As you can see from Figure The distributions discussed in this reading, as well as other distributions, can be combined to create unique PDFs. It may be helpful to create a new distribution if the underlying data you are working with does not currently fit a predetermined distribution.
In this case, a newly created distribution may assist with explaining the relevant data. Here, we have two normal distributions with the same mean, but different risk levels.
We then generate a random return from the selected distribution. By repeating this process several times, we will create a probability distribution that reflects both levels of volatility. Mixture distributions contain elements of both parametric and nonparametric distributions. The distributions used as inputs i. The more component distributions used as inputs, the more closely the mixture distribution will follow the actual data.
However, more component distributions will make it difficult to draw conclusions given that the newly created distribution will be very specific to the data. By mixing distributions, it is easy to see how we can alter skewness and kurtosis of the component distributions. Skewness can be changed by combining distributions with different means, and kurtosis can be changed by combining distributions with different variances.
Also, by combining distributions that have significantly different means, we can create a mixture distribution with multiple modes e. Creating a more robust distribution is clearly beneficial to risk managers.
By creating these mixture distributions, we can improve risk models by incorporating the potential for low-frequency, high-severity events. The t-distribution is the appropriate distribution to use when constructing confidence intervals based on A.
Which of the following statements about F- and chi-squared distributions is least accurate? Both distributions A. The normal probability distribution has the following characteristics: The normal curve is symmetrical and bell-shaped with a single peak at the exact center of the distribution. The normal distribution can be completely defined by its mean and standard deviation because the skew is always 0 and kurtosis is always 3. A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
The t-distribution is similar, but not identical, to the normal distribution in shape—it is defined by the degrees of freedom and has fatter tails. The t-distribution is used to construct confidence intervals for the population mean when the population variance is not known.
The F-distribution is right-skewed and is truncated at zero on the left-hand side. The shape of the F-distribution is determined by two separate degrees of freedom. The component distributions used as inputs are parametric while the weights of each distribution within the mixture are based on historical data, which is nonparametric.
A The probability of a defective car p is 0. C A lognormally distributed random variable cannot take on values less than zero. The return on a financial security can be negative. The other choices refer to variables that cannot be less than zero. D The t-distribution is the appropriate distribution to use when constructing confidence intervals based on small samples from populations with unknown variance that are either normal or approximately normal.
D There is no consistent relationship between the mean and standard deviation of the chi-squared or F-distributions. For the exam, be prepared to explain and calculate the mean and variance for bivariate random variables. The dependency between the components is important, and you should understand the calculation for covariance and correlation.
The marginal and conditional distributions are used to transform bivariate distributions and provide additional insights for finance and risk management. Be able to use these distributions to compute a conditional expectation and conditional moments that summarize the conditional distribution of a random variable.
A random variable is an uncertain quantity or number. Multivariate random variables are vectors of random variables where a vector is a dimension of n random variables.
Thus, the study of multivariate random variables includes measurements of dependency between two or more random variables.
In this reading, we will examine bivariate random variables or components, which is a special case of the n dimension multinomial distribution. The bivariate random variable X is a vector with two components: X1 and X2. A probability mass function PMF for a bivariate random variable describes the probability that two random variables each take a specific value.
All probabilities are positive or zero and are less than or equal to 1. The sum across all possible outcomes for X1 and X2 equals 1. A probability matrix is used to describe the relationship between discrete distributions defined over a finite set of values. The most common application of a discrete bivariate random variable is the trinomial distribution. In this type of example, there are n independent trials, and each trial has one of three discrete possible outcomes.
The trinomial distribution has three parameters: the number of trials n , the probability of observing outcome 1 p1 , and the probability of observing outcome 2 p2.
An analyst estimates the probability matrix in Figure Compute the probability of a negative earnings announcement. Marginal and Conditional Distributions LO A marginal distribution defines the distribution of a single component of a bivariate random variable i.
Summing across columns constructs the marginal distribution of the row variables in a probability matrix. Summing across rows constructs the marginal distribution for the column variables in a probability matrix. Note that the sum of all possible outcomes for the monthly stock return, X1, equals 1 at the bottom of Figure Similarly, the sum of all possible outcomes for the earnings announcements at the right of Figure When there is a negative earnings announcement, the three probabilities in the first row of the bivariate probability matrix illustrated in Figure These joint probabilities are then divided by the marginal probability of a negative earnings announcement.
This is summarized in the upper right-hand corner of Figure The fund manager constructs the following bivariate probability matrix for the stock. What is the marginal probability that the stock has a positive analyst rating? What are the conditional probabilities of the three monthly stock returns given that the analyst rating is positive? The first moment of a bivariate discrete random variable is referred to as an expectation of a function.
Expectations of bivariate random variables are used to describe relationships in the same way that they are used to define moments for univariate random variables. For example, the expected return for a stock is used to define the variance of the stock in a univariate random number. The second moment of a bivariate random X has two components and is calculated as a covariance.
Covariance is the expected value of the product of the deviations of the two random variables from their respective expected values. Covariance measures how two variables move with each other or the dependency between the two variables.
The covariance of a multivariate random variable X is a 2-by-2 matrix, where the values along one diagonal are the variances of X1 and X2. The values along the other diagonal are the covariance between X1 and X2. For bivariate random variables, there are two variances and one covariance.
These two returns will have a negative covariance because they move in opposite directions. In practice, covariance is difficult to interpret because it depends on the scales of X1 and X2. Thus, it can take on extremely large values, ranging from negative to positive infinity, and, like variance, these values are expressed in terms of squared units.
The resulting value is called the correlation coefficient, or simply, correlation. However, a correlation of 0 does not necessarily imply independence. A hedge fund manager computed the covariances between two bivariate random variables. However, she is having difficulty interpreting the implications of the dependency between the two variables as the scale of the two variables are very different. Which of the following statements will most likely benefit the fund manager when interpreting the dependency for these two bivariate random variables?
Disregard the covariance for bivariate random variables as this data is not relevant due to the nature of bivariate random variables. There are four important effects of a linear transformation on the covariance of bivariate random variables. The following example illustrates the effects. Therefore, location shifts have no impact on the variance or covariance calculations, because only the deviations from the respective means are relevant.
However, the scale of each component b and d contributes multiplicatively to the change in covariance. The fourth effect of linear transformations on covariance between random variables relates to coskewness and cokurtosis. Coskewness and cokurtosis are cross variable versions of skewness and kurtosis. Interpreting the meaning of coskewness and cokurtosis is not as clear as covariance. However, both coskewness and cokurtosis measure the direction of how one random variable raised to the first power is impacted when the other variable is raised to the second power.
For example, stock returns for one variable and volatility of the returns for another variable tend to have negative coskewness. In this case, negative coskewness implies that one variable has a negative return when the other variable has high volatility.
When measuring the variance of two random variables, the covariance or comovement between the two variables is a key component. The optimal weight of asset 1 with a correlation of 0. We can note a couple of observations from the graph in Figure The standard deviation is smallest with strong negative correlations. Second, the graph is asymmetrical because the larger positive correlations result in higher standard deviations right-hand side of graph than smaller negative correlations left-hand side of graph.
The reason for the larger correlations is because the optimal weight for the minimum risk portfolio is negative for the largest correlations. This results in the second asset having a weight greater than 1. Unfortunately, with high correlations, the benefits of diversification are limited with more exposure in one asset. In the context of portfolio risk management, a conditional expectation of a random variable is computed based on a specific event occurring.
A conditional PMF is used to determine the conditional expectation based on weighted averages. A conditional distribution is defined based on the conditional probability for a bivariate random variable X1 given X2. Suppose a portfolio manager creates a conditional PMF based on earnings announcements, X2.
We will return to our previous example in Figure What is the variance of a two-asset portfolio given the following covariance matrix and a correlation between the two assets of 0.
Suppose a portfolio manager creates a conditional PMF based on analyst ratings, X2. Independent and identically distributed i. Features of i. Variables are all from a single univariate distribution. Variables all have the same moments. Expected value of the sum of n i. Variance of the sum of n i. Variance of the sum of i.
Variance of the average of multiple i. Determining the mean and variance of i. The expected value of the sum of n i. All i. The expectation of a sum is always the sum of the expectations. In this case, we assume all variables are identical. This result is only true if the variables are independent of each other in addition to identical.
This can be illustrated with the following equations. The variance of i. Since all i. Thus, for two i. Which of the following statements regarding the sums of i. The sums of i. The expected value of a sum of three i. The variance of the sum of four i. The variance of the sum of i. The variance of the average of multiple i. All probabilities in the matrix are positive or zero, are less than or equal to 1, and the sum across all possible outcomes for X1 and X2 equals 1.
A conditional distribution sums the probabilities of the outcomes for each component conditional on the other component being a specific value. It measures how two variables move with each other.
The variance of the sum of n i. A A conditional distribution is defined based on the conditional probability for a bivariate random variable X1 given X2. C Correlation will standardize the data and remove the difficulty in interpreting the scale difference between variables. C The variance of the sum of n i. Thus, for four i. The covariance terms are all equal to zero because all variables are independent.
B The variance of the average of multiple i. The covariance of i. For the exam, be able to estimate these sample moments and explain the differences from population moments.
Also, be prepared to discuss what makes estimators biased, unbiased, and consistent. Lastly, be prepared to contrast the advantages of estimating quantiles to traditional measures of dispersion. It is used to make inferences about the population mean. The arithmetic mean is the only measure of central tendency for which the sum of the deviations from the mean is zero. The biased sample estimator of variance for a sample of n i. The variance and standard deviation measure the extent of the dispersion in the values of the random variable around the mean.
You have calculated the stock returns for Alpha Corporation over the last five years to develop the following sample data set. Given this information, calculate the sample mean, variance, and standard deviation. Xi Mean 0. In the third column, the mean is subtracted from the observed value, Xi. In the fourth column, the deviations from the mean in the third column are squared.
The sum of all squared deviations is equal to 0. This amount is then divided by the number of observations to compute the variance of 0. This will be discussed later in this reading. The unbiased estimate of the standard deviation is then 0.
Population and Sample Moments LO Measures of central tendency identify the center, or average, of a data set. This central point can then be used to represent the typical, or expected, value in the data set. The first moment of the distribution of data is the mean.
Note that the population mean is unique in that a given population has only one mean. Therefore, we create samples of data to estimate the true population mean. Thus, the sample mean is simply an estimate of the true population mean. Note the use of n, the sample size, versus N, the population size.
The population mean and sample mean are both examples of arithmetic means. The arithmetic mean is the sum of the observed values divided by the number of observations. It is the most widely used measure of central tendency and has the following properties: All interval and ratio data sets have an arithmetic mean. All data values are considered and included in the arithmetic mean computation.
A data set has only one arithmetic mean i. The following example illustrates the difference between the sample mean and the population mean. You have calculated the stock returns for Beta Corporation over the last 12 years to develop the following data set.
Your research assistant has decided to conduct his analysis using only the returns for the five most recent years, which are displayed as the bold numbers in the data set. Given this information, calculate the two sample means and discuss the population mean. Because all possible random observations of returns are not observable, sample estimates are used to estimate the true population mean. A larger sample size results in an estimate that is closer to the true unobservable population mean.
Unusually large or small values can have a disproportionate effect on the computed value for the arithmetic mean. For example, the mean of 1, 2, 3, and 50 is 14 and is not a good indication of what the individual data values really are. On the positive side, the arithmetic mean uses all the information available about the observations.
The arithmetic mean of a sample from a population is the best estimate of both the true mean of the sample and the value of the next observation. Variance and Standard Deviation The mean and variance of a distribution are defined as the first and second moments of the distribution, respectively.
This results in the second term in the brackets dropping out. If data is more variable, then it is more difficult to estimate the true variance. The variance of the mean estimator will decrease when the size of the sample or number of observations is increased. Therefore, a larger sample size helps to reduce the difference between the estimated variance and the true variance of the population.
Sample parameters can be used to draw conclusions about true population parameters which are unknown. The mean estimator is a formula that transforms data into an estimate of the true population mean using observed data from a sample of the population.
Biased Estimators LO When Xi consists of i. The following equation illustrates that the mean estimator bias is zero, because the expected mean estimator is equal to the true population mean. Conversely, the sample variance is a biased estimator. The bias for the estimator is based on the sample size n. This systematic underestimation causes the sample variance to be a biased estimator of the population variance.
The best linear unbiased estimator BLUE is the best estimator of the population mean available because it has the minimum variance of any linear unbiased estimator. When data is i. An unbiased estimator is one for which the expected value of the estimator is equal to the parameter you are trying to estimate. For example, the sample mean is an unbiased estimator of the population mean, because the expected value of the sample mean is equal to the population mean.
Note that there may be other nonlinear estimators that are better at estimating the true parameters of a distribution. For example, maximum likelihood estimators of the population mean may be more accurate. However, these estimators are nonlinear and are often biased in finite samples. He wants to compute the best unbiased estimator of the true population mean and standard deviation.
The sample mean is an unbiased estimator of the population mean because the A. The first property of a consistent estimator is that as the sample size increases, the finite sample bias is reduced to zero. The second property of a consistent estimator is as the sample size increases, the variance of the estimator approaches zero. The properties of consistency ensure that estimates from large samples have small deviations from the true population mean.
This is an important concept that ensures that the estimate of the mean and variance will be very close to the true mean and variance of the population in large sample sizes.
Thus, increasing the sample size results in better estimates of the true population distribution. The LLN only requires the assumption that the mean is finite. In addition, the CLT does not require assumptions about the distribution of the random variables of the population.
No assumption regarding the underlying distribution of the population is necessary because, when the sample size is large, the sums of i. The CLT is extremely useful because the normal distribution is easily applied in testing hypotheses and constructing confidence intervals. As the sample size increases, the sample distribution appears to be more normally distributed.
Important properties of the central limit theorem include the following: If the sample size n is sufficiently large, the sampling distribution of the sample means will be approximately normal.
Each of these random samples has its own mean, which is itself a random variable, and this set of sample means has a distribution that is approximately normal. Thus, it approaches zero as the sample size increases. Skewness and Kurtosis LO The skewness statistic is the standardized third central moment of the distribution. Skewness sometimes called relative skewness refers to the extent to which the distribution of data is not symmetric around its mean. Outliers are observations with extraordinarily large values, either positive or negative.
A positively skewed distribution is characterized by many outliers in the upper region, or right tail. A positively skewed distribution is said to be skewed right because of its relatively long upper right tail. A negatively skewed distribution has a disproportionately large amount of outliers that fall within its lower left tail.
A negatively skewed distribution is said to be skewed left because of its long lower tail. The mean is the arithmetic average, the median is the middle of the ranked data in order, and the mode is the most probable outcome. For a symmetrical distribution, the mean, median, and mode are equal. For a positively skewed, unimodal distribution, the mode is less than the median, which is less than the mean.
The mean is affected by outliers; in a positively skewed distribution, there are large, positive outliers that will tend to pull the mean upward, or more positive. An example of a positively skewed distribution is that of housing prices. Hence, the mean has been pulled upward to the right by the existence of one home outlier in the neighborhood. For a negatively skewed, unimodal distribution, the mean is less than the median, which is less than the mode.
In this case, there are large, negative outliers that tend to pull the mean downward to the left. Distributions with a kurtosis greater than 3 are referred to as heavy-tailed or fat-tailed. Thus, a normal distribution has excess kurtosis equal to zero. This means there is a relatively greater probability of an observed value being far from the mean.
With regard to an investment returns distribution, a greater likelihood of a large deviation from the mean return is often perceived as an increase in risk. Note that a distribution that has thinner tails than a normal distribution if referred to as a platykurtic distribution. Most research about the distribution of securities returns has shown that returns are not normally distributed. Actual securities returns tend to exhibit both skewness and kurtosis. Skewness and kurtosis are critical concepts for risk management because when securities returns are modeled using an assumed normal distribution, the predictions from the models will not take into account the potential for extremely large, negative outcomes.
In fact, most risk managers put very little emphasis on the mean and standard deviation of a distribution and focus more on the distribution of returns in the tails of the distribution—that is where the risk is. In general, greater positive kurtosis and more negative skew in return distributions indicates increased risk. Median and Quantile Estimates LO The median is the 50th percentile or midpoint of a data set when the data is arranged in ascending or descending order.
It is similar to the mean because both measure the central tendency of the data. If the data is symmetrical then the mean and median are the same when half the observations lie above the median and half are below. The median is important because the arithmetic mean can be affected by extremely large or small values outliers.
When this occurs, the median is a better measure of central tendency than the mean because it is not affected by extreme values that may possibly be errors in the data. Estimating quantiles To determine the median and other quantiles, arrange the data from the highest to the lowest value, or lowest to highest value, and find the middle observation. The middle of the observations will depend on whether the total sample size is an odd or even number.
Answer: First, arrange the returns in descending order. What is the median return? Thus, the median return for the six managers is Estimating quartiles In addition to the median, the two most commonly reported quantiles are the 25th and 75th quantiles. The estimation procedure for these quantiles is similar to the median process.
An interquartile range IQR is a measure of dispersion from the median similar to the measure of standard deviation from the mean. A common IQR is the range from the 25th to 75th quartile.
These measures are useful in determining the symmetry of the distribution and weight of the tails. There are two properties of quantiles that make them valuable in data analysis: The interpretation of the quantiles is easy because they have the same units as the sample data.
Quantiles are a robust measure for outliers or extreme values from the mean. In other words, the median and the IQR are not impacted by outliers. Conversely, the mean is impacted by outliers.
The mean of two random variables is estimated the same way as the mean for individual variables. The arithmetic average of the sample is determined by adding up all values and dividing by the number of observations in the sample, n. The covariance between two random variables is a statistical measure of the degree to which the two variables move together. The covariance captures the linear relationship between one variable and another. A positive covariance indicates that the variables tend to move together; a negative covariance indicates that the variables tend to move in opposite directions.
What is the covariance of the returns for Stock A and Stock B? Answer: First, the expected returns for each of the stocks must be determined. Also, the covariance may range from negative to positive infinity and it is presented in terms of squared units e.
Answer: First, it is necessary to convert the variances to standard deviations. Previously, the first and second moments of mean and variance were applied to pairs of random variables. We can also apply techniques to identify the third and fourth moments for pairs of random variables that are similar to the measurements of skewness and kurtosis for individual variables. The third cross central moment is known as coskewness and the fourth cross central moment is known as cokurtosis.
Coskewness measures are zero when there is no relationship between the sign of one variable when large moves occur with the other variable. Coskewness is always zero in a bivariate normal sample because the data is symmetrical and normally distributed. Note that the first cokurtosis measurement k X,X,Y,Y is for the symmetrical case where there are two measurements from each variable 2,2.
The asymmetric configurations are 1,3 and 3,1 where one of the variables measures to the third power and the other to the first power. The symmetrical case provides the sensitivity of the magnitude of one series to the magnitude of the other series.
The cokurtosis measure will be large if both series are large in magnitude at the same time. The other two asymmetrical cases indicate the agreement of the return signs when the power 3 return is large in magnitude. The cokurtosis of a bivariate normal depends on the correlation. When the correlation is zero, the returns are uncorrelated with one another because both random variables are normally distributed.
The cokurtosis then goes up symmetrically the further the correlation is away from zero. A junior analyst is assigned to estimate the first and second moments for an investment. Sample data was gathered that is assumed to represent the random data of the true population. Which of the following statements best describe the assumptions that are required to apply the central limit theorem CLT in estimating moments of this data set?
Only the variance is finite. Both the mean and variance are finite. The random variables are normally distributed. The mean is finite and the random variables are normally distributed. A distribution of returns that has a greater percentage of extremely large deviations from the mean A. The correlation of returns between Stocks A and B is 0.
The covariance between these two securities is 0. The variance of returns for Stock A is A. Use the following information to answer Question 4. Given this probability matrix, the covariance between Stock A and B is closest to A. An analyst is graphing the cokurtosis and correlation for a pair of bivariate random variables that are normally distributed. The shape of this graph should be best described as A. It is only an estimate of the true population mean. The central limit theorem CLT states that when the sample size is large, the sums of i.
Kurtosis is the fourth central moment of a distribution and refers to how fat or thin the tails are in the distribution of data. Coskewness is zero when there is no relationship between the sign of one variable when large moves occur with the other variable.
D The calculations for the sample mean and sample variance are shown in the following table: Xi Mean. Dividing this by the number of observations, 3, results in an unbiased estimate of the mean of 0. The third column subtracts the mean from the actual return for each year.
The last column squares these deviations from the mean. The sum of the squared deviations is equal to 0. The standard deviation is then 0. D The sample mean is an unbiased estimator of the population mean, because the expected value of the sample mean is equal to the population mean. B The CLT requires that the mean and variance are finite. The CLT does not require assumptions about the distribution of the random variables of the population.
C A distribution that has a greater percentage of extremely large deviations from the mean will be leptokurtic and will exhibit excess kurtosis positive. The distribution will have fatter tails than a normal distribution. B LOS D A symmetrical curved graph with the minimum cokurtosis of 1 when the correlation is 0. We first focus on hypothesis testing procedures used to conduct tests concerned with population means and population variances.
Specific tests reviewed include the z-test and the t-test. For the exam, you should be able to construct and interpret a confidence interval and know when and how to apply each of the test statistics discussed when conducting hypothesis testing. Hypothesis testing is the statistical assessment of a statement or idea regarding a population. A hypothesis is a statement about the value of a population parameter developed for the purpose of testing a theory or belief. For example, a researcher may be interested in the mean daily return on stock options.
Hence, the hypothesis may be that the mean daily return on a portfolio of stock options is positive. Hypothesis testing procedures, based on sample statistics and probability theory, are used to determine whether a hypothesis is a reasonable statement and should not be rejected or if it is an unreasonable statement and should be rejected. Any hypothesis test has six components: The null hypothesis, which specifies a value of the population parameter that is assumed to be true.
The alternative hypothesis, which specifies the values of the test statistic over which we should reject the null. The test statistic, which is calculated from the sample data. The size of the test commonly referred to as the significance level , which specifies the probability of rejecting the null hypothesis when it is true.
The critical value, which is the value that is compared to the value of the test statistic to determine whether or not the null hypothesis should be rejected. The decision rule, which is the rule for deciding whether or not to reject the null hypothesis based on a comparison of the test statistic and the critical value. However, on the exam, recognize that if you see test size, it simply means significance level. The Null Hypothesis and Alternative Hypothesis The null hypothesis, designated H0, is the hypothesis the researcher wants to reject.
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