# CFA I R9 Probability Concepts

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)
→ 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)!