Glossary: M06 — Simulation Methods

Module: M06 Formulas: Formula Sheet Concept page: Simulation Methods Concept

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Lognormal Distribution

A distribution of a variable whose natural logarithm is normally distributed. Used to model asset prices because it is bounded below by zero (prices cannot be negative).

$\text{If }\ln (X)\sim N(\mu ,{\sigma }^2),\text{ then }X\text{ is lognormally distributed.}$

$E(X)=e^{\mu +{\sigma }^2/2}\text{Var}(X)=e^{2\mu +{\sigma }^2}(e^{{\sigma }^2}-1)$

LOS: 6.a | Key: Lognormal distributions are right-skewed and bounded below at zero — consistent with stock prices. | Related: Continuously Compounded Return

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Continuously Compounded Return

The natural log of the ratio of ending price to beginning price. If asset prices are lognormally distributed, continuously compounded returns are normally distributed.

$r_{cc}=\ln \left(\frac{S_1}{S_0}\right)=\ln (1+HPR)$

LOS: 6.a | Key: Continuously compounded returns are additive over time; holding period returns are multiplicative. | Related: M01 — Continuously Compounded Return, Lognormal Distribution

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Student’s t-Distribution

A symmetric, bell-shaped distribution with heavier tails than the normal distribution. Used when the population variance is unknown and the sample size is small.

LOS: 6.b | Key properties: Defined by Degrees of Freedom; as degrees of freedom increase, the t-distribution approaches the standard normal. Heavier tails → more probability in extremes.

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Degrees of Freedom

The number of independent observations in a sample used to estimate a parameter. For a sample of $n$ observations, df = $n-1$ when estimating the mean.

LOS: 6.b | Role: Determines the shape of the Student’s t-Distribution, Chi-Square Distribution, and F-Distribution. More df → closer to normal.

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Chi-Square Distribution

A distribution formed by the sum of the squares of $k$ independent standard normal variables. Used for testing independence and variance.

${\chi }^2={\sum }_{i=1}^k{Z}_i^2\text{where }{Z}_i\sim N(0,1)$

LOS: 6.b | Properties: Right-skewed; always non-negative; defined by Degrees of Freedom $k$. | Related: Chi-Square Test

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F-Distribution

The ratio of two independent chi-square distributed variables, each divided by its degrees of freedom. Used to compare variances and in regression ANOVA.

$F=\frac{{\chi }_{d{f}_1}^2/d{f}_1}{{\chi }_{d{f}_2}^2/d{f}_2}$

LOS: 6.b | Properties: Non-negative; right-skewed; defined by two degrees of freedom parameters ($d{f}_1$, $d{f}_2$). | Related: F-Test

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Monte Carlo Simulation

A method that uses repeated random sampling from specified probability distributions to generate a large number of scenarios and estimate the probability distribution of an outcome variable.

LOS: 6.c | Steps: (1) Specify distributions for input variables; (2) draw random values; (3) compute outcome; (4) repeat many times; (5) analyze the resulting distribution. | Application: Estimating Value at Risk (VaR), option pricing, pension fund risk modeling.

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Value at Risk (VaR)

The minimum loss expected to be exceeded with probability $\alpha$ over a specified time horizon, at a given confidence level.

$P(\text{Loss}>VaR)=\alpha$

LOS: 6.c | Example: A daily VaR of $1M at 5% means there is a 5% chance of losing more than $1M in a day. | Key limitation: Does not describe the magnitude of losses exceeding VaR.

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Resampling

A class of statistical methods that repeatedly draw samples from observed data to make inferences about a population. Includes Bootstrap and Jackknife.

LOS: 6.d | Advantage: Makes no assumptions about the underlying population distribution. | Related: Simulation Methods Concept

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Bootstrap

A resampling method in which samples are drawn with replacement from the original observed data set. Used to estimate the sampling distribution of a statistic.

LOS: 6.d | Process: (1) Draw $B$ bootstrap samples of size $n$ (with replacement); (2) calculate the statistic for each; (3) use the empirical distribution of the $B$ statistics for inference. | Contrast: Jackknife

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Jackknife

A resampling method in which each observation is systematically removed one at a time, and the statistic is recomputed on the remaining $n-1$ observations. Used to estimate bias and variance of estimators.

LOS: 6.d | Key: More structured than Bootstrap (deterministic, not random). Less computationally flexible but useful for bias estimation.