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# Probability and Sampling Theory - PowerPoint PPT Presentation

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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Fall 2010

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

• Sampling

• Coin flips

• The birthday problem (a not so obvious problem)

• Random variables and probabilities

• Rainfall

• The portfolio (rainfall) problem

• sample

• count

• proportion

• quantile

• histogram

• multiples

• finboot

• coinflip.m

• birthday.m

• portfolio1.m

• portfolio2.m

• 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

• Try to figure out properties of these samples using math formulas

• Precise/Mathematical

• Complicated formulas

• For relatively complex problems becomes very difficult

• 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

• Coin flips

• The birthday problem (a not so obvious problem)

• Random variables and probabilities

• Rainfall

• A first portfolio problem

• 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

• Perform 1000 trials

• Each trial

• Flip 100 coins

• Write down how many heads

• Summarize

• Analyze the distribution of heads

• Specifically: Fraction < 40

• coinflip.m and the matlab editor

• Sampling

• Coin flips

• The birthday problem (a not so obvious problem)

• Random variables and probabilities

• Rainfall

• A portfolio problem

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

• Each trial

• days = sample(1:365,30);

• b = multiples(days);

• z(trial) = any(b>1)

• proportion (z == 1)

• on to code

• Sampling

• Coin flips and political polls

• The birthday problem (a not so obvious problem)

• Random variables and probabilities

• Rainfall

• A portfolio problem

• dailyrain = [80; 10 ; 5 ]

• probs = [0.25; 0.5; 0.25]

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

• 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)

• 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

• Coin flips

• The birthday problem (a not so obvious problem)

• Random variables and probabilities

• Rainfall

• The portfolio (rainfall) problem