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MIT ESD 70J - Engineering Economy - SESSION 3

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ESD 70J Engineering Economy Fall 2006 Session Three Alex Fadeev afadeev mit edu Link for this PPT http ardent mit edu real options ROcse Excel latest ExcelSession3 pdf ESD 70J Engineering Economy Module Session 3 1 Question from Session Two Last time we used uniformly distributed random variables to model the uncertain demand This implies identical 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 ESD 70J Engineering Economy Module Session 3 2 1 Modeling Uncertainty Generate random numbers from various distributions Normal LogNormal etc Random variables as time function stochastic processes Geometric Brownian Motion Mean Reversion S curve Statistical analysis to data mine distribution and its descriptive stats from historical data ESD 70J Engineering Economy Module Session 3 3 Random numbers generation redux Generate normally distributed random numbers Use norminv rand norminv stands for the inverse of the normal cumulative distribution is the mean is the standard deviation In the data table output formula cell B1 in Sim 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 ESD 70J Engineering Economy Module Session 3 4 2 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 Sim sheet of 1 xls press F9 see what happens ESD 70J Engineering Economy Module Session 3 5 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 log log is the log mean log 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 ESD 70J Engineering Economy Module Session 3 6 3 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 Histogram Histogram 300 250 200 Histogram 350 700 300 600 250 500 200 400 150 150 300 100 100 50 50 0 0 1 02 1 216 1 413 1 609 1 806 2 003 2 199 2 396 2 592 2 789 2 985 Year 1 200 100 0 3 734 4 835 5 935 7 036 8 136 9 237 10 34 11 44 12 54 13 64 14 74 Year 2 0 739 6 969 13 2 19 43 25 66 31 89 38 12 44 35 50 57 56 8 63 03 Year 3 Time Life is random in a non random way ESD 70J Engineering Economy Module Session 3 7 From probability to stochastic processes We have to 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 UNCERTAINTY ESD 70J Engineering Economy Module Session 3 8 4 Check the solution sheet Please ask questions now ESD 70J Engineering Economy Module Session 3 9 Three stochastic models Geometric Brownian Motion Mean reversion S Curve ESD 70J Engineering Economy Module Session 3 10 5 Geometric Brownian Motion Brownian motion aka random walk the motion of a pollen in water a drunk walks 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 follow Brownian motion but their returns do ESD 70J Engineering Economy Module Session 3 11 Simulate a stock price Google s stock price is 378 49 per class A common share on 9 8 06 see GOOG tab Using historical data we calculate monthly mean return and volatility of 6 and 14 These two values are key inputs into any forward looking simulation models We will be using them repeatedly so lets define their names ESD 70J Engineering Economy Module Session 3 12 6 Defining Excel variable names 1 Select sell with the historical mean value 6 16 and go to Insert Name 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 13 Simulate a stock price Cont Complete the following table for Google stock Time Stock Price Random Draw from standardized normal distribution1 Expected Return random draw volatility September 378 49 NORMINV RAND 0 1 drift vol C2 October B2 1 D2 November December 1 Standardized normal distribution with mean 0 and standard deviation 1 ESD 70J Engineering Economy Module Session 3 14 7 Simulating Google returns in Excel 1 Open a new worksheet name it GOOG forecast 2 Copy or input forecasting time frame i e Time in cell A1 September in A2 3 Type norminv rand 0 1 in cell C2 and drag down to cell C13 4 Type drift vol C2 in cell D2 and drag down to cell D13 5 Type B2 1 D2 in cell B3 and drag down to cell B13 6 Click Chart under Insert menu ESD 70J Engineering Economy Module Session 3 15 Simulating Google returns in Excel cont 7 Standard types select Line Chart sub type select whichever you like click Next 8 Data range select GOOG forecast 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 ESD 70J Engineering Economy Module Session 3 16 8 Check the solution sheet Please ask questions now ESD 70J Engineering Economy Module Session 3 17 Brownian Motion Theory This is the standard model for modeling stock price behavior in finance theory and lots of other uncertainties enter the Central Limit Theorem Mathematic form for Geometric Brownian Motion you do not have to know dS Sdt Sdz where S is the stock price is the expected return on the stock is the volatility of the stock price and dz is the basic Wiener process ESD 70J Engineering Economy Module Session 3 18 9 Mean 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 ESD 70J Engineering Economy Module Session 3 19 Simulating interest rate In finance people usually use mean reversion to model behavior of interest rates and asset volatilities Suppose the Fed rate r is 4 25 today the speed of mean reversion is 0 3 the long term mean r is 7 the volatility is 1 5 per year Expected mean reversion is dr r r dt ESD 70J Engineering


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