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Hi, I'm Sergey Savon and

I would like to welcome you to week 4 of our course on Modeling Risk and Realities.

Continues for building distributions like normal distribution or

uniform distribution over a convenient way of summarizing historical data and

describing uncertainty of future outcomes.

At the same time, using continuous distributions may make it difficult to

employ optimization toolkit for identifying the best decision.

In Week 4, we'll look at simulation as a way of enabling comparison

between different alternatives in settings where uncertainty is described

using continuous distributions.

The focus for our discussions this week will be the simulation toolkit.

In Session 1, we will talk about making decisions in high uncertainty settings

where random inputs are described by continuous probability distributions.

We'll use an example of a company that needs to design its new apartment building

in the face of the uncertain demand for different types of apartments.

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In Session 3, we will analyze simulation output and

discuss how we can use it to compare alternative decisions.

Just a reminder, when we looked at the decision making and uncertainty in Week 2,

we have used the scenario approach to modelling random variables.

Under the scenario approach, we have used a number of potential generalizations of

the random variable with the probability attached to each realization.

For example, we have used 20 equally likely scenarios,

to describe a daily return on the stock price.

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One of the attractive features of this approach to modeling random variables

is that it is easy under the scenario approach to calculate precise values of

various parameters that decision makers care about.

Such as the expected value we used as the reward measure or

the standard deviation, we used as a measure of risk.

2:27

But what do we do if the number of potential values that are random variable

modeling can take is infinite?

Such as when the random variable has a continuous distribution, normal,

uniform etcetera.

How do we calculate various performance measures, such as measures of reward and

risk, if the number of scenarios we have to account for is infinite?

Simulation is the approach that can be used in such cases.

Simulation works as follows.

We can use Excel to generate instances of random variables

coming from a number of continuous distributions, like normal distribution.

One instance, two instances, a thousand of instances if necessary.

If we use these instances as scenarios, we can generate estimates for the risk and

reward measures associated with any course of action.

3:18

For example, the value of the average profit calculated using a finite number of

scenarios generated from a continuous distribution will, of course,

be an approximation and the estimate of the true expected profit.

Because that true value can be obtained only if we use

infinite number of scenarios covering the entire continuous distribution.

But the largest of the scenarios we use,

the closer we should expect the estimate to be to the true expected value.

4:26

Apartments will be priced competitively and the company estimates

that the price that it plans to charge, the profit he will earn for

a regular apartment sold during the next year will be $500,000.

And the profit it will earn for their luxury apartment,

sold during the next year, will be $900,000.

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On the other hand, while the company can control the price that it charges for

the apartments, it cannot really control the demand for the apartments.

In particular, it is possible that it may not be able to sell all of the apartments

over the next year.

At that point, the company will sell all of the remaining apartments

to a real estate investor at the much reduced profits.

In particular, if there are regular apartments left,

they will be disposed of at a profit of $100,00 each.

5:18

And if there are luxury apartments left,

they will be sold at the profit of $150,000 each.

Based on the analysis of historical trends and

expert estimates, Stargrove believes that the demand for

the regular apartments can be modeled as a normally distributed

random variable with mean of 90 and a standard deviation of 25.

In other words, it expects to have 90 buyers for regular apartments, but

also thinks that the actual number of buyers they will see next year,

can be quite far away from that expectation.

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The two kinds of demand, I assume to be independent random variables.

This in particular means, that the correlation between random demands for

regular and luxury apartments is 0.

Now, there are a couple of caveats to use in a normal distribution

to model non negative integer demand values.

The instances of normal random variable can take

fractional as well as negative values and we need to be a bit careful

with those normal random values if we were to use them to model the demand.

We will look at this issue again when we set up our simulation.

The company assumes that if it runs out of the apartments to sell,

of either kind, the extra customers will be lost to competition.

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The company also thinks that its regular and

luxury customers are two very distinct groups.

In particular, Stargrove thinks that there will be no switching of regular customers

to luxury apartments or of the luxury customers to regular apartments.

The regular customers will not be able to afford a luxury apartment and

luxury customers will not settle for a regular apartment.

This assumptions will help Stargrove to calculate how many apartments of

each kind it will sell during the next year for

any combination of the demand and the number of apartments it decides to build.

It will also help the company to calculate the number of apartments,

if any, it will have to sell to the real estate investor at the reduced profit.

For regular apartments, if Stargrove builds R of them, and

the demand for the regular apartments turns out to be DR,

it can calculate both the numbers of apartments sold at the high profit

of $500,000 and at the low profit of $100,000.

For the high profit sales, Stargrove cannot sell more than what it builds, R.

And they cannot sell more than what's demanded, that's DR.

So, it will sell the minimum of the two numbers.

For example, if Stargrove builds 96 regular apartments,

that's 12 regular floors with 8 apartments on each floor, and

the number of buyers of regular apartments turns out to be 90.

The number of high profit sales the company will

make is minimum of (96, 90) = 90.

If on the other hand, the company builds 96 regular apartments and

the number of potential buyers turns out to be 100,

Stargrove will manage to sell 96, which is the minimum of 96 and 100.

The number of the regular apartments that the company will have to sell

at the low profit value, is the difference between what it builds R and

what it sells at the high profit value, which is the minimum of (R, Dr).

For example, if the company builds 96 regular apartments and the demand for

regular apartments is 90, then the company will have to sell 96- 90,

6 apartments to the real estate investor at low profit.

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The corresponding calculation for the luxury apartments is similar.

In particular the number of the apartment sold at the high profit of $900,000

is determined by the minimum of the number of apartments the company builds, L,

and the demand for those apartments, DL.

Also, the number of luxury apartments that have to be sold at the low profit of

$150,000 is the difference between the number of apartments

the company builds and the number of apartments it sells at high profit,

which is the minimum of L and DL.

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In order to see how to use simulation to make the best decisions,

we will first look at how simulation can be used to evaluate a particular decision.

Suppose that Stargrove decides to build 12 regular floors and 3 luxury floors,

so it will have 96 regular apartments, and 12 luxury apartments.

The company is interested in figuring out the profitability of this decision.

Given that the demand for each apartment type is random,

company's profit pi will also be random.

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So, let's select the expected profit as a reward that the company would like to

maximize, and the probability that the actual

profit falls below $45 million as a measure of risk.

As you can see, we have decided to use in our analysis not a standard deviation of

profits, but a different risk measure.

In other words, we're considering a situation where a company

does not really worry about the standard deviation of its profits, but

rather has a profitability goal it wants to meet, and

it wants to make sure that a chance that it will miss this goal is not too high.

Let's talk a little bit about the terminology associated with simulations.

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In algebraic terms, the profit as a function of these two random demands

can be expressed as follows.

Pi = $500,000 times the minimum of (DR,R)

+ $900,000 times minimum (DL,L) +

$100,000 times the remaining of regular apartments

+ $150,000 times the remaining luxury apartments.

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In this expression, we have four terms, two terms for each kind of demand.

The first two terms express the profits Stargrove gets from high profit sales.

$500,000 for each regular apartment it sells during the next year and

$900,000 for each luxury apartment it sells during the next year.

The third and fourth terms express the low profit sales to the real estate investor.

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The random variables for the demand values for DR and

DL, are called random inputs into a simulation.

They represent the factors that the decision maker does not fully control.

The random profit value is called random output of a simulation.

It represents the random quantity that a decision maker is interested in.

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The simulation is based on generating instances of the random inputs and

calculating the corresponding instances of their random outputs.

In other words,

simulation is a mechanism that uses the probability distribution of the random

inputs to approximate the probability distribution of the random output.

So, if we have an algebraic formula that expresses the random output pi

as the function of random inputs DR and DL, the task of the simulation is to

figure out what the distribution of pi is in particular.

We want to use simulation to figure out what is the reward,

the expected value of pi, and risk, the probability that pi falls below

a threshold associated with a particular decision, R and L.

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Well, we can look at that and

ask ourselves, if we want to know what the expected profit is,

can't we just plug in the expected values for DR and DL into that formula?

In other words, if we want to get the expected value of the random output,

can we just use the expected values of the random inputs in the formula

that connects random inputs and random outputs?

14:03

The answer is, in general, not really.

It is a tempting thing to do, as we do not need to run any simulation to do that but

the number we get as a result.

Maybe quite some distance away from the correct value.

The point I'm making here is that in general simulation is a necessary tool for

evaluating the reward and risk measures in uncertain settings.

And one should be very careful with attempting shortcuts.

Like, replacing random quantities by the expected values.

In order to appreciate this point, let's have a look at a simple example.

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Suppose that the demand for regular apartment's DR takes two values, 65,

and 115, each with probability of 0.5.

And the demand for luxury apartments deal takes the values of 7 and

13, each with probability 0.5.

Note that the expected demand on the standard deviation values for

both demand distributions are the same as the ones that Stargrove uses for

modeling demand distributions using normal distribution form.

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Since the demand value for regular apartments DR can take two values, 65 and

115, with equal probabilities, and the demand value for luxury apartments can

also take two values, 7 and 13, also with equal probabilities.

And those demand random variables take these values independently

from each other.

That's a total of four possibilities for the pair of demand values, DR and DL,

each possibility being realized with a probability of 0.5 * 0.5 = 0.25.

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We'll go with these four possibilities one by one.

The first possibility is that both demands simultaneously take the lowest values,

65 for DR, and 7 for DL.

In this case, 65 regular apartments are sold at the profit of $500,000 each,

and the remaining 31 at the profit of $100,000 each.

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In a similar fashion, 7 luxury apartments are sold at the profit of $900,000 each,

and the remaining 5 at the lower profit of $150,000 each.

The profit value in this scenario is $42,650,000.

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The second possibility is that the demand DR takes the high value,

of 115, and DL takes the low value, 7.

The profit value in this scenario is,

500,000*96 + 900,000*7

+ 100,000 *0+150,000*5 = $55,050,000.

The third possibility is for DR to take the low value 65 and

for DL to take the high value 13.

The profit value in this scenario is 500,000*65

+ 900,000*12 + 100,000*31

+ 150,000*0 = 46,400,000.

Finally, the fourth scenario is when both demands take their highest values.

The profit in this case is 500,000*96 +

900,000*12 + 100,000*0 +

150,000*0 which is 58,800,000.

The expected profit is the average of the profit values in

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four scenarios, and that's $50,725,000.

As you could see,

the true value of the expected profit is much lower than the value

we obtain by replacing random variables by the expected values in formula for profit.

We devoted session 1 of the fourth week to an introduction to a decision making in

settings where future rewards and

risks, must be evaluated using continuous probability distributions.

Next, we will focus on the mechanics of simulation.

We will set up the simulation and run it using Excel.