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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

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