Dispersion is the variability around the central tendency. Absolute dispersion is the amount of variability present without comparison to any reference point or benchmark. The Range The range is the difference between the maximum and minimum values in a dataset: Range = Maximum value – Minimum value Mean Absolute Deviation (MAD) where X‾\bar{X} is the…
Normal distribution, this symmetrical, bell-shaped distribution plays a central role in the mean-variance model of portfolio selection; it is also used extensively in financial risk management. The normal distribution has the following charactristics: Skewness A distribution that is not symmetrical is termed skewed. A return distribution with positive skew has frequent small losses and a…
Scatter Plot Covariance and Correlation Correlation is a measure of the linear relationship between two random variables. The sample covariance (sXYs_{XY}) is a measure of how two variables in a sample move together: The sample correlation coefficient is a standardised measure of how two variables in a sample move together. The sample correlation coefficient (rXYr_{XY})…
The expected value of a random variable is the probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of X is denoted E(X). where XiX_i is one of n possible outcomes of the discrete random variable X. The variance of a random variable is the expected…
Conditional Expected Values, the expected value of a random variable X given an event or scenario S denoted E(X|S). Suppose the random variable X can take on any one of n distinct outcomes X1,X2,…,XnX_1, X2,\dots,X_n (these outcomes form a set of mutually exclusive and exhaustive events). The expected value of X conditional on S is…
Bayes’ Formula is a rational method for adjusting our viewpoints as we confront new information. Bayes’ formula makes use of the total probability rule: In probability notation, this formula can be written concisely as follows:
The joint probability function of two random variables X and Y, denoted P(X,Y), gives the probability of joint occurrences of values of X and Y. A formula for computing the covariance between random variables RAR_A and RBR_B is The formula tells us to sum all possible deviation cross-products weighted by the appropriate joint probability. Independence…
Mean-variance analysis holds exactly when investors are risk averse; when they choose investments to maximise expected utility or satisfaction; and when either (assumption 1) returns are normally distributed or (assumption 2) investors have quadratic utility functions (a concept used in economics for a mathematical representation of risk and return trade-offs). Safety-first rules focus on shortfall…
The expected return on the portfolio (E(Rp))(E(R_p)) is a weighted average of the expected returns (R1 to Rn)R_1\ to\ R_n) on the component securities using their respective proportions of the portfolio in currency units as weights (w1 to w2)(w_1\ to\ w_2): Portfolio variance is as follows: Covariance Given two random variables RiR_i and RjR_j, the covariance between RiR_i and…
The Lognormal Distribution A random variable Y follows a lognormal distribution if its natural logarithm, ln Y, is normally distributed. The reverse is also true: If the natural logarithm of a random variable Y, ln Y, is normally distributed, then Y follows a lognormal distribution. Like the normal distribution, the lognormal distribution is completely described by two parameters. Unlike many other distributions,…