0:00

Hi, I'm Sergey Savon and

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

Â Continues for building distributions like normal distribution or

Â uniform distribution over a convenient way of summarizing historical data and

Â describing uncertainty of future outcomes.

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

Â employ optimization toolkit for identifying the best decision.

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

Â between different alternatives in settings where uncertainty is described

Â using continuous distributions.

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

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

Â where random inputs are described by continuous probability distributions.

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

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

Â 1:06

In Session 3, we will analyze simulation output and

Â discuss how we can use it to compare alternative decisions.

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

Â we have used the scenario approach to modelling random variables.

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

Â the random variable with the probability attached to each realization.

Â For example, we have used 20 equally likely scenarios,

Â to describe a daily return on the stock price.

Â 1:38

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

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

Â various parameters that decision makers care about.

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

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

Â 2:27

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

Â modeling can take is infinite?

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

Â uniform etcetera.

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

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

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

Â Simulation works as follows.

Â We can use Excel to generate instances of random variables

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

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

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

Â reward measures associated with any course of action.

Â 3:18

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

Â scenarios generated from a continuous distribution will, of course,

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

Â Because that true value can be obtained only if we use

Â infinite number of scenarios covering the entire continuous distribution.

Â But the largest of the scenarios we use,

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

Â 4:26

Apartments will be priced competitively and the company estimates

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

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

Â And the profit it will earn for their luxury apartment,

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

Â 4:47

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

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

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

Â over the next year.

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

Â to a real estate investor at the much reduced profits.

Â In particular, if there are regular apartments left,

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

Â 5:18

And if there are luxury apartments left,

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

Â Based on the analysis of historical trends and

Â expert estimates, Stargrove believes that the demand for

Â the regular apartments can be modeled as a normally distributed

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

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

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

Â can be quite far away from that expectation.

Â 6:11

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

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

Â regular and luxury apartments is 0.

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

Â to model non negative integer demand values.

Â The instances of normal random variable can take

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

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

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

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

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

Â 6:57

The company also thinks that its regular and

Â luxury customers are two very distinct groups.

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

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

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

Â luxury customers will not settle for a regular apartment.

Â This assumptions will help Stargrove to calculate how many apartments of

Â each kind it will sell during the next year for

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

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

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

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

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

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

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

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

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

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

Â For example, if Stargrove builds 96 regular apartments,

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

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

Â The number of high profit sales the company will

Â make is minimum of (96, 90) = 90.

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

Â the number of potential buyers turns out to be 100,

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

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

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

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

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

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

Â 6 apartments to the real estate investor at low profit.

Â 9:34

The corresponding calculation for the luxury apartments is similar.

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

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

Â and the demand for those apartments, DL.

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

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

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

Â which is the minimum of L and DL.

Â 10:07

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

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

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

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

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

Â Given that the demand for each apartment type is random,

Â company's profit pi will also be random.

Â 10:41

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

Â maximize, and the probability that the actual

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

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

Â profits, but a different risk measure.

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

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

Â rather has a profitability goal it wants to meet, and

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

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

Â 11:38

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

Â can be expressed as follows.

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

Â + $900,000 times minimum (DL,L) +

Â $100,000 times the remaining of regular apartments

Â + $150,000 times the remaining luxury apartments.

Â 12:08

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

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

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

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

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

Â 12:33

The random variables for the demand values for DR and

Â DL, are called random inputs into a simulation.

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

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

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

Â 12:54

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

Â calculating the corresponding instances of their random outputs.

Â In other words,

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

Â inputs to approximate the probability distribution of the random output.

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

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

Â figure out what the distribution of pi is in particular.

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

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

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

Â 13:42

Well, we can look at that and

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

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

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

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

Â that connects random inputs and random outputs?

Â 14:03

The answer is, in general, not really.

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

Â the number we get as a result.

Â Maybe quite some distance away from the correct value.

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

Â evaluating the reward and risk measures in uncertain settings.

Â And one should be very careful with attempting shortcuts.

Â Like, replacing random quantities by the expected values.

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

Â 14:38

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

Â and 115, each with probability of 0.5.

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

Â 13, each with probability 0.5.

Â Note that the expected demand on the standard deviation values for

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

Â modeling demand distributions using normal distribution form.

Â 15:40

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

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

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

Â And those demand random variables take these values independently

Â from each other.

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

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

Â 16:11

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

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

Â 65 for DR, and 7 for DL.

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

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

Â 16:33

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

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

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

Â 16:52

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

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

Â The profit value in this scenario is,

Â 500,000*96 + 900,000*7

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

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

Â for DL to take the high value 13.

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

Â + 900,000*12 + 100,000*31

Â + 150,000*0 = 46,400,000.

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

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

Â 900,000*12 + 100,000*0 +

Â 150,000*0 which is 58,800,000.

Â The expected profit is the average of the profit values in

Â 18:10

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

Â As you could see,

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

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

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

Â settings where future rewards and

Â risks, must be evaluated using continuous probability distributions.

Â Next, we will focus on the mechanics of simulation.

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

Â