Break even level of ebit

Lecture 7: Capital Budgeting Part II Break-even, Sensitivity, and Scenario Analysis Berk, DeMarzo 3rd edition, Chapter 8 Section 8.5

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Last lecture

Capital budgeting:

Estimate incremental cash flows in the project.

Determine a cost of capital used for discounting

Calculate NPV

Accept or reject the project

Question: There is significant uncertainty in estimating cash flows. How do we know whether the NPV result is accurate?

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Outline

Methods to assess uncertainty and evaluate project risk:

Sensitivity Analysis

Scenario Analysis

Break-Even Analysis

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In-class exercise

As the finance manager of a company, you are presented with the following project. The company is considering the purchase of a new piece of equipment which would cost $200,000. This equipment will have a five-year useful life and have a salvage value of $0 at the end of the five-year period. It is estimated that

the new equipment will be able to produce 10,000 shelves per year.

the incremental overhead for running the equipment will be $20,000 per year.

they can sell the shelves for $25 each.

the cost of sales is $15 per shelf.

Net Working Capital requirements for the project are as follows:

Year 0 = $10,000

Year 1 = $15,000

Year 2 = $17,000

Year 3 = $15,000

Year 4 = $10,000

The company has a 30% marginal tax rate and a cost of capital of 15%.

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Would you accept this project (support your answer with NPV)?

I use the example from our last lecture to illustrate how we can use the three methods to evaluate the uncertainty and the risk of the project.

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Sensitivity Analysis

Sensitivity analysis shows how the NPV varies with a change in one of the assumptions, holding the other assumptions constant.

Example: how does the NPV change when the sale price is $20 per unit? or $30?

Various cash flow assumptions are used.

Best and worst cases are developed and NPVs compared.

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Suppose your boss is more optimistic (or pessimistic) about the assumptions you made on sale price, the unit of sales, sales growth, or cost of capital (discount rate to discount cash flows), how does the NPV change in one of the assumptions, holding the other assumptions constant?

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Sensitivity Analysis

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Parameter Initial Assumption Worst Case Best Case
Sale price($/unit) 25 20 30
NPV 21,232 ? ?
Units sold (000s) 10
NPV 21,232
Sales growth 0%
NPV 21,232
Cost of capital 15%
NPV 21,232
Based on the baseline (initial) assumptions, NPV=$21,232 (see cell B31 in the sheet “Baseline” of EXCEL file “In-class exercise - DCF analysis”)

What is the NPV when sale price is $20 per unit? $30?

Answer: Just change the price from $25 (cell B3) to $20, then NPV (cell B31) becomes -96,093.

Similarly, when price = $30, then NPV = 138,557

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Sensitivity Analysis

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Parameter Initial Assumption Worst Case Best Case
Sale price($/unit) 25 20 30
NPV 21,232 -96,093 138,557
Units sold (000s) 10 8 12
NPV 21,232 ? ?
Sales growth 0% -5% 5%
NPV 21,232 ? ?
Cost of capital 15% 18% 12%
NPV 21,232 ? ?
For example, what is the NPV when units sold = 8,000, holding the other assumptions constant (i.e., the other values are based on initial or baseline assumptions: sale price =25, sales growth = 0%, cost of capital = 15%)?

Answer: just change the unit (cell C15 in sheet “Baseline”) from 10,000 to 8,000, then the NPV = -25,698.

Similarly, you can find the NPVs for all other cases. Again, in sensitivity analysis, we want to see how the NPV varies with a change in one of the assumptions, holding the other assumptions constant (the other values are based on initial assumptions)

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Sensitivity Analysis

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Parameter Initial Assumption Worst Case Best Case
Sale price($/unit) 25 20 30
NPV 21,232 -96,093 138,557
Units sold (000s) 10 8 12
NPV 21,232 -25,698 68,162
Sales growth 0% -5% 5%
NPV 21,232 1,934 42,405
Cost of capital 15% 18% 12%
NPV 21,232 5,141 39,342
For example, when sales growth is -5%, holding the other assumptions constant (i.e., the other values are based on initial assumptions: sale price =25, units sold= 10,000, cost of capital = 15%), just change the cell B2 from 0% to -5%, then NPV = 1,934.

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Sensitivity Analysis

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Units Sold

Sales Growth

$20

$25

$30

8K

12K

10K

-5%

0%

5%

18%

15%

12%

The most important parameter assumptions are ?

Red bars show the NPV under the best-case assumptions.

Blue bars indicate the NPV under the worst-case assumptions.

Orange bars represent the baseline assumptions.

The most important parameter assumptions are on the sale price per unit and the number of units sold, because NPVs change dramatically in different cases and NPVs dropped significantly in the worst case (e.g. NPV=-96,093 when sale price =$20 in worst-case).

These assumptions deserve the greatest scrutiny during the estimation process. In addition, as the most important drivers of the project’s value, these factors deserve close attention when managing the project.

In contrast, when sales growth and the cost of capital change, NPVs do not change that much and stay positive. These two factors are not as important as the sale price and units sold.

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Worst Case 5141.2707123555665 1934.0843084341795 -25698.172083122317 -96093.42914136179 Baseline 21231.999289037303 21231.999289037303 21231.999289037303 21231.999289037303 Best Case 39341.527734909505 42404.89203263411 68162.170661196957 138557.42771943641
Project NPV

Sensitivity Analysis

Benefit: identifies critical assumptions.

In this example, pay special attention to the assumptions on the sale price per unit and the number of units sold.

We can invest further resources to refine these assumptions.

e.g., market research to analyze market size, customer demand, and competitors.

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Scenario Analysis

Process – Test particular combinations of assumptions to see the result on NPV.

Example - A major competitor may react differently to your project and your assumptions change with their reactions.

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Strategy Sale Price ($/unit) Expected Units Sold(thousands) NPV ($thousands)
Current strategy 25 10 21,232
Price reduction 23 12 ?
Price increase 27 9 ?
In the sheet “Baseline” of EXCEL file “In-class exercise - DCF analysis”, change both the sale price and expected units, i.e. change the cell B3 from 25 to 23, and cell C15 from 10,000 to 12,000, then NPV = 11,846.

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Scenario Analysis

Price increase strategy generates the highest NPV.

Benefit: scenario analysis is appropriate when assumptions are interrelated.

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Strategy Sale Price ($/unit) Expected Units Sold(thousands) NPV ($thousands)
Current strategy 25 10 21,232
Price reduction 23 12 11,846
Price increase 27 9 40,004
Break-Even Analysis

Analysis of the level of sales (or other parameters) at which the company “breaks even”.

Two types of Break-Even:

Accounting Break-Even focuses on the level of a parameter for which a project’s EBIT=0 (sales cover costs; unlevered net income is zero).

Economic Break-Even focuses on the value of a parameter for which NPV is zero.

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Accounting Break-Even

Projects have Revenues, Variable Costs, Fixed Costs, and Depreciation.

Variable costs: costs that vary with the level of production.

Fixed costs: costs that do not vary with the level of production.

Accounting Break-Even measures the level of a parameter for which EBIT=0

(Revenues – Variable Costs – Fixed Costs – Depreciation= 0)

Example:

The Accounting Break-Even level of quantity sold indicates how many units you need to sell to satisfy EBIT=0

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A variable cost can be thought as cost of goods sold. It depends on production output. The variable cost of production is a constant amount per unit produced. As the volume of production and output increases, variable costs will also increase.

Fixed costs can be thought of as Selling, General, and Administrative expenses (SG&A).

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Accounting Break-Even

Example

Assumptions for a project with a 3-year life:

Revenue $100 per unit for a Printer

Variable Costs $70 per unit → Gross Profit $30 per unit

Fixed Costs $300,000

$900,000 Investment with a salvage value of 0, annual depreciation of $300,000 for three years

Taxes 35%

Question: how many units you need to sell such that EBIT=0? (EBIT = Revenues – Variable Costs – Fixed Costs – Depreciation)

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All sales and costs are incremental.

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Template for Accounting Break Even

Revenues ($100/unit)

Variable Costs ($70/unit)

Actual Gross Profit

Gross Profit Required ? ($30/unit)

Fixed Costs - 300,000

Depreciation - 300,000

EBIT 0

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← accounting break-even asks EBIT = 0

Question: how many units you need to sell such that EBIT=0?

Hint: work backwards from EBIT to find the gross profit.

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Accounting Break-Even Example

Revenues ($100/unit)

Variable Costs ($70/unit)

Actual Gross Profit

Gross Profit Required 600,000 ($30/unit)

Fixed Costs - 300,000

Depreciation - 300,000

EBIT 0

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Gross profit = Fixed Costs + Depreciation = 300,000+300,000 = 600,000

← accounting break-even asks EBIT = 0

If we can find the total amount of gross profit, then we just calculate the number of units to be sold by dividing that number by gross profit per unit ($30/unit).

So how to find the total amount of gross profit?

Step 1.

accounting break-even means EBIT = 0, i.e.,

Revenues – Variable Costs – Fixed Costs – Depreciation= 0

where gross profit = Revenues – Variable Costs

Gross profit – Fixed Costs – Depreciation= 0

Step 2.

Gross profit = Fixed Costs + Depreciation = 300,000+300,000 = 600,000

Step 3.

the number of units to be sold to satisfy accounting break-even EBIT=0 is

Quantity (accounting break-even) = 600,000/ 30 = 20,000.

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Accounting Break-Even Example

Accounting break-even for quantity

total Gross Profit /Gross Profit per Unit

$600,000/$30 = 20,000 units

Accounting break-even generates no tax liability.

What is the project’s NPV under accounting break-even?

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Accounting break-even generates no tax liability because EBIT=0.

This accounting break-even level of 20,000 units will be used in a few slides.

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Accounting Break-Even Example

Accounting break-even results in free cash flows equal to depreciation in year 1 to 3 (note ΔNWC = 0 in this example)

Investors will not be happy if the company simply returns their original investment over time.

Accounting break-even always results in a NEGATIVE NPV.

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Year 0 1 2 3
Unlevered Net Income 0 0 0
Add Back Depreciation 300,000 300,000 300,000
Subtract Change in NWC
Subtract Capital Expenditure -900,000
Free Cash Flow -900,000 300,000 300,000 300,000
Accounting break-even requires EBIT = 0, so unlevered net income = 0. After we add back depreciation, we can find the FCFs.

Because of time value of money, investors will not be happy if the company simply returns their original investment over time.

that is, not happy with the cash flow stream -900,000, 300,000, 300,000, 300,000

For simplicity, we ignore the cash impact of change in NWC. You can show that NPV is still negative when we consider the cash flow impact of NWC (you can calculate the present value of cash flow impact of NWC by discounting row 38 in the sheet “Baseline” of the Excel file “In-class exercise - DCF analysis”, which will give you a negative number).

Note if netting across row 38 from Year 0 to Year 5, we always get zero.

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Economic Break-Even

Economic Break-Even focuses on the level of a parameter for which NPV is zero.

Example:

Revenue $100 per unit for a Printer

Variable Costs $70 per unit. Gross Profit $30 per unit

Fixed Costs $300,000

Initial Investment $900,000, salvage value $0, useful life 3 years

Taxes 35%

Cost of capital= 10%

No cash flow effect of change in NWC (for simplicity)

Question: what is the economic break-even level of quantity sold? In other words, how many units a project need to sell such that NPV=0?

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Economic Break-Even

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Year 0 1 2 3
Unlevered Net Income ? ? ?
Add Back Depreciation 300,000 300,000 300,000
Subtract Change in NWC
Subtract Capital Expenditure -900,000
Free Cash Flow -900,000 C C C
Economic break-even requires NPV=0 (PV(FCFs from Year 1 to 3) + PV(FCF in Year 0) = 0)

PV(FCFs from Year 1 to 3) = 900,000

The FCF stream from Year 1 to 3 is a 3-year annuity with payment C. PV(FCFs from Year 1 to 3)= =

, so

Recall PV(Annuity)

In this example, we assume unlevered net incomes remain unchanged from Year 1 to 3. Since there are no other cash flow adjustments (e.g. no cash impact of change in NWC for simplicity), we have the same FCFs from Year 1 to 3.

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Economic Break-Even

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Year 0 1 2 3
Unlevered Net Income ? ? ?
Add Back Depreciation 300,000 300,000 300,000
Subtract Change in NWC
Subtract Capital Expenditure -900,000
Free Cash Flow -900,000 361,903 361,903 361,903
Economic break-even requires NPV=0

PV(FCFs from Year 1 to 3) = 900,000

The FCF stream from Year 1 to 3 is a 3-year annuity with payment C. PV(FCFs from Year 1 to 3)= =

, so

Now we have free cash flows, then we should be able to work backwards to find the gross profit.

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Template for Economic Break-Even

Revenues ($100/unit)

Variable Costs ($70/unit)

Gross Profit Required ? ($30/unit)

Fixed Costs - 300,000

Depreciation - 300,000

EBIT ?

Tax @ 35%

Unlevered net income ?

add back Depreciation +300,000

Free cash flow $

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Fill in the blanks (Hint below: Work backwards from free cash flow to the total gross profit)

Work backwards from free cash flow to the total gross profit. Once we calculate the total gross profit, then the economic break-even level of quantity sold is just:

total gross profit/gross profit per unit

Again, in this example, for simplicity, we ignore the cash impact of change in NWC.

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Template for Economic Break-Even

Revenues ($100/unit)

Variable Costs ($70/unit)

Gross Profit Required 695,235 ($30/unit)

Fixed Costs - 300,000

Depreciation - 300,000

EBIT 95,235 = 61,903/(1-0.35)

Tax @ 35%

Unlevered net income 61,903

add back Depreciation +300,000

Free cash flow $

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Fill in the blanks (see details below: Work backwards from free cash flow to the total gross profit)

Method 1: Work backwards from free cash flow to find the total gross profit:

Unlevered net income = FCF – Depreciation = - 300,000 = 61,903

EBIT= Unlevered net income / (1-tax rate) = 61,903/0.65 = 95,235

Gross profit = Pretax profit + Depreciation + Fixed Costs = 95,235 + 300,000 +300,000 = 695,235

Method 2: Work from the top to the bottom.

(Gross profit - Fixed Costs – Depreciation) * (1-tax) + Depreciation = FCF

that is :

(Gross Profit - 300,000 - 300,000) * (1-0.35) + 300,000 =361,903

so Gross profit = (361,903-300,000)/(1-0.35) + 300,000 + 300,000 = 95,235

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Economic Break-Even

The economic break-even level of quantity:

Gross profit / Gross profit per unit = 695,235/30 = 23,174.5

Result: the economic break-even level of sales quantity is 23,175.

Quantity (Economic break-even) =23,175

Quantity (Accounting break-even) =20,000 (see slide 18)

Economic break-even > Accounting break-even

Does this relation make sense? (see notes underneath this slide)

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Intuitively, at the accounting break-even level of quantity, the NPV is negative. To achieve the economic break-even level of quantity, that is to get NPV=0, we need to sell more products so that we have more net income and more free cash flows such that NPV becomes zero.

Of course, when we sell more than the economic break-even level of quantity, the NPV will become positive.

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Economic Break-Even

Quantity (Economic break-even) =23,175

Quantity (Accounting break-even) =20,000 (see slide 18)

Test: positive, negative, or zero?

When quantity=19,000, unlevered net income is ___, NPV is ___

When quantity=20,000, unlevered net income is ___, NPV is ___

When quantity=22,000, unlevered net income is ___, NPV is ___

When quantity=23,175, unlevered net income is ___, NPV is ___

When quantity=24,000, unlevered net income is ___, NPV is ___

See solutions underneath this slide.

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Assuming all other variables remain unchanged (e.g. sale price, sales growth, cost of capital)

When quantity=19,000, unlevered net income is NEGATIVE, NPV is NEGATIVE

When quantity=20,000, unlevered net income is ZERO, NPV is NEGATIVE

When quantity=22,000, unlevered net income is POSITIVE, NPV is NEGATIVE

When quantity=23,175, unlevered net income is POSITIVE, NPV is ZERO

When quantity=24,000, unlevered net income is POSITIVE, NPV is POSITIVE

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Economic Break-Even

Economic break-even level for other parameters:

Price per unit (see the example underneath the slide)

Cost of capital.

At the break-even level of cost of capital, the project’s NPV=0 (that is, the break-even level of cost of capital is the discount rate that sets the NPV equal to zero, which is also called the internal rate of return (IRR)).

In Excel, use IRR function

= IRR(-900,000, 361,903, 361,903, 361,903) = 10%

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Year 0 1 2 3
Free Cash Flow -900,000 361,903 361,903 361,903
ABC Corp. has decided to ask suppliers to bid on the 10,000 cartons of precision machine screws that ABC Corp. needs to purchase per year to support its manufacturing needs over the next five years. You have decided to submit a bid to supply the machine screws. Given the assumptions on the capital expenditure, costs, and cost of capital, what is the minimum price per carton that you should bid?

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Summary

In capital budgeting, to assess uncertainty and evaluate project risk:

Sensitivity Analysis

identifies critical assumptions.

Scenario Analysis

tests interrelated assumption.

Break-Even Analysis

accounting vs. economic break-even

calculate economic break-even level of quantity, price per unit, cost of capital

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