Reading 9 Probability Distributions Flashcards
Describe a tracking error, and express it mathematically.
A measure of how closely a portfolio’s returns match the returns of the benchmark or index. It is the difference between the total return on the portfolio (before deducting fees) and the total return on the benchmark or index.
Tracking error = Gross return on a portfolio − Total return on benchmark or index
Mathematically describe how to standardize a given observation of a normally distributed random variable.
z = (observed value − population mean) / standard deviation = (x − μ) / σ
Explain a continuous random variable.
One for which the number of possible outcomes cannot be counted (there are infinite possible outcomes), and therefore, probabilities cannot be attached to specific outcomes.
List 2 properties of all probability distributions.
- 0 ≤ p(x) ≤ 1: The probability of an outcome cannot be less than zero. If the outcome is not possible, its probability will equal zero; p(x) = 0. The probability of an outcome cannot exceed 1. If an outcome is the only outcome possible, the probability of the outcome equals 1; p(x) = 1.
- ∑p(x) = 1: The sum of the probabilities of all possible outcomes equals 1.
When is the binomial distribution skewed to the right and left?
It is skewed to the right when the probability of success is less than 0.50.
It is skewed to the left when the probability of success is more than 0.50.
Describe the Monte Carlo simulation, and list the 3 steps needed to specify the Monte Carlo simulation.
Monte Carlo simulation generates random numbers and operator inputs to synthetically create probability distributions for variables. It is used to calculate expected values and dispersion measures of random variables, which are then used for statistical inference.
Step 1: Specify quantity of interest (e.g., option price) along with underlying variables (e.g., stock price).
Step 2: Specify a period, and break it down into a number of subperiods.
Step 3: Specify distributional assumptions for risk factors affecting underlying variables.
Explain how the Safety First (SF) Ratio provides a new perspective on the Sharpe ratio.
Highest Sharpe ratio optimizes excess return over the risk-free rate, Rf. Highest Safety First ratio optimizes excess return over the lowest acceptable return, RL.
Identify the properties of a normal distribution.
It is described by its mean (μ) and variance (σ2). The distribution is stated as X ∼ N (μ,σ2).
The distribution has a skewness of 0, which means that it is symmetric about its mean. P(X≤mean) = P(X≥mean) = 0.5, and the mean, median, and mode are the same.
Kurtosis equals 3, and excess kurtosis equals 0.
A linear combination of normally distributed random variables is also normally distributed.
The probability of the random variable lying in ranges further away from the mean gets smaller and smaller, but never goes to zero.
Identify the investment applications of the Monte Carlo simulation, and describe its limitations.
To test models, investment tools, and strategies, or to experiment with a proposed policy before implementing it.
To provide a probability distribution used to estimate investment risk (e.g., VAR).
To estimate expected values of investments that are difficult to price.
The simulation is only as good as the assumptions and model used. It is also limited, because it does not provide cause-and-effect relationships.
Describe the probability statements that can be made about normal distributions.
Approximately 50% of all observations lie in the interval μ ± (2/3)σ
Approximately 68% of all observations lie in the interval μ ± 1σ
Approximately 95% of all observations lie in the interval μ±1.96σ
Approximately 99% of all observations lie in the interval μ ± 2.58σ
Define a continuous uniform distribution.
A continuous uniform distribution is described by a lower limit, a, and an upper limit, b. These limits serve as the parameters of the distribution. The probability of any outcome or range of outcomes outside these limits is 0. Being a continuous distribution, individual outcomes also have a probability of 0. The distribution is often denoted as U(a, b).
Explain what a confidence interval represents.
The range of values within which a population parameter is expected to lie a specified percentage of the time.
When is the lognormal distribution frequently used? List the 3 features that differentiate the lognormal distribution from the normal distribution.
The lognormal distribution is used to model the probability distribution of asset prices because it is bounded by zero on the lower side.
Three features are:
- It is bounded by zero on the lower end.
- The upper end of its range is unbounded.
- It is skewed to the right (positively skewed).
Describe a Bernoulli trial.
An experiment with 2 possible outcomes labeled “success” or “failure,” which are:
Mutually exclusive; the occurrence of one outcome precludes the occurrence of the other.
Collectively exhaustive; no other outcomes are possible.
Note: A Bernoulli random variable is different from a binomial distribution because a Bernoulli trial is carried out n times, the number of successes, X, is called a Bernoulli random variable, and the distribution that X follows is known as the binomial distribution.
Explain the probability density function (pdf) and the cumulative distribution function (cdf).
A pdf can be used to determine the probability that the outcome lies within a specified range of possible values.
A cdf is the probability that a random variable, X, takes on a value less than or equal to a specific value, x. It represents the sum of the probabilities of all outcomes that are less than or equal to the specified value of x.
Note: Below is the cdf function denoted mathematically.
F(x) = P(X ≤ x)
