the principal plus any previously earned interest.

For our problem scenario, the interest rate is compounded annually at 10%, meaning that interest is calculated once a year and added to the principal, making the total amount invested at the end of the year subject to interest again in the following year.

The Role of Present Value

Present value (PV) is a financial concept used to determine the current worth of a sum of money that is to be received in the future, discounted at a particular interest rate. Understanding present value helps in comparing different cash flows at different points in time.
The formula to calculate present value is: \[ PV = \frac{FV}{(1 + r)^t} \]where:

• FV is the future value of the money.
• r is the interest rate.
• t is the time period in years.

In our exercise, we calculated the present value of $100 to be paid in one, two, and five years using a 10% interest rate. A high interest rate decreases the present value, meaning the future money is worth less today. The longer the time period, the less present value the future money holds.

Exploring Compounding

Compounding refers to the process where the investment earns returns that are then reinvested to generate additional returns over time. It is one of the most powerful financial concepts that allows your money to grow at an exponential rate.
The formula used for compounding is:\[ A = P(1 + \frac{r}{n})^{nt} \]where:

• A is the amount after time t.
• P is the principal balance.
• r is the annual interest rate (decimal).
• n is the number of times interest is compounded per year.
• t is the number of years.

In this exercise, our interest was compounded annually. This means that each year the interest earned is added to the principal, and the new total acts as the principal for the next year. That's why, after one year, \(100 at 10% becomes \)110, and continues to grow more significantly over subsequent years due to compounding.