Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002

Sampling Outline (1) Sampling –Coin flips and political polls –The birthday problem (a not so obvious problem) Random variables and probabilities –Rainfall –The portfolio (rainfall) problem

Financial Bootstrap Commands sample count proportion percentile histogram multiples

Sampling Classical Probability/Statistics –Random variables come from static well defined probability distributions or populations –Observe only samples from these populations Example –Fair coin: (0 1) (1/2 1/2) populations –Sample = 10 draws from this coin

Old Style Probability and Statistics Try to figure out properties of these samples using math formulas Advantage: –Precise/Mathematical Disadvantage –Complicated formulas –For relatively complex problems becomes very difficult

Bootstrap (resample) Style Probability and Statistics Go to the computer (finboot toolbox) Example coin = [ 0 ; 1] % heads tails flips = sample(coin,100) flips = sample(coin,1000) nheads = count(flips == 0) ntails = count(flips == 1);

Monte-Carlo versus Bootstrap Monte-Carlo –Assume a random variable comes from a given distribution –Use the computer and its random number generators to generate draws of this random variable

Monte-Carlo versus Bootstrap Bootstrap –Assume that sample = population –Draw random variables from this sample itself –Advantage No assumption about the distribution –Disadvantage Small amounts of data can mess this up –Many examples coming

Sampling Outline (1) Sampling –Coin flips and political polls –The birthday problem (a not so obvious problem) Random variables and probabilities –Rainfall –A first portfolio problem

The Coin Flip Example What is the chance of getting fewer than 40 heads in a 100 flips of a fair (50/50) coin? Could use probability theory, but we’ll use the computer

Coin Flip Program in Words Perform 1000 trials Each trial –Flip 100 coins –Write down how many heads Summarize –Analyze the distribution of heads –Specifically: Fraction < 40

Now to the Computer coinflip.m and the matlab editor

Application: Political Polling Heads/Tails ->O’Brien/Reich Poll 100 people, 39 for O’Brien How likely is it that the distribution is 50/50? What is the probability of sampling less than 40 in the sample of 100? Remember: it is not zero!!! Try this with smaller samples

Sampling Outline (1) Sampling –Coin flips and political polls –The birthday problem (a not so obvious problem) Random variables and probabilities –Rainfall –A portfolio problem

Birthday If you draw 30 people at random what is the probability that more two or more have the same birthday?

Birthday in Matlab Each trial days = sample(1:365,30); b = multiples(days); z(trial) = any(b>1) proportion (z == 1) on to code

Sampling Outline (1) Sampling –Coin flips and political polls –The birthday problem (a not so obvious problem) Random variables and probabilities –Rainfall –A portfolio problem

Adding Probabilities: Rainfall Example dailyrain = [80; 10 ; 5 ] probs = [0.25; 0.5; 0.25]

Sampling annualrain = sum( sample(dailyrain,365,probs))

Portfolio Problem Distribution of portfolio of size 50 Return of each stock [ -0.05; 0.0; 0.10] Prob(0.25,0.5,0.25) Portfolio is equally weighted on to matlab code (portfolio1.m)

Portfolio Problem 2 1 Stock Return –[-0.05; 0.05] with probability [0.25; 0.75] Probabilities of runs of positives –5 days of positive returns –4/5 days of positive returns on to matlab code –portfolio2.m