# CFA I R10 Common Probability Distribution

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. Discrete distribution:
→ 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:
→ p(x) = 0 even though x can occur
→ P(x1<X<x2)
3. Binomial random variable
→ 2 dependent outcomes, mutually exclusive
→ discrete probability distribution
→ 1 trial: Bernoulli random variable
→ P(x) = P(X=x) = (Number of ways to choose x from n) (p^x) (1-p)^x
→ Number of ways to choose x from n = n! / (n-x)!x!
→ 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-µ) / σ
*Normal distribution: mean = 0, σ = 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)] / σp
a) If portfolio returns normally distributed, minimize SFRatio
b) Choose portfolio with largest SFRatio
*Sharpe ratio = [E(Rp-Rf)] / σp, where rf = Risk free rate
8. Discrete compounded returns
→ Compound returns given some discrete compounding period, such as semiannual or quarterly
→ The higher the frequency of compounding, the higher the effective annual return
9. Continuous compounding
→ Compounding periods get shorter and shorter
→ Effective annual rate = e^(Rcc) -1
→ ln[S1/S0] = ln (1+HPR) = Rcc, where S = Interest rate, Rcc = Earnings after compounding
10. HPR = e^(Rcc×T) – 1
11. Monte Carlo Simulation
→ Technique based on repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values
→ 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
→ 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
→ 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
→ It cannot address the sort of ‘what-if’ questions that Monte Carlo simulation can
→ 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) n × (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
→ 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
→ a: lower limit; b: upper limit of uniform distribution
→ a≤ x1 ≤ x2 ≤ b, P(x1≤X≤x2) = (x2-x1)/(b-a)
19. Returns from confidence intervals: µ ± 2σ
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 within ± one standard deviation of the mean;
34% of the area falls between 0 & +1 standard deviation from mean
b) 90% of observations fall within ±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:
→ Probability that the portfolio’s return over a given time period is less than a particular target return