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1. CFA Ethics & Quantitative Methods > Probability Distributions > Flashcards

Probability Distributions Flashcards

1
Q

What does a probability distribution indicate?

A
  1. Lists all of the possible outcomes of an experiment, along with their associated probabilities
  2. This range will be between the min and max statistically possible values.
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2
Q

What is a discrete random variable?

A
  1. Positive probabilities associated with a finite number of outcomes
  2. Ex: Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.
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3
Q

What is a continuous random variable?

A
  1. Has positive probabilities assocaited with a range of values.
  2. Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter’s weight could take on any value between 150 and 250 pounds.
  3. The probability of a single value is ZERO!
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4
Q

What is the probability function of a discrete random variable?

A
  1. P(X=x) = p(x)
  2. Two key properties:
    1. 0_<p(x)<_1
    2. Σp(x)=1
  3. Specifies the probability that a discrete random variable is equal to a specific value.
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5
Q

What is a probability density function(pdf)?

A
  1. The term for a function for a continuous random variable used to determine that it will fall in a particular range.
  2. Often denoted as f(x)
  3. the PDFs of futures exchange rates and equity prices can be employed in models in order to get a more complete picture regarding future market sentiment.
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6
Q

What is the cumulative distribution function?

A
  1. Gives the probability that a random variable will be less than or equal to specific values.
  2. Represented by the area under the probability distribution to the left of that value.
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7
Q

What is a discrete uniform distribution?

A
  1. A distribution where there are n discrete, equally likely outcomes.
  2. There are two types of uniform distributions: discrete and continuous. The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5 or 6, but it is not possible to roll a 2.3, 4.7 or 5.5.
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8
Q

What are the two types of uniform distributions?

A
  1. Discrete
  2. Continuous
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9
Q

What is a binomial distribution?

A

A probability distribution for a binomial (discrete) random variable that has two possible outcomes.

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10
Q

What is the probability for a discrete uniform distribution with n possible outcomes?

A

The probability for each outcome equals 1/n

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11
Q

What is a binomial tree?

A
  1. Illustration of all the possible values that a variable(such as a stock price) can take on, given the probability of an up-move and the magnitude of an up-move(up-move factor)
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12
Q

What is traking error?

A
  1. The total return on a portfolio minus the total return on a benchmark or index portfolio
  2. This measure reports the difference between the return an investor receives and that of the benchmark he or she was attempting to imitate.
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13
Q

What is a continuous uniform distribution?

A
  1. A distribution where the probability of X occuring in a possible range is the length of the range relative to the total of all possible values.
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14
Q

What are four characteristics of a normal distribution?

A
  1. Symmetrical and bell-shaped with a single peak at the exact center of the distrobution
  2. Mean=median=mode(all are in the exact center)
  3. Skew = 0
  4. Kurtosis = 3
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15
Q

What is a multivariate distribution?

A
  1. Describes probabilities for more than one random variable.
  2. Ex: A portfolio of 20 stocks could have return outcomes described in terms of 20 separate univariate distributions, or as one multivariate distribution.
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16
Q

What does a correlation of a multivariate distribution describe?

A

It describes the relationship between the outcomes of it’s variables relative to their expected values.

17
Q

What is a confidence interval?

A
  1. A range with which we have a given level of confidence of finding a point estimate.
  2. The probability that a value will fall between an upper and lower bound of a probability distribution.
  3. Ex: given a 99% confidence interval, stock XYZ’s return will fall between -6.7% and +8.3% over the next year.
18
Q

What is the standard deviation for a 90% confidence interval? (normally distributed random variable)

A

µ_+_ 1.65 standard deviations

19
Q

What is the standard deviation for a 95% confidence interval? (normally distributed random variable)

A

µ+ 1.96 standard deviations

20
Q

What is the standard deviation for a 99% confidence interval? (normally distributed random variable)

A

µ+ 2.58 standard deviations

21
Q

What is the mean & standard deviation of a normal probability distribution?

A
  1. Mean: 0
  2. Standard Distribution: 1
22
Q

What is the equation to standardize a normally distributed random variable “X”?

A

Z = (x-µ) / σ

23
Q

What is the z-table used for?

A

Used to find the probability that X will be less than or equal to a given value.

24
Q

What is the safety first ratio for a portfolio?

A

= E(r) - Threshold Return
Standard Deviation

25
Q

What is shortfall risk? How does it relate to the safety first ratio?

A
  1. The probability that a portfolio’s value (or return) will fall below a specific value over a given period of time.
  2. Greater safety-first ratios are preferred and indicate a smaller shortfall probability.
  3. Roy’s safety-first criterion states that the optimal portfolio minimizes shortfall risk.
26
Q
A
27
Q

Why is the lognormal distribution is used to model asset prices?

A
  1. Because a lognormal random variable cannot be negative and can take on any positive value.
  2. Useful because even though stock percentage returns can be negative, the acual stock cannot be worth less than $1.
28
Q

What is a lognormal distribution? How does it differ from a normal distribution?

A
  1. If x is normally distributed, ex follows a lognormal distribution.
  2. Statistical distribution of random variables which have a normally distributed logarithm.
29
Q

What is the difference between discretely & continuously compounded rates of return?

A
  1. Discrete compounding relates to measurable holding periods and a finite number of holding periods.
  2. Continuous compounding relates to holding periods so small they cannot be measured, with frequency of compounding so large it goes to infinity.
30
Q

Explain the effect that decreasing the length discrete compounding periods has on the effective annual rate:

A
  1. As we decrease the length of discrete compounding periods (e.g., from quarterly to monthly) the effective annual rate increases.
  2. Ex:
  • Annual holding periods, 12% compounded once = (1.12)1 - 1 = 12%.
  • Quarterly holding periods, 3% compounded 4 times = (1.03)4 - 1 = 12.55%
31
Q

What is the equation for the effective annual rate when compounding becomes continuous?

A

As the length of the compounding period in discrete compounding gets shorter and shorter, the compounding becomes continuous, where the effective annual rate = ei – 1.

32
Q

What is the calculation of a continuously compounded rate of return, given a specific holding period return?

A

For a holding period return (HPR) over any period, the equivalent continuously compounded rate over the period is ln(1 + HPR).

33
Q

What is Monte Carlo simulation?

A
  1. Monte Carlo simulation uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values.
  2. A problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables.
34
Q

What are the limitations of Monte Carlo simulations?

A

It is fairly complex and will provide answers that are no better than the assumptions used.

35
Q

Compare Monte Carlo simulation and historical simulation:

A
  1. Historical simulation uses randomly selected past changes in risk factors to generate a distribution of possible security values, in contrast to
  2. The Monte Carlo simulation(which) uses randomly generated values.
36
Q

What is a limitation of historical simulation?

A

It cannot consider the effects of significant events that did not occur in the sample period.

1. CFA Ethics & Quantitative Methods (8 decks)

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