1. 1 Chapter 3 Cost-Volume-Profit Analysis Decisions between discrete alternatives under uncertainty

2. 2 Fundamental Assumptions and Problem Cost is a function of exogenous demand quantity s either committed s or flexible without further managerial decisions required Revenue is a function of exogenous demand, too Problem: Determine the region of possible demand volumes for which an alternative is profitable Approach can be generalized to other uncertain variables that drive cost and revenue at a time, e.g. s life time of investments: payoff period s capitalization rate: internal rate of return

3. 3 General Approach Determine the set of demand volumes for which profit is positive, i.e. where Revenue > Cost. Example: Step cost function, constant selling price: Demand level: unprofitable profitable Revenue Cost Demand

4. 4 In general a probability distribution F(x) of demand x can be considered determine the probability that profit is positive: determine expected profit Revenue Cost x 1 x 2 x 3 x 4 x 5 x 6 x

5. 5 Other examples Convex cost, concave revenue: s the profitable region is a connected interval Revenue Cost Demand volume

6. 6 Indifference Points between alternatives Sometimes there are several alternatives with different fixed costs and different variable costs: One can save fixed costs by admitting higher variable costs and vice versa. Examples: s Two-part tariffs, common with electrical energy or telephone s Specific example: cell phone fees s Production processes: investing in devices or long term contracts add to committed costs but reduce variable costs

7. 7 Graphical Solution CheepCall TeleChat Gesamtkosten Critical Volume Total Cost Volume

8. 8 Mathematical Solution determine the cost function determine the indifference point Definition of a linear Cost function Total cost = Fixed cost + volume variable  cost per unit

9. 9 Stochastic Break-even-Analysis Let F denote the probability distribution of profit for an alternative. Aspiration criterion F(z 0 ) s determine the probability that an alternative‘s profit will fall below a certain aspiration level z 0, e.g.  probability of a loss Fractile criterion F -1 (p 0 )  the level of profit that is at least attained with a fixed probability of (1  p 0 ).

10. 10 Aspiration criterion z Prob{Z  z} Probability of a loss, alt.B Probability of a loss (Alternative A) A B z 0 = 0

11. 11 Fractile criterion z Prob{Z  z} lower 8% - quantile z|Z B < 0,08 z|Z A < 0,08 A B p 0 = 0,08 A riskier than B

12. 12 General critique of Aspiration and Fractile criterion only very few of the information from the whole distribution is considered comparison of alternatives depends on the critical levels for profit or the threshold probability but: you can consider several threshold levels: (sensitivity analysis)

13. 13 CC Problems 3-45 (10%) 3-47 ( 5%) 3-49 (10%) Draw a graph of the profit function for the three relevant alternatives in one coordinate system Extra problem (3-47 11 th ed.) (15%) determine also s the lower 10% fractile of the profit distribution, s the probability that gross profit exceeds $200 000, s and the conditional expectation of profit, given it falls below $200 000.

14. 14 Extra problem Jaro Comp. considers adding to new colors of umbrellas to ist product mix, fixed product cost: $400,000 per additional color, selling price: $10, variable cost per unit: $8. Demand distribution (probabilities): 1. Breakeven point in units, each color? 2. Product, maximizing the expected operating income? Calculations! 3. Assume now, demand for shocking pink = 300,000, while emerald green has the demand distribution above. Which product should be chosen? Why? What is the information benefit of having the entire distribution instead of just the expected value? 4. see questions on the previous slide! Demand in 1000 unitsEmerald greenShocking pink 50 100 200 300 400 500 0.1.2.4.2.1.2.4.1

15. 15 Problem 3-49: Comparison of alternatives Operating income Units Mn Mc Pc =Pn