Chapter 5, Problem 18P

a.

Summary Introduction

To calculate: Present value of cash flow stream at 8% discounting rate.

Present value of cash flow: It is also called as discounted value, it defines that amount of money that is invested at a given rate of interest, which will further increase the amount of future cash flow at that particular time in future.

Explanation of Solution

Solution:

Calculation of present value of cash flow stream at 8% discounting rate

Year	Discounting Rate	Cash Flows		Present value of cash flows
		Stream A	Stream B	Stream A $\text{B}\times \text{C}$	Stream B $\text{B}\times \text{D}$
A	B	C	D	E	F
0	1.000000	0	0	0	0
1	0.92592	100	300	$92.592	$277.77
2	0.85733	400	400	$342.932	$342.932
3	0.79383	400	400	$317.532	$317.532
4	0.73502	400	400	$294.008	$294.008
5	0.68058	300	100	$204.174	$68.058
		Present value for Stream A and Stream B		$1248.23	$1300.306

Table (1)

Working Note to calculate discounting rate

Formula to calculate discounting rate for year 1

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.08\right)}^1}\\ =0.92592\end{array}$

Formula to calculate discounting rate for year 2

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.08\right)}^2}\\ =0.85733\end{array}$

Formula to calculate discounting rate for year 3

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.08\right)}^3}\\ =0.79383\end{array}$

Formula to calculate discounting rate for year 4

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.08\right)}^4}\\ =0.73502\end{array}$

Formula to calculate discounting rate for year 5

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.08\right)}^5}\\ =0.68058\end{array}$

Conclusion

Present value for stream A and stream B is $1248.23 and $1300.306 respectively.

b.

Summary Introduction

To calculate: Present value of cash flow stream at 0% discounting rate.

Explanation of Solution

Solution:

Calculation of present value of cash flow stream at 0% discounting rate

Year	Discounting Rate	Cash Flows		Present value of cash flows
		Stream A	Stream B	Stream A $\text{B}\times \text{C}$	Stream B $\text{B}\times \text{D}$
A	B	C	D	E	F
0	1.000000	0	0	0	0
1	1.000000	100	300	$100	$300
2	1.000000	400	400	$400	$400
3	1.000000	400	400	$400	$400
4	1.000000	400	400	$400	$400
5	1.000000	300	100	$300	$100
		Present value for Stream A and Stream B		$1600	$1600

Table (2)

Working Note to calculate discounting rate

Formula to calculate discounting rate for year 1

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.00\right)}^1}\\ =1.0000\end{array}$

Formula to calculate discounting rate for year 2

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.00\right)}^2}\\ =1.0000\end{array}$

Formula to calculate discounting rate for year 3

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.00\right)}^3}\\ =1.0000\end{array}$

Formula to calculate discounting rate for year 4

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.00\right)}^4}\\ =1.0000\end{array}$

Formula to calculate discounting rate for year 5

$\begin{array}{c}\text{Discount}\text{Rate}=\frac{1}{{\left(1+\text{Interest}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{1}{{\left(1+0.00\right)}^5}\\ =1.0000\end{array}$

Conclusion

Present value for stream A and stream B is $1600 and $1600 respectively.