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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:

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 ГЧ (.1125 ГЧ sales Ј $1.02 million)

The project breaks even in present value terms (that is, has a zero NPV) if the present 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 ГЧ (.1125 ГЧ sales Ј $1.02 million) = $5.4 million Ј$7.69 million + .8478 ГЧ sales = $5.4 million sales = 5.4 + 7.69 = $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 superstore 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.

Break-Even Analysis

We have said that projects that break even on an accounting basis are really making a loss ¤they are losing the opportunity cost of their investment. Here is a dramatic example.

Lophead Aviation is contemplating investment in a new passenger aircraft, codenamed the Trinova. Lophead`s financial staff has gathered together the following estimates:

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 following 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 interchangeable.)

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 ГЧ planes sold) Ј 325 = 0 Planes sold = 325 = 46

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 breakeven level of revenue. The variable cost of each plane is $8.5 million, which is 54.8 percent of the $15.5 million price. Therefore, each dollar of sales increases pretax profits

by $1 Ј $.548 = $.452. So Break-even revenue = fixed costs including depreciation 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. Development of the Trinova costs $900 million. For each of the next 6 years the company expects a cash flow of $3.5 million ГЧ planes sold Ј $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 = Ј900 + 4.355(3.5 ГЧ planes sold Ј 12.5) = 15.24 ГЧ planes sold Ј 954.44 If the project has a zero NPV, 0 = 15.24 planes sold Ј 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 program 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 exceed 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