IFoA CM2 cheat sheet

A model is a simplified representation of reality; it must be fit for purpose, transparent, and its outputs interpreted with the assumptions in view.

Deterministic models give one output per input set; stochastic (probabilistic) models give a distribution of outcomes and are needed for risk quantification.

The actuarial control cycle: define problem, build model, estimate parameters, run the model, monitor experience, and feed back revisions.

A cash-flow model projects assets and liabilities over time; a profit-testing model adds a profit signature and measures profitability via NPV or IRR.

The principle of parsimony: prefer the simplest model that captures the material features; extra parameters must be justified by data.

Sensitivity and scenario testing vary one assumption at a time; stochastic modelling varies many jointly via simulation, capturing correlations.

Discounting: present value = sum over t of CF_t * v^t with v = 1/(1+i); for continuous discounting v^t = exp(-delta t).

Model risk arises from wrong structure, wrong parameters, or wrong data; document assumptions and reconcile against observed experience.

Binomial(n, p): P(X = x) = C(n,x) p^x (1-p)^(n-x), x = 0,...,n; mean np, variance np(1-p).

Binomial conditions: fixed n independent trials, constant success probability p, two outcomes per trial.

Poisson(lambda): P(N = n) = exp(-lambda) lambda^n / n!, n = 0,1,2,...; mean = variance = lambda.

Poisson limit of binomial: if n is large and p small with np tending to lambda, Binomial(n,p) tends to Poisson(lambda).

Poisson process: events occur singly and independently at constant rate lambda; waiting time to first event is Exponential(lambda).

Sum of independent Poissons is Poisson with summed rates; sum of independent Binomials with common p is Binomial with summed n.

Poisson as count of rare events: use for claim numbers per policy-year; overdispersion (variance > mean) signals a mixed or negative binomial model.

Recurrence: P(N = n+1) = P(N = n) * lambda/(n+1), useful for tabulating Poisson probabilities.

Normal(mu, sigma^2): density proportional to exp(-(x-mu)^2/(2 sigma^2)); symmetric, mean = median = mode = mu, variance sigma^2.

Standardise: Z = (X - mu)/sigma ~ N(0,1); use the standard normal cdf Phi for all probability calculations.

Sum of independent normals is normal: if X_i ~ N(mu_i, sigma_i^2) then sum X_i ~ N(sum mu_i, sum sigma_i^2).

Lognormal: if Y = exp(X) with X normal, Y is lognormal with positive skew; mean exp(mu + sigma^2/2), variance exp(2mu + sigma^2)(exp(sigma^2) - 1).

Central limit theorem: the mean of n iid variables with finite variance is approximately normal for large n, regardless of the parent distribution.

Chi-square with k degrees of freedom is the sum of k squared independent standard normals; used in goodness-of-fit and variance tests.

Student t with k degrees of freedom has heavier tails than normal and tends to normal as k grows; arises when sigma is estimated.

Exponential(lambda) has mean 1/lambda, variance 1/lambda^2, and is memoryless: P(T > s+t | T > s) = P(T > t).

An estimator is a function of the sample; an estimate is its realised value. Desirable properties: unbiasedness, efficiency, consistency, sufficiency.

Method of moments equates sample moments to theoretical moments and solves for parameters.

Maximum likelihood: choose parameters maximising the log-likelihood; MLEs are consistent, asymptotically normal, and asymptotically efficient.

Cramer-Rao lower bound: variance of any unbiased estimator is at least 1/I(theta), where I is the Fisher information.

Fisher information I(theta) = -E[d^2 log L / d theta^2]; for n iid observations it is n times the single-observation information.

Confidence interval for a mean with known variance: xbar +/- z * sigma/sqrt(n); replace z by t when sigma is estimated.

Delta method: if T estimates theta, then g(T) has approximate variance g'(theta)^2 Var(T), used to transform estimates and their standard errors.

Bayesian estimation combines a prior with the likelihood to give a posterior; the posterior mean or mode is the Bayes estimator under squared-error or zero-one loss.

Structure: null H0 and alternative H1; a test statistic; a rejection region; errors of type I (reject true H0, probability alpha) and type II (accept false H0, probability beta).

p-value is the probability, under H0, of a result at least as extreme as observed; reject H0 if p-value < alpha.

Power = 1 - beta; it increases with larger sample size, larger true effect, and larger alpha.

One-sample z-test for a mean with known variance: z = (xbar - mu0)/(sigma/sqrt(n)); use t with n-1 df when sigma is estimated.

Chi-square goodness-of-fit: statistic = sum (O - E)^2/E with df = number of categories - 1 - number of estimated parameters.

Likelihood ratio test: 2(log L(alternative) - log L(null)) is approximately chi-square with df equal to the number of restricted parameters.

Neyman-Pearson lemma: the most powerful test of a simple null against a simple alternative rejects for large values of the likelihood ratio.

Confidence interval and hypothesis test duality: a 100(1-alpha)% interval contains exactly the values not rejected by a two-sided test at level alpha.

Simple linear regression: y_i = alpha + beta x_i + e_i with e_i iid mean zero, constant variance; OLS minimises the residual sum of squares.

Slope estimate beta-hat = Sxy/Sxx; intercept alpha-hat = ybar - beta-hat xbar; both are linear in y and unbiased under the model.

R-squared = 1 - RSS/TSS measures the proportion of variation explained; adjusted R-squared penalises extra predictors.

Multiple regression in matrix form: beta-hat = (X'X)^{-1} X'y; Var(beta-hat) = sigma^2 (X'X)^{-1}, with sigma^2 estimated by RSS/(n-k).

GLM components: a random component from an exponential family, a linear predictor eta = X beta, and a link g with E[Y] = mu and g(mu) = eta.

Poisson GLM uses a log link: log(mu_i) = x_i' beta; the variance function is mu, and the mean-variance relationship drives the weighting.

Logistic regression uses a logit link with Bernoulli/binomial data: log(mu/(1-mu)) = x' beta; coefficients are log odds ratios.

Deviance = 2(log L(saturated) - log L(model)); compare nested models by the change in deviance, approximately chi-square with df equal to the difference in parameters.

Overdispersion: if deviance/df substantially exceeds 1 for Poisson, use a quasi-Poisson or negative binomial model.

Stationarity: mean and variance constant, autocovariance depends only on lag; weak stationarity is required for ARMA modelling.

Autocovariance gamma_k = Cov(X_t, X_{t+k}); autocorrelation rho_k = gamma_k/gamma_0, with rho_0 = 1.

AR(1): X_t = c + phi X_{t-1} + e_t; stationary if |phi| < 1, with mean c/(1-phi) and variance sigma_e^2/(1-phi^2).

MA(1): X_t = e_t + theta e_{t-1}; autocorrelation at lag 1 is theta/(1+theta^2) and zero beyond, giving a cut-off in the ACF.

ARMA(p,q) combines AR and MA terms; AR processes show ACF decay, MA processes show ACF cut-off, guiding order selection.

Random walk: X_t = X_{t-1} + e_t; non-stationary, variance grows with t; differencing once gives white noise.

Brownian motion: continuous path with independent normal increments, Var(B_t - B_s) = sigma^2 (t - s); used as the driver of asset models.

Geometric Brownian motion: dS = mu S dt + sigma S dB; log S is normal, giving lognormal prices; the basis of Black-Scholes.

Markov property: the future depends on the present state only, not on the past path; P(X_{n+1} = j | X_n = i, past) = p_ij.

Transition matrix P has non-negative entries with row sums 1; the n-step matrix is P^n.

Stationary distribution pi satisfies pi P = pi with sum of pi_i = 1; it is the long-run proportion of time spent in each state for an irreducible, aperiodic chain.

Communicating classes: a chain is irreducible if all states communicate; recurrent states are revisited with probability 1, transient states are not.

Continuous-time Markov jump process: holding time in state i is Exponential(q_i) and the jump chain uses p_ij = q_ij/q_i, where q_i = -q_ii.

Generator matrix Q has off-diagonal rates q_ij >= 0 and row sums zero; the transition matrix over time t is P(t) = exp(Qt).

Kolmogorov forward equations: P'(t) = P(t) Q; stationary distribution solves pi Q = 0 with sum of pi_i = 1.

Time-inhomogeneous chains replace constant Q with Q(t); the transition matrix is then a product integral, not a simple exponential.

Survival function S(t) = P(T > t); hazard mu(t) = f(t)/S(t) = -d/dt log S(t); cumulative hazard H(t) = integral of mu.

S(t) = exp(-H(t)); for constant hazard mu, T is exponential with S(t) = exp(-mu t).

Kaplan-Meier estimator: S-hat(t) = product over event times t_j <= t of (1 - d_j/n_j), with n_j at risk and d_j deaths.

Nelson-Aalen estimator of cumulative hazard: H-hat(t) = sum over t_j <= t of d_j/n_j; then S-hat = exp(-H-hat).

Actuarial (moment) estimator of q_x from n_x lives and d_x deaths: q-hat_x = d_x/(n_x - d_x/2) if deaths are assumed uniformly distributed over the year.

Exposure: central exposed to risk E_x = integral of l_{x+t} dt; initial exposed to risk equals E_x plus a half (or a fraction) of deaths.

Force of mortality relation to q: q_x = 1 - exp(-integral from 0 to 1 of mu_{x+t} dt); under constant mu, q_x = 1 - exp(-mu).

Binomial model for deaths: D_x ~ Binomial(E_x, q_x) in a single-year group, giving q-hat_x = d_x/E_x with variance q_x(1-q_x)/E_x.

Graduation fits a smooth curve to crude age-specific rates to remove sampling error while preserving the underlying pattern.

Two conflicting aims: smoothness (small variance) and adherence to data (small bias); balance is judged by comparing graduated and crude rates.

Standard tests: chi-square goodness-of-fit, the signs test (runs test) for systematic deviation, and the cumulative deviations test.

Chi-square graduation test: statistic = sum (observed - expected)^2/expected, with degrees of freedom reduced by the number of fitted parameters.

Gompertz law: mu_x = B c^x, so log mu_x is linear in x; Makeham adds a constant: mu_x = A + B c^x.

Heligman-Pollard law models mortality across all ages with a child term, a hump at young adult ages, and an exponential old-age term.

Lee-Carter model: log m_{x,t} = a_x + b_x k_t + e_{x,t}; a_x is the age profile, b_x age sensitivity, and k_t a time index typically modelled as a random walk with drift.

Projection via Lee-Carter: forecast k_t, then combine with a_x and b_x; uncertainty comes from the k_t forecast and from parameter estimation error.

Cohort effects: mortality improvement shared by a birth cohort, modelled by adding a cohort parameter to the age-period structure.

A model is a simplified representation of reality; it must be fit for purpose, transparent, and its outputs interpreted with the assumptions in view.

Deterministic models give one output per input set; stochastic (probabilistic) models give a distribution of outcomes and are needed for risk quantification.

The actuarial control cycle: define problem, build model, estimate parameters, run the model, monitor experience, and feed back revisions.

A cash-flow model projects assets and liabilities over time; a profit-testing model adds a profit signature and measures profitability via NPV or IRR.

The principle of parsimony: prefer the simplest model that captures the material features; extra parameters must be justified by data.

Sensitivity and scenario testing vary one assumption at a time; stochastic modelling varies many jointly via simulation, capturing correlations.

Discounting: present value = sum over t of CF_t * v^t with v = 1/(1+i); for continuous discounting v^t = exp(-delta t).

Model risk arises from wrong structure, wrong parameters, or wrong data; document assumptions and reconcile against observed experience.

Binomial(n, p): P(X = x) = C(n,x) p^x (1-p)^(n-x), x = 0,...,n; mean np, variance np(1-p).

Binomial conditions: fixed n independent trials, constant success probability p, two outcomes per trial.

Poisson(lambda): P(N = n) = exp(-lambda) lambda^n / n!, n = 0,1,2,...; mean = variance = lambda.

Poisson limit of binomial: if n is large and p small with np tending to lambda, Binomial(n,p) tends to Poisson(lambda).

Poisson process: events occur singly and independently at constant rate lambda; waiting time to first event is Exponential(lambda).

Sum of independent Poissons is Poisson with summed rates; sum of independent Binomials with common p is Binomial with summed n.

Poisson as count of rare events: use for claim numbers per policy-year; overdispersion (variance > mean) signals a mixed or negative binomial model.

Recurrence: P(N = n+1) = P(N = n) * lambda/(n+1), useful for tabulating Poisson probabilities.

Normal(mu, sigma^2): density proportional to exp(-(x-mu)^2/(2 sigma^2)); symmetric, mean = median = mode = mu, variance sigma^2.

Standardise: Z = (X - mu)/sigma ~ N(0,1); use the standard normal cdf Phi for all probability calculations.

Sum of independent normals is normal: if X_i ~ N(mu_i, sigma_i^2) then sum X_i ~ N(sum mu_i, sum sigma_i^2).

Lognormal: if Y = exp(X) with X normal, Y is lognormal with positive skew; mean exp(mu + sigma^2/2), variance exp(2mu + sigma^2)(exp(sigma^2) - 1).

Central limit theorem: the mean of n iid variables with finite variance is approximately normal for large n, regardless of the parent distribution.

Chi-square with k degrees of freedom is the sum of k squared independent standard normals; used in goodness-of-fit and variance tests.

Student t with k degrees of freedom has heavier tails than normal and tends to normal as k grows; arises when sigma is estimated.

Exponential(lambda) has mean 1/lambda, variance 1/lambda^2, and is memoryless: P(T > s+t | T > s) = P(T > t).

An estimator is a function of the sample; an estimate is its realised value. Desirable properties: unbiasedness, efficiency, consistency, sufficiency.

Method of moments equates sample moments to theoretical moments and solves for parameters.

Maximum likelihood: choose parameters maximising the log-likelihood; MLEs are consistent, asymptotically normal, and asymptotically efficient.

Cramer-Rao lower bound: variance of any unbiased estimator is at least 1/I(theta), where I is the Fisher information.

Fisher information I(theta) = -E[d^2 log L / d theta^2]; for n iid observations it is n times the single-observation information.

Confidence interval for a mean with known variance: xbar +/- z * sigma/sqrt(n); replace z by t when sigma is estimated.

Delta method: if T estimates theta, then g(T) has approximate variance g'(theta)^2 Var(T), used to transform estimates and their standard errors.

Bayesian estimation combines a prior with the likelihood to give a posterior; the posterior mean or mode is the Bayes estimator under squared-error or zero-one loss.

Structure: null H0 and alternative H1; a test statistic; a rejection region; errors of type I (reject true H0, probability alpha) and type II (accept false H0, probability beta).

p-value is the probability, under H0, of a result at least as extreme as observed; reject H0 if p-value < alpha.

Power = 1 - beta; it increases with larger sample size, larger true effect, and larger alpha.

One-sample z-test for a mean with known variance: z = (xbar - mu0)/(sigma/sqrt(n)); use t with n-1 df when sigma is estimated.

Chi-square goodness-of-fit: statistic = sum (O - E)^2/E with df = number of categories - 1 - number of estimated parameters.

Likelihood ratio test: 2(log L(alternative) - log L(null)) is approximately chi-square with df equal to the number of restricted parameters.

Neyman-Pearson lemma: the most powerful test of a simple null against a simple alternative rejects for large values of the likelihood ratio.

Confidence interval and hypothesis test duality: a 100(1-alpha)% interval contains exactly the values not rejected by a two-sided test at level alpha.

Simple linear regression: y_i = alpha + beta x_i + e_i with e_i iid mean zero, constant variance; OLS minimises the residual sum of squares.

Slope estimate beta-hat = Sxy/Sxx; intercept alpha-hat = ybar - beta-hat xbar; both are linear in y and unbiased under the model.

R-squared = 1 - RSS/TSS measures the proportion of variation explained; adjusted R-squared penalises extra predictors.

Multiple regression in matrix form: beta-hat = (X'X)^{-1} X'y; Var(beta-hat) = sigma^2 (X'X)^{-1}, with sigma^2 estimated by RSS/(n-k).

GLM components: a random component from an exponential family, a linear predictor eta = X beta, and a link g with E[Y] = mu and g(mu) = eta.

Poisson GLM uses a log link: log(mu_i) = x_i' beta; the variance function is mu, and the mean-variance relationship drives the weighting.

Logistic regression uses a logit link with Bernoulli/binomial data: log(mu/(1-mu)) = x' beta; coefficients are log odds ratios.

Deviance = 2(log L(saturated) - log L(model)); compare nested models by the change in deviance, approximately chi-square with df equal to the difference in parameters.

Overdispersion: if deviance/df substantially exceeds 1 for Poisson, use a quasi-Poisson or negative binomial model.

Stationarity: mean and variance constant, autocovariance depends only on lag; weak stationarity is required for ARMA modelling.

Autocovariance gamma_k = Cov(X_t, X_{t+k}); autocorrelation rho_k = gamma_k/gamma_0, with rho_0 = 1.

AR(1): X_t = c + phi X_{t-1} + e_t; stationary if |phi| < 1, with mean c/(1-phi) and variance sigma_e^2/(1-phi^2).

MA(1): X_t = e_t + theta e_{t-1}; autocorrelation at lag 1 is theta/(1+theta^2) and zero beyond, giving a cut-off in the ACF.

ARMA(p,q) combines AR and MA terms; AR processes show ACF decay, MA processes show ACF cut-off, guiding order selection.

Random walk: X_t = X_{t-1} + e_t; non-stationary, variance grows with t; differencing once gives white noise.

Brownian motion: continuous path with independent normal increments, Var(B_t - B_s) = sigma^2 (t - s); used as the driver of asset models.

Geometric Brownian motion: dS = mu S dt + sigma S dB; log S is normal, giving lognormal prices; the basis of Black-Scholes.

Markov property: the future depends on the present state only, not on the past path; P(X_{n+1} = j | X_n = i, past) = p_ij.

Transition matrix P has non-negative entries with row sums 1; the n-step matrix is P^n.

Stationary distribution pi satisfies pi P = pi with sum of pi_i = 1; it is the long-run proportion of time spent in each state for an irreducible, aperiodic chain.

Communicating classes: a chain is irreducible if all states communicate; recurrent states are revisited with probability 1, transient states are not.

Continuous-time Markov jump process: holding time in state i is Exponential(q_i) and the jump chain uses p_ij = q_ij/q_i, where q_i = -q_ii.

Generator matrix Q has off-diagonal rates q_ij >= 0 and row sums zero; the transition matrix over time t is P(t) = exp(Qt).

Kolmogorov forward equations: P'(t) = P(t) Q; stationary distribution solves pi Q = 0 with sum of pi_i = 1.

Time-inhomogeneous chains replace constant Q with Q(t); the transition matrix is then a product integral, not a simple exponential.

Survival function S(t) = P(T > t); hazard mu(t) = f(t)/S(t) = -d/dt log S(t); cumulative hazard H(t) = integral of mu.

S(t) = exp(-H(t)); for constant hazard mu, T is exponential with S(t) = exp(-mu t).

Kaplan-Meier estimator: S-hat(t) = product over event times t_j <= t of (1 - d_j/n_j), with n_j at risk and d_j deaths.

Nelson-Aalen estimator of cumulative hazard: H-hat(t) = sum over t_j <= t of d_j/n_j; then S-hat = exp(-H-hat).

Actuarial (moment) estimator of q_x from n_x lives and d_x deaths: q-hat_x = d_x/(n_x - d_x/2) if deaths are assumed uniformly distributed over the year.

Exposure: central exposed to risk E_x = integral of l_{x+t} dt; initial exposed to risk equals E_x plus a half (or a fraction) of deaths.

Force of mortality relation to q: q_x = 1 - exp(-integral from 0 to 1 of mu_{x+t} dt); under constant mu, q_x = 1 - exp(-mu).

Binomial model for deaths: D_x ~ Binomial(E_x, q_x) in a single-year group, giving q-hat_x = d_x/E_x with variance q_x(1-q_x)/E_x.

Graduation fits a smooth curve to crude age-specific rates to remove sampling error while preserving the underlying pattern.

Two conflicting aims: smoothness (small variance) and adherence to data (small bias); balance is judged by comparing graduated and crude rates.

Standard tests: chi-square goodness-of-fit, the signs test (runs test) for systematic deviation, and the cumulative deviations test.

Chi-square graduation test: statistic = sum (observed - expected)^2/expected, with degrees of freedom reduced by the number of fitted parameters.

Gompertz law: mu_x = B c^x, so log mu_x is linear in x; Makeham adds a constant: mu_x = A + B c^x.

Heligman-Pollard law models mortality across all ages with a child term, a hump at young adult ages, and an exponential old-age term.

Lee-Carter model: log m_{x,t} = a_x + b_x k_t + e_{x,t}; a_x is the age profile, b_x age sensitivity, and k_t a time index typically modelled as a random walk with drift.

Projection via Lee-Carter: forecast k_t, then combine with a_x and b_x; uncertainty comes from the k_t forecast and from parameter estimation error.

Cohort effects: mortality improvement shared by a birth cohort, modelled by adding a cohort parameter to the age-period structure.