PROBABILISTIC CASH FLOWS

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PROBABILISTIC CASH FLOWS

• As Engineering economy deals with the present and
  future there is a significant amount of
  uncertainty on the parameters being estimated.
• Timing of a cash flow n
• Revenue to be received at certain time Ft
• Salvage value S
• Initial cost P (or F0)
• Annual cost A
• Etc.

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RANDOM VARIABLES

• Probability theory consists of knowledge
  concerned with the quantitative treatment of
  uncertainty.
• The probability that an event will occur may be
  expressed by a number that represents the
  likelihood of the occurrence.

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PROBABILITY AXIOMS

• For any event A, the probability of A, P(A) 0.
• P(S) 1, where S is the set of all possible
  events or outcomes under consideration.
• If AnB ?, then P(AUB) P(A)P(B). That is for
  two mutually exclusive events the probability
  that event A or B occurs is equal to the sum of
  their probabilities.

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PROBABILITY FUNCTION

• A random variable is a function that assigns a
  value to each event included in the set of all
  events.
• Example If a coin is to be tossed twice, a
  random variable describing the number heads
  occurring can have values 0, 1, or 2.
• When the random variable is discrete as in the
  coin toss example, a probability function is used
  to describe the probability of the random
  variable being equal to a particular value.

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Coin toss probability table
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Continuous random variables

• When a random variable is continuous, a
  probability function is used (called probability
  density function)
• If X is the continuous random variable then its
  pdf is given by f(x). This f(x) has the following
  properties to satisfy the axiom of probability
• 0 f(x) 8,

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Continuous random variables

• Moreover, the probability that the random
  variable will be between two values a and b is
  given by

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Mean and variance of a random variable

• The mean (or expected value of a discrete random
  variable is given by
• The mean (or expected value of a discrete random
  variable is given by

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Variance of a RV

• Var(x) E(x2) E(x)2
• For discrete RVs
• For continuous RVs

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Expected value and variance calculations
Var(N) E(N2)-E(N)2 02(0.25)12(0.5)22(0.25)
12 0.5
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Engineering economy example

• A firm is planning to introduce a new product
  that is similar to an existing product.
• The marketing department made some estimates for
  possible future cash flows as related to varying
  market conditions.
• The table below shows that if this new product
  will produce cash flow A if demand decreases,
  cash flow B if demand remains constant and cash
  flow C if demand increases.
• It is believed that the probabilities of
  decreasing, steady and increasing demand will be
  0.1, 0.3, and 0.6, respectively.

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Cash flows for different possibilities
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Example continued

• The firm uses NPW for its economic decisions with
  MARR 10.
• By calculating the NPW for each level of demand,
  the firm develops a probability function for the
  present worth amount.
• The company uses expected net present worth
  amount to determine the acceptability of such
  ventures.

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NPW calculations

• EPW(10)(0.1)PW(10)A (0.3)PW(10)B
  (0.6)PW(10)C
• (0.1)-30,000
  11,000(P/A,10,3)
  -1,000(A/G,10,4)(P/A,10,4)
  (0.3)-30,000 11,000(P/A,10,3)
  (0.6)-30,000 4,000(P/A,10,3)
  3,000(A/G,10,4)(P/A,10,4)
  (0.1)(492)(0.3)(4,870)
  (0.6)(-4,185)
• -1,001 reject the
  project.

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Individual elements uncertain
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• The contract duration is uncertain and is
  estimated to be either 1 year, 2 years, or 3
  years. The probability of duration is given
  below

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Capital Costs

• Capital cost for A 70,000
• Capital Cost for B 20,000
• MARR 10

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Terminology

• Let
• xi annual OM cost, for outcome i
• yj annual capital recovery cost for contract
  duration j
• Pxi probability that the outcome is xi.
• Pyj probability that the outcome is yj.
• AEx annual equivalent cost for alternative x,
  where x is Alternative A or B.

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Expected value formulas
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Calculations

• EAEA (2,00077,000)(1/6)(1/4)
• (2,00040,334)(1/6)(1/2)
• (2,00028,147)(1/6)(1/4)
• (3,00077,000)(1/2)(1/4)
• (3,00040,334)(1/2)(1/2)
• (3,00028,147)(1/2)(1/4)
• (5,00077,000)(1/3)(1/4)
• (5,00040,334)(1/3)(1/2)
• (5,00028,147)(1/3)(1/4)
• 49,954

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Calculations

• EAEB (12,00022,000)(1/6)(1/4)
• (12,00011,524)(1/6)(1/2)
• (12,0008,042)(1/6)(1/4)
• (25,00022,000)(1/3)(1/4)
• (25,00011,524)(1/3)(1/2)
• (25,0008,042)(1/3)(1/4)
• (30,00022,000)(1/3)(1/4)
• (30,00011,524)(1/3)(1/2)
• (30,0008,042)(1/3)(1/4)
• (40,00022,000)(1/6)(1/4)
• (40,00011,524)(1/6)(1/2)
• (40,0008,042)(1/6)(1/4)
• 39,773
• CHOOSE B SINCE EAEB lt EAEA

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