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### Lecture5_stat107_spring2012v1_1up

Course: STATS 107, Spring 2012
School: Harvard
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107: Stat Introduction to Business and Financial Statistics Class 4: Monte Carlo Simulation 1 Historical Note The name Monte Carlo simulation comes from the computer simulations performed during the 1930s and 1940s to estimate the probability that the chain reaction needed for an atom bomb to detonate would work successfully. The physicists involved in this work were big fans of gambling, so they gave the...

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107: Stat Introduction to Business and Financial Statistics Class 4: Monte Carlo Simulation 1 Historical Note The name Monte Carlo simulation comes from the computer simulations performed during the 1930s and 1940s to estimate the probability that the chain reaction needed for an atom bomb to detonate would work successfully. The physicists involved in this work were big fans of gambling, so they gave the simulations the code name Monte Carlo. 2 Uses of Monte Carlo Simulation Ford, Proctor and Gamble, Pfizer, Bristol-Myers Squibb, and Eli Lilly use simulation to estimate both the average return and the risk factor of new products. Proctor and Gamble uses simulation to model and optimally hedge foreign exchange risk. Sears uses simulation to determine how many units of each product line should be ordered from suppliersfor example, the number of pairs of Dockers trousers that should be ordered this year. Financial planners use Monte Carlo simulation to determine optimal investment strategies for their clients retirement. 3 The Monte Carlo Method The method is usually used to understand how a process or estimator will perform in practice. The Monte Carlo method is often performed when the computational resources of a researcher are more abundant than the researchers mental resources. 4 Wikipedia fm3: Chapter 22 Monte Carlo 5 Warning Monte Carlo is often time-consuming, computationally wasteful Our first example (computing ) is a good illustration Analytical methods (formulas ) are better if available But Sometimes theres nothing else Monte Carlo is fun! 6 Example: Estimating Did you know that you can calculate (3.14159..) by throwing darts at a dartboard ? Unfortunately, you have to be pretty bad at darts darts for it to work, though. 7 Geometry Review Suppose we have a circle of radius 1, with a center at (0,0). Then the unit circle is given by the equation x2+y2=1. 8 The Set Up Suppose our dartboard is a 2x2 square and we inscribe a circle of radius 1 inside the square, with a center at (0,0). 9 Computing : Unit circle inscribed in unit square 1.0 The square has area 4 Unit circle has area 0.5 1.0 0.5 0.5 1.0 0.5 1.0 10 The Algorithm We randomly throw darts at this square and keep track of how many land inside the circle. That would give us a relative estimation of what percentage of the box is in the circle. The area of the box is 4 and the area of the circle is . So the ratio of circle area to box area is /4. Hence, the percentage of darts that land inside the circle should be approximately /4. 11 The runif() command To simulate throwing darts at the dartboard, we need to learn about the R command runif(): R Command: runif(n) uniform random number generator that returns n random numbers between 0 and 1. 12 Examples > runif(5) [1] 0.7000905 0.6717379 0.9857697 0.6081811 0.8611683 > > runif(5) [1] 0.1719788 0.5086680 0.6567634 0.1922734 0.8315596 > > foo=runif(1000) > hist(foo) > 13 Our circle is as follows: 1 How do we simulate darts thrown at this board ? The location of the dart is given by its (x,y) point. (0,0) y -1 x 1 14 Code Fragment Examine the following code fragment x = 2*runif(1)-1 y = 2*runif(1)-1 if (x*x+y*y<=1) numincircle = numincircle+1 What are the possible values for x and y ? 15 The Complete Function simulpi = function(n) { numincircle = 0 Keep count of darts inside circle do n simulations for(i in 1:n) { x = 2*runif(1)-1 y = 2*runif(1)-1 if (x*x+y*y<1) numincircle <- numincircle+1 } cat("Number of simulations = ",n,"\n") cat("Estimate of PI = ",4*numincircle/n,"\n") } 16 Output > simulpi(10) Number of simulations = Estimate of PI = 3.2 > simulpi(100) Number of simulations = Estimate of PI = 3.04 > simulpi(1000) Number of simulations = Estimate of PI = 3.136 > simulpi(1000) Number of simulations = Estimate of PI = 3.184 > simulpi(100000) Number of simulations = Estimate of PI = 3.1472 10 100 1000 1000 iterations-why arent the values the same each time ? 1000 1e+05 17 1.0 0.5 0.0 -0.5 -1.0 y1 -1.0 -0.5 0.0 x1 0.5 1.0 18 In Excel A B C D E COMPUTING PI USING MONTE CARLO METHODS INITIAL EXPERIMENT 1 2 Number of data points 3 Inside circle 4 Pi? 5 Each cell in these columns contains the Excel function =Rand() 6 7 8 9 10 30 <-- =COUNT(A:A) 23 <-- =COUNTIF(D:D,TRUE) 3.066666667 <-- =B3/B2*4 Experiment 1 2 3 Random1 0.23437 0.92322 0.95535 Random2 0.03080 0.61267 0.25644 In unit circle? TRUE FALSE TRUE <-- =(B8^2+C8^2<=1) Columns B and C have function Rand(). Column D uses Boolean function to determine if point is inside unit circle Cell B4 approximates p by Computing ratio of inside/total points Multiplying this ratio by 4 19 Different experiments give different approximations A B C D E COMPUTING PI USING MONTE CARLO METHODS INITIAL EXPERIMENT 1 2 Number of data points 3 Inside circle 4 Pi? 5 Each cell in these columns contains the Excel function =Rand() 6 7 8 9 10 30 <-- =COUNT(A:A) 21 <-- =COUNTIF(D:D,TRUE) 2.8 <-- =B3/B2*4 Experiment 1 2 3 Random1 0.88255 0.37558 0.81044 In unit circle? TRUE TRUE FALSE Random2 0.06206 0.41664 0.65635 A B <-- =(B8^2+C8^2<=1) C D E COMPUTING PI USING MONTE CARLO METHODS INITIAL EXPERIMENT 1 2 Number of data points 3 Inside circle 4 Pi? 5 Each cell in these columns contains the Excel function =Rand() 6 7 8 9 10 30 <-- =COUNT(A:A) 24 <-- =COUNTIF(D:D,TRUE) 3.2 <-- =B3/B2*4 Experiment 1 2 3 Random1 0.27544 0.81204 0.91731 Random2 0.04368 0.52793 0.90177 In unit circle? TRUE TRUE FALSE <-- =(B8^2+C8^2<=1) 20 Stock Returns Suppose you put \$1000 into an S&P 500 index fund on January 1. How much will you have at the end of the year? We can easily simulate this using Monte Carlo. How do we simulate yearly stock returns? There are several methods. 21 Simulating Stock Returns Assuming normality of returns is not that good an idea since stock returns demonstrate tail behavior, but it is less common than what the normal distribution would dictate. We will produce the returns using a procedure called resampling. We have a dataset with the yearly S&P 500 returns from 1926 to 2010. 22 The Historical Returns A snapshot 23 The Histogram of Historical Returns 24 Resampling To produce a return in the future for the index, we resample from our 83 historical observationsthat is, we just randomly pick one out of the 83 past yearly returns. This scheme doesnt build in any ideas like good years are more likely to be followed by good years, etc but it is a technique used by several software packages that do financial planning. To do this in R, we use the sample command 25 The SAMPLE command The command we use is sample(x, size=<<see below>>, replace=F) Collection of items to sample from How many times to sample Sample with or without replacement 26 Example Output > 0.2380297 > sample(rets,1) [1] sample(rets,1) [1] 0.09967052 > sample(rets,1) [1] 0.06146142 > sample(rets,1) [1] 0.3023484 > sample(rets,1) [1] -0.01188566 > sample(rets,1) [1] -0.01208205 27 Very basic idea This is called a nonparametric routine-it does not assume any structure on the returns. This method assumes the historical data are essential the population of possible returns. The sampled data looks very much like the historical data. > summary(rets) Min. 1st Qu. Median Mean 3rd Qu. -0.43840 -0.02136 0.14220 0.11310 0.25490 > summary(sample(rets,100000,replace=TRUE)) Min. 1st Qu. Median Mean 3rd Qu. -0.43840 -0.03064 0.14220 0.11220 0.25920 Max. 0.52560 Max. 0.52560 28 The Monte Carlo How many iterations We are now ready to code year.ret = function(rets,howmany) { store = 1:howmany Creates a place to to store our results for(i in 1:howmany) store[i] = 1000*(1+sample(rets,1)) return(store) } 29 Running the Monte Carlo Cut and paste the code into R > out=year.ret(rets,10000) > hist(out) The output is not normal. Should it be? 30 Summarize the Results > summary(out) Min. 1st Qu. 561.6 969.4 > Median 1142.0 Mean 3rd Qu. 1113.0 1259.0 Max. 1526.0 The probability of making a positive return is > sum(out>1000)/10000 [1] 0.708 This makes sense since > sum(rets>0)/83 [1] 0.7108434 71% of the historical returns are positive so it makes sense that we made more than \$1000 71% of the time. 31 Convert to CAGRWhat is CAGR? CAGR=compound annual growth rate The compound annual growth rate is calculated by taking the nth root of the total percentage growth rate, where n is the number of years in the period being considered. 32 Huh? CAGR isn't the actual return in reality. It's an imaginary number that describes the rate at which an investment would have grown if it grew at a steady rate. You can think of CAGR as a way to smooth out the returns. Suppose you take a look at a stock price over ten years, and the price price went from P0 (10 years ago) to P1 (today). Then the CAGR is defined so that 10 P1 = P0 (1 + CAGR) 33 Can Convert to CAGR if desired Recall CAGR > summary(out) Min. 1st Qu. 561.6 969.4 > CAGR= - 43.8% Median 1142.0 Mean 3rd Qu. 1113.0 1259.0 Max. 1526.0 CAGR=52.6% But this isnt an interesting example since it is just what happens in one year. 34 Growth over 30 years We can do the same idea for 30 years in a row money = 1000 for(i in 1:30) { ret = sample(rets,size=1) money = money*(1+ret) } cat("In 30 years you have ",round(money),"\n") 35 Lets make it into a function mysim = function(numyears) { money = 1000 for(i in 1:numyears) { ret = sample(rets,size=1) money = money*(1+ret) } cat("In ",numyears," years you have ",round(money),"\n") } 36 Running this code in R > mysim(30) In 30 years > mysim(30) In 30 years > mysim(30) In 30 years > mysim(30) In 30 years > mysim(30) In 30 years > mysim(30) In 30 years > you have 66089 you have 8522 you have 20847 you have 38044 you have 22745 you have 148696 37 Keeping Track of the Results simul=function(rets,howmany) { values = 1:howmany for(j in 1:howmany) { money = 1000 for(i in 1:30) { ret = sample(rets,size=1) money = money*(1+ret) This creates a vector to store our results This double looping makes the problem difficult to do in Excel Excel } values[j] = money } return(values) } 38 Run the code in R Cut and paste it into R > summary(out) Min. 1st Qu. Median 249.1 7229.0 15370.0 > CAGR= 9.5% > (15370/1000)^(1/30) - 1 [1] 0.09535727 > Mean 24810.0 3rd Qu. Max. 30870.0 489900.0 CAGR=22.9% > (489900/1000)^(1/30) - 1 [1] 0.229335 39 Run the code in R hist(out,breaks=50) 40 Go Normal What if we wanted to assume the returns were normal? Sample statistics > mean(rets) [1] 0.1131480 > sd(rets) [1] 0.2021068 41 The rnorm command simul=function(howmany) { values = 1:howmany for(j in 1:howmany) { money = 1000 for(i in 1:30) { ret=rnorm(1,mean=.1131,s=.2021) money = money*(1+ret) } values[j] = money } return(values) } 42 Different Results (of course) > summary(out) Min. 1st Qu. 119.2 7360.0 > > summary(out) Min. 1st Qu. 249.1 249.1 7229.0 > Normal Median 14490.0 Mean 24240.0 3rd Qu. Max. 28670.0 454000.0 Resampled Median 15370.0 Mean 24810.0 3rd Qu. Max. 30870.0 489900.0 Which model for returns allows for slightly more upside? Which is more realistic, which would we show a client? What would logspline routine show? 43 Go Semiparametric Density Estimation refers to determining a probability density for a given set of data. This can be as simple as assuming the data is normal, and using the sample mean and standard deviation as parameters. Or it can involve some fancy math. 44 What do densities require A probability density is a continuous function f(x) so that f(x) >=0 for all values of x The function f(x) integrates to 1. 45 The logspline package library(logspline) Performs a semi-parametric density estimate, then lets you sample from the estimate. Pretty cool routine. 46 The historical stock returns Just a few lines of code fit=logspline(rets) > hist(rets,probability=TRUE) > x=seq(-.6,.6,.01) > lines(x,dlogspline(x,fit)) 47 The resulting density 48 Compare with the Normal Longer tail 49 Can draw from this density A powerful ability of the logspline routine is that one can draw random values from the fitted density function. This is easily done using the rlogspline routine. 50 Another Simulation fit = logspline(rets) simul=function(fit,howmany) { values = 1:howmany for(j in 1:howmany) { money = 1000 for(i in 1:30) { ret=rlogspline(1,fit) money = money*(1+ret) } values[j] = money } return(values) } 51 Results More long tailed behavior Min. 1st Qu. -41290 6102 Median 14000 > plogspline(-.3,fit) [1] 0.03610169 > pnorm(-.3,mean(rets),sd(rets)) [1] 0.02046658 Mean 3rd Qu. 24950 29360 Max. 485700 Almost double the probability of a return being less than -0.3! 52 Example: Retirement Planning (hw) Joe Renk is a 35 year old freelance writer with \$30,000 in an IRA. He figures he can put in \$2000 (the maximum allowed every year) for the next 30 years. John wants to have a simple portfolio so he decides to put all his money into an S&P500 index fund, which has an annual management fee of 0.5%. How much money will John have in 30 years ? What is the probability he is a millionaire ? 53 A code fragment (modify for hw) money = 30000 for(i in 1:30) { ret = sample(rets,size=1) money = (money+2000)*(1+ret) fee = .005*money money = money-fee } cat("In 30 years Joe has ",round(money),"\n") Cut and pasting this into R, you get results such as In 30 years Joe has In 30 years Joe has In 30 years Joe has 197919 364987 1040814 54 Run this code many times values = 1:1000 for(j in 1:1000) { money = 30000 for(i in 1:30) { ret = sample(rets,size=1) money = (money+2000)*(1+ret) fee = .005*money money = money-fee } values[j] = money } 55 Results: > summary(values) Min. 1st Qu. Median Mean 3rd Qu. Max. 36930 207900 353700 501400 608200 9083000 > length(values) [1] 1000 > sum(values >= 1000000) [1] 108 > 108/1000 [1] 0.108 > sum(values<100000) [1] 65 > sum(values>1400000) [1] 46 11% chance john will have a million or more. 6.5% chance john will have less than 100000. 4.6% chance john will have more than 1.4 million 56 300000 200000 100000 Value 400000 500000 600000 Middle 50% of Final Values 0 5 10 15 20 25 30 Time 57 Our numbers are close to financialengines.com, a website founded by Nobel Prize winner Bill Sharpe: 58
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BILD 2, Multicellular LifeProblem set #5 1. What is the difference between &quot;food&quot; and &quot;nutrients&quot;? 2. What is the general structure of the digestive tract in a wide variety of animals? 3. Define each
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BILD 2, Multicellular LifeProblem set #6 1. Compare arteries and veins with respect to: a. Structure b. Contribution to circulatory function c. Pressure exerted on the blood in each kind of vessel
UCSD - BILD - 2
BILD 2, Multicellular LifeProblem set #7 1. What functions are associated with each of the two major regions of a plant body? 2. What are the three major kinds of plant tissues, and where are they located i
UCSD - BILD - 2
BILD 2, Multicellular LifeProblem set #8 INSTRUCTIONS: The machine will score only Scantron forms that are labeled with both a name AND an ID number. I will check every Scantron form to make certain it can be scor