The Investment Advisor Body of Knowledge + Test Bank. IMCA
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Название: The Investment Advisor Body of Knowledge + Test Bank

Автор: IMCA

Издательство: John Wiley & Sons Limited

Жанр: Зарубежная образовательная литература

Серия:

isbn: 9781118912348

isbn:

СКАЧАТЬ add the ages together, and divide by the number of people in the firm. Assuming there are n employees and ai is the age of the ith employee, then the mean, μ, is simply:

(3.21)

      It is important at this stage to differentiate between population statistics and sample statistics. In this example, μ is the population mean. Assuming nobody lied about his or her age, and forgetting about rounding errors and other trivial details, we know the mean age of people in your firm exactly. We have a complete data set of everybody in your firm; we've surveyed the entire population.

      This state of absolute certainty is, unfortunately, quite rare in finance. More often, we are faced with a situation such as this: estimate the mean return of stock ABC, given the most recent year of daily returns. In a situation like this, we assume there is some underlying data generating process, whose statistical properties are constant over time. The underlying process still has a true mean, but we cannot observe it directly. We can only estimate that mean based on our limited data sample. In our example, assuming n returns, we estimate the mean using the same formula as before:

(3.22)

      where

(pronounced “mu hat”) is our estimate of the true mean based on our sample of n returns. We call this the sample mean.

      The median and mode are also types of averages. They are used less frequently in finance, but both can be useful. The median represents the center of a group of data; within the group, half the data points will be less than the median, and half will be greater. The mode is the value that occurs most frequently.

      Sample Problem

      Question:

      Calculate the mean, median, and mode of the following data set:

      Answer:

      If there is an even number of data points, the median is found by averaging the two center-most points. In the following series:

      the median is 15 percent. The median can be useful for summarizing data that is asymmetrical or contains significant outliers.

      A data set can also have more than one mode. If the maximum frequency is shared by two or more values, all of those values are considered modes. In the following example, the modes are 10 percent and 20 percent:

In calculating the mean in Equation 3.21 and Equation 3.22, each data point was counted exactly once. In certain situations, we might want to give more or less weight to certain data points. In calculating the average return of stocks in an equity index, we might want to give more weight to larger firms, perhaps weighting their returns in proportion to their market capitalization. Given n data points, xi = x1, x2, … , xn, with corresponding weights, wi, we can define the weighted mean, μw, as:

      (3.23)

      The standard mean from Equation 3.21 can be viewed as a special case of the weighted mean, where all the values have equal weight.

      DISCRETE RANDOM VARIABLES

      For a discrete random variable, we can also calculate the mean, median, and mode. For a random variable, X, with possible values, xi, and corresponding probabilities, pi, we define the mean, μ, as:

      (3.24)

      The equation for the mean of a discrete random variable is a special case of the weighted mean, where the outcomes are weighted by their probabilities, and the sum of the weights is equal to one.

      The median of a discrete random variable is the value such that the probability that a value is less than or equal to the median is equal to 50 percent. Working from the other end of the distribution, we can also define the median such that 50 percent of the values are greater than or equal to the median. For a random variable, X, if we denote the median as m, we have:

      (3.25)

      For a discrete random variable, the mode is the value associated with the highest probability. As with population and sample data sets, the mode of a discrete random variable need not be unique.

      Sample Problem

      Question:

      At the start of the year, a bond portfolio consists of two bonds, each worth $100. At the end of the year, if a bond defaults, it will be worth $20. If it does not default, the bond will be worth $100. The probability that both bonds default is 20 percent. The probability that neither bond defaults is 45 percent. What are the mean, median, and mode of the year-end portfolio value?

      Answer:

      We are given the probability for two outcomes:

      At year-end, the value of the portfolio, V, can only have one of three values, and the sum of all the probabilities must sum to 100 percent. This allows us to calculate the final probability:

      The mean of V is then $140:

      The mode of the distribution is $200; this is the most likely single outcome. The median of the distribution is $120; half of the outcomes are less than or equal to $120.

      CONTINUOUS RANDOM VARIABLES

      We can also define the mean, median, and mode for a continuous random variable. To find the mean of a continuous random variable, we simply integrate the product of the variable and its probability density function (PDF). In the limit, this is equivalent to our approach to calculating the mean of a discrete random variable. For a continuous random variable, X, with a PDF, f(x), the mean, μ, is then:

      (3.26)

      The median of a continuous random variable is defined exactly as it is for a discrete random variable, such that there is a 50 percent probability that values are less than or equal to, or greater than or equal to, the median. If we define the median as m, then:

      (3.27)

      Alternatively, we can define the СКАЧАТЬ