# CFA I R10 Common Probability Distribution

Thank you for visiting visiting WhitneyDiary CFA I Notes series,Â 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. Discrete distribution:
â†’ p(x) = 0 when x cannot occur, or p(x) > 0 if x can occur
*A discrete uniform random variable has a finite set of possible outcomes, each with an equal probability
2. Continuous distribution:
â†’ p(x) = 0 even though x can occur
â†’ P(x1<X<x2)
3. Binomial random variable
â†’ 2 dependent outcomes, mutually exclusive
â†’ discrete probability distribution
â†’ 1 trial: Bernoulli random variable
â†’ P(x) = P(X=x) = (Number of ways to choose x from n) (p^x) (1-p)^x
â†’ Number of ways to choose x from n = n! / (n-x)!x!
â†’ Probability of exactly x successes in n trials: p(x) = {n!/[(n-x)!x!]} (p^x) [(1-p)^(n-x)]
*A binomial random variable is the number of successes in a given number of trials of a Bernoulli random variable
4. Expected value of X = E(X) = np
5. Variance of X = np(1-p)
6. Standard normal distribution (Z) = (Observation-population mean) / standard deviation = (x-Âµ) /Â Ïƒ
*Normal distribution: mean = 0,Â Ïƒ = 1
7. Roy’s safety-first criterion: The optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level, P(Rp<Rl)
*Rp = Portfolio return, Rl = Threshold level return
*SFRatio = [E(Rp-Rl)] /Â Ïƒp
a) If portfolio returns normally distributed, minimize SFRatio
b) Choose portfolio with largest SFRatio
*Sharpe ratio = [E(Rp-Rf)] /Â Ïƒp, where rf = Risk free rate
8. Discrete compounded returns
â†’ Compound returns given some discrete compounding period, such as semiannual or quarterly
â†’ The higher the frequency of compounding, the higher the effective annual return
9. Continuous compounding
â†’ Compounding periods get shorter and shorter
â†’ Effective annual rate = e^(Rcc) -1
â†’ ln[S1/S0] = ln (1+HPR) = Rcc, where S = Interest rate, Rcc = Earnings after compounding
10. HPR = e^(RccÃ—T) – 1
11. Monte Carlo Simulation
â†’ Technique based on repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values
â†’ Simulation procedure:
a) Specify probability distributions of stock prices & of the relevant interest rate, as well as the parameters (mean, variance, possibly skewness) of the distributions
b) Randomly generate values for both stock prices & interest rates
c) Value the options for each pair of risk factor values
d) After many iterations, calculate the mean option value & use that as your estimate of the option’s value
12. Uses of Monte Carlo simulation:
a) Value complex securities
b) Simulate the profits/losses from a trading strategy
c) Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets & liabilities
d) Simulate pension fund assets & liabilities over time to examine the variability of the difference between the two
e) Value portfolios of assets that have non-normal returns distributions
13. Limitations of Monte Carlo simulation:
a) Fairly complex
b) Provide answers that are no better than the assumptions about the distributions of the risk factors & the pricing/valuation model that is used
c) Not an analytical method, but a statistical one
14. Historical simulation
â†’ Based on actual changes in value or actual changes in risk factors over some prior period
a) Advantage: Using the actual distribution of risk factors
b) Disadvantage: Past changes in risk factors may not be a good indication of future changes
â†’ Event that occur infrequently may not be reflected in historical simulation results unless the events occurred during the period from which the values for risk factors are drawn
â†’ It cannot address the sort of ‘what-if’ questions that Monte Carlo simulation can
â†’ It cannot investigate the effect on the distribution of security/portfolio values if we increase the variance of one of the risk factors by 20% like Monte Carlo simulation
15. Multivariate distributions describe the probabilities for more than one random variable, whereas a univariate distribution is for a single random variable
16. A multivariate normal distribution for the returns of n stocks is defined by
a) n means
b) n variances
c) nÂ Ã— (n-1)/2 distinct correlations
17. The correlation(s) of a multivariate distribution describes the relation between the outcomes of its variables relative to their expected values
18. Continuous uniform distribution
â†’ One where the probability of X occurring in a possible range is the length of the range relative to the total of all possible values
â†’ a: lower limit; b: upper limit of uniform distribution
â†’ aâ‰¤ x1 â‰¤ x2 â‰¤ b, P(x1â‰¤Xâ‰¤x2) = (x2-x1)/(b-a)
19. Returns from confidence intervals:Â ÂµÂ Â± 2Ïƒ
20. Lognormal distribution vs normal distribution
 Lognormal distribution Normal distribution Univariate distribution Symmetrical Skewed to the right Positive & negative values Positive values More useful for modeling asset prices

*If a stock’s continuously compounded return is normally distributed, then the distribution of future stock price is lognormal.
*Given Y is lognormally distributed, then ln y is normally distributed

21. Annual rate of compounding = (e^r) – 1
22. General approximations of normal distribution:
a) 68% of observations fall withinÂ Â± one standard deviation of the mean;
34% of the area falls between 0 & +1 standard deviation from mean
b) 90%Â of observations fall withinÂ Â±1.65 standard deviation of the mean;
34% of the area falls between 0 & +1 standard deviation from mean
23. Confidence interval
 Probability (Z-tailed) Z-score One-tail 0.95 1.65 1.28 0.975 1.96 1.65 0.99 2.575 2.33 0.999 3
24. A portfolio’s shortfall risk:
â†’ Probability that the portfolio’s return over a given time period is less than a particular target return