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TABLE 5.4
Item \$ Thousands
Income statement, break-even
sales volume Revenues 13,067
Variable costs 10,617 (81.25 percent of sales)
Fixed costs 2,000
Depreciation 450
Pretax profit 0
Taxes 0
Profit after tax 0
Project Analysis 475

FIGURE 5.1
Accounting break-even
analysis Revenue

Costs and revenue, \$ million
Total costs

13.067
Variable costs

2.45
Fixed costs
13.067

Revenue exceeds costs
Costs exceed revenue
Sales revenue, \$ million

are not sure about the answer, hereвЂ™s a possibly easier question. Would you be happy
about an investment in a stock that after 5 years gave you a total rate of return of zero?
We hope not. You might break even on such a stock but a zero return does not com-
pensate you for the time value of money or the risk that you have taken.

A project that simply breaks even on an accounting basis gives you your
money back but does not cover the opportunity cost of the capital tied up in
the project. A project that breaks even in accounting terms will surely have a
negative NPV.

LetвЂ™s check this with the superstore project. Suppose that in each year the store has
sales of \$13.067 millionвЂ”just enough to break even on an accounting basis. What
would be the cash flow from operations?
Cash flow from operations = profit after tax + depreciation
= 0 + \$450,000 = \$450,000
The initial investment is \$5.4 million. In each of the next 12 years, the firm receives a
cash flow of \$450,000. So the firm gets its money back:
Total cash flow from operations = initial investment
12 Г— \$450,000 = \$5.4 million
But revenues are not sufficient to repay the opportunity cost of that \$5.4 million in-
vestment. NPV is negative.

NPV BREAK-EVEN ANALYSIS
Instead of asking how bad sales can get before the project makes an accounting loss, it
is more useful to focus on the point at which NPV switches from positive to negative.
The cash flows of the project in each year will depend on sales as follows:
476 SECTION FIVE

1. Variable costs 81.25 percent of sales
2. Fixed costs \$2 million
3. Depreciation \$450,000
(.1875 Г— sales) вЂ“ \$2.45 million
4. Pretax profit
.40 Г— (.1875 Г— sales вЂ“ \$2.45 million)
5. Tax (at 40%)
.60 Г— (.1875 Г— sales вЂ“ \$2.45 million)
6. Profit after tax
\$450,000 + .60 Г— (.1875 Г— sales вЂ“ \$2.45 million)
7. Cash flow (3 + 6)
= .1125 Г— sales вЂ“ \$1.02 million

This cash flow will last for 12 years. So to find its present value we multiply by the
12-year annuity factor. With a discount rate of 8 percent, the present value of \$1 a year
for each of 12 years is \$7.536. Thus the present value of the cash flows is
PV (cash flows) = 7.536 Г— (.1125 Г— sales вЂ“ \$1.02 million)
The project breaks even in present value terms (that is, has a zero NPV) if the pres-
ent value of these cash flows is equal to the initial \$5.4 million investment. Therefore,
break-even occurs when
PV (cash flows) = investment
7.536 Г— (.1125 Г— sales вЂ“ \$1.02 million) = \$5.4 million
вЂ“\$7.69 million + .8478 Г— sales = \$5.4 million
5.4 + 7.69
sales = = \$15.4 million
.8478
This implies that the store needs sales of \$15.4 million a year for the investment to have
a zero NPV. This is more than 18 percent higher than the point at which the project has
zero profit.
Figure 5.2 is a plot of the present value of the inflows and outflows from the super-
store as a function of annual sales. The two lines cross when sales are \$15.4 million.
This is the point at which the project has zero NPV As long as sales are greater than
.
this, the present value of the inflows exceeds the present value of the outflows and the
project has a positive NPV.

FIGURE 5.2
NPV break-even analysis PV of
Project values, millions of dollars

project
cash flows

Investment
5.4

Sales
0
15.4 revenue,
millions
of
dollars
7.69

NPV is positive
NPV is negative
Project Analysis 477

What would be the NPV break-even level of sales if the capital investment was only \$5
Self-Test 3
million?

Break-Even Analysis
EXAMPLE 3
We have said that projects that break even on an accounting basis are really making a
lossвЂ”they are losing the opportunity cost of their investment. Here is a dramatic ex-
ample. Lophead Aviation is contemplating investment in a new passenger aircraft, code-
named the Trinova. LopheadвЂ™s financial staff has gathered together the following esti-
mates:
1. The cost of developing the Trinova is forecast at \$900 million, and this investment
can be depreciated in 6 equal annual amounts.
2. Production of the plane is expected to take place at a steady annual rate over the fol-
lowing 6 years.
3. The average price of the Trinova is expected to be \$15.5 million.
4. Fixed costs are forecast at \$175 million a year.
5. Variable costs are forecast at \$8.5 million a plane.
6. The tax rate is 50 percent.
7. The cost of capital is 10 percent.
How many aircraft does Lophead need to sell to break even in accounting terms?
And how many does it need to sell to break even on the basis of NPV? (Notice that the
break-even point is defined here in terms of number of aircraft, rather than revenue. But
since revenue is proportional to planes sold, these two break-even concepts are inter-
changeable.)
To answer the first question we set out the profits from the Trinova program in rows
1 to 7 of Table 5.5 (ignore row 8 for a moment).
In accounting terms the venture breaks even when pretax profit (and therefore net
profit) is zero. In this case
(7 Г— planes sold) вЂ“ 325 = 0
325
Planes sold = = 46
7

TABLE 5.5
Year 0 Years 1вЂ“6
Forecast profitability for
production of the Trinova Investment \$900
airliner (figures in millions
15.5 Г— planes sold
1. Sales
of dollars)
8.5 Г— planes sold
2. Variable costs
3. Fixed costs 175
4. Depreciation 900/6 = 150
(7 Г— planes sold) вЂ“ 325
5. Pretax profit (1 вЂ“ 2 вЂ“ 3 вЂ“ 4)
(3.5 Г— planes sold) вЂ“ 162.5
6. Taxes (at 50%)
(3.5 Г— planes sold) вЂ“ 162.5
7. Net profit (5 вЂ“ 6)
(3.5 Г— planes sold) вЂ“ 12.5
8. Net cash flow (4 + 7) вЂ“\$900
478 SECTION FIVE

Thus Lophead needs to sell about 46 planes a year, or a total of about 280 planes over
the 6 years to show a profit.
Notice that we obtain the same result if we attack the problem in terms of the break-
even level of revenue. The variable cost of each plane is \$8.5 million, which is 54.8 per-
cent of the \$15.5 million price. Therefore, each dollar of sales increases pretax profits
by \$1 вЂ“ \$.548 = \$.452. So
fixed costs including depreciation
Break-even revenue =
additional profit from each additional dollar of sales
\$325 million
= = \$719 million
.452
Since each plane cost \$15.5 million, this revenue level implies sales of 719/15.5 = 46
planes per year.
Now let us look at what sales are needed before the project has a zero NPV. Devel-
opment of the Trinova costs \$900 million. For each of the next 6 years the company ex-
pects a cash flow of \$3.5 million Г— planes sold вЂ“ \$12.5 million (see row 8 of Table 5.5).
If the cost of capital is 10 percent, the 6-year annuity factor is 4.355. So
NPV = вЂ“900 + 4.355(3.5 Г— planes sold вЂ“ 12.5)
= 15.24 Г— planes sold вЂ“ 954.44
If the project has a zero NPV,
0 = 15.24 planes sold вЂ“ 954.44
planes sold = 63
Thus Lophead can recover its initial investment with sales of 46 planes a year (about
280 in total), but it needs to sell 63 a year (or about 375 in total) to earn a return on this
investment equal to the opportunity cost of capital.

Our example may seem fanciful but it is based loosely on reality. In 1971 Lockheed
was in the middle of a major program to bring out the L-1011 TriStar airliner. This pro-
gram was to bring Lockheed to the brink of failure and it tipped Rolls-Royce (supplier
of the TriStar engine) over the brink. In giving evidence to Congress, Lockheed argued
that the TriStar program was commercially attractive and that sales would eventually ex-
ceed the break-even point of about 200 aircraft. But in calculating this break-even point
Lockheed appears to have ignored the opportunity cost of the huge capital investment
in the project. Lockheed probably needed to sell about 500 aircraft to reach a zero net
present value.3

What is the basic difference between sensitivity analysis and break-even analysis?
Self-Test 4

OPERATING LEVERAGE
A projectвЂ™s break-even point depends on both its fixed costs, which do not vary with
sales, and the profit on each extra sale. Managers often face a trade-off between these
3 Thetrue break-even point for the TriStar program is estimated in U. E. Reinhardt, вЂњBreak-Even Analysis for
LockheedвЂ™s TriStar: An Application of Financial Theory,вЂќ Journal of Finance 28 (September 1973), pp.
821вЂ“838.
Project Analysis 479

variables. For example, we typically think of rental expenses as fixed costs. But super-
market companies sometimes rent stores with contingent rent agreements. This means
that the amount of rent the company pays is tied to the level of sales from the store. Rent
rises and falls along with sales. The store thus replaces a fixed cost with a variable cost
that rises along with sales. Because a greater proportion of the companyвЂ™s expenses will
fall when its sales fall, its break-even point is reduced.
Of course, a high proportion of fixed costs is not all bad. The firm whose costs are
largely fixed fares poorly when demand is low, but it may make a killing during a boom.
Let us illustrate.
Finefodder has a policy of hiring long-term employees who will not be laid off ex-
cept in the most dire circumstances. For all intents and purposes, these salaries are fixed
costs. Its rival, Stop and Scoff, has a much smaller permanent labor force and uses ex-
pensive temporary help whenever demand for its product requires extra staff. A greater
proportion of its labor expenses are therefore variable costs.
Suppose that if Finefodder adopted its rivalвЂ™s policy, fixed costs in its new superstore
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