Notation
This page gathers the notation used across the subject in one place, so you can quickly check what a symbol means. It is grouped by topic. Where a symbol could mean different things in different textbooks, the meaning given here is the one we use.
A few conventions used throughout:
- Expectation is written \(\mathrm{E}[\cdot]\) and variance \(\mathrm{Var}(\cdot)\).
- Reserving triangles are indexed from 0: period of origin \(i = 0,\dots,I\) and development period \(j = 0,1,\dots\) (not from 1).
- “Incremental” refers to a single cell; “cumulative” means summed across development periods.
1 Reserving (M1–M4)
| Symbol | Meaning |
|---|---|
| \(i\) | period of origin (accident period), \(i=0,\dots,I\) |
| \(j\) | development period, \(j=0,1,\dots\) |
| \(k\) | experience / calendar period, \(k=i+j\) |
| \(C(i,j)\) | incremental paid losses in cell \((i,j)\) |
| \(C^*(i,j)\) | losses before inflation adjustment; \(C(i,j)=C^*(i,j)\,\lambda(k)/\lambda_0\) |
| \(D(i,j)=\sum_{m=0}^{j}C(i,m)\) | cumulative paid losses to development \(j\) |
| \(\overline{P}(i,k)\equiv P(i,j)=\sum_{m=j+1}^{\infty}C(i,m)\) | outstanding loss liability for origin \(i\) at end of \(k\) |
| \(\overline{P}(k)=\sum_{i=0}^{I}\overline{P}(i,k)\) | total outstanding liability at end of \(k\) |
| \(\lambda(t),\ \lambda_0\) | claims inflation index and its base value |
| \(d(\cdot)\) | discount factor — note \(\lambda \neq d\) (claims inflation is not the investment rate) |
| \(e(i,m)\) | superimposed-inflation adjustment factor \([1+e(i,m)]\) |
Related terms: development factors (age-to-age), age-to-ultimate factor, chain ladder, Bornhuetter–Ferguson (prior = premium × loss ratio), loss ratio, PPCI, separation method, IBNR, ULAE. Chain-ladder factors are computed on cumulative data.
2 Utility, risk and insurance (M5)
| Symbol | Meaning |
|---|---|
| \(u(\cdot)\) | utility function; families used include \(u(x)=\tfrac{x^{\gamma}-1}{\gamma}\), \(u(x)=\log x\), \(u(x)=1-e^{-x}\) |
| \(W_0\) | initial wealth |
| \(I\) | certainty equivalent of a prospect |
| \(P = I - W_0\) | indifference price (often negative) |
| \(\text{risk premium} = \mathrm{E}[X]-P\) | loading over the fair price |
Related terms: expected utility theory (EUT) and its axioms, first- and second-order stochastic dominance, absolute risk aversion (ARA), relative risk aversion (RRA), HARA utilities.
3 Ruin theory (M7–M9)
| Symbol | Meaning |
|---|---|
| \(S\) | aggregate claims, compound Poisson: \(S=\sum_{i=1}^{N}X_i\) |
| \(N\) | claim count, Poisson\((\lambda)\); \(X_i\) individual claim amounts |
| \(M_X(t)=\mathrm{E}[e^{tX}]\) | moment generating function; for the compound Poisson \(M_S(r)=\exp(\lambda(M_X(r)-1))\) |
| \(p,\ P_R\) | insurance premium, reinsurance premium |
| \(M\) | retention level (excess-of-loss); retained proportion for proportional reinsurance |
| \(\theta\) | loading in the expected-value premium principle |
| \(A\) | parameter of the exponential premium principle; \(P_R=\log M_X(A)/A\) |
| \(U(t),\ u,\ c\) | surplus process, initial surplus, premium rate |
| \(R\) | adjustment coefficient; Lundberg’s inequality \(\psi(u)\le e^{-Ru}\) |
| \(\psi(u)\) | probability of ultimate ruin from initial surplus \(u\) |
Related terms: net adjustment coefficient (with reinsurance), optimal reinsurance under a budget constraint.
4 Stochastic interest (M10–M12)
| Symbol | Meaning |
|---|---|
| \(i_t\) | rate of return in period \([t-1,t]\) |
| \(S_n=\prod_{t=1}^{n}(1+i_t)\) | accumulation of 1 over \(n\) periods |
| \(V_n=\prod_{t=1}^{n}(1+i_t)^{-1}\) | present value of 1 due in \(n\) periods |
| \(A_n=\sum_{t=1}^{n}\prod_{k=t}^{n}(1+i_k)\) | accumulated value of an annuity |
| \(P_n=\sum_{t=1}^{n}\prod_{k=1}^{t}(1+i_k)^{-1}\) | present value of an annuity |
| \(\mu,\ \sigma^2\) | mean and variance of a single-period rate \(i_t\) |
| \(F_t\) | accumulated fund at time \(t\) |
Useful facts: fixed vs varying rate models; under independence \(\mathrm{E}[S_n^k]=\prod_{t=1}^{n}\mathrm{E}[(1+i_t)^k]\), and \(\mathrm{E}[V_n]\neq \mathrm{E}[S_n]^{-1}\); the lognormal model (\(\mu,\sigma\)); Value-at-Risk (VaR); inverse-transform simulation and Monte Carlo estimation.