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,…
A characteristic of Monte Carlo simulation is the generation of a very large number of random samples from a specified probability distribution or distributions to obtain the likelihood of a range of results. Another important use of Monte Carlo simulation in investments is as a tool for valuing complex securities for which no analytic pricing…