Chapter 14, Problem 27P

To determine

Calculate the present worth.

Explanation of Solution

The cash flow in Year 0 (CF0) is -$10,000 and in Year 1 (CF1) is $2,000. From Year 2 to 4 is $5000. The time period (n) is 3.

The market rate (MR) is 20%. The inflation rate (f) is 5%.

The interest rate (if) can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{i}+\text{f}+\text{i}\times \text{f}\\ 0.2=\text{i}+0.05+\text{i}\times 0.05\\ 1.05\text{i}=0.2-0.05\\ \text{i}=\frac{0.15}{1.05}\\ =0.1429\end{array}$

The interest rate is 14.29%.

The present worth (PW) without considering the inflation can be calculated as follows:

$\begin{array}{c}\text{PW}=\text{CF0}+\text{CF1}\left(\frac{1}{{\left(1+\text{MR}\right)}^{\text{1}}}\right)+\text{CF2}\left(\frac{{\left(1+\text{MR}\right)}^{\text{n}}-1}{\text{MR}{\left(1+\text{MR}\right)}^{\text{n}}}\right)\left(\frac{1}{{\left(1+\text{MR}\right)}^{\text{1}}}\right)\\ =-\text{10,000}+\text{2,000}\left(\frac{1}{{\left(1+\text{0}\text{.2}\right)}^{\text{1}}}\right)+\text{5,000}\left(\frac{{\left(1+\text{0}\text{.2}\right)}^{\text{3}}-1}{\text{0}\text{.2}{\left(1+\text{0}\text{.2}\right)}^{\text{3}}}\right)\left(\frac{1}{{\left(1+\text{0}\text{.2}\right)}^{\text{1}}}\right)\\ =-\text{10,000}+\text{2,000}\left(0.8333\right)+\text{5,000}\left(2.1065\right)\left(0.8333\right)\\ =-\text{10,000}+1,666.6+8,776.73\\ =443.33\end{array}$

The present worth is $443.33.

The present worth (PW) with considering the inflation can be calculated as follows:

$\begin{array}{c}\text{PW}=\text{CF0}+\left(\frac{\text{CF1}}{{\left(1+\text{f}\right)}^{\text{1}}{\left(1+\text{i}\right)}^1}\right)+\left(\frac{\text{CF2}}{{\left(1+\text{f}\right)}^{\text{2}}{\left(1+\text{i}\right)}^2}\right)+\left(\frac{\text{CF3}}{{\left(1+\text{f}\right)}^{\text{3}}{\left(1+\text{i}\right)}^{\text{3}}}\right)+\left(\frac{\text{CF4}}{{\left(1+\text{f}\right)}^{\text{4}}{\left(1+\text{i}\right)}^{\text{4}}}\right)\\ =\left(\begin{array}{l}-\text{10,000}+\left(\frac{\text{2,000}}{{\left(1+\text{0}\text{.05}\right)}^{\text{1}}{\left(1+\text{0}\text{.1429}\right)}^1}\right)+\left(\frac{\text{5,0000}}{{\left(1+\text{0}\text{.05}\right)}^{\text{2}}{\left(1+\text{0}\text{.1429}\right)}^2}\right)\\ +\left(\frac{\text{5,0000}}{{\left(1+\text{0}\text{.05}\right)}^{\text{3}}{\left(1+\text{0}\text{.1429}\right)}^3}\right)+\left(\frac{\text{5,000}}{{\left(1+\text{0}\text{.05}\right)}^{\text{4}}{\left(1+\text{0}\text{.1429}\right)}^4}\right)\end{array}\right)\\ =\left(-\text{10,000}+\left(\frac{\text{2,000}}{1.2}\right)+\left(\frac{\text{5,0000}}{1.4401}\right)+\left(\frac{\text{5,0000}}{1.7282}\right)+\left(\frac{\text{5,000}}{2.0739}\right)\right)\\ =\left(-\text{10,000}+1,666.67+3,471.98+2,893.18+2,410.92\right)\\ =442.75\end{array}$

The present worth is $442.75.

The present worth can be calculated using a spreadsheet as follows:

The present worth (CV) of individual cash flow can be found using a spreadsheet function as follows:

= $B7/(1 + $B$2)^$A7

The above function is for calculating the present value of the cash flow in Year 2.