Question

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

Short answer

$100 invested today will be worth $110 after one year, $121 after two years, and $161.05 after five years.

The value of $100 today paid after one year will be $90.91, after two years will be 82.64, and after five years will be $62.09.

Step-by-step solution

Calculating the future value

The future value is the amount an individual will get if invested at a specific interest rate and for a given time.

The future value of the investment is calculated below:

$$\begin{gathered} P=\$100 \\ r=\frac{10}{100} \\ =0.1 \\ n=1 \\ FV=P{\left(1+r\right)}^n \\ =100{\left(1+0.1\right)}^1 \\ =100\times 1.1 \\ =\$110 \end{gathered}$$

The future value for $100 after one year will be $110.

Similarly, after two years and five years will be:

$$\begin{gathered} n=2; \\ FV=P{\left(1+r\right)}^n \\ =100{\left(1+0.1\right)}^2 \\ =100\times 1.21 \\ =\$121 \\ n=5; \\ FV=P{\left(1+r\right)}^n \\ =100\left(1+0.1\right)5 \\ =100\times 1.61051 \\ =\$161.05 \end{gathered}$$

The future value for $100 after two years will be $121, and after five years will be $161.05.

Calculating the present value

The present value states the worth of an investment today but received at a later time.

The present value of the investment is calculated below:

$$\begin{gathered} FV=\$100 \\ r=\frac{10}{100} \\ PV=FV{\left(1+r\right)}^{-n} \\ =100{\left(1+0.1\right)}^{-1} \\ =100\times 0.9091 \\ =\$90.91 \end{gathered}$$

The present value for $100 paid after one year will be $90.91.

Similarly, after two years and five years will be:

$$\begin{gathered} n=2; \\ PV=FV{\left(1+r\right)}^{-n} \\ =100{\left(1+0.1\right)}^{-2} \\ =100\times 0.8264 \\ =\$82.64 \\ n=5; \\ PV=FV{\left(1+r\right)}^{-n} \\ =100{\left(1+0.1\right)}^{-5} \\ =100\times 0.6209 \\ =\$62.09 \end{gathered}$$

The present value for $100 paid after two years will be $82.64, and paid after five years will be $62.09.