A grocery store manager must decide whether to buy four rug cleaners to rent to customers. The manager estimates that the first would yield $200 a year, the second $150, the third $75, and the fourth $20. If the interest rate is 12 percent and each rug cleaner costs $500, how many should the manager buy? a. none b. two c. one d. four e. three
The answer is three, but how do I get it?
To determine the number of rug cleaners the manager should buy, we need to compare the annual revenue from each cleaner with the cost of the cleaner.
Let's calculate the annual revenue from each cleaner:
Cleaner 1: $200/year
Cleaner 2: $150/year
Cleaner 3: $75/year
Cleaner 4: $20/year
Now let's calculate the present value (PV) of the revenue generated by each cleaner using the formula:
PV = Annual Revenue / (1 + interest rate)^n
Where:
Annual Revenue = revenue from each cleaner
Interest rate = 12% (expressed as a decimal: 0.12)
n = number of years (we assume n = 1 as we are calculating present value for a single year)
PV of Cleaner 1 = $200 / (1 + 0.12)^1 = $200 / 1.12 = $178.57
PV of Cleaner 2 = $150 / (1 + 0.12)^1 = $150 / 1.12 = $133.93
PV of Cleaner 3 = $75 / (1 + 0.12)^1 = $75 / 1.12 = $66.96
PV of Cleaner 4 = $20 / (1 + 0.12)^1 = $20 / 1.12 = $17.86
Now let's analyze the cost of purchasing the cleaners:
The cost of each cleaner is $500.
Now let's calculate the present value (PV) of the cost of a cleaner:
PV of Cleaner Cost = $500 / (1 + 0.12)^1 = $500 / 1.12 = $446.43
To decide how many cleaners the manager should buy, we need to compare the present value (PV) of the revenue generated by each cleaner with the present value of the cost of a cleaner.
PV of Cleaner 1 < PV of Cleaner Cost
PV of Cleaner 2 < PV of Cleaner Cost
PV of Cleaner 3 < PV of Cleaner Cost
PV of Cleaner 4 < PV of Cleaner Cost
Based on the calculations, the present value of the revenue from all the cleaners is less than the present value of the cost of a cleaner. Therefore, the manager should not buy any cleaners, so the correct answer is (a) none.
To determine how many rug cleaners the grocery store manager should buy, we need to compare the returns from renting each cleaner to the cost of purchasing them. Here are the steps to calculate this:
1. Calculate the Present Value of the cash flows for each rug cleaner. The Present Value (PV) formula is given by:
PV = CF / (1 + r)^n
where CF is the cash flow, r is the interest rate (0.12 in this case), and n is the number of years.
For the four rug cleaners, the Present Values would be:
PV1 = $200 / (1 + 0.12)^1 = $178.57
PV2 = $150 / (1 + 0.12)^1 = $133.93
PV3 = $75 / (1 + 0.12)^1 = $66.96
PV4 = $20 / (1 + 0.12)^1 = $17.86
2. Calculate the Total Present Value by adding up the present values of all the rug cleaners:
Total PV = PV1 + PV2 + PV3 + PV4
3. Compare the Total Present Value to the total cost of purchasing all the rug cleaners. Each rug cleaner costs $500, so the total cost is $500 x 4 = $2000.
If the Total PV is greater than the total cost, it would be profitable to purchase that number of rug cleaners. However, if it is lower, it would be better not to buy any.
In this case, let's calculate the Total PV:
Total PV = $178.57 + $133.93 + $66.96 + $17.86 = $397.32
Since the Total PV is greater than the total cost of purchasing all the rug cleaners ($397.32 > $2000), the grocery store manager should buy some rug cleaners. However, the manager should only purchase as many as required to maximize the Total PV.
To determine the optimal number to purchase, compare the marginal present value of each additional rug cleaner with its cost.
In this case, the marginal present values are:
Marginal PV1 = $133.93 + $66.96 + $17.86 = $218.75
Marginal PV2 = $66.96 + $17.86 = $84.82
Marginal PV3 = $17.86
Comparing these values with the cost of $500 for each rug cleaner, we find that the first 3 rug cleaners provide a higher marginal present value ($218.75, $84.82, and $17.86, respectively) than their cost of $500. However, the fourth rug cleaner would have a marginal present value lower than its cost, as there is no cash inflow associated with it.
Therefore, the grocery store manager should buy three rug cleaners (option e) to maximize the total present value and profitability.
Popioo
I'm thinking it's a function of saving the money in an account that earns the 12% (compounded monthly) over the 4 year period.
$500 saved @ an annual rate of return of 12% (again, compounded monthly) yields $806.11 - or $306.11 of interest, which comes to $76.73 of interest returned per year - $1.73 better than the $75 per year yield on the fourth rug cleaner.
I used a simple savings calculator to help calculate my answer.