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

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

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

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

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

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)

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

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

Discrete versus Continuous Random Variables Discrete: –[ ] die –[H T] coin Continuous –Uniform –Normal

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

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

Normal Distribution Gaussian Bell Curve

Picture

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

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

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

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

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

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

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

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

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

Monte-Carlo/Sampling Bootstrapping/Resampling

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

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);

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

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

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