Chapter 13 Appendix: Conventions for Notation

13.1 Risk Retention Conventions

Here is a set of symbols used in this book. The symbols are common although their use is not universal.

\[ {\small \begin{array}{c|l}\hline \hline \textbf{Symbol} & \textbf{Description} \\ \hline n, m & n \text{ risks with potentially } m \text{ constraints} \\ \hline \boldsymbol \theta & \text{a } p \text{ - dimensional vector of risk retention parameters,} \\ & ~~~~~~~~~~~~~ \text{ such as} \textit{ d, c, u}\\ \textit{d} & \text{deductible, a risk retention parameter} \\ \textit{c} & \text{coinsurance, a risk retention parameter} \\ \textit{u} & \text{upper limit, a risk retention parameter} \\ g(\cdot) & \text{a generic risk retention function} \\ S(u_1, \ldots, u_p)& \text{sum limited by } u_1, \ldots, u_p, \\ & ~~~~~~~~~~~~~ \text{a type of risk retention function} \\ \hline \mathbf{X} & \text{a } n \text{ - dimensional vector of random risks}, X_1, \ldots, X_n\\ RM & \text{a generic risk measure, such as }VaR, ~ES \\ VaR & \text{value at risk}\\ ES & \text{expected shortfall} \\ RVaR & \text{range value at risk} \\ \alpha & \text{confidence level for risk measures} \\\hline LA & \text{Lagrangian} \\ RTC & \text{risk transfer cost, the risk owner's cost of offloading a risk} \\ RTC_{max} & \text{maximal risk transfer cost, a fixed constant} \\ LME & \text{Lagrange multiplier for equality constraints} \\ LMI & \text{Lagrange multiplier for inequality constraints} \\ CON_{eq}, CON_{in} & \text{sets of equality, inequality constraints} \\ \hline {\bf z} & \text{the decision variables, a }p_z \text{ - dimensional vector} \\ f_0 & \text{an objective function to be minimized} \\ f_{con} & \text{a generic constraint function} \\ \hline \lambda_{SL}, \lambda & \text{security loading, generic penalty parameter} \\ U(\cdot) & \text{utility function} \\ \partial_{x} & \text{short-hand notation for a partial derivative with} \\ & ~~~~~~~~~~~~~ \text{respect to }x, \text{that is, } \frac{\partial}{\partial x} \\ \wedge & \text{wedge operator, take the minimum of two quantities} \\ (\cdot)_+ & \text{take the positive part, e.g., } (-4)_+ = 0 \\ \boldsymbol \Sigma & \text{matrix of association or dependence parameters} \\ \hline \hline \end{array} } \]

13.2 General Conventions

Random variables are denoted by upper-case italicized Roman letters, with \(X\) or \(Y\) denoting a claim size variable, \(N\) a claim count variable, and \(S\) an aggregate loss variable. Realizations of random variables are denoted by corresponding lower-case italicized Roman letters, with \(x\) or \(y\) for claim sizes, \(n\) for a claim count, and \(s\) for an aggregate loss.
Probability events are denoted by upper-case Roman letters, such as \(\Pr(\mathrm{A})\) for the probability that an outcome in the event \(\mathrm{A}\) occurs.
Cumulative probability functions are denoted by \(F(z)\) and probability density functions by the associated lower-case Roman letter: \(f(z)\).
For distributions, parameters are denoted by lower-case Greek letters. A caret or ‘’hat’’ indicates a sample estimate of the corresponding population parameter. For example, \(\hat{\beta}\) is an estimate of \(\beta\) .
Upper-case boldface Roman letters denote a matrix other than a vector. Lower-case boldface Roman letters denote a (column) vector. Use a superscript prime ’‘\(\prime\)’’ for transpose. For example, \(\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}\) is a quadratic form.

13.2.1 Common Statistical Symbols and Operators

Here is a list of commonly used statistical symbols and operators.

\[ {\small \begin{array}{cl} \hline I(\cdot) & \text{binary indicator function. For example, }I(A) \text{ is one} \\ & \ \ \ \ \ \text{if an outcome in event } A \text{ occurs and is 0 otherwise.} \\ \Pr(\cdot) & \text{probability function.}\\ F(\cdot) & \text{distribution function of the random variable } X. \text{ For example, } \\ & \ \ \ \ \ F(4) = \Pr(X \le 4). \\ F_{(\cdot)}(\cdot) & \text{distribution function of a complex random variable or vector}. \\ & \ \ \ \ \ \text{For example, } F_{g(X)}(4) = \Pr[g(X) \le 4]. \\ F^{-1}_{\alpha} & \text{quantile function of the random variable } X \text{ at confidence level } \alpha. \\ & \ \ \ \ \ \text{ For example, } F^{-1}_{0.95} \text{ is the 95}^{th} \text{ percentile of the distribution}\\ & \ \ \ \ \ \ F. \text{ Also denoted as } F^{-1}(0.95). \\ F^{-1}_{(\cdot)} & \text{quantile function of a complex random variable or vector }. \\ & \ \ \ \ \ \text{ For example, } F^{-1}_{g(X)}(0.95) \text{ is the 95}^{th} \text{ percentile of the} . \\ & \ \ \ \ \ \text{ distribution function for the random variable } g(X). \\ \mathrm{E}(\cdot) & {\text{expectation operator}}. {\text{ For example, }} \mathrm{E}(X) {\text{ is the expected}} \\ & \ \ \ \ \ {\text{value of the random variable }}X,{\text{ commonly denoted by }}\mu. \\ \mathrm{Var}(\cdot) & \text{variance operator}. \text{ For example, } \mathrm{Var}(X)\text{ is the variance} \\ & \ \ \ \ \ \text{of the random variable } X, \text{commonly denoted by } \sigma^2. \\ \mathrm{Cov}(\cdot,\cdot) & \text{covariance operator}.\text{ For example, } \\ & \ \ \ \ \ \mathrm{Cov}(X,Y)=\mathrm{E}\left\{(X -\mathrm{E}~X)(Y-\mathrm{E}~Y)\right\} =\mathrm{E}(XY) -(\mathrm{E}~X)(\mathrm{E}~Y)\\ & \ \ \ \ \ \text{ is the covariance between random variables }X\text{ and }Y. \\ \mathrm{E}(X | \cdot) & \text{conditional expectation operator. For example, }\mathrm{E}(X |Y=y) \text{ is the}\\ & \ \ \ \ \ \text{ conditional expected value of a random variable }X\text{ given that }\\ & \ \ \ \ \ \text{ the random variable }Y\text{ equals y. }\\ \Phi(\cdot) & \text{standard normal cumulative distribution function }\\ \phi(\cdot) & \text{standard normal probability density function }\\ \sim & \text{means 'is distributed as' }. \text{ For example, }X\sim F \text{ means that } \\ & \ \ \ \ \ \text{the random variable } X \text{ has distribution function }F. \\ \hline \end{array} } \]

Notes on Notation

\(\mathrm{Var}(\cdot)\) is the variance operator in contrast to \(VaR\) which stands for value at risk
\(\mathrm{E}(\cdot)\) is the expectation operator in contrast to \(ES\) which stands for expected shortfall.