Cfa Quant Methods 2

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Probability distribution

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Describes the probabilities of all the possible outcomes for a random variable

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Discrete random variable

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A random variable for which the number of possible outcomes can be counted, and for each possible outcome there is a measurable and positive probaility

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Example of discrete random variable

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The number of days it rains in a given month, because the number of days it can rain is defined by the number of days in the month

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What are the two key properties of a probability function?

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1. 0 ≤ p(x) ≤ 1
2.Σp(x) = 1

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Continuous random variable

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A random variable for which the number of possible outcomes is infinite, even if lower and upper bounds exist

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Example of discrete random variable

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The actual amount of daily rainfall between 0 and 100 inches - beause the actual amount of rainfall can take on an infinite number of values

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Cumulative distribution function

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A distribution that represents the sum or cumulative value of the probabilities for the outcomes up to and including a specified outcome. May be expressed as F(x) = P(X
x)

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Discrete uniform random variable

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A random variable for which the probabilities for all possible outcomes for a discrete random variable are equal.

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Example of discrete uniform random variable

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X = {1,2,3,4,5}, p(x) = 0.2. The probability for each outcome is 0.2

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Binomial random variable

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A random variable where the number of "successes" in a given number of trials, where the outcome can either be a "success" or a "failure"

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Bernoulli random variable

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A binomial random variable where the number of trials is 1

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Binomial random variable formula

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p(x) = [n!/(n-x)!x!](p^x)(1-p)^n-x

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Expected value of X in a binomial random variable

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X = E(X) = np

where n = number of trials, p = probability of each trial

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Variance of a binomial random variable

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variance of X = np(1-p)

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Binomial tree

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A tool to express a binomial random variable, that shows all the possible combinations of up moves and down moves over a number of successive periods

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Tracking error

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The difference between the total retrun on a portfolio and the return on the benchmark against which its performance is measured

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Continuous uniform distribution

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A distribution that is defined over a range that spans between some lower limit, a and some upper limit b. Outcomes can only occur between a and b

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Probability of outcomes formula for continuous uniform distribution

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P(x1 ≤ X ≤ x2) = (x2-x1)/ (b-a)

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Properties of the normal distribution

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1. Completely described by mean and variance
2. Skewness = 0
3. Kurtosis = 3
4. A linear combination of normally distributed random variables is also normally distributed
5. The probabilities of outcomes further above and further below the mean get smaller and smaller but do not go to zero

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Univariate Distribution

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The distribution of a single random variable

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Multivariate Distribution

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Specifies the probabilities associated with a group of random variables and is meaningful only when the behavior of each random variable in the group is in some way dependent upon the behavior of the others.

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What distinguishes a multivariate distribution from a univariate normal distribution?

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The correlation is the feature that distinguishes the two distributions

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Confidence interval

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A range of values around the expected outcome within which we expect the actual outcome to be some specified percentage of the time

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Common Confidence Intervals

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90% confidence interval for X is -1.65 to +1.65
95% confidence interval for X is -1.96 to +1.96
99% confidence interval for X is -2.58 to +2.58

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Standard Normal Distribution

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A normal distribution that has been standardized so that it has a mean of 0 and a standard deviation of 1

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How to standardize a normal distribution

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Calculate the z -value of the observations

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Formual for z value

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z = (observation - population mean) / standard deviation

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Shortfall risk

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The probability that a portfolio value or return will fall below a particular (target) value or return over a given time period

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Roy's safety first criterion

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States that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level

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Threshold level

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The minimum acceptable level in Roy's safety first criterion

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Roy's safety first criterion formula

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maximize the SFRatio where SFRation = [E(Rp) - RL]/ portfolio standard deviation

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Relationship between SFR and probability of returns below the threshold return

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The larger the SFR the lower the probability of returns below the threshold return

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Lognormal distribution

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A normal distribution generated by the function e^x where x is normally distributed

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Properties of the lognormal distribution

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1. The lognormal distribution is skewed to the right
2. The lognormal distribution is bounded from below by zero so that it is useful in modeling asset prices which never take negative values

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Discretely compounded

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A compound return given some discrete compounding period, such as semiannual or quarterly

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Relationship between compunding frequency and effective annual return

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The higher the compounding frequency the greater the effective annual return

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Continuius compounding

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The limit of compounding periods

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Effective annual rate for continuous compounding formula

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Effective annual rate = eRcc - 1

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Continuously compounded rate of return fomula

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ln(S1/S0) = ln (1+HPR) = Rcc

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Monte Carlo simulation

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A technique based on the repeated generation of one or more risk factors that affect security values, n order to generate a distribution of security values

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What is Monte Carlo simulation used for

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1. Value complex securities
2. Simulate the profits/ losses from a trading strategy
3. Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities
4. Simulate pension fund assets and liabilities over time to examine the variability of the differences of the two
5. Valye portfolios of assets that have non normal returns distributions

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Limitations of monte carlo simulation

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It is fairly complex and will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/ valuation model that is used

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Historical simulation

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A simulaiton based on the actual changes in value or actual changes in risk factors over some prior period.

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Advantages of historical simulation

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Uses the actual distribution of risk factors so that the distribution of changes in the risk factors does not have to be estimated

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Limitations of historical simulation

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Past changes in risk factors may not be a good indication of future changes, and this simulation cannot answer the "What if" questions that the Monte Carlo simulation can