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

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FIN285a: Section 2.2.2 Fall 2010. Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1). Sampling Outline (1). Sampling Coin flips The birthday problem (a not so obvious problem) Random variables and probabilities Rainfall The portfolio (rainfall) problem.

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sampling outline 1
Sampling Outline (1)
  • Sampling
    • Coin flips
    • The birthday problem (a not so obvious problem)
  • Random variables and probabilities
    • Rainfall
    • The portfolio (rainfall) problem
financial bootstrap commands
Financial Bootstrap Commands
  • sample
  • count
  • proportion
  • quantile
  • histogram
  • multiples
software
Software
  • finboot
  • coinflip.m
  • birthday.m
  • portfolio1.m
  • portfolio2.m
sampling
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
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
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);
sampling outline 18
Sampling Outline (1)
  • Sampling
    • Coin flips
    • The birthday problem (a not so obvious problem)
  • Random variables and probabilities
    • Rainfall
    • A first portfolio problem
the coin flip example
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
    • This is a classic binomial distribution (see Jorion 2.4.5)
    • The computer is not really necessary for this problem
coin flip program in words
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
Now to the Computer
  • coinflip.m and the matlab editor
sampling outline 112
Sampling Outline (1)
  • Sampling
    • Coin flips
    • The birthday problem (a not so obvious problem)
  • Random variables and probabilities
    • Rainfall
    • A portfolio problem
birthday
Birthday
  • If you draw 30 people at random what is the probability that two or more have the same birthday?
birthday in matlab
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 115
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
Adding Probabilities:Rainfall Example
  • dailyrain = [80; 10 ; 5 ]
  • probs = [0.25; 0.5; 0.25]
sampling17
Sampling
  • annualrain = sum(sample(dailyrain,365,probs))
portfolio problem
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
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
sampling outline 120
Sampling Outline (1)
  • Sampling
    • Coin flips
    • The birthday problem (a not so obvious problem)
  • Random variables and probabilities
    • Rainfall
    • The portfolio (rainfall) problem
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