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### Enterprise Risk Management Formula Book

Gamma, incomplete gamma, beta, incomplete beta, regularised incomplete beta, binomial

Exponential, natural logarithm, binomial, Taylor series

3.        Calculus

Integration by parts, changing order of integration, differentiating an integral

Probability distribution terminology, Bayes’ theorem, compound distributions

Sample moments, parametric inference (with normal underlying distributions), maximum likelihood estimators, method-of-moments estimators, goodness of fit, linear regression, generalised linear models, correlations, analysis of variance, Bayesian priors and posteriors

Creation of normal random variables, Cholesky decomposition

Spot and forward rates, duration, modified duration, gross redemption yield (yield to maturity), credit spread, option-adjusted spread, annualisation conventions

Forward prices, Black-Scholes formulae

Value-at-Risk, tail Value-at-Risk, expected shortfall, expected worst outcome, tracking error, drawdown, marginal VaR, incremental VaR, estimating VaR

Mean-variance optimisation, capital asset pricing model

Maximum domain of attraction, Fisher-Tippett theorem, Pickands-Balkema-de Hann theorem, estimating tail distributions

12.      Copulas

Definition, properties, Sklar’s theorem, example copulas, tail dependence, simulating copulas

Combining solvency capital requirements using correlations, credit risk modelling, GARCH modelling, linear algebra and principal components, central limit theorem, Cornish-Fisher asymptotic expansion, Euler capital allocation principle, equiprobable outcomes for a multivariate normal distribution, RAROC and EVA

Appendix A: Probability distributions

Discrete: Binomial (and Bernoulli), Poisson

Continuous: Normal, uniform, chi-squared, exponential, F, generalised extreme value (GEV) (and Frechét, Gumbel and Weibull), generalised Pareto, lognormal, Student’s t

Other: Distributional mixtures, location and scale adjusted distributions, multivariate distributions, distributional families

Tables: cumulative distribution function and quantile function for normal distribution

Note: In this note, denotes the standard cumulative normal distribution function, denotes logarithms to base , etc. but if is a cumulative distribution function then is the corresponding inverse cumulative distribution function.