Chapter 5, Problem 20P

Summary Introduction

To calculate: The best stream for present value of cash flow

Introduction:

Future Value of Cash Flow:

If single cash flow put in an investment today which pays us compound interest how much does it grow over the period of time is known as future value of cash flow.

Explanation of Solution

Calculation of present value of cash flow stream at 7% compounding rate

	Contract 1		Contract 2		Contract 3
Year	Cash in flow	Present Value	Cash in flow	Present Value	Cash in flow	Present Value
1	3,000,000	2,803,738	2,000,000	1,869,159	7,000,000	6,542,056
2	3,000,000	2,620,316	3,000,000	2,620,316	1,000,000	873,439
3	3,000,000	2,448,894	4,500,000	3,673,340	1,000,000	816,298
4	3,000,000	2,288,686	5,500,000	4,195,924	1,000,000	762,895
	Total PV	10,161,634		12,358,739		8,994,688

Table (1)

Working Note for present value:

Present value for year 1 and contract 1

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$3,000,000}{{\left(1+0.07\right)}^1}\\ =\frac{\$3,000,000}{1.07}\\ =\$2,803,738.3\end{array}$

Present value for year 2 and contract 1

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$3,000,000}{{\left(1+0.07\right)}^2}\\ =\$2,620,316.1\end{array}$

Present value for year 3 and contract 1

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$3,000,000}{{\left(1+0.07\right)}^3}\\ =\$2,448,893.6\end{array}$

Present value for year 4 and contract 1

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$3,000,000}{{\left(1+0.07\right)}^4}\\ =\$2,288,685.6\end{array}$

Present value for year 1 and contract 2

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$2,000,000}{{\left(1+0.07\right)}^1}\\ =\$1,869,158.8\end{array}$

Present value for year 2 and contract 2

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$3,000,000}{{\left(1+0.07\right)}^2}\\ =\$2,620,316.1\end{array}$

Formula to calculate present value for year 3 and contract 2

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$4,500,000}{{\left(1+0.07\right)}^3}\\ =\$3,673,340.4\end{array}$

Formula to calculate present value for year 4 and contract 2

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$5,500,000}{{\left(1+0.07\right)}^4}\\ =\$4,195,923.7\end{array}$

Formula to calculate present value for year 1 and contract 3

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$7,000,000}{{\left(1+0.07\right)}^1}\\ =\$6,542,056.07\end{array}$

Formula to calculate present value for year 2 and contract 3

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$1,000,000}{{\left(1+0.07\right)}^2}\\ =\$873,438.7\end{array}$

Formula to calculate present value for year 3 and contract 3

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$1,000,000}{{\left(1+0.07\right)}^3}\\ =\$816,297.9\end{array}$

Formula to calculate present value for year 4 and contract 3

$\begin{array}{c}\text{Present}\text{value}\text{factor}=\frac{\text{Cash}\text{Flow}}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$1,000,000}{{\left(1+0.07\right)}^4}\\ =\$762,895.2\end{array}$

Conclusion

So, the contract 2 is the best option as total present value is highest for contract 2.