Enterprise Risk Management Formula Book

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1.        Function definitions

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


2.        Series expansions

Exponential, natural logarithm, binomial, Taylor series


3.        Calculus

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


4.        Statistical distributions

Probability distribution terminology, Bayes’ theorem, compound distributions


5.        Statistical methods

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


6.        Monte Carlo methods

Creation of normal random variables, Cholesky decomposition


7.        Interest rates and bond pricing

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


8.        Financial derivatives

Forward prices, Black-Scholes formulae


9.        Risk measures

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


10.      Portfolio optimisation

Mean-variance optimisation, capital asset pricing model


11.      Extreme value theory

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


13.      Miscellaneous

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.


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