 Engineering 90
 Dr. Gregory Crawford
Four Ways to do Project Analysis  Statistical / Regression Analysis (forecasting)
 Sensitivity Analysis
 Monte Carlo Simulations
 Decision Trees
What’s the difference?  Each shows a manager different aspects of the decision he/she faces:
 Regression / Statistical Forecasting is a way to estimate future sales growth based on current or past performances.
 Sensitivity Analysis shows her how much each variable affects the NPV.
 Monte Carlo gives a statistical breakdown of the possible outcomes.
 Decision Trees are visual representations of the average outcome.
Regression and Statistical Forecasting  Mathematically model past sales of either same product or similar product
 Projects future sales as a function of these past sales with respect to time
 We will talk about two types of regression
 Linear Regression
 Polynomial Regression
 (but there are many more, logarithmic, exponential, etc)
 Quick primer on Statistics and Probability
 Definitions:
 Expected Value of x: E(x) = ; as P(x) represents the probability of x.
 (Note that = 1 and that the because P(x) represents a probability density function)

 Variance of x:

 Standard Deviation = the sq. root of the variance

 Median = “the center of the set of numbers”; or the point m such that P(x < m)< ½ and P(x > m)> ½ .
Simple Example – Widget Sales Widgets (cont.)  Suppose Greg plans on releasing the next generation widget. (old widget data on previous page)
 He already has sales of:
 Year 1 = $0.5 million
 Year 2 = $5.1 million
 Year 3 = $13.0 million
 What should he estimate his future sales to be?
Mmmm… more widgets  Propose that sales is:
 Assume f(x) = 6t  5, where t = number of years
Regression – Least Squares  Is there a formal way to get this estimation function?
 Fit a line such that the square of the vertical deviations between the function and the data points is minimized
 Derivation of Least Squares Regression
 Assume you have an arbitrary straight line:
 y = B1 + B2x [note, this is simply y = mx + b]
 Let q = the distance between the function point and the actual data point; therefore
 q = y – (B1 + B2x)
 The square of q is = [ y – (B1 + B2)]2
 The sum of all of the squares of q we will denote Q
Derivation Continued…  Recall, we want to minimize Q, so using partial derivatives and setting them = 0 we get
 Setting these equations equal to zero and solving for B1 and B2 gives us...
 Which will yield the equation y = B1 + B2x ?
 x = Average x, y = Average y
Using Microsoft Excel for Regression  Of course, no one really does this by hand any more…
 Plot your data points in adjacent columns
 Use “=forecast(x, previous data f(x), previous data x)”
 This is a linearfit regression command
What’s wrong with this picture?  First, it is unrealistic to have infinitely rising sales
 Second, it doesn’t fit with Greg’s previous widget product’s sales, which eventually decline
 Let’s try to find a function that takes the first set of widget sales into account.
F(x) = ax2 + bx + c  In fact, the function is f(x) = .8(x4)2 + 13
Least Squares Regression for Polynomials  (You are not responsible for this material)
 Minimize the sum Q of the squares of these differences:

 This will yield a (k+1)x(k+1) matrix of equations that can be solved for Bi, yielding the equation:
 f(x) = B1 + B2x + B3x2 + … + Bnx(n1)
Summary  Least squares regression is a common scientific & engineering practice.
 In business, it can be used to forecast possible future trends.
 You’re responsible for linear least squares regression only.
Sensitivity Analysis  Set up an Excel spreadsheet that will calculate your projects NPV
 Individually change your assumptions to see how the NPV changes with respect to different variables
 Helps to determine how much to spend on additional information
Jalopy Motor’s Example  Suppose that you forecast the following for an electric scooter project:
 Market Size of .9 (worst case)– 1.1 million (best case) customers
 Market Share of between 4% (wc)and 16% (bc) after the first year
 Unit price between $3,500 (wc) and $3,800 (bc)
 Unit cost (variable) between $3,600 (wc) and $2,750 (bc)
 Fixed costs between $40 (wc) and 20 million (bc).
 From Principles of Corporate Finance, (c) 1996 Brealey/Myers
Jalopy Example (cont.) Explanations  NPV is calculated by subtracting the initial investment from the sum of yearly $30M net cash flow.
 NPV =  150 + 30 [1 – (1.1)10 / .1] = $34.3
 Net Cash Flow is defined as net profit plus the tax savings you get from depreciation
Jalopy Example (cont.) Monte Carlo Simulations  Simulations are a tool for considering all possibilities
 Step 1 – Model the project (where are choices made, where are the chances)
 Step 2 – Assign Probabilities to outcomes (assumption)
 Step 3 – Simulate the Cash Flows (use a computer simulation program)
 The result will be a probability distribution.
Monte Carlo Simulation (cont.) (test scores example) Equations (Mmmm… Math)  Normal Distribution: f(x  and )

 Standard Normal Distributions have a mean (x) of 0 and a variance (2) of 1
Monte Carlo Simulations (projected cash flow)  The distribution shows the percentage of times the program predicts NVP above cost of project.
Summary Monte Carlo  You are not responsible for this on the test.
 Statistical breakdown of possible outcomes.
 Dealing with continuous distribution.
What is a Decision Tree?  A Visual Representation of Choices, Consequences, Probabilities, and Opportunities.
 A Way of Breaking Down Complicated Situations Down to EasiertoUnderstand Scenarios.
Easy Example  A Decision Tree with two choices.
 Go to Graduate School to get my MBA.
 Go to Work “in the Real World”
Notation Used in Decision Trees  A box is used to show a choice that the manager has to make.
 A circle is used to show that a probability outcome will occur.
 Lines connect outcomes to their choice or probability outcome.
 What are some of the costs we should take into account when deciding whether or not to go to business school?
 Tuition and Fees
 Rent / Food / etc.
 Opportunity cost of salary
 Anticipated future earnings
Simple Decision Tree Model  Go to Graduate School to get my MBA.
 Go to Work “in the Real World”
 2 Years of tuition: $55,000, 2 years of Room/Board: $20,000; 2 years of Opportunity Cost of Salary = $100,000 Total = $175,000.
 PLUS Anticipated 5 year salary after Business School = $600,000.
 NPV (business school) = $600,000  $175,000 = $425,000
 First two year salary = $100,000 (from above), minus expenses of $20,000.
 Final five year salary = $330,000
 NPV (no bschool) = $410,000
 Is this a realistic model?
 What is missing?
The Yeaple Study (1994)  According to Ronald Yeaple, it is only profitable to go to one of the top 15 Business Schools – otherwise you have a NEGATIVE NPV!
 (Economist, Aug. 6, 1994)
 Benefits of Learning
 School Net Value ($) Harvard $148,378 Chicago $106,378 Stanford $97,462 MIT (Sloan) $85,736 Yale $83,775 Northwestern $53,526 Berkeley $54,101 Wharton $59,486 UCLA $55,088 Virginia $30,046 Cornell $30,974 Michigan $21,502 Dartmouth $22,509 Carnegie Mellon $18,679 Texas $17,459 Rochester  $307 Indiana  $3,315 North Carolina  $4,565 Duke  $17,631 NYU  $3,749
Things he may have missed  Future uncertainty (interest rates, future salary, etc)
 Cost of Living differences
 Type of Job [utility function = f($, enjoyment)]
 Girlfriend / Boyfriend / Family concerns
 Others?
 Utility Function = f ($, enjoyment, family, location, type of job / prestige, gender, age, race) Human Factors Considerations
Mary’s Factory  Mary is a manager of a gadget factory. Her factory has been quite successful the past three years. She is wondering whether or not it is a good idea to expand her factory this year. The cost to expand her factory is $1.5M. If she does nothing and the economy stays good and people continue to buy lots of gadgets she expects $3M in revenue; while only $1M if the economy is bad.
 If she expands the factory, she expects to receive $6M if economy is good and $2M if economy is bad.
 She also assumes that there is a 40% chance of a good economy and a 60% chance of a bad economy.
 (a) Draw a Decision Tree showing these choices.
Decision Tree Example  Expand Factory
 Cost = $1.5 M
 Don’t Expand Factory
 Cost = $0
 40 % Chance of a Good Economy
 Profit = $6M
 60% Chance Bad Economy
 Profit = $2M
 Good Economy (40%)
 Profit = $3M
 Bad Economy (60%)
 Profit = $1M
 NPVExpand = (.4(6) + .6(2)) – 1.5 = $2.1M
 NPVNo Expand = .4(3) + .6(1) = $1.8M
 $2.1 > 1.8, therefore you should expand the factory
Example 2 – Joe’s Garage  Joe’s garage is considering hiring another mechanic. The mechanic would cost them an additional $50,000 / year in salary and benefits. If there are a lot of accidents in Providence this year, they anticipate making an additional $75,000 in net revenue. If there are not a lot of accidents, they could lose $20,000 off of last year’s total net revenues. Because of all the ice on the roads, Joe thinks that there will be a 70% chance of “a lot of accidents” and a 30% chance of “fewer accidents”. Assume if he doesn’t expand he will have the same revenue as last year.
 Draw a decision tree for Joe and tell him what he should do.
Example 2  Answer  Hire new mechanic
 Cost = $50,000
 Don’t hire new mechanic
 Cost = $0
 70% chance of an increase in accidents
 Profit = $70,000
 30% chance of a decrease in accidents
 Profit =  $20,000
 Estimated value of “Hire Mechanic” = NPV =.7(70,000) + .3( $20,000)  $50,000 =  $7,000
 Therefore you should not hire the mechanic
Mary’s Factory – With Options  A few days later she was told that if she expands, she can opt to either (a) expand the factory further if the economy is good which costs 1.5M, but will yield an additional $2M in profit when economy is good but only $1M when economy is bad, (b) abandon the project and sell the equipment she originally bought for $1.3M, or (c) do nothing.

 (b) Draw a decision tree to show these three options for each possible outcome, and compute the NPV for the expansion.
Decision Trees, with Options  Expand further – yielding $8M (but costing $1.5)
 Stay at new expanded levels – yielding $6M
 Reduce to old levels – yielding $3M (but saving $1.3  sell equipment)
 Expand further – yielding $3M (but costing $1.5)
 Stay at new expanded levels – yielding $2M
 Reduce to old levels – yielding $1M (but saving $1.3 in equipment cost)
Present Value of the Options  Good Economy
 Expand further = 8M – 1.5M = 6.5M
 Do nothing = 6M
 Abandon Project = 3M + 1.3M = 4.3M
 Bad Economy
 Expand further = 3M – 1.5M = 1.5M
 Do nothing = 2M
 Abandon Project = 1M + 1.3M = 2.3M
 So the NPV of Expanding the factory is:
 NPVExpand = [.4(6.5) + .6(2.3)]  1.5M = $2.48M
 Therefore the value of the option is
 2.48 (new NPV) – 2.1 (old NPV) = $380,000
 You would pay up to this amount to exercise that option.

Mary’s Factory – Discounting  Before Mary takes this to her boss, she wants to account for the time value of money. The gadget company uses a 10% discount rate. The cost of expanding the factory is borne in year zero but the revenue streams are in year one.

 (c) Compute the NPV in part (a) again, this time account the time value of money in your analysis. Should she expand the factory?
Time Value of Money  Expand Factory
 Cost = $1.5 M
 Don’t Expand Factory
 Cost = $0
 40 % Chance of a Good Economy
 Profit = $6M
 60% Chance Bad Economy
 Profit = $2M
 Good Economy (40%)
 Profit = $3M
 Bad Economy (60%)
 Profit = $1M
Time Value of Money  Recall that the formula for discounting money as a
 function of time is: PV = S (1+i)n
 [where i = interest / discount rate; n = number of years /
 S = nominal value]

 So, in each scenario, we get the Present Value (PV) of the
 estimated net revenues:
 a) PV = 6(1.1)1 = $5,454,454
 b) PV = 2(1.1)1 = $1,818,181
 c) PV = 3(1.1)1 = $2,727,272
 d) PV = 1(1.1)1 = $0.909,091
Time Value of Money  Therefore, the PV of the revenue streams (once you account for the time value of money) are:
 PVExpand =.4(5.5M) + .6(1.82M) = $3.29M
 PVDon’t Ex. = 0.4(2.73) + 0.6(.910) = 1.638
 So, should you expand the factory?
 Yes, because the cost of the expansion is $1.5M, and that means the NPV = 3.29 – 1.5 = $1.79 > $1.64
 Note that since the cost of expansion is borne in year 0, you don’t discount it.
 Stephanie has a hardware store and she is deciding whether or not to buy Adler’s Hardware store on Wickendon Street. She can buy it for $400,000; however it would take one year to renovate, implement her computer inventory system, etc.
 The next year she expects to earn $600,000 if the economy is good and only $200,000 if the economy is bad. She estimates a 65% probability of a good economy and a 35% probability of a bad economy. If she doesn’t buy Adler’s she knows she will get $0 additional profits.
 Taking the time value of money into account, find the NPV of the project with a discount rate of 10%
Answer to Stephanie’s Problem  Buy Adler’s
 Cost = $400,000
 65 % Chance of a Good Economy
 Profit = $600,000
 35% Chance Bad Economy
 Profit = $200,000
Should she buy?  NPV of purchase =
 .65(600,000/1.1) + .35(200,000/1.1) – 400,000 = $18,181.82
 Therefore, she should do the project!
 What happens if the discount rate = 15%?
 The NPV = 0, so it probably is not worth it.
 What happens if the discount rate = 20%?
 The NPV =  $16,666.67; so you should not buy!
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