Statistics In Finance

# Statistics In Finance

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## Statistics In Finance

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1. Statistics In Finance Probability

2. Probability • A Simple Event • is an outcome of an experiment which cannot be decomposed into a simpler outcome. • An Event • is a collection of one or more simple events. • Random Sample • is taken in such a way that any possible sample of specific size has the same probability as any other of being selected.

3. Probability • An Experiment • Results in ... • Outcomes • which are made up of Simple Events and Events… • Classical Probability attempts to assess the whole population assigning probabilities by calculating relative likely frequencies.

4. Probability Notation • The probability of event A is written… • P(A) • with P being “the probability” and • the parentheses being “of event” • All the probabilities of events in a sample space add up to 1, and • No event can occur less than zero times so… • 0 >= P(Event) >= 1

5. Probability Notation • No event can occur less than zero times so… • 0 >= P(Event) >= 1 • Don’t ever, ever, ever answer a probability question with a number less than 0 or greater than 1

6. Probability • If an Experiment • Can results in only two outcomes... • P(A) + P(B) = 1 • P(B) = 1 - P(A) • and • P(A) = 1 - P(B)

7. Statistics In Finance Samples and Populations

8. Basics • Population is the whole group of interest • Sample is a subset of the population that you may have data about. • Elements are the individual members of the population or sample studied. • Variable is used to refer to a particular characteristic of an element which can take on different values for each element of the population.

9. Statistics In Finance Mean, Average or Expected Value of a Sample

10. Mean, Average or Expected Value • Ten students • Ages • If you threw a stone, what age would you expect the person it hit, to be? • Simple answer, 21 • No variability in the values 21 21 21 21 21 21 21 21 21 21

11. Mean, Average or Expected Value • A formula Number of values, n Index, i 2 1 3 4 7 8 9 10 5 6 21 21 Value 21 21 21 21 21 21 21 21 + + + + + + + + +

12. Mean, Average or Expected Value • Ten more students • Ages • If you threw a stone! • Less simple answer, 21 or 22 • Some variability in the values 22 21 21 22 22 21 22 21 22 21 Say, 21½

13. Mean, Average or Expected Value • A formula Number of values, n Index, i 2 1 3 4 7 8 9 10 5 6 22 21 Value 21 22 22 21 22 21 22 21 + + + + + + + + +

14. Mean, Average or Expected Value • Ten more students • Ages • If you threw a stone! • How to answer? Guess? • A good deal of variability in the values 27 6 12 22 30 3 29 8 28 21 Say, 14½

15. Mean, Average or Expected Value • A formula Number of values, n Index, i 2 1 3 4 7 8 9 10 5 6 27 6 Value 12 22 30 3 29 8 28 21 + + + + + + + + +

16. Statistics In Finance Variability, Volatility and Variance of a Sample

17. Variability, Volatility and Variance Index, i 2 1 3 4 7 8 9 10 5 6 • How far is each value from the mean value? Value 22 30 3 29 8 28 21 27 6 12

18. Variability, Volatility and Variance Index, i 2 1 3 4 7 8 9 10 5 6 • Use deviation² Value 22 30 3 29 8 28 21 27 6 12

19. Variability, Volatility and Variance

20. Variability, Volatility and Variance Index, i 2 1 3 4 7 8 9 10 5 6 • Use deviation² Value 22 22 21 22 21 22 21 22 21 21

21. Variability, Volatility and Variance

22. Statistics In Finance Standard Deviation of a Sample

23. Sample Standard Deviation • The Standard Deviation is the square root of the Variance • The Sample Standard Deviation is denoted by S

24. Statistics In Finance Population Statistics

25. Population Mean • The Population Mean, μ, is: • Where n is the size of the population

26. Population Variance • The Population Variance is: • Where n is the size of the population

27. Population Standard Deviation • The Population Standard Deviation is the square root of the Population Variance • The Population Standard Deviation is denoted by σ • Where n is the size of the population

28. Statistics In Finance Covariance and Correlation

30. Covariance – Varying Apart

31. Covariance – Varying Together

32. Covariance – No Clear Pattern

33. Covariance – A Formula

34. Covariance – Numeric Results • Covariance Between A and B • Covariance Between A and C • Covariance Between A and D Positive Negative Positive Relatively Small

35. Covariance – Has No Scale • Covariance has no scale • What did 386, minus 386 and 10.1 mean? • Comparing two covariances is therefore no very meaningful. • Correlation is a “normalized” covariance • Correlation is covariance on a scale from minus one to plus one.

36. Correlation – A Formula

37. Correlation– Numeric Results • Correlation Between A and B • Correlation Between A and C • Correlation Between A and D Plus One Minus One Positive Nearly Zero

38. Statistics In Finance Using Probabilities in Calculations

39. Mean, Average or Expected Value Number of values, n Index, i 2 1 3 4 7 8 9 10 5 6 22 21 Value 21 22 22 21 22 21 22 21 + + + + + + + + +

40. Mean, Average or Expected Value 22 21 Value 21 22 22 21 22 21 22 21

41. Statistics In Finance The End