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Probability and Sampling Theoryand 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