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Book 1: Quantitative Methods > Reading 10: Common Probability Distributions > Flashcards

Reading 10: Common Probability Distributions Flashcards

1
Q

Probability Distribution

PoORV 1

A

Lists all the possible outcomes of an experiments, along with their associated probabilities

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2
Q

Discrete Random Variable (PPFO)

A

Positive probabilities associated with a finite number of outcomes

E.g. number of days it will rain in a month

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3
Q

Probability Function (Px) (Definition)

A

Probability that a random variable is equal to a specific value

E.g. Probability that random variable X = x

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4
Q

Continuous Random Variable
(PoINO)

P(x1 < X < x2)

A

Positive probabilities associated with an infinite number of outcomes

P(x) = 0

e.g. if a variable can take on any value between two specified values (rain)

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5
Q

Distributions

A

The assignment of probabilities to the possible outcomes

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6
Q

Cumulative Distribution Function (“CDF”) (SoP)

A

Cumulative probability that a random variable will be less than or equal to a specific value e.g. sum of all probabilities up to a specific point

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7
Q

Cumulative Distribution Formula

A

F(x) = P(X < x)
Where:
< = less than or equal to

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8
Q

Discrete uniform distribution (“NDE”)

A

One where there are n discrete, equally likely outcomes

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9
Q

Cumulative Distribution Function (Fxn) for the “nth” outcome (Formula)

A

F(x) = n*P(x)
Where:
n = number of possible outcomes

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10
Q

Binomial Random Variable (Definition) (PxNt)

A

Probability of “x” successes in “n” trials where the outcomes can either be ‘success’ or ‘failure’

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11
Q

Bernoulli Random Variable

A

A binomial random variable for which the number of trials is 1

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12
Q

Binomial Random Variable (Formula) (CFS)

A

P(x) = n! / (n-x)!x! * p^x * (1-p)^n-x
Where:
p = probability of success in each trial (don’t confuse with px!)

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13
Q

Expected Value of X (Formula)

A

E(X) = np
Where:
n = number of trials
p = probability of success on each trial

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14
Q

Variance of a binomial random variable (Formula)

A

Variance of X = np(1-p)

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15
Q

Binomial Tree (“Binomial Stock Price Model”) is constructed by…

A

Showing all the possible combinations of up-moves and down-moves over a number of successive periods

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16
Q

Node

A

Each of the possible values along a binomial tree

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17
Q

Binomial Tree (“Binomial Stock Price Model”) applications

A

Pricing Options.
We can make it more realistic by shortening the length of the periods and increasing the number of periods and possible outcomes

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18
Q

Continuous Uniform Distribution (Definition)

A

Where the probability of X occurring between a and b e.g. if the range is one half of the whole distribution then the probability of x occurring in the range will be 50%

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19
Q

Continuous Uniform Distribution (Formulas)

A

P(x1 < X < x2) = (x2 - x1)/(b-a)
Where:
X’s = probability between these two
A/B = total distribution

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20
Q

Continuous Uniform Distribution (Prob of x’s between a and b expression)

A

a < x1 < x2 < b

For all x’s between a and b

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21
Q

Continuous Uniform Distribution (Prob of x outside boundaries expression)

A

P(x < a x > b) = 0

Probability of x outside of the boundaries is zero

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22
Q

Continuous Uniform Distribution (Prob of outcomes between x1 and x2 formula)

A

= (x2 - x1) / (b-a)

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23
Q

Normal Distribution (Key Properties x5)

A
  1. Symmetrical and bell-shaped curve with a single peak at the exact center of the distribution
  2. Mean = median = mode, and all are in the exact center of the distribution
  3. Skew = 0, Kurtosis = 3. Can be completely defined by its mean as a result
  4. Tails get thin but extend infinitely
  5. Linear combination of normally distributed random variables is normally distributed as well
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24
Q

Univatiate Distribution

A

Distribution of a single random variable

25
Q

Multivariate Distribution

A

Probabilities associated with a group of random variables.

Only meaningful when the behavior of each variable in the group is dependent upon the behavior of the others

26
Q

Correlation (Definition relating to multi/univariates)

A

Feature that distinguishes a multivariate distribution from a univariate normal

It indicates the strength of the linear relationship between a pair of random variables

27
Q

Correlation (Formula for multi/univariates)

A

0.5n(n-1)
Where:
N = Number of assets

28
Q

Confidence Interval (Definition + example)

A

Range within which we have a given level of confidence of finding a point estimate (mean) e.g. a 95% confidence interval is a range that we expect a random variable to be in 95% of the time

29
Q

Confidence Interval (Formula)

A

Mean + / - (1 + CI)(SD) = x
Where:
CI = Specified confidence interval
+ / - - Use plus for the top of the range and minus for the bottom

30
Q

Standard Normal Distribution (Definition)

A

Normal distribution that has been standardized so that it has a mean of zero and a standard deviation of 1

31
Q

Z-Value (Definition)

A

Number of standard deviations a given observation is from the population mean

32
Q

Standardization (Definition)

A

Process of converting an observed value for a random variable to it z-value

33
Q

Z Value (Formula)

A

Z = (observation - population mean) / Standard Deviation

34
Q

Z Table (Important characteristic)

A

Contains values generated using the cumulative density function e.g. values in the z-table are the probabilities of observing a z-value that is LESS THAN a given value

35
Q

Shortfall Risk (Definition)

A

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

36
Q

Roy’s Safety-First Criterion (Definition)

A

States that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level (“threshold level”)

37
Q

Roy’s Safety-First Criterion (Expression)

A 
Minimize P(Rp < RL)
Where:
Rp = portfolio return
Rl = threshold level return
38
Q

What other rule is Roy’s Safety-First Criterion similar to?

A

Sharpe Ratio. Although Sharpe ratio measures excess returns over a risk free rate

39
Q

Shortfall Risk Ratio (Formula)

A

SFRatio = [E(Rp) - RL] / SD p
Where:
Rp = portfolio return
RI = threshold level return

Larger SFR ratio the better

40
Q

Explain the two steps used when choosing portfolio’s with Roy’s safety-first criterion

A
  1. Determine the shortfall risk ratio

2. Choose the portfolio with the largest SFR Ratio

41
Q

Lognormal Distribution (Definition)

A

if x is normally distributed, e^x follows a lognormal distribution. A lognormal distribution is often used to model asset prices, since a lognormal random variable cannot be negative and can take on any positive value

42
Q

Lognormal Distribution (Characteristics)

A
  1. Skewed to the right

2. Bounded from below by zero so that it is useful for modeling asset prices which never take on negative values

43
Q

Price Relative (Definition)

A

End-of-period price of the asset divided by the beginning price (S1/S0) and is equal to (1+HPR)

Price relative of zero indicates a HPR of -100% as the asset has gone down to zero

44
Q

Ending Price (Formula)

A

Price Relative x Beginning Price

45
Q

Discretely Compounded Returns

A

Compound returns given some discrete compounding period, such as semiannual or quarterly

46
Q

Compound Returns (Formula)

A

[(1+ I/Y)^compounding periods] - 1

47
Q

EAR based on continuous compounding

A

[x^Rcc]-1
Where:
Rcc = stated annual rate

48
Q

EAR to continuous compounding rate

A

1+EAR + [LN key in the calculator]

49
Q

HPR to continuous compounding rate

A

1+HPR + [LN key in the calculator]

50
Q

Continuous compounding rate to HPR

A

HPRt = [e^Rcc x t] -1
Where:
t = holding period years

51
Q

Monte Carlo Simulation (Definition)

A

Uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values

52
Q

Monte Carlo Simulation (Limitations)

A
  1. Fairly Complex
  2. Answers are no better than the assumptions used
  3. Not an analytic method, but a statistical one, and cannot provide the insights that analytical methods can
53
Q

Monte Carlo Simulation (Uses)

A
  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/liabilities to examine the variability of the difference between the two
  5. Value portfolios of assets that have non-normal returns distributions
54
Q

Historical Simulation (Definition)

A

Historical simulation is based on actual changes in value or actual changes in risk factors over some prior period

55
Q

Historical Simulation (Limitations)

A
  1. Past changes in risk factors may not be a good indication of future changes
  2. Infrequent events may not be reflected in historical simulation results
  3. Cannot address the ‘what of’ scenario that the MC simulation can
56
Q

Historical Simulation (Advantage)

A

Distribution of changes in the risk factors does not have to be estimated

57
Q

90% Confidence Interval for X is…

A

1.65

58
Q

95% Confidence Interval for X is…

A

1.96

59
Q

99% Confidence Interval for X is…

A

2.58

Book 1: Quantitative Methods (10 decks)

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