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22. To calculate the sensitivity of the NPV to changes in the price of the new club, we need to change the
price of the new club. We will choose $855, but the choice is irrelevant as the sensitivity will be the
same no matter what price we choose.
We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs
Sales
New clubs $855 60,000 = $51,300,000
For the variable costs, we must include the units gained or lost from the existing clubs. Note that the
variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will
save these variable costs, which is an inflow. So:
Variable Costs
New clubs –$405 60,000 = –$24,300,000
The pro forma income statement will be:
Sales $44,770,000
Variable costs 20,500,000
Using the bottom-up OCF calculation, we get:
OCF = NI + Depreciation
And the NPV is:
So, the sensitivity of the NPV to changes in the price of the new club is:
To calculate the sensitivity of the NPV to changes in the quantity sold of the new club, we need to
We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs
Sales
New clubs $845 61,000 = $51,545,000
For the variable costs, we must include the units gained or lost from the existing clubs. Note that the
variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will
save these variable costs, which is an inflow. So:
Variable costs
New clubs –$405 61,000 = –$24,705,000
The pro forma income statement will be:
Sales $45,015,000
Variable costs 20,905,000
Using the bottom up OCF calculation, we get:
OCF = NI + Depreciation
The NPV at this quantity is:
So, the sensitivity of the NPV to changes in the quantity sold is:
23. a. First we need to determine the total additional cost of the hybrid. The hybrid costs more to
purchase and more each year, so the total additional cost is:
Next, we need to determine the cost per mile for each vehicle. The cost per mile is the cost per
gallon of gasoline divided by the miles per gallon, or:
So, the savings per mile driven for the hybrid will be:
We can now determine the break-even point by dividing the total additional cost by the savings
per mile, which is:
So, the miles you would need to drive per year is the total break-even miles divided by the
number of years of ownership, or:
b. First, we need to determine the total miles driven over the life of either vehicle, which will be:
Since we know the total additional cost of the hybrid from part a, we can determine the
necessary savings per mile to make the hybrid financially attractive. The necessary cost savings
per mile will be:
Now we can find the price per gallon for the miles driven. If we let P be the price per gallon,
the necessary price per gallon will be:
P/30 – P/39 = $.08444
c. To find the number of miles it is necessary to drive, we need the present value of the costs and
savings to be equal to zero. If we let MDPY equal the miles driven per year, the break-even
equation for the hybrid car is:
The savings per mile driven, $.021923, is the same as we calculated in part a. Solving this
equation for the number of miles driven per year, we find:
To find the cost per gallon of gasoline necessary to make the hybrid break even in a financial
sense, if we let CSPG equal the cost savings per gallon of gas, the cost equation is:
Solving this equation for the cost savings per gallon of gas necessary for the hybrid to break
even from a financial sense, we find:
Now we can find the price per gallon for the miles driven. If we let P be the price per gallon,
the necessary price per gallon will be:
d. The implicit assumption in the previous analysis is that each car depreciates by the same dollar
amount and has identical resale value.
24. a. The cash flow per plane is the initial cost divided by the break-even number of planes, or:
b. In this case the cash flows are a perpetuity. Since we know the cash flow per plane, we need to
determine the annual cash flow necessary to deliver a 20 percent return. Using the perpetuity
equation, we find:
PV = C/R
This is the total cash flow, so the number of planes that must be sold is the total cash flow
divided by the cash flow per plane, or:
c. In this case the cash flows are an annuity. Since we know the cash flow per plane, we need to
determine the annual cash flow necessary to deliver a 20 percent return. Using the present value
of an annuity equation, we find:
PV = C(PVIFA20%,10)
This is the total cash flow, so the number of planes that must be sold is the total cash flow
divided by the cash flow per plane, or:
Challenge
25. a. The tax shield definition of OCF is:
Rearranging and solving for Q, we find:
(OCF – TCD)/(1 – TC) = (P – v)Q – FC
Q = {FC + [(OCF – TCD)/(1 – TC)]}/(P – v)
b. The cash break-even is:
And the accounting break-even is:
The financial break-even is the point at which the NPV is zero, so:
So:
QF = [FC + (OCF – TCD)/(1 – TC)]/(P – v)
c. At the accounting break-even point, the net income is zero. Thus using the bottom-up definition
of OCF:
OCF = NI + D
We can see that OCF must be equal to depreciation. So, the accounting break-even is:
The tax rate has cancelled out in this case.
26. The DOL is expressed as:
The OCF for the initial period and the first period is:
The difference between these two cash flows is:
Dividing both sides by the initial OCF we get:
Rearranging we get:
27. a. Using the tax shield approach, the OCF is:
And the NPV is:
b. In the worst-case, the OCF is:
And the worst-case NPV is:
The best-case OCF is:
And the best-case NPV is:
28. To calculate the sensitivity to changes in quantity sold, we will choose a quantity of 31,000 tons. The
OCF at this level of sales is:
The sensitivity of changes in the OCF to quantity sold is:
The NPV at this level of sales is:
And the sensitivity of NPV to changes in the quantity sold is:
You wouldn’t want the quantity to fall below the point where the NPV is zero. We know the NPV
For a zero NPV, we need to decrease sales by 6,436 units, so the minimum quantity is:
29. At the cash break-even, the OCF is zero. Setting the tax shield equation equal to zero and solving for
the quantity, we get:
The accounting break-even is:
From Problem 28, we know the financial break-even is 23,564 tons.
30. Using the tax shield approach to calculate the OCF, the DOL is:
Thus a 1 percent rise in Q leads to a 1.35 percent rise in OCF. If Q rises to 31,000, then
the percentage change in quantity is:
So, the percentage change in OCF is:
From Problem 28:
In general, if Q rises by 1,000 tons, OCF rises by 4.51%.
