Term
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Definition
The weakest level of measurement that categorizes data but does not rank them.
An example would be assigning numbers to investment funds with the numbers not being correlated to the performance of the funds. |
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Term
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Definition
A variable where the order of the data points is defined (and important), but not the difference between data points.
For example, assigning a 1 to the top ten firms in an index and a 2 to the next 10. |
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Term
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Definition
A variable where the measurement between two values is defined and meaningful.
For example, thermometers have markings describing the value between two points. |
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Term
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Definition
The most meaningful data scale that includes a true 0 origin point.
An example is height, weight, etc. |
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Term
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Definition
| A measure used to describe a characteristic of a population. |
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Term
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Definition
| A measure used to describe a characteristic of a sample set. |
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Term
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Definition
| The interval with the higher frequency of observations. |
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Term
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Definition
| A graphical representation of the absolute (i.e. data count) frequency distribution. It allows users to see where most of the data points are located. |
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Term
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Definition
| A chart connecting the mid points of each interval. Allows the user to see the shape of a data set. |
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Term
| Cumulative Relative Frequency |
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Definition
| Identifies the fraction of observations that are less than the upper limit of the given interval. It is determined by summing the relative frequencies from the lower interval up to and including the given interval. |
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Term
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Definition
| A mean used to calculate the average cost per share of a stock. See notes for equation. |
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Term
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Definition
A general term for a value at or below which a stated proportion of the data in a distribution lies. Divides a data set into fifths.
Can also be expressed as a percentile. For example the third quintile is the same as the 75% percentile.
Ly = (x+1)*(y/100)
Where X is the size of the same size and Y is the quintile you are looking for. |
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Term
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Definition
| The average of the absolute values of the deviations of individual observations from the mean. |
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Term
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Definition
| A way to standardize standard deviations to allow comparison across data sets with vastly different means or units of measurement. |
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Term
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Definition
A widely used measure of excess return per unit of portfolio risk. The formula takes into account the extra reward over the risk free rate that investors receive for choosing to invest in a risky portfolio.
Limitations include:
1) As the SR approaches 0, it gains more risk per unit of risk.
2) When returns are non linear (i.e. w/ options).
3) When returns are not normally dist. |
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Term
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Definition
| A measure of dispersion within a data set, expressed in the same units as our data. The higher the figure the more spread out for the mean data points are, indicating lack of consistency or volatility. |
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Term
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Definition
| States that for any set of observations (regardless if its a population or a sample) in any shape of distribution, the percentage of observations that lie within K standard deviations of the mean is at least 1-1/k^2 for all scenarios where K>1. |
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Term
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Definition
The distortion on a symmetric distribution which is the result of outliers. Skews affect the measures of central tendency.
Positive Skew: Mode < Median < Mean
Negative Skew: Mode > Median > Mean
Symmetrical: Mode = Median = Mean |
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Term
| Measures of Central Tendency |
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Definition
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Term
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Definition
| The measure of the degree to which a distribution is more or less peaked than a normal distribution. |
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Term
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Definition
| A situation where a distribution is more peaked than a normal distribution. |
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Term
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Definition
| A situation where a distribution is more less peaked than a normal distribution. |
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Term
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Definition
A situation where a distribution has the same level of peak as a normal distribution.
Normal distribution as a kurtosis of 3. |
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Term
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Definition
| Value of a security at maturity taking into account variables including interest rates. |
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Term
| Mutually Exclusive Events |
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Definition
| Events that cannot both happen at the same time. |
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Term
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Definition
| Probability established through analyzing past data. |
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Term
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Definition
| Probability established using formal reasoning and inspection. Still rooted in 'reality'. |
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Term
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Definition
| The least formal method of projecting a probability of occurrence - involves personal judgment. |
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Term
| Unconditional Probability |
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Definition
| Probability of an event regardless of past of future occurrence of other events. |
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Term
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Definition
A situation where the occurrence of one event affects the probability of the occurrence of another event.
P(A!B) |
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Term
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Definition
| Two events that cannot happen at the same time. |
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Term
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Definition
| The probability two things will happen simultaneously. |
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Term
| Total Probability Rule for Expected Value |
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Definition
| Used to estimate an expected value based on mutually exclusive and exhaustive scenarios. E.g. the value of a firm given probabilities for bankruptcy and survivorship. |
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Term
| Addition Rule of Probability |
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Definition
| The additional rule determines the probability of at least one of the events occurring. |
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Term
| Multiplication Rule of Probability |
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Definition
Multiplication rule is used to determine the joint probability of two events.
Can be used to find the probability of two independent (not related) or dependent events. |
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Term
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Definition
| Two events for which their probabilities of occurring are not affected by one another. |
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Term
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Definition
| The total probability rule determines the unconditional probability of an event in terms of probabilities conditional on scenarios. This is the total probability of event A occuring under all scenarios. |
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Term
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Definition
The measure of how two assets move in relation to one another. It is calculated as the product of the deviations of two random value from their expected values.
The Covariance of one asset is equal to its variance. |
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Term
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Definition
| Measures the strength of the linear relationship between two variables. Correlation has no units and can range between -1 and 1. A Corr of 1 means perfect correlation while a corr of -1 means perfectly inversely correlated. |
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Term
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Definition
| A formulaic approach to finding how many possible combinations can be made when order matters. |
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Term
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Definition
| A formulaic approach to finding how many possible combinations can be made when order does not matter. |
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Term
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Definition
| Describes the probabilities of all possible outcomes for a random variable. All probabilities of all outcomes must sum to 1. |
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Term
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Definition
| A variable for which the number of possible outcomes can be counted, and for each outcome, there is a measurable and positive portability. |
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Term
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Definition
| Specifies the probability that a random variable is equal to a specific value. THe probability that X takes on the value x. |
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Term
| Continuous Random Variable |
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Definition
| A variable for which the number of possible outcomes is infinite, even if a lower and upper bound exist. |
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Term
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Definition
| The P(x) = 0 when x cannot occur, and P(x)>0 if it can. |
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Term
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Definition
| P(X)=0 even if X can occur. We can only consider P(x1<= X <=X2) where x1 and x2 are actual numners. |
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Term
| Cumulative Distribution Function |
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Definition
| Defines the probability that a random variable x takes on a value equal to or less than a specified value x. The probability of a number includes all the probabilities below it. |
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Term
| Discrete Uniform Random Variable |
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Definition
| The simplest of all probability distributions where the probabilities for all possible outcomes of a discrete random variable are equal. |
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Term
| Binomial Random Variable and Distribution |
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Definition
| Defined as the number of successes in a given number of tries, whereby the outcomes can be either success or failure. These trials use Bernoulli random variables. Note that the P(x) is constant and all trails are deemed independent. |
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Term
| Bernoulli Random Variable |
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Definition
| A random variable that has two outcomes and trials that can be repeated. An example is a coin flip. |
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Term
| Expected Value of a Binomial Random Variable |
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Definition
| For a given series of N Trials, the expected value is equal to E(x) = n*p |
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Term
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Definition
| The difference between the total return of a portfolio and the total return on the buckram by which its performance is measured. |
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Term
| Continuous Uniform Distributions |
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Definition
Defined over a range that spans between some upper limit (a) and some lower limit (b). All outcomes can only occur between A and B.
For all A < X1 < X2 < B
P(x b) = 0
P (x1 < X < x2) = (X2 - X1)/(b-a). This defines the probability of an outcome between X2 and X1. |
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Term
| Normal Probability Distribution |
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Definition
The most extensively used probability distribution that plays key roles in modern portfolio theory and risk management technologies.
It is described by its mean and variance and is stated as X~N (mean, variance). Meaning that X is normally distributed with mean of __ and variance of __.
Skewness = 0 means that the normal distribution is symmetric about its mean so that P(x>mean) = P(X
Kurtosis = 3
Finally, the tail probabilities approach but never reach 0. |
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Term
| Multivariate Normal \ Distribution |
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Definition
Specifies the probabilities associated with a group of random variables, and is meaningful only when the behavior of each random variable is the group is dependent on the behavior of the others.
Described using the mean and variance of data, as well as the correlation between data points.
For a MND, N = x means that the MND will have X means and variances.
For a MND with x data points, the number of pairwise correlations is equal to 0.5n(n-1). |
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Term
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Definition
A range for which one can asset with a given probability (1-p), called the degree of confidence, that it will contain the parameter it is intended to estimate. That interval is often referred to as the 100(1-p) confidence interval for that parameter.
In the probabilistic interpretation, we interpret a x percent confidence interval for the population mean as: In repeating sampling, 95% of such confidence intervals will, in the long run, include or bracket, the population mean. |
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Term
| Standard Normal Distribution |
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Definition
| A normal distribution that has been standardizes so it has a population mean of 0 and a population variance of 1. To do this a Z value must be calculated which finds the number of standard deviations an observation is from the population mean. This is also known as standardization. |
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Term
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Definition
| The number of standard deviations an observation is from the population mean in a standardized normal distribution. |
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Term
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Definition
A table that shows the probability of observing a Z value that is less than the given value, z. P(Z < z).
For example, if the Z table gives you a P = .9743%, there is a 97.43% change a variable would fall to the left of the observation. To find the probability that it would fall to right of the observation, simply take the inverse of the probability (1-P). This relationship holds true only for standard normal distributions. |
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Term
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Definition
| The probability that a portfolio value or return will fall below the target value or return over a given period of time. |
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Term
| Roys Safety First Criterion |
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Definition
States that the optimal portfolio minimizes the probability that the return falls below some minimum acceptable level. This is referred to as the threshold level.
Only applies to a normally distributed portfolio.
The SFR is the number of standard deviations below the mean, thus the portfolio with the larger SFR has the lower probability of returns below the threshold amount.
To find the probability of ending with a value less than the threshold, plug the SFR into the Z table. |
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Term
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Definition
Generated by the function E^x where x is normally distributed. Widely used in the modeling of options prices (black Sholes, etc.).
Skewed to the right.
Bounded below by 0 (i.e. not data points below 0) and is skewed to the right. |
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Term
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Definition
A technique based on the repeated generation of more risk factors that affect security values in order to generate the distribution of security values.
Limitations include the complexity of the mode and that answers are no better predictors that assumptions about the distributions or risk and pricing model uses.
Advantages include being able to price complex securities w/o comparable valuation methods. |
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Term
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Definition
| A subset of a population (which is a random variable) which in terms of statistics mirrors the population. IN certain situations, evaluation of a whole population may be impossible. |
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Term
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Definition
A method of selecting a sample in a way that each item or person has the same likelihood of being included in the sample.
EX: Placing 5 objects in a hat and drawing 2. Each has a 1/5 chance of being drawn. |
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Term
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Definition
| A method of simple random sampling that involves selecting every nth object/person. |
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Term
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Definition
| THe difference between a sample statistic (mean, median, or mode) and its corresponding population parameter. Measures the accuracy of the sample. |
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Term
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Definition
| THe probability distribution of al possible sample statistics computed from a set of equal-sized samples that were randomly drawn from the same population set. |
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Term
| Stratified Random Sampling |
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Definition
| USes a classification system to separate the population into smaller groups based on a set of characteristics. From each subgroup, a random sample is taken and the results are pooled. |
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Term
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Definition
Consists of observations taken over a period of time at specific and equally spaced intervals.
EX: Yearly Stock performance. |
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Term
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Definition
| A sample of data taken at a single point in time. AN example would be the EPS of all NASDAQ stocks as of June 1st, 1993. |
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Term
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Definition
A set containing the characteristics of a time series and a cross-sectional data set. It is data that tracks the same data points over an extended period of time.
EX: GPD, inflation, and Employment data of a country over a set period of time. |
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Term
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Definition
For simple random samples of size 'n' from a population with a mean 'μ' and a finite variance 'σ2', the sampling distribution of the same mean approaches a normal probability distribution with a mean μ and variance equal to σ2/n as the sample becomes large.
If the same size n is sufficiently large (n>30) then the sample distribution of x̅ will be approximately normal.
The population mean and the mean of the distribution of all possible x̅ are all equal. |
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Term
| Standard Error of a Sample Mean |
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Definition
| This is the standard deviation of the sample mean, xBar, and describes its accuracy as an estimate of the population mean, mu. When the sample size increases, the estimator is based on more information and becomes more accurate, so its standard error decreases. |
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Term
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Definition
One for which E(x) of the estimator is equal to the perimeter you are trying to estimate.
EX: Since the E(sample mean) = E(pop. mean) the sample mean is an unbiased estimater of the population mean. |
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Term
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Definition
| An estimator for which the variance of its sampling distribution is smaller than all other unbiased estimators of the parameters you are trying to estimate. |
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Term
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Definition
| An estimator for which the accuracy of the parameter estimate increases as the sample size increases, the standard error (standard deviation of the population distribution) falls, and the sampling distribution bunches more closely around the population mean. |
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Term
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Definition
| Single (sample) values are used to estimate population parameters. |
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Term
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Definition
A bell-shaped probability distribution that is symmetrical about its mean. It is used when constructing confidence intervals based on small sample sizes (n<30) from populations with unknown variance and a normal, or approximately normal distribution. It can also be used if the variance is unknown but the sample size is large (n>30) so we can assume the central limit theorem holds true.
Properties: Symmetrical about 0. Defined by degrees of freedom (df) where the df are equal to the number of sample observations minus 1. Compared to a normal distribution, the T-Distribution has a larger probability in its tails. As the df increase, the shape of the T-Distribution more closely approaches a standard distribution. |
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Term
| Sample Standard Deviation (S) |
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Definition
| This describes the spread of values in the sample. The sample standard deviation, s, is a random quantity -- it varies from sample to sample -- but it stays the same on average when the sample size increases |
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Term
| Confidence Intervals for Non-normal Dist. |
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Definition
If the dist. is non-normal but the population variance is known, the z-statistic can be used aslong as n>30. This is because the CLT assures us that the dist. of the sample mean is approximately normal when the sample is large.
If the dist. is non-normal and the population variance is unknown, the t-statistic can be used as long as N>30. You can also use the Z-Statistic but the T-Statistic will be more conservative.
You cannot construct CI's if N<30. |
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Term
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Definition
| Using large databases to search for patterns or "rules". Some people argue that some conclusions are the result of data mining vs. being grounded in fact. |
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Term
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Definition
A situation which results when the statistical significant of the pattern is over estimated due to the result coming form data mining.
Avoid common pitfalls such as reporting conclusions until a patter was found, or the lack of supporting economic theory.
Best way to avoid the bias is to test on a separate data set for accuracy. |
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Term
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Definition
| When data is removed form the analysis to achieve a goal or presentation. This prevents data from being truly random. |
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Term
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Definition
| The elimination of "failure data" and only showing active data. For example the elimination of defunct fund returns in a sample of mutual fund returns. |
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Term
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Definition
| When a study tests data that was not available on the test date. |
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Term
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Definition
| When the data used in a study is too long or too short. To short time spans may capture anomalies, while to long sets might contains fundamental shifts in economic trends. |
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Term
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Definition
| The statistical statement or idea regarding a population. It is developed for the purpose of testing a theory or belief. |
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Term
| Hypothesis Testing Procedure |
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Definition
1) State Hypothesis
2) Select Appropriate Test Statistic
3) Specify level of significance
4) State the decision rule
5) Collect the sample and calculate data
6) Make a decision on hypothesis
7) Make a final decision. |
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Term
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Definition
| Designated H0, this is the hypothesis the researcher wants to reject. It is actually tested and is the basis for the selection of the test statistic. This is generally stated as a simple statement about a population. |
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Term
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Definition
Designate Ha, is what is concluded if there is sufficient evidence to rejct the null hypothesis. This is the hypothesis we are really trying to assess. This is b/c since you can't prove anything w/ statistics, when the null is discredited the implication is that Ha is valid.
Ha can be either one or two tailed. The classification depends on the population being tested.
Considered true until disproved. |
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Term
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Definition
The rejection of H0 when it is actually true.
Increasing the power of a test will increase the P(type I). |
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Term
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Definition
The failure to reject the H0 when it is false.
The sample size and the choice of the significant level will together determine the probability of a type II error.
Increasing the SL will increase the P(type II error) and therefor the power of the test. |
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Term
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Definition
The probably of making a Type I error (rejecting a null when it is true).
For a given SL, we can decrease P(type II) and increase the power of the test by increase the sample size. |
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Term
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Definition
| The probability of correcting rejecting the null when it is false. Also written as: 1-P(Type II Error). |
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Term
| Relationship between Hypothesis tests and CI |
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Definition
The CI is the range of value within with the researches believes the true population parameter may lie.
A CI of 95% has a LOS of 5%. |
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Term
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Definition
The probability of obtaining a test statistic that would lead to a rejection of the null hypothesis, assuming the H0 is true. It is the smallest level of significant for which the null can be rejected.
For one tailed P tests, the P value is the probability that lies above the computed test statistic for the upper tail test or below the lower tail test.
For two tail tests the p-value is the probably that lies above the positive value test plus the prob. that lies below the negative value test.
P values address variances across samples of a population which are the result of sampling errors. The lower the score the better. |
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Term
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Definition
Used for hypothesis tests concerning the variance of a normally distributed population. It is used to test a hypothesis on a specific value of the population variance.
Testing the variance requires a chi-squared test (X^2). The X^2 distribution is asymmetrical and approaches the normal distribution as the df increases. |
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Term
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Definition
An F-test is any statistical test in which the sampling distribution of test statistic has an F-distribution when the null hypothesis is true. Similarly, any statistical test that uses the F distribution can be called F test.
It is used to compare the variances of two quantitative variables. |
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Term
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Definition
| The point which is compared to the test statistic to determine if we can reject the null hypothesis. |
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Term
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Definition
The study of collective market sentiment, as expressed in buying and selling assets. It is based on the idea that prices are based on supply and demand. Only participants who actually trade effect prices.
A key assumption is that market prices reflect both relational and irrational investor behavior. This further implies that the efficient market theory does not hold.
Based on observable data vs. assumptions where are used in fundamental anlaysis (i.e. projections).
Can be applied to assets classes which do not have cash flows such as commodities. |
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Term
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Definition
The simplest technical analysis chart that shows closing prices for each period as a continuous line.
[image] |
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Term
| Bar Chart (Tech. Analysis) |
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Definition
| A chart which adds the highs and lows for each trading period and often including op[image]en prices as well. Closing prices are on the right and open on the left. |
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Term
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Definition
USes the same data as bar charts but displays a box bounded by the opening and closing prices. The box is filled if the closing price is lower than the open and clear if the inverse is true.
[image] |
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Term
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Definition
Charts drawn on a piece of graph paper w/ price on the vertical axis. Unlike other charts the horizontal axis does not represent time, but the number of directional changes.
[image] |
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Term
| Relative Strength Analysis |
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Definition
| An analysis that compares the ratio of an assets closing price to a benchmark and is plotted on a line. Increase trends point to outperformance or positive relative value. |
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Term
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Definition
| A price level where market buying is suppose to emerge to prevent further decreases in the stock price. |
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Term
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Definition
| The price level where selling is said to occur which prevents stock prices from passing. |
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Term
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Definition
| Refers to the belief that breach's in the resistance level become resistance levels and the breaches of the support level become resistance levels. |
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Term
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Definition
| Occurs when a trend approaches a range of prices but [image]fails to continues beyond that range. |
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Term
| Head and Shoulders Pattern |
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Definition
On the technical analysis chart, the Head and shoulders formation occurs when a market trend is in the process of reversal either from a bullish or bearish trend; a characteristic pattern takes shape and is recognized as reversal formation.
Technical analysts use the size of the h&S pattern to project a price target for the ensuing downtrend. The size is the difference between the head (highest price) and the resistance level (shoulder). The price target for a post H&S pattern is the difference between the head and the neckline. |
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Term
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Definition
| Similar to the H&S pattern, they signal a weakening in the buying pressure that has being active during an uptrend. Capped by the resistance level. [image] |
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Term
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Definition
| Suggest a pause in a trend rather than a reversal. Triangles form when prices reach lower highs and higher lower over a period of time. |
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Term
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Definition
| When trading temporarily forms a range between support level and resistance level. |
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Term
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Definition
| When the ST moving average crosses above the LT moving average. Indicates a change in trends upwards. |
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Term
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Definition
| When the STMA crosses below the LTMA. Indicates a change in trends down. |
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Term
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Definition
Constructed based one the standard deviation of closing prices over the last n periods. Analysts typically draw them above/below the moving average. The bands converge during periods of low vol and widen during high vol.
[image] |
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Term
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Definition
| Doing the opposite of what most traders are doing. Contrarian's believe that people will sell at the wrong time. |
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Term
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Definition
| Indicators used to identify if markets are overbought or oversold. This indicators are scaled down so they oscillate around one or two points. Extremely high values are indicators that the market is overbought. |
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Term
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Definition
| Based on the ratio of total price increases to total prices declines over a select number of periods. The ratio is scaled to 100 with high values indicating overbought market. |
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Term
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Definition
Calculated from the latest closing price and the highest and lower prices reached in the recent period.
In a sustainable uptrend, price tend to move closer to recent highs. SPS uses two lines that are bound by - and 100. |
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Term
| Mutual Fund Cash Position |
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Definition
| The ratio of MF's cash to assets. During an uptrend, fund managers want to invest cash b/c it only earns the Rf and decreases fund returns. |
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Term
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Definition
| Based on the belief that financial market prices can be described by an interconnected set of cycles. Cycles can range from a few minutes to centuries. |
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Term
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Definition
| The measure of funds flowing into advancing and declining stocks. An index value of >1 mean more vol in declining stocks while a value of <1 means more value in advancing stocks. |
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Term
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Definition
| States that years ending with a 5 will have the best performance of any of the 10 years in a decade. Those ending in a 0 will have the worth. |
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Term
| The significance level of a test |
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Definition
| the probability that a true null hypothesis will be rejected by chance because the test statistic is from a sample and may take on a value that is outside the range of critical values because of sampling error. |
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