IFoA CB2 cheat sheet

The actuarial control cycle has four stages: specify the problem, develop a solution, monitor experience, and revise the model; feedback loops connect every stage.

A model is a simplified representation of reality; its purpose determines the appropriate balance of accuracy, transparency, and computational cost.

Deterministic models give a single outcome per scenario; stochastic (probabilistic) models give a distribution of outcomes, enabling quantification of risk.

The principle of parsimony: prefer the simplest model that captures material features; unnecessary parameters add error and reduce transparency.

Materiality governs model choice: a simplification is acceptable if its effect on decisions is small relative to the cost of greater precision.

Data quality checks before modelling: completeness, accuracy, consistency, and relevance of the data to the population being modelled.

A model's output is only as reliable as its assumptions; document each assumption, its source, and its sensitivity to change.

Communication of results must suit the audience: technical detail for peers, clear decision-focused summaries for non-specialists.

Mean of a random variable X: E[X] = sum xp(x) for discrete, or integral xf(x) dx for continuous; variance Var(X) = E[X^2] - (E[X])^2.

For independent X and Y: E[X+Y] = E[X] + E[Y]; Var(X+Y) = Var(X) + Var(Y); for dependent variables add 2*Cov(X,Y).

Binomial(n,p): mean np, variance np(1-p); Poisson(lambda): mean = variance = lambda; Poisson is the limit of Binomial as n grows and p shrinks with np fixed.

Exponential distribution: f(x) = lambdaexp(-lambdax); memoryless property P(X > s+t | X > s) = P(X > t).

Central limit theorem: the sum (or mean) of n independent identically distributed variables tends to Normal as n increases, regardless of the underlying distribution.

Maximum likelihood estimation: choose parameters maximising the likelihood; the log-likelihood is maximised instead because it is easier to differentiate and sums rather than multiplies.

Confidence interval for a mean with known variance: x_bar +/- z * sigma / sqrt(n); use t-distribution when variance is estimated from a small sample.

Hypothesis testing: a p-value is the probability of observing data at least as extreme as the sample, given the null hypothesis is true; it is not the probability the null is true.

Survival function S(x) = P(X > x); force of mortality mu(x) = -d/dx ln S(x); density of age at death f(x) = S(x)*mu(x).

Survival from age x to x+t: tp_x = S(x+t)/S(x); tq_x = 1 - tp_x; deferred probability u|tq_x = up_x * tq_{x+u}.

Under a constant force of mortality mu, tp_x = exp(-mu*t) and curtate future lifetime is geometric.

Whole life annuity-due a_x = sum over k>=0 of v^k * kp_x; whole life insurance A_x = sum over k>=0 of v^(k+1) * kp_x * q_{x+k} (payable at end of year of death).

Relationship linking insurance and annuity: A_x = 1 - d * a_x, where d = i/(1+i) is the discount rate.

Net premium for a whole life policy: P = A_x / a_x; equivalence principle sets expected present value of premiums equal to expected present value of benefits.

Net future loss at issue L = v^K * B - P * a_due(K), where K is curtate future lifetime; the net premium makes E[L] = 0.

Select mortality: the force of mortality depends on duration since underwriting, so q_[x]+t differs from q_{x+t}; select period is the number of years this effect persists.

A Markov chain satisfies the Markov property: the future state depends only on the current state, not on the path taken to reach it.

Transition probability matrix P has rows summing to 1; the n-step matrix is P^n; stationary distribution pi satisfies pi*P = pi and sums to 1.

Kolmogorov forward equations: P'(t) = P(t)*Q, where Q is the generator (transition intensity) matrix with off-diagonal entries >= 0 and each row summing to 0.

For a two-state alive/dead model, the transition intensity is the force of mortality; probabilities solve the forward equations.

A0/A1 model: state 0 = healthy, state 1 = disabled, state 2 = dead; transitions 0->1 (sickness), 1->0 (recovery), and both to death.

Expected present value in a multi-state model: solve Thiele's differential equation or use a recursive matrix approach discounting each transition.

The Chapman-Kolmogorov equations P(s,u) = P(s,t)*P(t,u) link transition probabilities across intermediate times for time-inhomogeneous chains.

Semi-Markov models relax the Markov property by allowing transition intensities to depend on duration in the current state.

Gross premium = net premium + loading for expenses, profit, and contingency margins; the gross premium reserve uses gross (not net) cash flows.

Prospective reserve V_t = expected present value of future benefits + expenses minus expected present value of future premiums, conditional on being in force at t.

Retrospective reserve accumulates past premiums and benefits at the valuation rate; equals the prospective reserve if assumptions are consistent.

Recursive reserve: (V_t + P)*(1+i) = q_{x+t} * S + p_{x+t} * V_{t+1}, where S is the sum assured and P the premium.

Profit testing: project cash flows including premiums, claims, expenses, interest, and reserves; discount to give the net present value at the risk discount rate.

Internal rate of return (IRR) of a policy is the discount rate making NPV zero; compare with the required return to judge profitability.

Profit signature is the stream of emerging profits by policy year; its present value at the risk discount rate is the expected profit.

Reserves are a liability; a negative reserve (asset) may arise where premiums exceed risk cost early in the contract.

Solvency capital must cover adverse deviations in assumptions; a common basis is a value-at-risk at a specified confidence level over a one-year horizon.

Risk types: insurance (mortality, morbidity, longevity), market (interest rate, equity, credit), operational, and concentration risk.

Diversification benefit: total capital for combined risks is less than the sum of standalone capitals because risks are not perfectly correlated.

Discounting for reserves uses a risk-free or valuation rate; the choice of rate materially affects reserve and capital estimates.

Sensitivity analysis varies one assumption at a time; scenario testing varies several jointly; stochastic modelling produces a full distribution.

Embedded value = net assets + present value of future profits from in-force business; it excludes the value of future new business.

Risk-based capital ratios compare available capital to required capital; a ratio above 100% indicates capital adequacy on that basis.

Reinsurance reduces retained risk and required capital but transfers profit; net retained risk equals gross risk less reinsured risk.