In a nutshell, Monte Carlo simulation uses random numbers to approximate the solutions to a variety of problems.  For our purposes, these problems will generally involve trying to approximate complicated probability distributions for such problems as calculating: A portfolio’s value at risk (VaR) The probability that an investor will outlive her assets The probability that […]

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Before reading this article, make sure that you have read the article on probability distributions in general. Suppose that X is a normally distributed random variable with a mean of μ and a variance of σ2 (\(X\ \sim\ N\left(\mu,\ \sigma^2\right)\)).  Then the random variable \(Y\ =\ e^X\) has what is known as a lognormal distribution.  […]

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Man, that title sounds imposing! Multivariate.  Seriously: how often do you use a word with five or more syllables?  You might occasionally say “inconsequential”, and at Level I you’ll say “heteroskedasticity”, but that’s about it. So . . . about what are we really talking here? Probability distributions with more than one variable.  If x […]

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Binomial trees for stock prices (or prices for pretty much any asset: stocks, commodities, real estate, baseball cards, whatever) are pretty simple (i.e., too simple to be realistic): they assume that: Time (into the future) can reasonably be divided into periods of equal length For each (future) time period, there are only two (hence: binomial) […]

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If you thought that you learned all you needed to know about counting in elementary school, think again.  I’ll run through several counting formulae (with examples) used in the CFA curriculum, but rest assured that this barely touches on the subject of counting; indeed, it is a main branch of mathematics (combinatorics) with active research […]

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Terminology It’s probably (sorry) best to define some terms at the outset, so here goes: Random variable: a variable whose future value is as yet unknown. Examples of random variables are the: Value of a risky portfolio one year from today Number of cars that will pass through a given intersection tomorrow Number of times […]

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If a rational investor wants to earn a higher return on their investment, they should expect to bear a higher risk.  Conversely, if a rational investor accepts more risk on an investment, they should expect a higher return. So, at least, goes the theory. The questions, then, are: How does an investor quantify the risk […]

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When describing a data set, two obvious questions arise: Where is it located? How big is it? The article on central tendency focuses on the first question.  This article will address the second. The CFA curriculum claims to focus on four* measures of the dispersion (or size) of a data set: Range Mean Absolute Deviation […]

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All of the isles (or iles) of CFA fall under the broad heading of quantiles (or fractiles); values that divide a data set into a certain number of subsets with the same number of data points in each subset.  For example, the median divides a data set into two subsets: half of the values are […]

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When describing a data set, two obvious questions arise: Where is it located? How big is it? This article focuses on the first question.  The second is covered here. A couple of preliminaries: The word “mean” can be ambiguous, as there are several types of means, four of which are included in the CFA curriculum: […]

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