1. ESD.70J Engineering Economy Module - Session 31 ESD.70J Engineering Economy Module Fall 2005 Session Three Alex Fadeev - afadeev@mit.edu Link for this PPT: http://ardent.mit.edu/real_options/ROcse_Excel_latest/ExcelSession3.pdf

2. ESD.70J Engineering Economy Module - Session 32 One note on Session Two If you Excel crashes during simulation runs, try to input numbers (0’s or whatever) into the dummy input values in a column (or row). Do not leave the area of input values blank in the data table.

3. ESD.70J Engineering Economy Module - Session 33 Question from Session Two Last time we used evenly distributed random variables to model the demand uncertainty. This implies equal probability of median as well as extreme high and low outcomes. It’s not too hard to imagine why this is not very realistic. What alternative models for demand uncertainties should we try?

4. ESD.70J Engineering Economy Module - Session 34 Session three – Modeling Uncertainties Generate random numbers from various distributions Random variables as time function (stochastic processes) –Geometric Brownian Motion –Mean Reversion –S-curve Statistical analysis to obtain key parameters from a data set

5. ESD.70J Engineering Economy Module - Session 35 Generate random numbers from various distributions How to generate random numbers from normal distribution? –Using norminv(rand(), μ, σ) (norminv stands for “the inverse of the normal cumulative distribution”) –μ is the mean –σ is the standard deviation In the data table output formula cell (B1 in “Simu” sheet of 1.xls) type in “=norminv(rand(), 5, 1)”. Press “F9”, see what happens) Link for Excel: http://ardent.mit.edu/real_options/ROcse_Excel_latest/Session3-1.xls

6. ESD.70J Engineering Economy Module - Session 36 How to generate random numbers from triangular distribution Triangular distribution could work as an approximation of other distribution (e.g. normal, Weibull, and Beta) Try “=rand()+rand()” in the data table output formula cell (B1 in “Simu” sheet of 1.xls), press “F9”, see what happens.

7. ESD.70J Engineering Economy Module - Session 37 How to generate random numbers from lognormal distribution A random variable X has a lognormal distribution if its natural logarithm has a normal distribution Using loginv(rand(), log_μ, log_σ) –log_μ is the log mean –log_σ is the log standard deviation In the data table output formula cell (B1 in “Simu” sheet of 1.xls) type in “=loginv(rand(), 2, 0.3)”. Press “F9”, see what happens)

8. ESD.70J Engineering Economy Module - Session 38 From probability to stochastic processes We can describe the probability density function (PDF) of random variable x, or f(x) Apparently, the distribution of a random variable in the future is not independent from what happens now Life is random in a non-random way… Time Year 1Year 2 Year 3

9. ESD.70J Engineering Economy Module - Session 39 From probability to stochastic processes (Cont) We have to study the time function of distribution of random variable x, or f(x,t) That is a stochastic process, or in language other than mathematics jargon: TREND + UNCERTAINTY

10. ESD.70J Engineering Economy Module - Session 310 Three stochastic models Geometric Brownian Motion Mean-reversion S-Curve

11. ESD.70J Engineering Economy Module - Session 311 Geometric Brownian Motion Brownian motion is a random walk –the motion of a pollen in water –a drunk walks in Boston Common Geometric means the change rate is Brownian, not the subject itself –For example, in Geometric Brownian Motion model, the stock price itself is not a random work, but the return on the stock is

12. ESD.70J Engineering Economy Module - Session 312 Simulate a stock price Google’s stock price is $288.45 per class A common share on 9/2/05, assuming volatility of the stock price is 20% per quarter Volatility can be approximately taken as the standard deviation of quarterly return on stock Assume quarterly expected return of Google stock is 4%

13. ESD.70J Engineering Economy Module - Session 313 Simulate a stock price (Cont) TimeStock PriceRandom Draw from standardized normal distribution 1) Realized return (expected return + random draw * volatility) Sep 05$288.45 Dec 05 Mar 06 Jun 06 Sep 06 Complete the following table for Google stock: 1). Standardized normal distribution with mean 0 and standard deviation 1

14. ESD.70J Engineering Economy Module - Session 314 Using Spreadsheet to simulate Google stock Follow the instructions, step by step: 1.Open a new worksheet, name it “GBM” 2.Copy or input the table in the previous slide into Excel, with “Time” as cell A1 3.Type “=norminv(rand(),0,1)” in cell C2, and drag down to cell C6 4.Type “=0.04+0.20*C2” in cell D2, and drag down to cell D6 5.Type “=B2*(1+D2)” in cell B3, and drag down to cell B6 6.Click “Chart” under “Insert” menu

15. ESD.70J Engineering Economy Module - Session 315 Using Spreadsheet to simulate Google stock (Cont) 7.“Standard types” select “Line”, “Chart sub-type” select whichever you like, click “Next” 8.“Data range” select “=GBM!$A$1:$B$6”, click “Next” 9.“Chart options” select whatever pleases you, click “Next” 10.Choose “As object in” and click “Finish” 11.Press “F9” several times to see what happens.

16. ESD.70J Engineering Economy Module - Session 316 Brownian Motion (Again) This is the standard model for modeling stock price behavior in finance theory, and lots of other uncertainties (because of the Central Limit Theorem) Mathematic form for Geometric Brownian Motion (you do not have to know) where S is the stock price, μ is the expected return on the stock, σ is the volatility of the stock price, and dz is the basic Wiener process

17. ESD.70J Engineering Economy Module - Session 317 Mean-reversion Unlike Geometric Brownian Motion that grows forever, some processes have the tendency to –fluctuate around a mean –the farther away from the mean, the better the possibility to revert to the mean –the speed of mean reversion can be measured by a parameter η

18. ESD.70J Engineering Economy Module - Session 318 Simulate interest rate In finance, people usually use mean reversion to model the behavior of interest rate Suppose the interest rate r is 4% now, the speed of mean reversion η is 0.3, the long-term mean is 7%, the volatility σ is 1.5% per year Expected mean reversion is:

19. ESD.70J Engineering Economy Module - Session 319 Simulate interest rate (Cont) TimeInterest rateRandom Draw from standardized normal distribution Realized return (expected reversion + random draw * volatility) 20044% 2005 2006 2007 2008 Complete the following table for interest rate:

20. ESD.70J Engineering Economy Module - Session 320 Using Spreadsheet to simulate interest rate Follow the instructions, step by step: 1.Open a new worksheet, name it “Int” 2.Copy or input the table in the previous slide into Excel, with “Time” as cell A1 3.Type “=norminv(rand(),0,1)” in cell C2, and drag down to cell C6 4.Type “=0.3*(0.07-B2)+C2*0.015” in cell D2, and drag down to cell D6 5.Type “=B2+D2” in cell B3, and drag down to cell B6 6.Click “Chart” under “Insert” menu

21. ESD.70J Engineering Economy Module - Session 321 Using Spreadsheet to simulate interest rate (Cont) 7.“Standard types” select “XY(Scatter)”, “Chart sub-type” select any one with line, click “Next” 8.“Data range” select “=GBM!$A$1:$B$6”, click “Next” 9.“Chart options” select whatever pleases you, click “Next” 10.Choose “As object in” and click “Finish” 11.Press “F9” several times to see what happens.

22. ESD.70J Engineering Economy Module - Session 322 Mean reversion (Again) Mean reversion has many applications besides modeling interest rate behavior in finance theory Mathematic form (you do not have to know) where r is the interest rate, η is the speed of mean reversion, is the long-term mean, σ is the volatility, and dz is the basic Wiener process

23. ESD.70J Engineering Economy Module - Session 323 S-curve Many uncertain variables display the S- curve shape Time For example, demand for a new technology initially grows slowly, then the demand explodes exponentially and finally decays as it approaches a natural saturation limit

24. ESD.70J Engineering Economy Module - Session 324 Modeling S-curve Deterministically Parameters: –Demand at year 0 –Demand at year T –The limit of demand, or demand at time Model: α and β can be derived from demand at year 0 and year T

25. ESD.70J Engineering Economy Module - Session 325 Modeling S-curve dynamically We can estimate incorrectly the initial demand, demand at year T, and the limit of demand, so all of these are random variables The growth every year is subject to an additional annual volatility

26. ESD.70J Engineering Economy Module - Session 326 S-curve example Demand(0) = 80 (may differ plus or minus 20%) Demand(10) = 1000 (may differ plus or minus 40%) Limit of demand = 1600 (May differ plus or minus 40%, not less than (Demand(10)+100)) Annual volatility is 10% Link for Excel: http://ardent.mit.edu/real_options/ROcse_Excel_latest/Session3-2.xls

27. ESD.70J Engineering Economy Module - Session 327 Big vs. small? We talked about the following models today –Normal –LogNormal –Geometric Brownian Motion –Mean Reversion –S-curve Which one is more appropriate for our demand modeling problem? Why?

28. ESD.70J Engineering Economy Module - Session 328 Obtaining key parameters from data set Knowing the models is only a start, how to obtain good parameters is critical Otherwise – GIGO In many cases, data is scarce for interesting decision modeling problems. A good everyday habit to note good sources of data helps.

29. ESD.70J Engineering Economy Module - Session 329 Example We simulated the movement of Google stock price using the expected quarterly return of 4% and quarterly volatility of 20%. Is it reasonable? When Google IPO-ed last fall, there was no historical data to analyze. Solution - use a comparable stock, like Yahoo, to estimate expected Google volatility.

30. ESD.70J Engineering Economy Module - Session 330 Example (Cont) 1.Go to “Yahoo” sheet of 1.xls 2.Since what we have is the stock price, we need to get the quarterly returns 3.Use function “Average(number1, number2,…)” to get the mean of quarterly returns 4.Use function “Stdev(number1, number2,…)” to get quarterly volatility 5.What are your results?

31. ESD.70J Engineering Economy Module - Session 331 Issues in modeling Do not trust the model – this is the presumption for using any model. –Highly manipulative models are prone (if not doomed) to be misleading, always think how easy the model can generate the opposite conclusion –Check sensitivity of input parameters extensively Nevertheless, dynamic models offer great insights, though we should be very cautious of their numerical results In some sense, it is more of a way of thinking and communication

32. ESD.70J Engineering Economy Module - Session 332 Summary Generate random numbers from various distributions Random variables as time function (stochastic processes) –Geometric Brownian Motion –Mean Reversion –S-curve Statistical analysis to obtain key parameters from data set

33. ESD.70J Engineering Economy Module - Session 333 Next class… The course so far has covered ways to model the uncertainty. Modeling is passive. As human being, we have the capacity to manage uncertainties proactively. This capacity is called flexibility and contingency planning. The next class we’ll finally explore way to model the flexibility.