Chapter 7.S, Problem 39P

a)

Summary Introduction

To determine: The present value profit or loss of the deal.

Introduction:

Present value (PV):

It is the value that is in present form of money in contrast to some future amount. This is achieved when it is invested in compound interest.

Net present value (NPV):

The NPV is the measurement of profit that is calculated by subtracting the present values of cash outflows and cash inflows. The NPV is used as a tool to measure the profitability of investing in a project.

Answer to Problem 39P

The present value profit or loss of the deal is $77,750.

Explanation of Solution

Given information:

Initial cost = $1,000,000

Salvage cost= $50,000

Interest rate = 10%

Yearly maintenance= $75,000

Yearly dues = $300,000

Number of members = 500

$\begin{array}{c}\text{Annual dues per member}=\frac{\text{Yearly dues}}{\text{Number of members}}\\ =\frac{\text{\$300,000}}{\text{500}}\\ =\text{\$600}\end{array}$

From the present value of $1 table,

PV annuity factor @ 10% for 0 year= 1.000

PV annuity factor @ 10% for 1 year= 0.909

PV annuity factor @ 10% for 2 year= 0.826

PV annuity factor @ 10% for 3 year= 0.751

PV annuity factor @ 10% for 4 year= 0.683

PV annuity factor @ 10% for 5 year= 0.621

Calculation of cost at year 0:

The cost at year 0 is calculated summing the initial investment and yearly maintenance and expenses.

$\begin{array}{c}\text{Cost}=\text{Initial investment}+\text{Yearly maintenance}\\ =\$1,000,000+\$75,000\\ =\$1,075,000\end{array}$

Calculation revenue for year 5:

The revenue of year 5 is calculated by adding the yearly dues and the salvage cost.

$\begin{array}{c}\text{Revenue}=\text{Yearly dues}+\text{Salvage cost}\\ =\$300,000+\$50,000\\ =\$350,000\end{array}$

The cost for year 1 to year 4 is $75,000. The revenue for year 1 to year 4 is $300,000.

Formula to calculate profit for each year:

$\text{Profit}=\text{Revenues}-\text{Cost}$

Formula to calculate net present value (NPV) for each year:

$\text{Net Present value}=\text{Annuity factor}\times \text{Profit}$

Formula to calculate total profit:

$\text{Total profit}={\displaystyle \sum \text{Individual profits of all the years}}$

Formula to calculate total net present value (NPV):

$\text{Total NPV}={\displaystyle \sum \text{Individual NPV of all the years}}$

Calculation of profit for each year:

The profit is calculated by subtracting the yearly revenues with yearly cost.

Year 0 Profit:

$\begin{array}{c}\text{Profit}=\$300,000-\$1,075,000\\ =-\$775,000\end{array}$

Year 1 Profit:

$\begin{array}{c}\text{Profit}=\$300,000-\$75,000\\ =\$225,000\end{array}$

Year 2 Profit:

$\begin{array}{c}\text{Profit}=\$300,000-\$75,000\\ =\$225,000\end{array}$

Year 3 Profit:

$\begin{array}{c}\text{Profit}=\$300,000-\$75,000\\ =\$225,000\end{array}$

Year 4 Profit:

$\begin{array}{c}\text{Profit}=\$300,000-\$75,000\\ =\$225,000\end{array}$

Year 5 Profit:

$\begin{array}{c}\text{Profit}=\$350,000-\$75,000\\ =\$275,000\end{array}$

Calculation of total profit:

The total profit is calculated by summing all the individual year profits.

$\begin{array}{c}\text{Total profit}=-\$775,000+\$225,000+\$225,000+\$225,000+\$225,000+\$275,000\\ =\$400,000\end{array}$

Calculation of net present value (NPV):

The net present value for each year is calculated by multiplying the annuity factor with the respective profit.

Year 0 NPV:

$\begin{array}{c}\text{Net Present value}=1.000\times \left(-\$775,000\right)\\ =-\$775,000\end{array}$

Year 1 NPV:

$\begin{array}{c}\text{Net Present value}=0.909\times \$225,000\\ =\$204,525\end{array}$

Year 2 NPV:

$\begin{array}{c}\text{Net Present value}=0.826\times \$225,000\\ =\$185,850\end{array}$

Year 3 NPV:

$\begin{array}{c}\text{Net Present value}=0.751\times \$225,000\\ =\$168,975\end{array}$

Year 4 NPV:

$\begin{array}{c}\text{Net Present value}=0.683\times \$225,000\\ =\$153,675\end{array}$

Year 5 NPV:

$\begin{array}{c}\text{Net Present value}=0.621\times \$275,000\\ =\$139,725\end{array}$

Calculation of total net present value (NPV):

The total net present value (NPV) is calculated by summing all the individual NPV values.

$\begin{array}{c}\text{Total NPV}=-\$775,000+\$204,525+\$185,850+\$168,975+\$153,675+\$139,725\\ =\$77,750\end{array}$

Hence, the present value profit of the deal is $77,750.

b)

Summary Introduction

To determine: The special deal should be made or not made.

Introduction:

Present value (PV):

It is the value that is in present form of money in contrast to some future amount. This is achieved when it is invested in compound interest.

Net present value (NPV):

The NPV is the measurement of profit that is calculated by subtracting the present values of cash outflows and cash inflows. The NPV is used as a tool to measure the profitability of investing in a project.

Answer to Problem 39P

The special deal should be made.

Explanation of Solution

Given information:

Initial cost = $0

Salvage cost= $0

Interest rate = 10%

Yearly maintenance= $0

Yearly dues = $600

From the present value of $1 table,

PV annuity factor @ 10% for 0 year= 1.000

PV annuity factor @ 10% for 1 year= 0.909

PV annuity factor @ 10% for 2 year= 0.826

PV annuity factor @ 10% for 3 year= 0.751

PV annuity factor @ 10% for 4 year= 0.683

PV annuity factor @ 10% for 5 year= 0.621

The cost for year 0 to year 5 is $0. The revenue for year 0 to year 5 is $600.

Formula to calculate profit for each year:

$\text{Profit}=\text{Revenues}-\text{Cost}$

Formula to calculate net present value (NPV) for each year:

$\text{Net Present value}=\text{Annuity factor}\times \text{Profit}$

Formula to calculate total profit:

$\text{Total profit}={\displaystyle \sum \text{Individual profits of all the years}}$

Formula to calculate total net present value (NPV):

$\text{Total NPV}={\displaystyle \sum \text{Individual NPV of all the years}}$

Calculation of profit for each year:

The profit is calculated by subtracting the yearly revenues with yearly cost.

Year 0 Profit:

$\begin{array}{c}\text{Profit}=\$600-\$0\\ =\$600\end{array}$

Year 1 Profit:

$\begin{array}{c}\text{Profit}=\$600-\$0\\ =\$600\end{array}$

Year 2 Profit:

$\begin{array}{c}\text{Profit}=\$600-\$0\\ =\$600\end{array}$

Year 3 Profit:

$\begin{array}{c}\text{Profit}=\$600-\$0\\ =\$600\end{array}$

Year 4 Profit:

$\begin{array}{c}\text{Profit}=\$600-\$0\\ =\$600\end{array}$

Year 5 Profit:

$\begin{array}{c}\text{Profit}=\$600-\$0\\ =\$600\end{array}$

Calculation of total profit:

The total profit is calculated by summing all the individual year profits.

$\begin{array}{c}\text{Total profit}=\$600+\$600+\$600+\$600+\$600+\$600\\ =\$3,600\end{array}$

Calculation of net present value (NPV):

The net present value for each year is calculated by multiplying the annuity factor with the respective profit.

Year 0 NPV:

$\begin{array}{c}\text{Net Present value}=1.000\times \$600\\ =\$600\end{array}$

Year 1 NPV:

$\begin{array}{c}\text{Net Present value}=0.909\times \$600\\ =\$545\end{array}$

Year 2 NPV:

$\begin{array}{c}\text{Net Present value}=0.826\times \$600\\ =\$496\end{array}$

Year 3 NPV:

$\begin{array}{c}\text{Net Present value}=0.751\times \$600\\ =\$451\end{array}$

Year 4 NPV:

$\begin{array}{c}\text{Net Present value}=0.683\times \$600\\ =\$410\end{array}$

Year 5 NPV:

$\begin{array}{c}\text{Net Present value}=0.621\times \$600\\ =\$373\end{array}$

Calculation of total net present value (NPV):

The total net present value (NPV) is calculated by summing all the individual NPV values.

$\begin{array}{c}\text{Total NPV}=\$600+\$545+\$496+\$451+\$410+\$373\\ =\$2,274\end{array}$

The present value profit of the deal is $2,274. The PV profit is less than $3,000. This deal is more worth than the health club deal.

Hence, the special must be made.