2017 Study Session # 3, Reading # 10 “COMMON PROBABILITY DISTRIBUTIONS” Probability Distribution Describes the probabilities of all possible outcomes for a random variable Sum of probabilities of all possible outcomes is Random Variable Distribution Discrete Finite (measurable) # of possible outcomes P(x) cannot be if ‘x’ can occur We can find the probability of a specific point in time Probability Density Function (PDF) It is used for continuous distribution Denoted by f(x) Discrete uniform random variable Probability Function Probability of a random variable being equal to a specific value Properties: ≤ p(x) ≤ Σ p(x) = Continuous Infinite (immeasurable) # of possible outcomes P(x) can be zero even if ‘x’ can occur We cannot find the probability of a specific point in time Cumulative Distribution Function (CDF) Calculates the probability of a random variable ‘x’ taking on the value less than or equal to a specific value of ‘x’ F(x) = P (X ≤ x) All outcomes have the same probability Uniform Probability Distribution Discrete Has a finite number of specified outcomes P(x)×k K is the probability for ‘k’ number of possible outcomes in a range cdf: F(xn) = n.p(x) Continuous Defined over a range with parameters ‘b’ (upper limit) & ‘a’ (lower limit) cdf: It is linear over the variable’s range Properties: P ( x ≤ a) = & P (x ≥ b) = ି P( a < x < b) = ௫మ ି௫భ Binomial Distribution Properties: Two outcomes (success & failure) ‘n’ number of independent trials Probability of success remains constant p(x) = ݊! ௫ (1 − )ି௫ ሺ݊ − ݔሻ! !ݔ Binomial Tree Shows all possible combinations of up & down moves over a number of successive periods Node: Each of the possible values along the tree U is up-move factor D is down-move factor (1/U) p is probability of up move (1-p) is probability of down move Copyright © FinQuiz.com All rights reserved 2017 Study Session # 3, Reading # 10 Normal Distribution Properties of Normal Distribution: Symmetric distribution Mean = Median = Mode Skewness = Kurtosis = & Excess Kurtosis = Range of possible outcomes lie between -∞ to + ∞ Asymptotic to the horizontal axis Described by two parameters i.e Mean and Variance or (standard deviation) When S.D ↑ (↓), the curve flattens (steepens) Smaller the S.D, more the observations are centered around mean Not appropriate to use for options Not appropriate to use to model asset prices Central Limit Theorem ⇒ Sum and mean of large no of independent variables in approximately normally distributed Linear combination of two or more normal random variables is also normally distributed Confidence Interval Range of values around the expected value within which actual outcome is expected to be some specified percentage of time Confidence Interval x ± 1s x ± 1.65s x ± 1.96s x ± 2s x ± 2.58s x ± 3s % 68.% 90.% 95% 95.45% 99% 99.73% Applications of Normal Distribution Shortfall Risk Risk that portfolio value will fall below some minimum level at a future date Safety First Rule focuses on Shortfall Risk Roy’s Safety First Criterion Optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level Minimize P(RP < RL) SFRatio = [ܧሺܴ ) − ܴ ሿ σ Choose the portfolio with greatest SFRatio Sharpe Ratio = [E (Rp) – Rf] / σp Portfolio with the highest Sharpe ratio minimizes the probability that its return will be less than the Rf (assuming returns are normally distributed) Managing Financial Risk value ofvariable losses (in money terms) expected over Lognormal Value at risk (VAR) ⇒minimum A random of probability Distribution a specified time period at a specified whoselevel natural log has normal distribution ⇒use of of techniques to estimate losses in Properties Stress testing/scenario analysiscannot beset negative extremely worst combinations of or scenarios is events completely described by mean and variance Log Normal distribution is more appropriate to use to model asset prices is used in Black Scholes Merton Model Compounds Rate of Return Discrete: Daily, annually, weekly, monthly compounding Continuous ln(S1/S0) = ln(1+HPR) These are additive for multiple periods Effective annual rate based on continuous compounding is given as: EAR = e Rcc-1 Copyright © FinQuiz.com All rights reserved 2017 Study Session # 3, Reading # 10 Monte Carlo Simulation Use of a computer to generate a large number of random samples from a probability distribution Uses It is used to: Plan and manage financial risk Value complex securities Estimate VAR Examine model's sensitivity to changes in the assumptions Simulation Procedure for Stock Option Valuation Step 1: Specify underlying variable Step 2: Specify beginning value of underlying variable Limitations Complex procedure Highly dependent on assumed distributions Based on a statistical rather than an analytical method Random Number Generator An algorithm that generates uniformly distributed random numbers between and Step 3: Specify a time period Step 4: Specify regression model for changes in stock price Step 5: K random variables are drawn for each risk factor using computer program/ spreadsheet Step 6: Estimate underlying variables by substituting values of random observations in the model specified in Step Step 7: Calculate value of call option at maturity and then discount back that value at time period Step 8: This process is repeated until a specified number of trials ‘I’ is completed Step 9: Finally, mean value and S.D for the simulation are calculated Historical Simulation or Back Simulation Based on actual values & actual distribution of the factors i.e., based on historical data Drawbacks Cannot be used to perform “what if’ analysis History does not repeat itself Copyright © FinQuiz.com All rights reserved ...2 017 Study Session # 3, Reading # 10 Normal Distribution Properties of Normal Distribution: Symmetric distribution... ln(S1/S0) = ln (1+ HPR) These are additive for multiple periods Effective annual rate based on continuous compounding is given as: EAR = e Rcc -1 Copyright © FinQuiz.com All rights reserved 2 017 Study. .. compounding is given as: EAR = e Rcc -1 Copyright © FinQuiz.com All rights reserved 2 017 Study Session # 3, Reading # 10 Monte Carlo Simulation Use of a computer to generate a large number of random samples