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MIT ESD 70J - Lecture Notes

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1ESD.70J Engineering Economy Module - Session 3 1ESD.70J Engineering EconomyFall 2007Session ThreeMichel-Alexandre Cardin – [email protected]. Richard de Neufville – [email protected] Engineering Economy Module - Session 3 2Question from Session TwoYesterday we used uniformly distributed random variables to model uncertain demand. This implies identical probability of median as well as extreme high and low outcomes. This is not very realistic…⇒ What alternative probability distribution models should we use to sample demand?2ESD.70J Engineering Economy Module - Session 3 3Session three – Modeling Uncertainty• Objectives:– Generate random numbers from various distributions (Normal, Lognormal, etc)• So you can incorporate in your model as you wish– Generate and understand random variables that evolve through time (stochastic processes)• Geometric Brownian Motion, Mean Reversion, S-curveESD.70J Engineering Economy Module - Session 3 4Open ESD70session3-1Part1.xls(Two parts because RAND() calls and graphs take long to compute and update for every Data Table iteration…)http://ardent.mit.edu/real_options/ROcse_Excel_latest/ESD 70 2007/ESD70session3-1Part1.xlsAbout random number generation3ESD.70J Engineering Economy Module - Session 3 5About random number generation• Generate normally distributed random numbers:– Use NORMINV(RAND(), μ, σ) (NORMINV stands for “the inverse of the normal cumulative distribution”) – μ is the mean– σ is the standard deviation• In cell B1 in “Sim” sheet, type in “=NORMINV(RAND(), 5, 1)”• Create the Data Table for 2000 samples• Press “F9”, see what happensESD.70J Engineering Economy Module - Session 3 6Random 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• Press “F9”, see what happens4ESD.70J Engineering Economy Module - Session 3 7Random numbers from lognormal distribution• A random variable X has a lognormal distribution if its natural logarithm has a normal distribution• Using LOGINV(RAND(), ln_μ, ln_σ)–ln_μ is the mean of ln(X)–ln_σ is the standard deviation of ln(X)• In the Data Table output formula cell B1, type “=LOGINV(RAND(), 2, 0.3)”• Press “F9”, see what happensESD.70J Engineering Economy Module - Session 3 8Give it a try!Check with your neighbors…Check the solution sheet…Ask me questions…5ESD.70J Engineering Economy Module - Session 3 9• We have just described the probability density function (PDF) of random variable x, or f(x)• We can now study the time function of distribution of random variable x across time, or f(x,t)• That is a stochastic process, or in plain English language: TREND + UNCERTAINTYFrom probability to stochastic processesESD.70J Engineering Economy Module - Session 3 10Three stochastic models• Geometric Brownian Motion• Mean-reversion•S-Curve6ESD.70J Engineering Economy Module - Session 3 11Geometric Brownian Motion• Brownian motion (also called random walk)– The motion of a pollen in water– A drunk walk in Boston Common– S&P500 return• Rate of change of the geometric mean is Brownian, not the underlying observations– For example, the stock prices do not necessarily follow Brownian motion, but their returns do!ESD.70J Engineering Economy Module - Session 3 12• This is the standard model for modeling stock price behavior in finance theory, and lots of other uncertainties• Mathematic form for Geometric Brownian Motion (you do not have to know):SdzSdtdSσμ+=where S is the stock price, μis the expected return on the stock, σis the volatility of the stock price, and dz is the basic Wiener process Brownian motion theorytrend uncertainty7ESD.70J Engineering Economy Module - Session 3 13Open ESD70session3-1Part2.xls http://ardent.mit.edu/real_options/ROcse_Excel_latest/ESD 70 2007/ESD70session3-1Part2.xlsSimulate a stock priceESD.70J Engineering Economy Module - Session 3 14Simulate a stock price• Google’s common stock price as of 9/5/07 is $527.80 (see “GOOG” tab)• Using historical data, we calculate monthly mean return and volatility of 5.15% and 11.97%• These two values are key inputs into any forward-looking simulation models. We will be using them repeatedly, so lets define their names…8ESD.70J Engineering Economy Module - Session 3 15Defining Excel variable names1. Select cell with the historical mean value (5.15%) and go to: “Insert” Æ “Name” Æ“Define”2. Enter field name “drift” and hit “OK”3. Repeat the same for historical standard deviation and call that variable “vol”ESD.70J Engineering Economy Module - Session 3 16Simulate a stock price (Cont)DecemberNovember=B2*(1+D2)October=drift+vol*C2=NORMINV(RAND(),0,1)$527.80SeptemberExpected Return + random draw * volatilityRandom Draw from standardized normal distribution1Stock PriceTimeComplete the following table for Google stock in tab “GOOG forecast”:1) Standardized normal distribution with mean 0 and standard deviation 19ESD.70J Engineering Economy Module - Session 3 17Simulating Google returns in Excel1. In worksheet “GOOG forecast”, type “=NORMINV(RAND(),0,1)” in cell C2, and drag down to cell C132. Type “=drift+vol*C2” in cell D2, and drag down to cell D133. Type “=B2*(1+D2)” in cell B3, and drag down to cell B134. Click “Chart” under “Insert” menuESD.70J Engineering Economy Module - Session 3 187. “Standard Types” select “Line”, “Chart sub-type”select whichever you like, click “Next”8. “Data Range” select “= ‘GOOG forecast’!$A$2:$B$13”, click “Next”9. “Chart options” select whatever pleases you, click “Next”10. Choose “As object in” and click “Finish”11. Press “F9” several times, see what happensSimulating Google returns in Excel10ESD.70J Engineering Economy Module - Session 3 19Give it a try!Check with your neighbors…Check the solution sheet…Ask me questions…ESD.70J Engineering Economy Module - Session 3 20Mean reversion• Unlike Geometric Brownian Motion that grows forever at the rate of drift, some processes have the tendency to – Fluctuate around a mean– The farther away from the mean, the high the probability of reversion to the mean– The speed of mean reversion can be measured by a parameter η11ESD.70J Engineering Economy Module -


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