Chapter 3 CostVolumeProfit Analysis

1
Chapter 3Cost-Volume-Profit Analysis

• Decisions between discrete alternatives under
  uncertainty

2
Fundamental Assumptions and Problem

• Cost is a function of exogenous demand quantity
• either committed
• 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.
• life time of investments payoff period
• capitalization rate internal rate of return

3
General Approach

• Determine the set of demand volumes for which
  profit is positive, i.e. where
• Revenue gt Cost.
• Example Step cost function, constant selling
  price

Revenue Cost
Demand level
unprofitable
profitable
Demand
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
x1 x2 x3 x4 x5
x6 x
5
Other examples

• Convex cost, concave revenue
• the profitable region is a connected interval

Cost
Revenue
Demand volume
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
• Two-part tariffs, common with electrical energy
  or telephone
• Specific example cell phone fees
• Production processes investing in devices or
  long term contracts add to committed costs but
  reduce variable costs

7
GraphicalSolution
Gesamtkosten
Total Cost
Volume
8
MathematicalSolution

• determine the cost function
• determine the indifference point

Definition of a linear Cost functionTotal cost
Fixed cost volume variable ? cost per unit
9
Stochastic Break-even-Analysis

• Let F denote the probability distribution of
  profit for an alternative.
• Aspiration criterion F(z0)
• determine the probability that an alternatives
  profit will fall below a certain aspiration level
  z0 , e.g.
• probability of a loss
• Fractile criterion F-1(p0)
• the level of profit that is at least attained
  with a fixed probability of (1 - p0).

10
Aspiration criterion
B
ProbZ ? z
A
Probability of a loss (Alternative A)
Probability of a loss, alt.B
z
z0 0
11
Fractile criterion
B
ProbZ ? z
A
A riskier than B
lower 8 - quantile
p0 0,08
z
zZB lt 0,08
zZA lt 0,08
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
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 11th ed.) (15)
• determine also
• the lower 10 fractile of the profit
  distribution,
• the probability that gross profit exceeds 200
  000,
• and the conditional expectation of profit, given
  it falls below 200 000.

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)
• Breakeven point in units, each color?
• Product, maximizing the expected operating
  income? Calculations!
• 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?
  
• see questions on the previous slide!

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