1. Probability and Sampling Theory and the Financial Bootstrap Tools (Part 2) IEF 217a: Lecture 2.b Fall 2002 Jorion chapter 4

2. Common Theme Estimated statistics from samples are random variables too!! Use bootstrap to get their distributions

3. Sampling Outline (2) Uncertainty in the mean (light bulbs) Uncertainty in the extremes (light bulbs) Continuous random variables Portfolio examples again (portfolio2) Medians Terminology

4. The Mean as a Random Variable The light bulb problem (bootstrap) Light bulb population = data Assume: population = data (equal probability) 1000 Trials –Sample light bulbs (sample = 4) –Estimate mean –Store mean Explore means (percentiles, histogram) (Matlab code) bulbmn.m

5. Sampling Outline (2) Uncertainty in the mean (light bulbs) Uncertainty in the extremes (light bulbs) Continuous random variables Portfolio examples again (portfolio2) Medians Terminology

6. Uncertainty in Extremes Sampling –Draw 4 bulbs from population –Record sum of bulb life Report the lower tail of the distribution (percentile 0.01) Might put a guarantee on this lifetime matlab code (bulbmin.m)

7. Sampling Outline (2) Uncertainty in the mean (light bulbs) Uncertainty in the extremes (light bulbs) Continuous random variables Portfolio examples again (portfolio2) Medians Terminology

8. Continuous Random Variables Discrete versus continuous random variables Two main examples –Uniform –Normal

9. Discrete versus Continuous Random Variables Discrete: –[ 1 2 3 4 5 6] die –[H T] coin Continuous –Uniform –Normal

10. Uniform Distribution Uniform 0 to 3 U(0,3) 0123

11. Uniform Matlab Function rand(m,n) –Uniform [0,1] random variables – m rows, n columns matrix over to matlab test histogram

12. Normal Distribution Gaussian Bell Curve

13. Picture

14. Parameters Mean, Variance matlab –normal(n, mean, std)

15. Why Do We Care? Financial return series close to normal –We’ll look at the details of how close Central limit theorem – Let x be a random variable (any) –Assume that the variance of x exists –Let y = sum(x) for some length –Then: y (eventually) follows a normal distribution

16. Corollary z = mean(x) z eventually follows a normal since – mean(x) = (1/n) * sum(x) clt.m matlab example

17. When is Normality a Problem? Derivatives Real investments (option like) Higher frequency financial data Very large moves

18. Sampling Outline (2) Uncertainty in the mean (light bulbs) Uncertainty in the extremes (light bulbs) Continuous random variables Portfolio examples again Medians Terminology

19. Portfolio Left Tail Generate distribution – z = normal(1000,0.05,0.02) Report percentile(z,0.05) An early picture of –Value at Risk

20. Sampling Outline (2) Uncertainty in the mean (light bulbs) Uncertainty in the extremes (light bulbs) Continuous random variables Portfolio examples again (portfolio2) Medians Terminology

21. Medians Confidence on the median Like mean to matlab: mediandist.m

22. Sampling Outline (2) Uncertainty in the mean (light bulbs) Uncertainty in the extremes (light bulbs) Continuous random variables Portfolio examples again (portfolio2) Medians Terminology

23. Monte-Carlo/Sampling Bootstrapping/Resampling

24. Monte-Carlo Draw from theoretical distribution as population coin = [ 0 1] flips = sample(coin,10)

25. Resampling = Bootstrapping Get a sample Use sample as population for new draw –Prob 1/n on each element [30; 100; 6] with prob [ 1/3 1/3 1/3] data = [30; 100; 6]; samp = sample(data,2);

26. Monte-Carlo Advantages –Real statistical sampling –No sample limitation Disadvantages –Distribution assumptions

27. Bootstrap Advantages –No distributional assumptions Disadvantages –Small sample issues –Representative?? –Overlaps

28. Sampling Outline (2) Uncertainty in the mean (light bulbs) Uncertainty in the extremes (light bulbs) Continuous random variables Portfolio examples again (portfolio2) Medians Terminology