IFoA CB2 flashcards

39 cards. Read the term and the definition here. The deck at the top of this page is optional.

Actuarial control cycle

A four-stage framework: specify the problem, develop a solution, monitor experience, and revise the model. Feedback from monitoring informs the next iteration of the cycle.

Deterministic versus stochastic model

A deterministic model produces one outcome per set of inputs. A stochastic model produces a probability distribution of outcomes, allowing risk to be quantified rather than a single point estimate.

Principle of parsimony

Choose the simplest model that captures the material features of the problem. Extra parameters add estimation error and reduce transparency without necessarily improving decisions.

Materiality in modelling

A simplification is acceptable when its effect on the decision is small relative to the cost of greater precision. Materiality depends on the purpose and the decision-maker.

Data quality checks

Before modelling, check data for completeness, accuracy, consistency, and relevance. Poor data invalidates results regardless of the sophistication of the model.

Variance of a random variable

Var(X) = E[X^2] - (E[X])^2. It measures the spread of a distribution around its mean and is central to quantifying insurance risk.

Binomial distribution moments

For X ~ Binomial(n,p): mean = np and variance = np(1-p). It models the number of successes in n independent trials each with success probability p.

Poisson as binomial limit

As n grows and p shrinks with np = lambda fixed, Binomial(n,p) tends to Poisson(lambda). The Poisson has mean and variance both equal to lambda.

Memoryless property

For an exponential distribution, P(X > s+t | X > s) = P(X > t). The remaining lifetime has the same distribution regardless of age already survived.

Central limit theorem

The sum or mean of many independent identically distributed random variables is approximately Normal, regardless of the underlying distribution, as the sample size grows.

Maximum likelihood estimation

Estimates parameters by maximising the likelihood of the observed data. The log-likelihood is maximised instead because it converts products into sums and is easier to differentiate.

Confidence interval for a mean

With known variance, x_bar +/- z*sigma/sqrt(n). Use the t-distribution when variance is estimated from a small sample, which widens the interval.

Interpretation of a p-value

The probability of observing data at least as extreme as the sample, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true.

Force of mortality

mu(x) = -d/dx ln S(x), the instantaneous rate of death at age x given survival to x. It relates to the survival function by f(x) = S(x)*mu(x).

Survival probability tp_x

The probability a life aged x survives to age x+t, equal to S(x+t)/S(x). Its complement tq_x is the probability of death within t years.

Deferred mortality probability

u|tq_x = up_x * tq_{x+u}: the probability of surviving u years from age x and then dying within the following t years. It combines survival and subsequent death.

Constant force of mortality

If mu(x) = mu for all x, then tp_x = exp(-mu*t). Under this assumption the curtate future lifetime follows a geometric distribution.

Whole life annuity-due

a_x = sum over k>=0 of v^k * kp_x, the expected present value of one unit paid at the start of each year while the life survives.

Insurance-annuity relationship

A_x = 1 - d * a_x, where d = i/(1+i). This links the expected present value of a whole life insurance payable at end of year of death to the annuity-due.

Net premium via equivalence principle

The net premium P sets the expected present value of premiums equal to that of benefits, so P = A_x / a_x for whole life. Expected future loss at issue is zero.

Select mortality

Mortality depends on duration since underwriting, so q_[x]+t differs from q_{x+t}. The select period is the number of years over which this effect persists.

Markov property

The future state depends only on the current state, not on the path taken to reach it. This allows transition probabilities to be summarised by a transition matrix.

Transition probability matrix

A matrix P whose rows sum to 1, where entry p_ij is the probability of moving from state i to state j. The n-step matrix is P raised to power n.

Stationary distribution

A probability vector pi satisfying pi*P = pi and summing to 1. It gives the long-run proportion of time spent in each state of a Markov chain.

Kolmogorov forward equations

P'(t) = P(t)*Q, where Q is the generator matrix of transition intensities with rows summing to zero. It determines transition probabilities over continuous time.

Generator matrix Q

A matrix of transition intensities with non-negative off-diagonal entries and each row summing to zero. Diagonal entries are minus the total intensity of leaving each state.

Chapman-Kolmogorov equations

P(s,u) = P(s,t)*P(t,u): the transition probability from time s to u equals the sum over intermediate states at time t of the two-step probabilities.

Semi-Markov model

Relaxes the Markov property by allowing transition intensities to depend on duration spent in the current state, not only on the current state itself.

Prospective reserve

The expected present value of future benefits and expenses minus future premiums, conditional on the policy being in force at the valuation date.

Recursive reserve formula

(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. It rolls the reserve forward one year.

Gross premium

The net premium plus loadings for expenses, profit, and contingency margins. The gross premium reserve uses gross cash flows rather than net ones.

Risk discount rate

The rate used to discount projected profit cash flows in profit testing. It reflects the required return on capital and the riskiness of the business.

Profit signature

The stream of emerging profits by policy year. Its present value at the risk discount rate gives the expected profit from the policy.

Internal rate of return

The discount rate at which the net present value of projected cash flows is zero. Comparing it with the required return indicates profitability.

Solvency capital requirement

Capital held to cover adverse deviations in assumptions, often set as a value-at-risk at a specified confidence level over a one-year horizon.

Diversification benefit

Total capital for combined risks is less than the sum of standalone capitals because risks are not perfectly correlated. It increases with lower correlation between risks.

Embedded value

Net assets plus the present value of future profits from in-force business. It excludes the value of future new business.

Sensitivity and scenario testing

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

Effect of reinsurance

Reinsurance reduces retained risk and required capital but transfers some profit to the reinsurer. Net retained risk equals gross risk less reinsured risk.