Term
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Definition
-variable that can take on a countable number of values
-each outcome has a specific probability of occurring, which can be measured |
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Term
| Continuous Random Variable |
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Definition
-variable for which the number of possible outcomes cannot be counted (there are infinite possible outcomes)
-probabilities cannot b attached to specific outcomes
-ex) rate of return |
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Term
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Definition
| -identifies the probability of each of the possible outcomes of a random variable |
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Term
| Probability Function p(x) |
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Definition
-expresses the probability that 'X', the random variable takes on a specific value of 'x'
-P(X=x)
-discrete random variables |
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Term
| Probability Density Function (pdf) |
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Definition
-used to determine the probability that the outcomes lies within a specified range of possible values
-used to interpret the probability structure of CONTINUOUS random variables
-f(x) |
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Term
| Cumulative Distribution Function |
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Definition
-the probability that a random variable, X, takes on a value less than or equal to a specific value x
-F(x) = P(X <(or = to) x) |
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Term
| Possible outcomes of specified discrete random variable |
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Definition
-p(x) = 0
--> random variable cannot take particular value of x
-p(x) > 0
--> specified value of x is present in the set of possible outcomes that the random variable can take
-p(x) = 1
--> x is the only possible outcome |
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Term
| Discrete Uniform Distribution |
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Definition
-distribution in which the probability of each of the possible outcomes is identical
-ex) probability diet of outcomes from roll of a fair die |
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Term
| Bernoulli Random Variable |
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Definition
-two possible outcomes
-probability of success is always the same no matter how many times the trial is performed
-P(1) = P(Y=1) = p
P(0) = P(Y=0) = 1-p
--> where Y is the random variable |
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Term
| Binomial Random Variable (X) |
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Definition
-the number of successes (Y=1) from a Bernoulli trial that is carried out 'n' times
-Probability of x success in n trials is given by:
P(X=x) = nCx * (p)^x * (1-p)^(n-x)
--> where p=probability of success on each trial
-B(n,p) --> mean = np(1-p) |
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Term
| Binomial Tree to describe stock price movements |
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Definition
-define two possible outcomes and the prob that each outcomes will occur
-binomial tree is constructed by showing all possible combinations |
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Term
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Definition
-measure of how closely a portfolio's returns match the returns of the index to which it is benchmarked
-Tracking Error = Gross return on portfolio - Total return on benchmark index |
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Term
| Continuous Uniform Distribution |
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Definition
-defined over a range that spans between some lower limit 'a' and some upper limit 'b'
-U(a,b)
-probability of any outcome outside this interval is 0
-individual outcomes also have a probability of 0
--> P(X=x) = 0
-probability that random variable will take a value that falls between x1 and x2 that both lie within the range, a to b, is the proportion of the total area taken up by the range x1 to x2 |
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Term
| Normal Distribution (Characteristics) |
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Definition
-bell shaped
-symmetric --> median, mean and mode is the same
-X~N(mean, var)
-skewness = 0
-kurtosis = 3; excess kurtosis = 0
-linear combination of normally distributed random variables is also normally distributed
--> if the returns on each stock in a portfolio are normally distributed the returns on the portfolio will also be normally distributed
-tail on either side extends to infinity
-total area/probability under curve = 1
-68% of all values will lie within +/- 1 st dev from the mean |
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Term
| Multivariate Distribution |
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Definition
-specifies probabilities for a group of RELATED random variables
-multivariate normal distribution for the return on a portfolio with n stocks is completely defined by:
1) mean returns on each n stock
2) variance of returns of each n stocks
3) pair-wise correlations. there will be n(n-1)/2 pairwise correlations in total |
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Term
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Definition
| -describes the distribution of a SINGLE random variable |
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Term
| Probability that a normally distributed random variable lies inside a given interval |
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Definition
-90% confidence interval for X is +/- 1.65 st dev
-95% confidence interval for X is +/- 1.98 st dev
-99% confidence interval for X is +/- 2.58 st dev |
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Term
| Standard Normal Distribution |
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Definition
| -X~(mean, var) --> Z~(0,1) |
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Term
| How to standardize a random variable |
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Definition
-Z=(X - mean) / st dev
ex) The probability that we'll observe a value smaller than 9.5 for X~N(5,1.5) is exactly that same as the probability that we'll observe a value smaller than 3 for Z~N(0,1) |
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Term
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Definition
| -probability that a portfolio's return will fall below a particular target return over a given period |
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Term
| Roy's Safety-First Criterion |
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Definition
=[E(Rp) - Rt] / st dev of portfolio
-the optimal portfolio minimezs the probability that the return of the portfolio, Rp, falls below some minimum acceptable level, Rt
-minimum acceptable level is called threshold level
-essentially calculating z-score
-the HIGHER SF ratio is preferred
-higher the SF ratio the further to the left it will be on the prob distribution so probability will be less |
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Term
| Lognormal Distribution (relationship w normal distribution) |
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Definition
-generated by the function e^x, where X is normally distributed
-lne^x=x --> the logarithms of log normally distributed random variables are normally distributed
-mathematic rule: e^x will always be positive
-LOG IS NORMAL (phrase to remember) |
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Term
| Lognormal Distribution (properties) |
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Definition
-skewed to the right
-bounded by zero on the lowered
-upper end is unbounded
-useful to model distribution of asset prices bc asset prices never take negative values |
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Term
| Continuously Compounded Rate |
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Definition
-continuously compounded rate is the stated rate where as effective annual rate is the effective return received over the year
-(P1/P0)-1 = EAR
-ln(EAR + 1) = continuously compounded rate
OR ln(HPR + 1) = continuously compounded rate
-(e^continuously compounded rate) - 1 = EAR |
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Term
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Definition
-generates random numbers and operator inputs to synthetically create probability distributions for variables
-used to calculate expected values and dispersion measures for random variables which are then used for statistical inferences
-for each of the risk factor inputs, analyst specifies the parameters of the probability distribution
-based on all the information, computer will generate a output which represents the distribution of possible values for the security that you are analyzing |
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Term
| Monte Carlo Simulation (Investment Applications) |
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Definition
-experiment with proposed policy before actually implementing it
-provide a probability distribution to estimate investment risk
-provide expected values of investments that can be difficult to price
-to test models and investment tools and strategies
-estimating the distribution of the return of a portfolio composed of assets that do NOT have normally distributed returns or that has assets w features such as embedded options, call features, and parameters that change w market conditions |
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Term
| Monte Carlo Simulation (Limitations) |
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Definition
-Answers are as good as the assumptions and model used
-does not provide cause-and-effect relationships |
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Term
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Definition
-uses historical data to generate the sets of realized random variables (as opposed to a random # generator as in Monte Carlo)
-assumes random var dist in future depends on its past dist
-advantage: distribution of risk factors does not need to be estimated |
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Term
| Limitations of Historical Simulations |
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Definition
-risk factor that was not represented in historical data will not be considered in simulation
-does not facilitate "what if" analysis if the "if" factor has not occurred in the past
-assumes future will be similar to the past
-does not provide cause-and-effect relationship information |
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