CFA I R9 Probability Concepts

Dear fellow CFA candidates/readers,

Thank you for visiting visiting WhitneyDiary CFA I Notes series, I have been using the notes below for my revision and have passed the CFA Level I on June 2018 Exam. Hereby, I share my notes with you and hope that they are helpful:

  1. Random variable: An uncertain quantity/number
  2. Outcome: An observed value of a random variable
  3. Event: A single outcome or a set of outcomes
  4. Mutually exclusive events: Events that cannot both happen at the same time
  5. Exhaustive events: Those that include all possible outcomes
  6. Probability of occurrence: 0 ≤ P(E) ≤ 1
  7. Sum of ∑P(E) = 1 when the set of events is mutually exclusive and exhaustive
  8. Empirical probability: Established by analyzing past data
  9. Priori probability: Determined by using a formal reasoning and inspecting process
  10. Subjective probability: The least formal method of developing probability and involves the use of personal judgement
  11. Objective probability: Contrast of empirical and priori probabilities
  12. Odds for and against the event
    → Odds = P(Occur) / P(Not occur)
    → P(Occur) = 1/13: If probability against event occurring is 12/1, meaning in 13 occurrences of events, it is expected that it will occur once and not 12 times
  13. Unconditional probability (marginal probability)
    → Probability of an event regardless of the past or future occurrence of other events
    → Weighted average of conditional probability
  14. Conditional probability (likelihood)
    → One where the occurrence of one event affects the probability of the occurrence of another event
    → Keyword: “given”
  15. Multiplication rule of probability/Joint probability
    → To determine the joint probability of at least two events
    → P(AB) = P(A|B) × P(B)
  16. Addition rule of probability
    → To determine the probability that at least one of two events will occur
    → Not mutually exclusive: P(A or B) = P(A) +P(B) – P(AB)
    → Mutually exclusive: P(A or B) = P(A) + P(B)
  17. Total probability rule
    → To determine the unconditional probability of an event, given conditional probabilities
    → P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + … + P(A|BN)P(BN), where B1, B2, …, BN is a mutually exclusive and exhaustive set of outcomes.
  18. Joint probability
    → Probability of the joint events will occur
    → P(A|B) = P(AB) / P(B)
  19. Independent events
    → Events for which the occurrence of one has no influence on the occurrence of the others
    → P(A and B) = P(A) × P(B)
    → P(A|B) = P(A), P(B|A) = P(B)
  20. Expected value
    → Weighted average of the possible outcomes of a random variable, where the weights are the possibilities that the outcomes will occur
    → E(x) = ∑P(x)x = P(x1)x1 + P(x2)x2 + … + P(xn)xn
  21. Expected values or returns can be calculated using conditional probabilities
  22. Covariance
    → Measure of how two assets move together
    → Cov(Ri,Rj) = E{[Ri-E(Ri)][Rj-E(Rj)} = σ(Ri)σ(Rj)ρ(Ri, Rj)
    → Indicates positive/negative linear relationship
    → Does not provide magnitude of covariance
  23. Properties of covariance
    General representation of same concept as variance
    → Cov(RA,RA) = Var(RA)
    → Covariance may range from negative infinity to positive infinity
  24. Correlation coefficient ρ(Ri, Rj) or ρij
    → Corr(Ri, Rj) = Cov(Ri, Rj) / σ(Ri)σ(Rj), where Cov(Ri, Rj) = Corr(Ri, Rj)σ(Ri)σ(Rj)
    → Measures the strength of the linear relationship between 2 random variables
    → No units
    → Ranges from -1 to +1, -1≤ Corr(Ri, Rj) ≤ +1
    → Corr(Ri, Rj) = 1.0, random variables have perfect positive correlation (proportional)
    → Corr(Ri, Rj) = -1.0, random variables have perfect negative correlation (proportional)
    → Corr(Ri, Rj) = 0, no linear relationship between variables, cannot predict Ri on basis of Rj using linear methods
  25. Weight of portfolio asset i (wi) = Market value of investment in asset i / market value of the portfolio
  26. Portfolio expected value, E(Rp) = ∑wi E(Ri) = w1 E(R1) + w2 E(R2) + … + wn E(Rn)
  27. Portfolio variance, Var(Rp)
    → 2-asset portfolio: Var(Rp) = wA²σ²(RA) + wB²σ²(RB) + 2 wAwB Cov(RA, RB) or Var(Rp) = wA²σ²(RA) + wB²σ²(RB) + 2 wAwB σ(RA)σ(RB)ρ(RA, RB)
    → 3-asset portfolio: Var(Rp) = wA²σ²(RA) + wB²σ²(RB) + wC²σ²(RC) + 2 wAwB Cov(RA, RB) + 2 wAwC Cov(RA, RC) + 2 wBwC Cov(RB, RC)
  28. Variance of asset portfolio:
    n(n-1)/2, to find the no. of 2 wiwj Cov(Ri, Rj) term
    → eg. 2-asset: 2 wi²σ²(Ri) terms, 1[2 wiwj Cov(Ri, Rj)]
    3-asset: 3 wi²σ²(Ri) terms, 3[2 wiwj Cov(Ri, Rj)]
               4-asset: 4 wi²σ²(Ri) terms, 6[2 wiwj Cov(Ri, Rj)]
               5-asset: 5 wi²σ²(Ri) terms, 10[2 wiwj Cov(Ri, Rj)]
  29. Bayes’ formula
    → To update a given set of prior probabilities for a given event in response to the arrival of new information
    → Update probability = Probability of new information for a given event / unconditional probability of new information × prior probability of event
    → P(I|O) = P(O|I) / P(O) × P(I)
  30. Labeling (3 or more sub-groups)
    → Situation where there are n items that can each receive one of k different labels
    → Total number of ways labels can be assigned = n! / [(n1!)×(n2!)×…×(nk!)]
    → ‘!’ = factorial (when no groups), eg. 4! = 4 × 3 × 2 × 1
    → If k=n, n!/1 = n!
  31. Combination formula (nCr) = Binomial formula (only 2 groups)
    → nCr = n! / [(n-r)!r!]
    → When k = 2, n1 + n2 = n, let r=n1 & n2 = n-r
    → Keywords: “Choose” & “Combination”
  32. Permutation formula (nPr) (only 2 groups)
    → Specific ordering of a group of objects
    → How many different groups of size r in specific order can be chosen from n objects
    → Number of permutations of r objects from n objects (nPr) = n! / (n-r)!

Thanks again for reading the notes, please like and share this post if you think my notes are helpful! Do leave your comments and let me know about your thoughts!

Good luck to your exam!

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