Question

Suppose the interest rate is 10 percent. If $\mathrm{\$}100$ is invested 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 $\mathrm{\$}100$ paid one year from now? Paid two years from now? Paid five years from now?

Short answer

After one year, the investment will be worth $110. After two years, it will be worth $121. After five years, it will amount to $161.05. The present value of $100 to be received one year from now is $90.91, in two years it is $82.64, and in five years it is $62.09.

Step-by-step solution

Calculate Future Value After 1 Year

To calculate future value, use the formula $FV=PV\ast (1+r)$ where PV is the present value (initial investment), r is the interest rate in decimal. Here, PV = $100 and r = 0.10 or 10%. Therefore, $FV=\mathrm{\$}100\ast (1+0.10)=\mathrm{\$}110$

Calculate Future Value After 2 and 5 Years

The formula for future value, when compounded annually over n years, is $FV=PV\ast (1+r{)}^n$. For 2 years, n=2 and for 5 years, n=5. Thus, after 2 years, $FV=\mathrm{\$}100\ast (1+0.10{)}^2=\mathrm{\$}121$, and after 5 years, $FV=\mathrm{\$}100\ast (1+0.10{)}^5=\mathrm{\$}161.05$

Calculate Present Value 1 Year From Now

Present value can be calculated using the formula $PV=FV/(1+r)$. Here, FV = $100 and the present value of $100 paid one year from now would be $PV=\mathrm{\$}100/(1+0.10)=\mathrm{\$}90.91$

Calculate Present Value 2 and 5 Years From Now

The present value formula for n years in the future is $PV=FV/(1+r{)}^n$. Therefore, the present value of $100 to be received in 2 and 5 years respectively would be $PV=\mathrm{\$}100/(1+0.10{)}^2=\mathrm{\$}82.64$ and $PV=\mathrm{\$}100/(1+0.10{)}^5=\mathrm{\$}62.09$

Key concepts

Future Value

The future value is the amount of money an investment will grow to over a period of time at a specified interest rate. When you invest money today, it increases in value due to interest—commonly known as earning interest on your investment.
For instance, if you invest \(100 at an interest rate of 10%, you can calculate how much this will grow in one year using the formula:

• $FV=PV\times (1+r)$

Where:

• $FV$ is the future value
• $PV$ is the present value (initial investment of \)100)
• $r$ is the interest rate (0.10 for a 10% interest rate)

After one year, the future value is $(110$. This means your investment grows \)10 from the initial \(100 due to the interest.
To calculate the future value over a more extended period, such as two or five years, the formula modifies slightly to account for the compounding effect:

• $FV=PV\times (1+r{)}^n$

Here, $n$ represents the number of years. After two years, $)100$ grows to $(121$, and after five years, it becomes approximately $)161.05$. Compounding means you earn interest not only on your initial investment but also on any interest already earned.

Present Value

Present Value (PV) is the current worth of a future sum of money given a specific rate of return. This concept helps to determine how much a future amount is worth today, considering an interest rate that reflects the time value of money.
To understand present value, imagine you expect to receive $(100$ one year from now. If we apply a 10% interest rate, you can calculate the present value using the formula:

• $PV=\frac{FV}{1+r}$

Where:

• $FV$ is the future value (\)100 in this example)
• $r$ is the interest rate (0.10 for 10%)

The result, $(90.91$, signifies that $)100$ received one year from now is worth $(90.91$ today.
When calculating for two years or five years into the future, the formula is:

• $PV=\frac{FV}{(1+r{)}^n}$

Here, $n$ represents the number of years. Therefore, a $)100$ sum expected in two years is worth $(82.64$ today, and in five years, it is $)62.09$. This concept shows how the present value decreases with time as the period for receiving the future amount increases.

Compound Interest

Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. This means your investment grows faster because you earn interest on both your original investment and on the interest that has been added to it.
For example, if you invest $(100$ at 10% annual interest, after the first year, you will have $)110$. In the second year, interest is calculated not just on your $(100$, but also on the $)10$ interest earned during the first year. This compounding effect is mathematically represented as:

• $A=P\times (1+r/n{)}^{nt}$

Where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (\(100).
• $r$ is the annual interest rate (decimal).
• $n$ is the number of times interest is compounded per year.
• $t$ is the time the money is invested for, in years.

In our scenario, assuming annually compounded interest, $A$ for two years would become $)121$ and for five years $\mathrm{\$}161.05$. It's evident that compound interest can significantly increase the future value of an investment.

Time Value of Money

The time value of money is a key financial concept that explains how the value of money you have today is worth more than the same amount in the future due to its earning potential. This fundamental principle illustrates why it is better to have money now rather than later.
This concept rests on the potential earning capacity of money—also understood through the concepts of present value and future value. The difference in value is primarily due to inflation, risk, and the opportunity to earn interest.
For someone receiving $(100$ in the future, they can calculate its present value today, as shown in previous sections, and see how it diminishes over time. Similarly, investing $)100$ today at a 10% interest rate, and observing future values after one, two, or five years, demonstrates the potential to grow.
The time value of money emphasizes not only the potential of earning interest but also underscores the importance of early investment, which takes advantage of the compounding effect to maximize returns over time.