Level I

Parametric Tests vs. Nonparametric Tests

Parametric tests are straightforward and are the focus of most of . . . well . . . all of the other hypothesis testing articles I’ve written here. They’re concerned, not surprisingly, with questions about the parameters of a particular population or populations: What’s the mean monthly return on this ETF? What’s the standard deviation […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Sampling and Estimation

Often it’s impractical (or even impossible) to collect data on every member of a population in which you are interested.  When it is, an alternative approach that may prove adequate is to collect data on a sample from that population: a subset of the population which you hope will have characteristics sufficiently close to those […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Central Limit Theorem

It’s probably (sorry) not an exaggeration (or, at least, not much of one) to say that the central limit theorem is the single most important theorem in probability theory.  It’s so important that it has its own abbreviation: CLT.  (Clever, eh?) In a nutshell, the CLT says that if you add up a bunch of […]

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Monte Carlo Simulation

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 […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Lognormal Distribution

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.  […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Multivariate Normal Distributions

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 […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Binomial Trees (for Stocks)

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) […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Using Your Fingers and Toes: Counting

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 […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Basics of Probability

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 […]

CFA® Level I Membership, CFA® Level I Quantitative Methods Membership

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Risk vs. Return – Quant

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 […]