Chapter 11
Simulation Models
Introduction
• In the previous chapter, we introduced most of the
important concepts for developing and analyzing
spreadsheet simulation models.
• We also discussed many of the features available
in the powerful simulation add-in, @RISK, that you
receive with this book.
• Now we apply the tools to a wide variety of
problems that can be analyzed with simulation.
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Introduction continued
• For convenience, we group the applications into four
general areas:
1. operations models,
2. financial models,
3. marketing models,
4. games of chance.
• The only overriding theme in this chapter is that simulation
models can yield important insights in all of these areas.
• You do not need to cover all of the models in this chapter or
cover them in any particular order.
• You can cover the ones of most interest to you in practically
any order.
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Operations models
• Whether we are discussing the operations of a
manufacturing or a service company, there is likely
to be uncertainty that can be modeled with
simulation.
• In this section we look at examples of bidding for a
government contract (uncertainty in the bids by
competitors), warranty costs (uncertainty in the
time until failure of an appliance), and drug
production (uncertainty in the yield and timing).
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Bidding for contracts
• In situations where a company must bid against
competitors, simulation can often be used to
determine the company’s optimal bid.
• Usually the company does not know what its
competitors will bid, but it might have an idea
about the range of the bids its competitors will
choose.
• In this section we show how to use simulation to
determine a bid that maximizes the company’s
expected profit.
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Warranty costs
• When you buy a new product, it usually carries a warranty.
A typical warranty might state that if the product fails within
a certain period such as one year, you will receive a new
product at no cost, and it will carry the same warranty.
• However, if the product fails after the warranty period, you
have to bear the cost of replacing the product.
• Due to random lifetimes of products, we need a way to
estimate the warranty costs (to the manufacturer) of a
product.
• Example 11.2 illustrates how this can be accomplished with
simulation.
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Drug production with
uncertain demand
• In many manufacturing settings, products are
produced in batches, and the usable yields from
these batches are uncertain.
• This is particularly true in the drug industry.
• Example 11.3 illustrates how a drug manufacturer
can take this uncertainty into account when
planning production.
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Deming’s funnel experiment
• Edwards Deming was an American statistician whose
views on quality management revolutionized the way
companies do business across the world.
• Deming has been given much of the credit for Japan’s
spectacular post–World War II economic recovery.
• He traveled around the United States giving a famous fourday seminar on quality management.
• An important component of Deming’s seminar was his
famous funnel experiment.
• The funnel experiment is designed to show how businesses
often greatly overadjust “stable” processes.
• We illustrate how it works in Example 11.4.
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Financial models
• There are many financial applications where
simulation can be applied.
• Future cash flows, future stock prices, and future
interest rates are some of the many uncertain
variables financial analysts must deal with.
• In every direction they turn, they see uncertainty.
• In this section we analyze a few typical financial
applications that can benefit from simulation
modeling.
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Financial planning models
• Many companies, such as GM, Eli Lilly, Procter & Gamble, and
Pfizer, use simulation in their capital budgeting and financial
planning processes.
• Simulation can be used to model the uncertainty associated with
future cash flows. In particular, simulation can be used to answer
questions such as the following:
– What are the mean and variance of a project’s net present value
(NPV)?
– What is the probability that a project will have a negative NPV?
– What are the mean and variance of a company’s profit during the
next fiscal year?
– What is the probability that a company will have to borrow more
than $2 million during the next year?
• Example 11.5 illustrates how simulation can be used to evaluate an
investment opportunity.
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Cash balance models
• All companies track their cash balance over time.
• As specific payments come due, companies
sometimes need to take out short-term loans to
keep a minimal cash balance.
• Example 11.6 illustrates one such application.
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Investment models
• Individual investors typically want to choose
investment strategies that meet some prespecified
goal.
• Example 11.7 is typical.
• Here, a person wants to meet a retirement goal,
starting at an early age.
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Simulating stock prices and
options
• In this section, we illustrate how @RISK can be
used to simulate stock prices.
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Modeling the price of a stock
• An enormous amount of research has been
devoted to discovering the way stock prices
change.
• Although few agree on the best model of stock
price changes, one popular model states that price
changes follow a lognormal distribution.
• Essentially, this means that the logarithm of a
stock’s price at any time is a normally distributed
random variable.
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Modeling the price of a stock
continued
• To be more specific, the stock price pt at any time t
in the future is related to the current price p0 by the
formula
𝑝𝑡 = 𝑝0 exp[ 𝜇 − 0.5𝜎 2 𝑡 + 𝜎𝑍 𝑡]
• Here, μ is the mean percentage growth rate of the
stock; σ is the standard deviation of the growth
rate, usually called the volatility; and Z is a normal
random variable with mean 0 and standard
deviation 1.
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Modeling the price of a stock
continued
• The spreadsheet
here illustrates how
to estimate the
parameters μ and σ
in from monthly
returns. (See the file
Stock Returns.xlsx.)
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Valuing a European call option
• A European option on a stock gives the owner of
the option the right to buy (if the option is a call
option) or sell (if the option is a put option) 100
shares of a stock on a particular date for a
particular price.
• The price at which an option holder can buy or sell
the stock is called the exercise price (or strike
price) of the option. The date by which the option
must be used (or “exercised”) is called the exercise
date.
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Valuing a European call option
• You must pay for the option in the first place.
• The question is, what is a fair price for such an
option? Because option trading is a multibilliondollar business, this is an important question.
• Black and Scholes (1973) were the first to derive a
formula for pricing options.
• Shortly after that, Cox et al. (1979) derived a
different but equivalent method for pricing options.
• We use their method, which is based on the
following extremely important result.
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Option pricing result
• The price of an option on a nondividend-paying
stock must be the expected discounted value of
the cash flows from an option on a stock having
the same standard deviation as the stock on which
the option is written and growing at the risk-free
rate of interest.
• Here, discounting is done continuously at the riskfree rate. (If the stock pays dividends, the risk-free
rate should be replaced by the difference between
the risk-free rate and the dividend rate in what
follows.)
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Option pricing result
continued
• One surprising implication of this result is that the
price of the option does not depend on the mean
growth rate of the stock itself, only on the risk-free
rate and the standard deviation of the growth rate
of the stock.
• Example 11.9 illustrates how @RISK can be used
to estimate the price of a European option.
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Return on a portfolio with a stock
and an option on the stock
• We now extend the previous example by
simulating a portfolio that includes a company’s
stock and a call option on that stock.
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Value of a more exotic call
option
• The European call option is fairly simple. A variety
of other derivative securities are currently
available.
• In fact, their variety and complexity are what make
them attractive – and dangerous for the
unsuspecting investor.
• We illustrate one variation of the basic call option,
an Asian option.
• Its payoff depends, not on the price at expiration of
the underlying stock, but on the average price of
the stock over
the lifetime of the option.
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Value of a more exotic call
option continued
• To price an Asian option (or any number of other
exotic options), you again find the expected
discounted value of the cash flow from the option,
assuming that the stock grows at the risk-free rate.
• Example 11.10 illustrates how to approximate this
expected value with simulation.
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Asian option
• This option is a variation of the call option.
• Its payoff depends not on the price at expiration of
the underlying stock, but on the average price of
the stock over the lifetime of the option.
• To price an Asian option, we again need to find the
expected discounted value of the payoff from the
option, assuming that the stock grows at the riskfree rate.
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Marketing models
• There are plenty of opportunities for marketing
departments to use simulation.
• They face uncertainty in the brand-switching
behavior of customers, the entry of new brands
into the market, customer preferences for different
attributes of products, the effects of advertising on
sales, and so on.
• We examine some interesting marketing
applications of simulation in this section.
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Models of customer loyalty
• What is a loyal customer worth to a company? This is an
extremely important question for companies.
• It is an important part of customer relationship
management, or CRM, currently one of the hottest topics in
marketing.
• Companies know that if customers become dissatisfied with
the company’s product, they are likely to switch and never
return.
• Marketers refer to this customer loss as churn.
• Example 11.11 uses a reasonable model of customer
loyalty and simulation to estimate the worth of a customer
to a company.
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Models of customer loyalty
continued
• Example 11.12 is a variation of the previous
example. We now investigate the effect of offering
a customer an incentive to remain loyal.
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Marketing and sales models
• We conclude this marketing section with a model
of marketing and selling condos.
• The main issue is the timing of sales, and we
demonstrate how a deterministic model of this
timing can provide very misleading results.
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Simulating games of chance
• We realize that this is a book about business
applications.
• However, it is instructive (and fun) to see how
simulation can be used to analyze games of
chance, including sports contests.
• Indeed, many analysts refer to Monte Carlo
simulation, and you can guess where that name
comes from—the gambling casinos of Monte
Carlo.
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Simulating the game of craps
• Most games of chance are great candidates for
simulation because they are, by their very nature,
driven by randomness.
• In this section we examine one such game that is
extremely popular in the gambling casinos: the
game of craps.
• In its most basic form, the game of craps is played
as follows.
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Simulating the game of craps
continued
– A player rolls two dice and observes the sum of the two
sides turned up. If this sum is 7 or 11, the player wins
immediately.
– If the sum is 2, 3, or 12, the player loses immediately.
– Otherwise, if this sum is any other number (4, 5, 6, 8, 9,
or 10), that number becomes the player’s point.
– Then the dice are thrown repeatedly until the sum is the
player’s point or 7.
– In case the player’s point occurs before a 7, the player
wins. But if a 7 occurs before the point, the player loses.
– Example 11.14 uses simulation to determine the
properties of this game.
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Simulating the NCAA
basketball tournament
• Each year the suspense reaches new levels as
“March Madness” approaches, the time of the
NCAA Basketball Tournament.
• Which of the 64 teams in the tournament will reach
the “Sweet Sixteen,” which will go on to the
prestigious “Final Four,” and which team will be
crowned champion?
• We share this simulation in the following example.
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Using TopRank with @RISK
for powerful modeling
• In this section, we illustrate how another Palisade
Decision Tools add-in, TopRank, can be used
together with @RISK as a very powerful modeling
combination.
• As you have seen, @RISK introduces uncertainty
explicitly into a spreadsheet model by allowing
several inputs to have probability distributions.
• TopRank is a what-if tool that allows you to see
which of many inputs have large effects on an
output variable.
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Using TopRank continued
• Example 11.16, which illustrates how TopRank and @RISK
can work in tandem, is an extremely important one.
• Simulation in the business world is often used to analyze
potential products.
• The profitability of a new product is highly uncertain
because it depends on many uncertain quantities.
• Many companies begin the analysis of every new product
by determining the uncertain quantities that can affect the
profitability of the product.
• This analysis is often the deciding factor in whether the
product is developed and marketed.
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Tornado charts
• Perhaps the best way to understand TopRank
results is through a tornado chart.
• Each bar in the chart indicates the variation in NPV
as an individual input varies its minimum and
maximum.
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Tornado charts continued
• Because the longer bars are always on the top and
the shortest ones are always on the bottom, the
inputs at the top of the chart are always the most
important ones.
• In this case the five most important inputs are
product lifetime, unit price, initial demand, discount
rate, and unit production cost.
• Clearly, if SimTex is going to simulate the
product’s NPV, it should spend most of its time
accurately assessing the distribution of these five
key inputs.
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Tornado charts continued
• In contrast, the tornado chart indicates that annual
fixed cost and salvage value have virtually no
effect on NPV. Therefore, little effort should be
spent trying to estimate their values accurately the base-case value will suffice.
• Before proceeding to a simulation, we mention two
other chart type available in TopRank: spider
charts and sensitivity charts. Let’s look at a spider
chart first.
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Spider charts
• A spider chart for the SimTex model appears below.
• This chart is fairly straightforward.
• For each of the five inputs, there is a curve that shows the
percentage change in NPV as a function of the percentage
change in the input - over the range we specified for the
input.
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Spider charts continued
• From the spider chart we learn (not surprisingly) that
changes in unit price, unit cost, and initial demand result in
linear changes in NPV.
• Also, a 1% increase in unit price results in a larger
percentage increase in NPV than does a 1% percentage
increase in demand.
• As the discount rate increases, NPV decreases, but the
rate of decrease slows; after a while increases in the
discount rate cannot decrease NPV much further.
Increases in product lifetime appear to increase product
NPV in a complex nonlinear fashion.
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Sensitivity charts
• This chart type is similar to the spider chart, except that it
shows one input only. Also it shows actual values rather
than percentage changes.
• To get a sensitivity chart for any input/output combination,
click on the desired input and output in TopRank Results
window, and then click on the Graph button and select
Sensitivity option.
• For example, this graph shows NPV versus product
lifetime.
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@RISK simulation
• The sensitivity analysis with TopRank has
indicated that the five key drivers of NPV are
product lifetime, unit price, unit cost, initial product
demand, and discount rate.
• We will run an @RISK simulation of this model to
estimate the distribution of NPV earned by
Biathnon.
• We will keep all inputs other than the five key
inputs fixed at their base values, and we will use
@RISK functions for the key inputs.
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@RISK simulation continued
• Actually we will use random functions for a product
lifetime, unit price, unit cost and initial demand,
and will vary discount rate systematically with the
RISKSIMTABLE function.
• Which probability distributions should we use to
model the product lifetime, unit price, unit cost, and
initial demand inputs?
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@RISK simulation continued
• There are several ways to proceed in general.
– First, if we have a lot of historical data on any input, we
can use the fitting capabilities of @RISK to fit a
distribution to the historical data.
– Second, we can use @RISK model window to examine
shapes of potential distributions that look like good
candidates.
– Finally, we can choose a simple distribution that
management has confidence in and assess its
parameters.
• We chose the latter approach, using a triangular
distribution for each of the random inputs.
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@RISK simulation continued
• The use of a triangular random variable is common at many
companies. The triangular distribution is often used
because, unlike the normal distribution, it makes no
assumption that the distribution of the uncertain quantity is
symmetric about the mean or most likely value.
• To assess a triangular distribution for any input, all we need
are the minimum, most likely, and maximum values for the
input. We use the same values of these that we used in the
TopRank analysis. They are shown in columns E-G of the
table on the next slide.
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Conclusion
• We claimed in the previous chapter that
spreadsheet simulation, especially together with
an add-in like @RISK, is a very powerful tool.
• After seeing the examples in this chapter, you
should now appreciate how powerful and flexible
simulation is.
• Unlike Solver optimization models, where you
often make simplifying assumptions to achieve
linearity, say, you can allow virtually anything in
simulation models.
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Conclusion continued
• All you need to do is relate output cells to input
cells with appropriate formulas, where any of the
input cells can contain probability distributions to
reflect uncertainty.
• The results of the simulation then show the
distribution of any particular output.
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Summary of key terms
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End of Chapter 11