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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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FIN285a: Section 2.2.2

Fall 2010

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


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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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Financial Bootstrap Commands

  • sample

  • count

  • proportion

  • quantile

  • histogram

  • multiples


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Software

  • finboot

  • coinflip.m

  • birthday.m

  • portfolio1.m

  • portfolio2.m


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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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Sampling Outline (1)

  • Sampling

    • Coin flips

    • 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 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


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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 < 40


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Now to the Computer

  • coinflip.m and the matlab editor


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Sampling Outline (1)

  • Sampling

    • Coin flips

    • 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 two or more have the same birthday?


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Birthday in Matlab

  • Each trial

    • days = sample(1:365,30);

    • b = multiples(days);

    • z(trial) = any(b>1)

  • 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 Probabilities:Rainfall 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


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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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