Probability Distributions

Question 1

What does a probability distribution indicate?

Answer 1

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.

Question 2

What is a discrete random variable?

Answer 2

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.

Question 3

What is a continuous random variable?

Answer 3

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!

Question 4

What is the probability function of a discrete random variable?

Answer 4

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.

Question 5

What is a probability density function(pdf)?

Answer 5

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.

Question 6

What is the cumulative distribution function?

Answer 6

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.

Question 7

What is a discrete uniform distribution?

Answer 7

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.

Question 8

What are the two types of uniform distributions?

Answer 8

1. Discrete
2. Continuous

Question 9

What is a binomial distribution?

Answer 9

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

Question 10

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

Answer 10

The probability for each outcome equals 1/n

Question 11

What is a binomial tree?

Answer 11

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)

Question 12

What is traking error?

Answer 12

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.

Question 13

What is a continuous uniform distribution?

Answer 13

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.

Question 14

What are four characteristics of a normal distribution?

Answer 14

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

Question 15

What is a multivariate distribution?

Answer 15

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.

Question 16

What does a correlation of a multivariate distribution describe?

Answer 16

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

Question 17

What is a confidence interval?

Answer 17

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.

Question 18

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

Answer 18

µ_+_ 1.65 standard deviations

Question 19

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

Answer 19

µ+ 1.96 standard deviations

Question 20

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

Answer 20

µ+ 2.58 standard deviations

Question 21

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

Answer 21

1. Mean: 0
2. Standard Distribution: 1

Question 22

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

Answer 22

Z = (x-µ) / σ

Question 23

What is the z-table used for?

Answer 23

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

Question 24

What is the safety first ratio for a portfolio?

Answer 24

= E(r) - Threshold Return
Standard Deviation

Question 25

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

Answer 25

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.

Question 27

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

Answer 27

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.

Question 28

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

Answer 28

1. If x is normally distributed, e^x follows a lognormal distribution. 2. Statistical distribution of random variables which have a normally distributed logarithm.

Question 29

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

Answer 29

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.

Question 30

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

Answer 30

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%

Question 31

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

Answer 31

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.**

Question 32

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

Answer 32

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

Question 33

What is Monte Carlo simulation?

Answer 33

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.

Question 34

What are the limitations of Monte Carlo simulations?

Answer 34

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

Question 35

Compare Monte Carlo simulation and historical simulation:

Answer 35

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.

Question 36

What is a limitation of historical simulation?

Answer 36

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