Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1)

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Probability and Sampling Theoryand the Financial
Bootstrap Tools(Part 1)

• IEF 217a Lecture 2.b
• Jorion, Chapter 4
• Fall 2002

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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

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Financial Bootstrap Commands

• sample
• count
• proportion
• percentile
• histogram
• multiples

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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

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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

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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)

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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

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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

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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

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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 well use the
  computer

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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 lt 40

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Now to the Computer

• coinflip.m and the matlab editor

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Application Political Polling

• Heads/Tails -gtOBrien/Reich
• Poll 100 people, 39 for OBrien
• 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

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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

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Birthday

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

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Birthday in Matlab

• Each trial
• days sample(1365,30)
• b multiples(days)
• z(trial) any(bgt1)
• proportion (z 1)
• on to code

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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

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Adding ProbabilitiesRainfall Example

• dailyrain 80 10 5
• probs 0.25 0.5 0.25

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Sampling

• annualrain sum( sample(dailyrain,365,probs))

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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)

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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