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2. Breakeven Analysis. A typical alternative has an initial cost, operating/maintenance costs, revenues, and a salvage value when the asset is disposedRevenues in turn could depend on

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## Slide Set 6: Ch. 5 Breakeven

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**1. **1 Slide Set 6: (Ch. 5) Breakeven/Sensitivity Analysis

**2. **2 Breakeven Analysis A typical alternative has an initial cost, operating/maintenance costs, revenues, and a salvage value when the asset is disposed
Revenues in turn could depend on #units sold, while operating costs could be made up of a fixed over-head and a variable component proportional to #units produced
“A typical break-even question” follows...
If everything except the #units per year is known, how many units per year need to be made for the alternative to be “just viable” (PW=0, no profit-no loss)

**3. **3 Fixed Costs Essentially constant for all values of the variable (parameter) in question (say volume for example)
Fixed Costs – Costs that do not vary with production or activity levels. Some examples:
Costs of buildings
Insurance
Equipment capital recovery

**4. **4 Variable Costs Variable costs change with the level of activity - usually proportional to activity
Costs that vary with the level of activity
Direct labor such as wages
Materials
Marketing

**5. **Example A contractor experiences a seasonal activity for compressors. The contractor owns eight compressors and is considering purchasing one additional compressor to meet the demand when it exceeds availability.
A local equipment rental firm will rent compressors at $50/day. Compressors can be purchased for $6000. The difference in M&O costs between owned and rented compressors id $3000/year. 5

**6. **Example Let X denote the number of days per year when a 9th compressor is needed. Assume a planning horizon of 5yrs, zero salvage value and 20% MARR.
AW(purchase) = -6000(A/P, 20%,5)-3000 = -5006.40
AW(rent) = -50X
Setting AW(rent)=AW(purchase) we get X = 100.128
If X > 100.128, AW(purchase) > AW(rent), so purchase
If X < 100.128, AW(rent) > AW(purchase), so rent 6

**7. **7 Cost, Revenue Relationships Linear or non-linear models capture the cost-volume and revenue-volume relationships
Linear and non-linear models are used as approximations to reality
Typical cost, revenue relationships are shown on following slides

**8. **8

**9. **9

**10. **10 Breakeven point The breakeven (BE) point QBE is the point where the revenue and total cost relationships intersect, R=TC, or profit, P=0
For non-linear relations, it is possible to have more than one QBE point

**11. **11 Non-linear BE Analysis For non-linear analysis the point of maximum profit is of interest
And, multiple BE points may exist

**12. **12

**13. **13 Example Suppose TC = 500/Q + Q + 1000, R = 2000, find QBE and Qmax
P = R-TC = 1000 – 500/Q –Q
P = 0 ? QBE = 0.5 and 999.5
dP/dQ = 500/Q2 -1 = 0 ? Q = ±22.36
d2P/dQ2 = -1000/Q3 ? Qmax = 22.36 is the maximizer as the second derivative is negative at that point

**14. **14 Breakeven Analysis: Two Alternatives Given two alternatives determine a variable economic parameter common to both alternatives that serves as the BE parameter, and choose the measure of worth PW/AW
BE parameter could be production/sales volume, interest rate, initial investment, M&O costs, yearly revenues, salvage value...
If the measure of worth depended linearly on the BE parameter, then BE analysis can use a “ranking” approach
e.g. PW on salvage value, how much larger should the second alternative’s salvage value be for it to be chosen? Plot both PW as a function of the extra salvage value, and find the region where one is larger than the other.
If the measure of worth dependence on the BE parameter is nonlinear, then look at the incremental measure of worth as a function of that parameter.
e.g. Incremental IRR analysis we’ve seen was a break-even analysis on MARR between 2 alternatives using a PW expression

**15. **15 Example Automatic Manual
Initial cost $23,000 $8000
Salvage $4000 $0
AOC $3500 $1500
Life 10 yrs 5 yrs
Labor cost $12/hr $8/hr
M/C Output 8tons/hr 6tons/hr
#operators reqd. 1 3
MARR = 10%
What is the break-even production quantity?

**16. **Let x denote production in tons/year
Automatic:
$/ton = ($12/hr)/(8tons/hr) = $1.5/ton
labor cost = $1.5x per year
AWauto = 23000(A/P,10%,10) -4000(A/F,10%,10) +3500 + 1.5x
AWauto = 6992 + 1.5x
Manual:
$/ton = 3 ($8/hr)/(6tons/hr) = 4
labor cost = $4x per year
AWmanual = 8000(A/P,10%,5) + 1500 +4x
AWmanual = 3610 + 4x 16

**17. **AWmanual = 3610 + 4x
AWauto = 6992 + 1.5x
AWauto = AWmanual ? x = QBE = 1353 tons per year
If x > QBE ? choose automatic (why?)
If x < QBE ? choose manual (why?) 17

**18. **18

**19. **19 Three Alternative Analysis For an example like the previous problem, PW depends linearly on the parameter and we can plot the present worth or annual worth over a specified range of values, i.e., use ranking in comparing, i.e., “which line is below?”
A typical three-alternative BE plot might look like ….

**20. **20

**21. **Example- two parameter, single alternative 21

**22. **Sensitivity analysis Warehouse alternative is viable if PW = 0,
PW = -2.5M -3.5M[1 + (P/F,15%,15)]*(1+x) -150K(P/A,15%,30) +1.26M(P/A,15%,30)*(1+y) + 800K = 0,
or if,
PW = 1,658110 -3930150x + 8273160y = 0,
or if,
-0.47505x +y +0.2004 = 0
Plot the line PW=0, the break-even line, then identify the favorable/viable region PW > 0 22

**23. **23

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