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…
Resampling, repeatedly draws samples from the original observed data sample for the statistical inference of population parameters. Boostrap, one of the most popular resampling methods, uses computer simulation for statistical inference without using an analytical formula such as a z-statistic or t-statistic. Both the bootstrap and the Monte Carlo simulation build on repetitive sampling. Bootstrapping…
Probability sampling gives every member of the population an equal chance of being selected. Non-probability sampling depends on factors other than probability considerations, such as a sampler’s judgment or the convenience of access data. Consequently, there is a significant risk that non-probability sampling might generate a non-representative sample. Simple Random Sampling A sampling plan is…
The Central Limit Theorem Central Limit Theorem. Given a population described by any probability distribution having mean µ and finite variance σ2, the sampling distribution of the sample mean X‾\bar{X} computed from random samples of size n from this population will be approximately normal with mean µ (the population mean) and variance σ2/n (the population variance divided by n) when the…
where: SX‾S_{\bar{X}} = the estimate of the standard error of the sample mean, B = the number of resamples drawn from the original sample, θ^b\hat\theta_b = the mean of a resample, and θ‾\bar\theta = the mean across all the resample means.
In hypothesis testing, we test to determine whether a sample statistic is likely to come from a population with the hypothesised value of the population parameter. The Process of Hypothesis Testing A hypothesis is a statement about one or more populations that we test using sample statistics. Stating the Hypotheses Null hypothesis (or null), designated…