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Probability and Sampling Theory and the Financial Bootstrap Tools Part 1






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Sampling Outline (1). SamplingCoin flipsThe birthday problem (a not so obvious problem)Random variables and probabilitiesRainfallThe portfolio (rainfall) problem. Financial Bootstrap Commands. sample count proportion quantile histogram multiples. Software. finbootcoinflip.mbirthday.mpo
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1. Probability and Sampling Theory and 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


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