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

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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 Theoryand 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**Financial Bootstrap Commands**• sample • count • proportion • quantile • histogram • multiples**Software**• finboot • coinflip.m • birthday.m • portfolio1.m • portfolio2.m**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);**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**• 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**• 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**Sampling Outline (1)**• Sampling • Coin flips • 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 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**Sampling Outline (1)**• Sampling • Coin flips • The birthday problem (a not so obvious problem) • Random variables and probabilities • Rainfall • The portfolio (rainfall) problem