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Some definitions and properties of Probability

  • Definitions
    • Mutually Exclusive: If one event occurs, then other cannot occur
    • Exhaustive: All exhaustive events taken together form the complete sample space (Sum of probability = 1)
    • Independent Events: One event occurring has no effect on the other event
  • The probability of any event A:
  • If the probability of happening of event A is P(A), then the probability of A not happening is (1-P(A))
  • For example, if the probability of a company going bankrupt within one year period is 20%, then the probability of company surviving within next one year period is 80% 


For a bond with “B” rating, assume 1 year probability of default for each issuer is 6%, and that default probability of each issuer are independent. What is the probability that both issuers avoid default during the 1st year.


Both would avoid default only if None defaults
This implies that first does not default AND second does not default
= (1 – PD (first)) x (1 – PD (second))
= (1 – 0.06) x (1 – 0.06) = 0.884 = 88.4%


Some Properties of Probability

The probability of happening of event A or event B can be given as the sum of the three portions defined by the figure below:




Jensen, a portfolio manager is managing two portfolios. One for High Net Worth Individuals (HNI) and second for Low Net Worth Individuals (LNI)
HNI portfolio contains 5 bonds and 7 stocks and LNI contains 6 bonds and 11 stocks
One instrument from HNI is transferred to LNI portfolio
Now Jensen selects an instrument from LNI, what is the probability that instrument selected is stock?

  1. 0.5382
  2. 0.7821
  3. 0.6435
  4. None of these

Here required probability = [P(stock transferred from HNI) AND P(Stock selected from LNI)] OR
[P(bond transferred from HNI) AND P(Stock selected from LNI)]
So, the required probability = (7/12) × (12/18) + (5/12) × (11/18) = 139/216 = 0.6435
Hence option ‘C’ is correct



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