IAI CP1 cheat sheet

Present value: PV = sum over t of CF_t * v^t, where v = 1/(1+i); for continuous discounting v^t = exp(-delta*t) with delta = ln(1+i).

Level annuity-immediate a_n = (1 - v^n)/i; annuity-due a-double-dot_n = (1 - v^n)/d with d = i/(1+i); increasing annuity (Ia)_n = (a-double-dot_n - n*v^n)/i.

Net present value decision rule: accept a project when NPV > 0 at the risk-adjusted discount rate; between mutually exclusive projects choose the higher NPV, not the higher IRR.

IRR is the rate r solving NPV(r) = 0; it is unreliable when cash flows change sign more than once (multiple roots) or when comparing projects of different scale or duration.

Equation of value: accumulate or discount all cash flows to a common date and set inflows equal to outflows; the solution rate is the yield. Use linear interpolation only as an approximation between two bracketing rates.

Expected present value of a life contingent benefit: EPV = sum over t of (benefit_t) * v^t * (probability of payment at t), with probabilities taken from the appropriate decrement table.

Premium principle: equivalence principle sets the premium so that EPV of premiums equals EPV of benefits plus expenses, i.e. EPV(premiums) - EPV(benefits) - EPV(expenses) = 0.

Reserve (net premium reserve) at duration t: prospective reserve = EPV(future benefits) - EPV(future net premiums); it equals the retrospective accumulated value of past premiums less past claims costs.

Profit signature: model cash flows per policy in force at outset, multiply by the expected in-force probability each year, then discount at the risk discount rate to obtain the NPV of the contract.

Thiele's differential equation for a policy with state-dependent reserve V: dV/dt = delta*V + P - b - (S - V)*mu, where b is the benefit outflow rate and S the sum at risk on death.

Survival model: mu_x is the force of mortality; tPx = exp(-integral from 0 to t of mu_{x+s} ds); the probability of death in year t is (tPx)*q_{x+t}.

Under the uniform distribution of deaths assumption, force of mortality in year of age x is mu_{x+t} = q_x / (1 - tq_x), and l_{x+t} = l_x(1 - t*q_x).

Constant force of mortality mu over a year gives q = 1 - exp(-mu) and tP = exp(-mu*t); this assumption is convenient for fractional ages in multi-decrement work.

Multiple decrement: independent rates q' relate to dependent rates q by q^(j) = integral over the year of tP^(tau) * mu^(j)_{x+t} dt; with constant forces, q^(j) = (mu^(j)/mu^(tau))*(1 - exp(-mu^(tau))).

Select mortality: rates depend on duration since underwriting, so q_[x]+t < q_{x+t} for early durations; select period is typically shown before the ultimate table is entered.

Expected cash flow in a unit-linked contract separates the non-unit reserve (expenses less charges) from the unit fund; the unit reserve equals the bid value of units allocated to the policyholder.

Asset-liability matching: immunise by matching present value, duration and convexity of assets and liabilities; Redington immunisation requires PV(A) = PV(L), duration(A) = duration(L) and convexity(A) >= convexity(L).

Risk classification decision rule: a factor is worth using when the reduction in adverse selection cost exceeds the cost of underwriting and the regulatory or fairness constraints on its use.

Capital requirement logic: required capital covers the risk that assets fall short of liabilities at a chosen confidence level over a chosen horizon; SCR-type measures use a 1-in-200 annual shock, economic capital uses the firm's own target.

Free surplus = assets - liabilities - required capital; distributable earnings change each year by investment return plus releases less new capital and tax.

Profit test output: NPV at the risk discount rate, the internal rate of return of the profit signature, and the break-even year when cumulative discounted profit first becomes positive.

Risk discount rate reflects the shareholders' required return; raising it reduces NPV and can change the ranking of products, so it must be set consistently with the risks being modelled.

Sensitivity analysis varies one assumption at a time (mortality, lapse, expenses, interest) and reports the change in NPV; scenario testing varies several assumptions together, and stochastic modelling generates a distribution of outcomes.

Experience analysis compares actual against expected claims, lapses and expenses; the ratio actual/expected (A/E ratio) drives the assumption review, with credibility weighting for small volumes.

Profit emergence is front-loaded by acquisition expenses and reserves; new business strain equals the capital needed to write the business and is recovered as the block runs off.

Embedded value = net assets plus value of in-force business, where value of in-force is the present value of future shareholder profits after allowing for required capital and the cost of holding it.

Reinsurance decision: quota share cedes a fixed proportion of premium and claims and reduces new business strain; excess of loss cedes claims above an attachment point and protects against large individual losses.

Professional judgement must be documented: state the assumptions chosen, the evidence and rationale, and the sensitivity of results to those assumptions, so another actuary could review the work.

When communicating results, lead with the decision-relevant conclusion, quantify uncertainty, and separate facts from judgement; avoid unexplained technical jargon for non-actuarial audiences.

Peer review and independent checking are expected controls; a model should be validated against simple hand calculations and its output reconciled to accounting or experience data.

Conflicts of interest must be disclosed and managed; where the actuary's own remuneration depends on the result, independent oversight or review is required.

Data quality: check completeness, accuracy, consistency and relevance; document any adjustments for missing or unreliable data and consider the effect on the results.

Model risk controls include version control, documented change logs, input and output checks, and periodic recalculation of a sample of policies by an independent method.

Regulatory reporting usually requires a best-estimate basis for reserves plus a prudential margin, and a separate demonstration that the firm holds adequate capital against adverse experience.

Ethical decision rule: if a course of action would not bear public explanation to clients, regulators and peers, it should not be adopted even where it is technically permitted.

Option pricing: for a non-dividend share, the Black-Scholes call value is SN(d1) - Kexp(-rT)N(d2), with d1 = (ln(S/K) + (r + 0.5sigma^2)T)/(sigmasqrt(T)) and d2 = d1 - sigmasqrt(T).

Put-call parity: C - P = S - K*exp(-rT); it holds for European options and provides a no-arbitrage check on any pricing model output.

Risk-neutral valuation: the price equals the discounted expectation of the payoff under the risk-neutral measure, so expected returns on the underlying equal the risk-free rate.

Binomial model: p = (exp(r*dt) - d)/(u - d) is the risk-neutral up-probability; price by backward induction, discounting each expected value one step at the risk-free rate.

Greeks: delta = dV/dS, gamma = d2V/dS2, vega = dV/dsigma, theta = dV/dt and rho = dV/dr; delta-hedging removes first-order price risk but leaves gamma and vega exposure.

Interest rate models: Vasicek and Hull-White are mean-reverting short-rate models with analytic bond prices; the Black model prices caps and swaptions using a lognormal forward rate.

Forward and futures prices: F = S*exp((r - q)T) for an asset with continuous yield q; for a futures contract, daily settlement changes the value but not the initial no-arbitrage price.

Duration and convexity: modified duration D = -(1/P)dP/dr, and the price change is approximately -Dchange in r + 0.5convexity(change in r)^2.

Stochastic models for equity returns: geometric Brownian motion gives lognormal prices with expected value S0exp(mut) and variance S0^2exp(2mut)(exp(sigma^2*t) - 1).