Monte Carlo Simulation
Monte Carlo Simulation
• Monte Carlo Simulation involves the use of pseudo random
numbers to model systems where time plays no substantive role
(i.e., static models)
• Generation of artificial data through the use of a random number
generator and the cumulative distribution of interest
• Random number generator
– Generates random variables that are uniformly distributed on the
interval from 0 to 1 (U(0, 1)) – Excel’s rand() function is an example
– Not possible to generate truly random numbers with a computer
algorithm
• Random numbers, U(0, 1), are then transformed so that they follow
the desired probability distribution.
– Uniform (a, b)
– Normal (µ, σ)
– Symmetric triangular (a, b)
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Example 1 – Investment Value
• You are planning to invest a total of $15,000 and you have three
investment vehicles from which to choose
• The return for each investment vehicle is a random variable (RL,
RM, and RH, respectively) and the distribution for each of these
random variables is given in the table below
• Use Monte Carlo simulation to characterize the distribution of the
investment value at the end of one year based on a user-given
allocation of the initial investment
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Investment Option
Distribution of return (in %)
Low risk
RL ~ Normal (3, 1)
Medium Risk
RM ~ Normal (5, 5)
High Risk
RH ~ Normal (10, 15)
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Example 1 – Investment Value
• The year-end value of the investment is given by the following
expression:
V = S L (1 + RL ) + S M (1 + RM ) + S H (1 + RH )
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Example 2 – Expected Profit
• Based on a model from Anthony Sun (used with permission)
– http://www.geocities.com/WallStreet/9245/vba12.htm
• A firm is considering producing and selling a new product under a
pure/perfect competition market and the firm wants to know the
probability distribution for the profit associated with this product.
• The total profit is given by the equation:
TP = (Q × P ) − (Q × V + F )
• Where:
–
–
–
–
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Q is the quantity sold
V is the variable cost per unit
P is the sales price per unit
F is the fixed cost for producing the product
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Example 2 – Expected Profit
• For this product, Q, P, and V are random variables with the
following distributions:
– Q: uniform (80,000, 120,000)
– P: normal(22, 5)
– V: normal(12, 8)
• F is estimated to be 300,000
• Want to use Monte Carlo simulation to characterize the distribution
of total profit for the proposed product
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Example 3 – Furniture Promotion
• This problem is from Section 2.2 of Seila et al. (2003)
• A large catalog merchandiser is planning to have a special
furniture promotion a year from now. To do this, the company
must place its order for furniture now. It plans to sign a contract
with the manufacturer for 3000 chairs at a cost of $175 per unit,
which the company plans to offer initially for $250 per unit. The
promotion will last for 8 weeks, after which all remaining units will
be offered for sale at half the original price, or $125 per unit. The
company believes that 2000 units will be sold during the first 8
weeks.
• While the number of chairs ordered and the ordering price are set
contractually, the number chairs sold during the first 8 weeks and
the initial selling price are actually random variables that depend
on a number of environmental conditions.
• The company would like to characterize the distribution of the
expected profit.
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Example 3 – Furniture Promotion
• Assume the following distributions for the two stochastic inputs:
– Demand for chairs – symmetric triangular (500, 3500)
– Initial selling price – uniform (200, 300)
• The profit is given by:
R

P = ( R − C )V +  − C  ( S − V )
2

• Use Monte Carlo simulation to characterize the expected profit for
the furniture promotion
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