ESD.70J Engineering Economy Module Fall 2004 Session Three Link for PPT: http://web.mit.edu/tao/www/esd70/s3/p.ppt ESD.70J Engineering Economy Module - Session 3 1 One note for Session Two If you Excel keeps crashing when simulating, 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. By doing that, crashes should be much less frequent. ESD.70J Engineering Economy Module - Session 3 2
Questions for Big or small From the simulation in the last session, we know the distribution of the NPV. We used evenly distribute random variables to model the demand uncertainty. Evenly distributed, however, means the probability of very high or low demand is the same as those near the expected demand. Arguably, it has obvious inadequacy for many real cases. What are other possible models for demand uncertainties? ESD.70J Engineering Economy Module - Session 3 3 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 data set ESD.70J Engineering Economy Module - Session 3 4
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 Open web.mit.edu/tao/www/esd70/s3/1.xls, or establish a simulation spreadsheet by your self 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) ESD.70J Engineering Economy Module - Session 3 5 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. Asymmetric triangular distribution more complex to generate (if interested, check link: http://www.sics.se/~ali/teaching/sysmod/e05.pdf) ESD.70J Engineering Economy Module - Session 3 6
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) ESD.70J Engineering Economy Module - Session 3 7 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 Histogram 300 250 200 150 100 50 0 1.02 1.216 1.413 1.609 1.806 2.003 2.199 2.396 2.592 2.789 2.985 350 300 250 200 150 100 50 0 3.734 4.835 5.935 7.036 8.136 9.237 10.34 11.44 12.54 13.64 14.74 700 600 500 400 300 200 100 0 0.739 6.969 13.2 19.43 25.66 31.89 38.12 44.35 50.57 56.8 63.03 Year 1 Year 2 Year 3 Time Life is random in a non-random way ESD.70J Engineering Economy Module - Session 3 8
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 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
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 ESD.70J Engineering Economy Module - Session 3 11 Simulate a stock price Google s stock price is $105.33 per share on 9/10/04, 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% ESD.70J Engineering Economy Module - Session 3 12
Simulate a stock price (Cont) Complete the following table for Google stock: Time Sep 04 Dec 04 Mar 05 Jun 05 Sep 05 Stock Price $105.33 Random Draw from standardized normal distribution 1) Realized return (expected return + random draw * volatility) 1). Standardized normal distribution with mean 0 and standard deviation 1 ESD.70J Engineering Economy Module - Session 3 13 Using Spreadsheet to simulate Google stock Follow the instructions, step by step: 1. Open a new worksheet, name it GBM 2. Copy or input the table in the previous slide into Excel, with Time as cell A1 3. Type =norminv(rand(),0,1) in cell C2, and drag down to cell C6 4. Type =0.04+0.20*C2 in cell D2, and drag down to cell D6 5. Type =B2*(1+D2) in cell B3, and drag down to cell B6 6. Click Chart under Insert menu ESD.70J Engineering Economy Module - Session 3 14
Using Spreadsheet to simulate Google stock (Cont) 7. Standard types select Line, Chart sub-type select whichever you like, click Next 8. Data range select =GBM!$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. ESD.70J Engineering Economy Module - Session 3 15 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) ds = µ Sdt + σsdz 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 ESD.70J Engineering Economy Module - Session 3 16
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 ESD.70J Engineering Economy Module - Session 3 17 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 r is 7%, the volatility σ is 1.5% per year Expected mean reversion is: dr = η( r r) dt ESD.70J Engineering Economy Module - Session 3 18
Simulate interest rate (Cont) Complete the following table for interest rate: Time 2004 2005 2006 2007 2008 Interest rate 4% Random Draw from standardized normal distribution Realized return (expected reversion + random draw * volatility) ESD.70J Engineering Economy Module - Session 3 19 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 ESD.70J Engineering Economy Module - Session 3 20
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. ESD.70J Engineering Economy Module - Session 3 21 Mean reversion (Again) Mean reversion has many applications besides modeling interest rate behavior in finance theory Mathematic form (you do not have to know) dr = η ( r r) dt + σdz where r is the interest rate, η is the speed of mean reversion, r is the long-term mean, σ is the volatility, and dz is the basic Wiener process ESD.70J Engineering Economy Module - Session 3 22
S-curve Many uncertain variables display the S- curve shape Time For example, demand for a new technology grows slow when the new technology is just introduced, then the demand explodes, finally it approaches a natural limit and saturated ESD.70J Engineering Economy Module - Session 3 23 Modeling S-curve Deterministically Parameters: Demand at year 0 Demand at year T The limit of demand, or demand at time Model: Demand ( t ) α and β can be derived from demand at year 0 and year T α = Demand( ) Demand(0) Demand( ) Demand(10) β = ln( ) /10 α ESD.70J Engineering Economy Module - Session 3 24 t = Demand ( ) αe β
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 ESD.70J Engineering Economy Module - Session 3 25 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% ESD.70J Engineering Economy Module - Session 3 26
S-curve example (Cont) All the functions and tools used in the S-curve simulation sheet have been taught Check the solution sheet at http://web.mit.edu/tao/www/esd70/s3/2.xls Let me know any questions ESD.70J Engineering Economy Module - Session 3 27 Big or small? Among the three models taught today Geometric Brownian Motion Mean Reversion S-curve which is the best for the problem of Big or small? ESD.70J Engineering Economy Module - Session 3 28
Obtaining key parameters from data set Knowing the models is only a start, how to obtain good parameters is critical Otherwise, garbage into the best model only generates garbage In many cases, data are scarce for decisions. A good everyday habit to collect data is essential. ESD.70J Engineering Economy Module - Session 3 29 Example We simulated the movement of Google stock price using the expected quarterly return of 4% and quarterly volatility of 20%. Is it reasonable? Since Google just did IPO, no historical data for analysis. Let us use a comparable stock, Yahoo, to get some clue. ESD.70J Engineering Economy Module - Session 3 30
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? ESD.70J Engineering Economy Module - Session 3 31 Issues in modeling Do not trust the model this is the presumption for using any model. Highly manipulatable models are prone (if not doom) 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 ESD.70J Engineering Economy Module - Session 3 32
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 ESD.70J Engineering Economy Module - Session 3 33 Next class The course so far has taught you the skills to model uncertainties. Modeling is passive. As human being, we have the capacity to manage uncertainties proactively. This capacity is called flexibility. The next class will show you how to model flexibility. And in the end, the class will give you an overview of Excel and the basics for self-learn Excel, for your deeper exploration of Excel in the future. ESD.70J Engineering Economy Module - Session 3 34