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Quantitative Methods: Reading #4, #5, #6: Probability Trees &…

- - - - An Event: An outcome or a specified set of outcomes
      - Phân loại
        
        Mutually Exclusive Events: Events that cannot both happen at the same time
        
        Exhaustive events: Those that include all possible outcomes
    - - Defining properties of probability
        
        Probability of occurrence of any event (Ei): P(Ei) thuộc [ 0 ; 1 ]
        
        If a set of events, E1, E2,... En, is mutually exclusive and exhaustive => sum P(Ei) = 1
      - Phân loại
        
        Unconditional (Marginal) Probability: The probability of an event regardless of the past or future occurrence of other events
        Ex: Toss a coin/dice Lottery
        
        Conditional Probability: The occurrence of one event affects the probability of the occurrence of another event
        
        P(A/B) = the probability of A given the occurrence of B
        EX: Pass CFA Exam. What's the condition of this event?
      - Probability Rules
        
        Multiplication Rule
        
        Determine the joint probability of two events
        
        P(AB) = P(A/B) x P(B) = P(B/A) x P(A)
        
        Addition Rule
        
        Determine the probability that at least one of 2 events will occur
        
        P(A or B) = P(A) + P(B) - P(AB)
        
        Total probability rule
        
        Determine the unconditional probability of an event
        
        P(A) = P(A/B1) P(B1) + P(A/B2) P(B2) + .... + P(A/Bn) P(Bn)
    - - Definition: Events for whihc the occurrence of one has no influence on the occurrence of the others
      - Events A and B are independent if and only if:
        
        P(A/B) = P(A) or equivalently, P(B/A) = P(B)
        EX: The return of Vinhomes 2021 and the temparature in London
- - - - It is completeoly descrebed by its mean, u and variance, o^2, stated as X ~ N(u,o^2)
        u: arithmetic mean of population
        o^2 : variance of population
      - Skewness = 0: normal distribution is symmetric about its mean P(X < = u) = P(u < = X) = 0.5, and mean = median = mode.
      - Kurtosis = 3: a measure of how flat the distribution
      - A linear combinaiton of 2 or more normal random variables is also normally distributed.
      - The probabilities of outcomes further above and below the mean get smaller and smaller but do not go to 0 (the tails get very thin but extend infinitely)
    - - An Univarate Distribution: The distribution of a single random variable
      - A Multivariate Distribution:
        
        Specifies the probabilities associated with a group of random variables
        
        Meaningful when the behavior of each random variable is dependent upon the behavior of the others.
      - The role of correlation in the Multivariate Normal Distribution:
        using asset returns as random variables, the multivariate normal distribution for the returns on n assets can be completely defined by:
        
        n means of the n series of returns (u1, u2, u3.....un)
        
        n variances of the n series of returns (o1^2, o2^2,......on^2)
        
        0.5n(n-1) pair-wise correlations
    - - Definition of confidence intervals: A confidence interval is a range of calues around the expected outcome within which we expect the actual outcome to be some specified percentage of time
      - Common confidence interval for normal distributions:
        
        The 90% confidence interval for X is u - 1.65s to u + 1.65s
        
        The 95% confidence interval for X is u - 1.96s to u + 1.96s
        
        The 99% confidence interval for X is u - 2.58s to x + 2.58s
    - - Standard normal distribution is a normal distribution that has been standardized so that it has a mean of 0 and a standard deviation of 1
      - The z-value represents the number of standard deviations away from the population mean, a given observations lies, the formula to standardize a random variable:
        
        z = ( observation - population mean) / standard deviation
        
        z = ( x - u ) / o
      - Calculating probabilities using Standard normal distribution
        
        Value of mean (u) & Standard deviation (o) => Determine z-value => Use z-table => Determine probability
    - - Roy's Safety-First Criterion: the optimal portfolio minimizes the probability that the return of the portfolio (Rp) falls below some minimum acceptable level (threshold level - Rl):
        
        also equal -Z-value the higher SFR, the more optimal a portfolio
        - Assumption:
        
        Return are normally distributed
        
        Investors are risk averse and rational
      - Stress testing and scenario analysis: estimating losses in extremely unfavorable combinations of events or scenarios.
        
        Value at Risk (VaR) is a money measure of the minimum value of losses expected over a specified time period (e.g., a day, a quarter, or a year) at a given level of probability (often 0.05 or 0.01)