1. 1 Managerial Finance Professor Andrew Hall Statistics In Finance Probability

2. 2 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 3 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 4 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 5 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 6 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 7 Managerial Finance Professor Andrew Hall Statistics In Finance Samples and Populations

8. 8 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 9 Managerial Finance Professor Andrew Hall Statistics In Finance Mean, Average or Expected Value of a Sample

10. 10 Managerial Finance I Managerial Finance Professor Andrew Hall 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

11. 11 Managerial Finance I Managerial Finance Professor Andrew Hall Mean, Average or Expected Value  A formula 1 2 34 56 78910 21 Index, i Value Number of values, n + + ++ ++ +++

12. 12 Managerial Finance I Managerial Finance Professor Andrew Hall 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 2221 2221 22212221 Say, 21 ½

13. 13 Managerial Finance I Managerial Finance Professor Andrew Hall Mean, Average or Expected Value  A formula 1 2 34 56 78910 22 21 2221 2221 22212221 Index, i Value Number of values, n + + ++ ++ +++

14. 14 Managerial Finance I Managerial Finance Professor Andrew Hall 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 22 12 303 276 2982821 Say, 14 ½

15. 15 Managerial Finance I Managerial Finance Professor Andrew Hall Mean, Average or Expected Value  A formula 1 2 34 56 78910 22 12 303 276 2982821 Index, i Value Number of values, n + + ++ ++ +++

16. 16 Managerial Finance Professor Andrew Hall Statistics In Finance Variability, Volatility and Variance of a Sample

17. 17 Managerial Finance I Managerial Finance Professor Andrew Hall Variability, Volatility and Variance  How far is each value from the mean value? 22 12 303 276 2982821 Value 1 2 34 56 78910 Index, i

18. 18 Managerial Finance I Managerial Finance Professor Andrew Hall Variability, Volatility and Variance  Use deviation ² 22 12 303 276 2982821 Value 1 2 34 56 78910 Index, i

19. 19 Managerial Finance I Managerial Finance Professor Andrew Hall Variability, Volatility and Variance

20. 20 Managerial Finance I Managerial Finance Professor Andrew Hall Variability, Volatility and Variance  Use deviation ² 22 21 2221 2221 22212221 Value 1 2 34 56 78910 Index, i

21. 21 Managerial Finance I Managerial Finance Professor Andrew Hall Variability, Volatility and Variance

22. 22 Managerial Finance Professor Andrew Hall Statistics In Finance Standard Deviation of a Sample

23. 23 Managerial Finance I Managerial Finance Professor Andrew Hall Sample Standard Deviation  The Standard Deviation is the square root of the Variance  The Sample Standard Deviation is denoted by S

24. 24 Managerial Finance Professor Andrew Hall Statistics In Finance Population Statistics

25. 25 Managerial Finance I Managerial Finance Professor Andrew Hall Population Mean  The Population Mean, μ, is:  Where n is the size of the population

26. 26 Managerial Finance I Managerial Finance Professor Andrew Hall Population Variance  The Population Variance is:  Where n is the size of the population

27. 27 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 28 Managerial Finance Professor Andrew Hall Statistics In Finance Covariance and Correlation

29. 29 Managerial Finance I Managerial Finance Professor Andrew Hall Covariance – Start with a time series

30. 30 Managerial Finance I Managerial Finance Professor Andrew Hall Covariance – Varying Apart

31. 31 Managerial Finance I Managerial Finance Professor Andrew Hall Covariance – Varying Together

32. 32 Managerial Finance I Managerial Finance Professor Andrew Hall Covariance – No Clear Pattern

33. 33 Managerial Finance I Managerial Finance Professor Andrew Hall Covariance – A Formula

34. 34 Managerial Finance I Managerial Finance Professor Andrew Hall Covariance – Numeric Results  Covariance Between A and B  Covariance Between A and C  Covariance Between A and D Positive Negative Positive Relatively Small

35. 35 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 36 Managerial Finance I Managerial Finance Professor Andrew Hall Correlation – A Formula

37. 37 Managerial Finance I Managerial Finance Professor Andrew Hall 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. 38 Managerial Finance Professor Andrew Hall Statistics In Finance Using Probabilities in Calculations

39. 39 Managerial Finance I Managerial Finance Professor Andrew Hall Mean, Average or Expected Value 1 2 34 56 78910 22 21 2221 2221 22212221 Index, i Value Number of values, n + + ++ ++ +++

40. 40 Managerial Finance I Managerial Finance Professor Andrew Hall Mean, Average or Expected Value 22 21 2221 2221 22212221 Value

41. 41 Managerial Finance Professor Andrew Hall Statistics In Finance The End