Probability and Sampling Theory and the Financial Bootstrap ...

**1. **Probability and Sampling Theoryand the Financial Bootstrap Tools(Part 1) FIN285a: Section 2.2.2
Fall 2010

**2. **Sampling Outline (1) Sampling
Coin flips
The birthday problem (a not so obvious problem)
Random variables and probabilities
Rainfall
The portfolio (rainfall) problem

**3. **Financial Bootstrap Commands sample
count
proportion
quantile
histogram
multiples

**4. **Software finboot
coinflip.m
birthday.m
portfolio1.m
portfolio2.m

**5. **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

**6. **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

**7. **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);

**8. **Sampling Outline (1) Sampling
Coin flips
The birthday problem (a not so obvious problem)
Random variables and probabilities
Rainfall
A first portfolio problem

**9. **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

**10. **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

**11. **Now to the Computer coinflip.m and the matlab editor

**12. **Sampling Outline (1) Sampling
Coin flips
The birthday problem (a not so obvious problem)
Random variables and probabilities
Rainfall
A portfolio problem

**13. **Birthday If you draw 30 people at random what is the probability that two or more have the same birthday?

**14. **Birthday in Matlab Each trial
days = sample(1:365,30);
b = multiples(days);
z(trial) = any(b>1)
proportion (z == 1)
on to code

**15. **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

**16. **Adding Probabilities:Rainfall Example dailyrain = [80; 10 ; 5 ]
probs = [0.25; 0.5; 0.25]

**17. **Sampling annualrain = sum(sample(dailyrain,365,probs))

**18. **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)

**19. **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

**20. **Sampling Outline (1) Sampling
Coin flips
The birthday problem (a not so obvious problem)
Random variables and probabilities
Rainfall
The portfolio (rainfall) problem