Chapter 7, Problem 40P

(a):

To determine

Future value of the project with different years.

Explanation of Solution

Table -1 shows the cash flow.

Table -1

Year	Net cash flow (NC) (in 1,000 units)
0	2,000
1	1,200
2	-4,000
3	-3,000
4	2,000

The investment rate (i) is 30%.

The initial year future value (FW0) of the cash flow is calculated as follows:

$\begin{array}{c}\text{FW0}=\text{NC0}{\left(1+\text{i}\right)}^{\text{n}-0}\\ =2,000{\left(1+\text{0}\text{.3}\right)}^{0-0}\\ =2,000\left(1\right)\\ =2,000\end{array}$

The future worth of the project at initial year is 2,000.

The first year future value (FW1) of the cash flow is calculated as follows:

$\begin{array}{c}\text{FW1}=\text{FW0}{\left(1+\text{i}\right)}^1+\text{NCF1}\\ =2,000{\left(1+\text{0}\text{.3}\right)}^1+1,200\\ =2,000\left(1.3\right)+1,200\\ =2,600+1,200\\ =3,800\end{array}$

The future worth of the project at year 1 is 3,800.

The second year future value (FW2) of the cash flow is calculated as follows:

$\begin{array}{c}\text{FW2}=\text{FW1}{\left(1+\text{i}\right)}^1+\text{NCF2}\\ =3,800{\left(1+\text{0}\text{.3}\right)}^1+\left(-4,000\right)\\ =3,800\left(1.3\right)-4,000\\ =4,940-4,000\\ =940\end{array}$

The future worth of the project at year 2 is 940.

The third year future value (FW3) of the cash flow is calculated as follows:

$\begin{array}{c}\text{FW3}=\text{FW2}{\left(1+\text{i}\right)}^1+\text{NCF3}\\ =940{\left(1+\text{0}\text{.3}\right)}^1+\left(-3,000\right)\\ =940\left(1.\text{3}\right)-3,000\\ =1,222-3,000\\ =-1,778\end{array}$

The future worth of the project at year 3 is -1,778.

The third year future value becomes negative. Thus, the next year cash flow should use the external rate of return (E). The fourth year future value (FW4) of the cash flow is calculated as follows:

$\begin{array}{c}\text{FW4}=\text{FW3}{\left(1+\text{E}\right)}^1+\text{NCF4}\\ =-1,778\left(1+\text{E}\right)+2,000\\ =-1,778-1,778\text{E}+2,000\\ =222-1,778\text{E}\end{array}$

The future worth of the project at year 4 is 222-1,778E.

The external rate of return is calculated by equating the fourth year cash flow with zero.

$\begin{array}{c}222-1,778\text{E}=0\\ 1,778\text{E}=222\\ \text{E}=\frac{222}{1,778}\\ =0.1249\end{array}$

Thus, the external rate of return is 12.49%.

(b):

To determine

Calculation of EROR (expected rate of return).

Explanation of Solution

The borrowing rate (b) is 10%. The negative cash flows are borrowing amount. The present worth of the negative cash flow (PW) is calculated as follows:

$\begin{array}{c}\text{PW}=\frac{\text{NC2}}{{\left(1+\text{b}\right)}^2}+\frac{\text{NC3}}{{\left(1+\text{b}\right)}^3}\\ =\frac{-\text{4,000}}{{\left(1+\text{0}\text{.1}\right)}^2}+\frac{-3,000}{{\left(1+\text{0}\text{.1}\right)}^3}\\ =\frac{-\text{4,000}}{1.21}+\frac{-3,000}{1.331}\\ =3,305.79+2,253.94\\ =-5,559.73\end{array}$

The present worth of the negative cash flow (borrowing) is -$5,559.73.

The investment rate (i) is 30%. The positive cash flows are investment amount. The time period (n) is 4. The future worth of the positive cash flow (FW) is calculated as follows:

$\begin{array}{c}\text{FW}=\text{NC0}{\left(1+\text{i}\right)}^{\text{n}-0}+\text{NC1}{\left(1+\text{i}\right)}^{\text{n}-1}+\text{NC4}{\left(1+\text{i}\right)}^{\text{n}-4}\\ =\text{2,000}{\left(1+\text{0}\text{.3}\right)}^{4-0}+\text{1,200}{\left(1+\text{0}\text{.3}\right)}^{4-1}+\text{2,000}{\left(1+\text{0}\text{.3}\right)}^{4-4}\\ =\text{2,000}\left(2.8561\right)+\text{1,200}\left(2.197\right)+\text{2,000}\left(1\right)\\ =5,712.2+\text{2,636}\text{.4+2,000}\\ =\text{10,348}\text{.6}\end{array}$

The future worth of the positive cash flow (investment) is $10,348.8.

The time period (n) is 4. The value of EROR (E) is calculated as follows:

$\begin{array}{c}-\text{PW}{\left(1+\text{E}\right)}^{\text{n}}=\text{FW}\\ -\left(-\text{5,559}\text{.73}\right){\left(1+\text{E}\right)}^{\text{4}}=10,348.6\\ {\left(1+\text{E}\right)}^{\text{4}}=\frac{10,348.6}{\text{5,559}\text{.73}}\\ \left(1+\text{E}\right)={\left(1.8613\right)}^{\frac{1}{4}}\\ \text{E}=1.168-1\\ =0.168\end{array}$

The value of EROR is 16.8%.