Chapter 16, Problem 21P

To determine

The benefit to cost ratio using present worth method, future worth method and annual worth method is the same.

Answer to Problem 21P

The benefit to cost ratio using present worth method, future worth method and annual worth method is $1.86$ which is same for all.

Explanation of Solution

Given:

The annual benefit is $\$500000$.

The annual disbenefit is $\$25000$.

The first cost is $\$1500000$.

The annual cost is $\$180000$.

The interest rate is $4\%$.

The time period is $40$ years.

Concept used:

Write the expression for present worth of benefits.

$P{W}_B=\left(A_B-A_{DB}\right)\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]$ ...... (I)

Here, the present worth is PW, annual benefit is $A_B$, annual disbenefits is $A_{DB}$, interest rate is $i$ and time period is $n$.

Write the expression for present worth of costs.

$P{W}_C=F_C+A_C\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]$ ...... (II)

Here, first cost is $F_C$ and annual cost is $A_C$.

Write the expression for benefit to cost ratio using the present worth method.

$\text{Benefit cost ratio}=\left(\frac{P{W}_B}{P{W}_C}\right)$ ...... (III)

Write the expression for future worth of benefits.

$F{W}_B=\left(A_B-A_{DB}\right)\left[\frac{{\left(1+i\right)}^n-1}{i}\right]$ ...... (IV)

Here, the future worth is FW.

Write the expression for future worth of costs.

$F{W}_C=F_C{\left(1+i\right)}^n+A_C\left[\frac{{\left(1+i\right)}^n-1}{i}\right]$ ...... (V)

Write the expression for benefit to cost ratio using the future worth method.

$\text{Benefit cost ratio}=\left(\frac{F{W}_B}{F{W}_C}\right)$ ...... (VI)

Write the expression for annual worth of benefits.

$A{W}_B=A_B-A_{DB}$ ...... (VII)

Here, the annual worth is AW.

Write the expression for annual worth of costs.

$A{W}_C=F_C\left[\frac{i{\left(1+i\right)}^n}{{\left(1+i\right)}^n-1}\right]+A_C$ ...... (VIII)

Write the expression for benefit to cost ratio using the annual worth method.

$\text{Benefit cost ratio}=\left(\frac{A{W}_B}{A{W}_C}\right)$ ...... (IX)

Calculations:

Calculate the present worth of benefits.

Substitute $\$500000$ for $A_B$, $\$25000$ for $A_{DB}$, $4\%$ for $i$ and $40$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_B=\left(\$500000-\$25000\right)\left[\frac{{\left(1+0.04\right)}^{40}-1}{0.04{\left(1+0.04\right)}^{40}}\right]\\ =\$475000\left[\frac{{\left(1.04\right)}^{40}-1}{0.04{\left(1.04\right)}^{40}}\right]\\ =\$475000\left(\frac{3.80}{0.192}\right)\\ =\$9399043\end{array}$

Calculate the present worth of costs.

Substitute $\$1500000$ for $F_C$, $\$180000$ for $A_C$, $4\%$ for $i$ and $40$ for $n$ in Equation (II).

$\begin{array}{c}P{W}_C=\$1500000+\$180000\left[\frac{{\left(1+0.04\right)}^{40}-1}{0.04{\left(1+0.04\right)}^{40}}\right]\\ =\$1500000+\$180000\left[\frac{{\left(1.04\right)}^{40}-1}{0.04{\left(1.04\right)}^{40}}\right]\\ =\$1500000+\$180000\left(\frac{3.80}{0.192}\right)\\ =\$5062500\end{array}$

Calculate the benefit to cost ratio using $PW$ method.

Substitute $\$9399043$ for $P{W}_B$, $\$5062500$ for $P{W}_C$ in Equation (III).

$\begin{array}{c}\text{Benefit cost ratio}=\left(\frac{\$9399043}{\$5062500}\right)\\ =1.86\end{array}$

The benefit to cost ratio using $PW$ method is $1.86$.

Calculate the future worth of benefits.

Substitute $\$500000$ for $A_B$, $\$25000$ for $A_{DB}$, $4\%$ for $i$ and $40$ for $n$ in Equation (VI).

$\begin{array}{c}F{W}_B=\left(\$500000-\$25000\right)\left[\frac{{\left(1+0.04\right)}^{40}-1}{0.04}\right]\\ =\left(\$475000\right)\left[\frac{{\left(1.04\right)}^{40}-1}{0.04}\right]\\ =\left(\$475000\right)\left(\frac{3.80}{0.04}\right)\\ =\$45137120\end{array}$

Calculate the future worth of costs.

Substitute $\$1500000$ for $F_C$, $\$180000$ for $A_C$, $4\%$ for $i$ and $40$ for $n$ in Equation (V).

$\begin{array}{c}F{W}_C=\$1500000{\left(1+0.04\right)}^{40}+\$180000\left[\frac{{\left(1+0.04\right)}^{40}-1}{0.04}\right]\\ =\$1500000\left(4.80\right)+\$180000\left(\frac{3.80}{0.04}\right)\\ =\$7200000+\$17100000\\ =\$24300000\end{array}$

Calculate the benefit to cost ratio using $FW$ method.

Substitute $\$45137120$ for $F{W}_B$, $\$24300000$ for $F{W}_C$ in Equation (VI).

$\begin{array}{c}\text{Benefit cost ratio}=\left(\frac{\$45137120}{\$24300000}\right)\\ =1.86\end{array}$

The benefit to cost ratio using $FW$ method is $1.86$.

Calculate the annual worth of benefits.

Substitute $\$500000$ for $A_B$ and $\$25000$ for $A_{DB}$ in Equation (VI).

$\begin{array}{c}A{W}_B=\$500000-\$25000\\ =\$475000\end{array}$

Calculate the annual worth of costs.

Substitute $\$1500000$ for $F_C$, $\$180000$ for $A_C$, $4\%$ for $i$ and $40$ for $n$ in Equation (V).

$\begin{array}{c}A{W}_C=\$1500000\left[\frac{0.04{\left(1+0.04\right)}^{40}}{{\left(1+0.04\right)}^{40}-1}\right]+\$180000\\ =\$1500000\left(\frac{0.192}{3.80}\right)+\$180000\\ =\$75785.23+\$180000\\ =\$255785.23\end{array}$

Calculate the benefit to cost ratio using $AW$ method.

Substitute $\$475000$ for $F{W}_B$, $\$255785.23$ for $F{W}_C$ in Equation (IX).

$\begin{array}{c}\text{Benefit cost ratio}=\left(\frac{\$475000}{\$255785.23}\right)\\ =1.86\end{array}$

The benefit to cost ratio using $AW$ method is $1.86$.

Conclusion:

Thus, the benefit to cost ratio using present worth method, future worth method and annual worth method is $1.86$ which is same for all.