probability and sampling theory and the financial bootstrap tools (part 1) ief 217a: lecture 2.b...
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![Page 1: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/1.jpg)
Probability and Sampling Theoryand the Financial Bootstrap Tools
(Part 1)IEF 217a: Lecture 2.b
Jorion, Chapter 4
Fall 2002
![Page 2: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/2.jpg)
Sampling Outline (1)
• Sampling– Coin flips and political polls– The birthday problem (a not so obvious
problem)
• Random variables and probabilities– Rainfall– The portfolio (rainfall) problem
![Page 3: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/3.jpg)
Financial Bootstrap Commands
• sample
• count
• proportion
• percentile
• histogram
• multiples
![Page 4: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/4.jpg)
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
![Page 5: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/5.jpg)
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
![Page 6: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/6.jpg)
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);
![Page 7: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/7.jpg)
Monte-Carlo versus Bootstrap
• Monte-Carlo– Assume a random variable comes from a given
distribution– Use the computer and its random number
generators to generate draws of this random variable
![Page 8: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/8.jpg)
Monte-Carlo versus Bootstrap
• Bootstrap– Assume that sample = population– Draw random variables from this sample itself– Advantage
• No assumption about the distribution
– Disadvantage• Small amounts of data can mess this up
– Many examples coming
![Page 9: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/9.jpg)
Sampling Outline (1)
• Sampling– Coin flips and political polls– The birthday problem (a not so obvious
problem)
• Random variables and probabilities– Rainfall– A first portfolio problem
![Page 10: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/10.jpg)
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
![Page 11: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/11.jpg)
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
![Page 12: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/12.jpg)
Now to the Computer
• coinflip.m and the matlab editor
![Page 13: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/13.jpg)
Application: Political Polling
• Heads/Tails ->O’Brien/Reich• Poll 100 people, 39 for O’Brien• How likely is it that the distribution is
50/50?• What is the probability of sampling less
than 40 in the sample of 100?• Remember: it is not zero!!!• Try this with smaller samples
![Page 14: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/14.jpg)
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
![Page 15: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/15.jpg)
Birthday
• If you draw 30 people at random what is the probability that more two or more have the same birthday?
![Page 16: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/16.jpg)
Birthday in Matlab
• Each trial• days = sample(1:365,30);
• b = multiples(days);
• z(trial) = any(b>1)
• proportion (z == 1)
• on to code
![Page 17: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/17.jpg)
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
![Page 18: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/18.jpg)
Adding Probabilities:Rainfall Example
• dailyrain = [80; 10 ; 5 ]
• probs = [0.25; 0.5; 0.25]
![Page 19: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/19.jpg)
Sampling
• annualrain = sum( sample(dailyrain,365,probs))
![Page 20: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/20.jpg)
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)
![Page 21: Probability and Sampling Theory and the Financial Bootstrap Tools (Part 1) IEF 217a: Lecture 2.b Jorion, Chapter 4 Fall 2002](https://reader036.vdocuments.net/reader036/viewer/2022062804/56649d495503460f94a25f23/html5/thumbnails/21.jpg)
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