Probability Concepts

1
Probability Concepts

• Quantitative Methods
• Ali Nejadmalayeri

2
Basics

• Random Variable
• A quantity with uncertain outcome (i.e.,
  possible value)
• Event
• A set of outcomes singular (one outcome) or
  multiple
• a stock returns 10
• a stock returns more than 10
• Probability
• A number between 0 and 1 which denotes changes of
  an event happening
• For even E, 0 P(E) 1
• For all mutually exclusives events sum of
  probability is one!

3
Probabilities

• Empirical Probabilities
• Data tells us what are likelihoods of events
• How many days a year stock market drops?
• Subjective Probabilities
• Not necessarily based on data but believe
• A Priori Probabilities
• Based on prior analysis but not data
• Objective Probabilities
• Probabilities that dont change no matter who
  assess them like empirical and a priori

4
Odds

• Probability stated as Odds
• For event E with probability P(E), odds are
• With odds for E of a to b then P(E)
  a/(ab)
• With odds against E of a to b then P(E)
  b/(ab)

5
Conditioning

• Unconditional Probabilities
• Also known as marginal is independent of any
  other ancillary events or states of nature
• Probability SP500 loosing in a given month
• Conditional Probabilities
• Dependent of ancillary events or states of nature
• Probability SP500 loosing in a given month during
  boom

6
Independence

• Union Rule
• Independence
• Total probability Rule

7
Expectation and Variance

• Expected Value
• Variance

8
Conditional Expectation

• Conditional Expected Value
• Total probability and expected value

9
Bond Rating Migration Matrix

• Courtesy of Moodys Inc., Default Recovery
  Rates of Corporate Bond Issuers A Statistical
  Review of Moody's Ratings Performance, 1920-2003

10
Credit Migration Matrix (contd)

• Wharton School of Business, Metrics for
  Comparing Credit Migration Matrices
• By Yusuf Jafry and Til Schuermann, 2003