IFoA CM1 cheat sheet

Accumulated value: A(t) = A(0)·a(t); present value: PV = A(t)·v(t) with v(t) = 1/a(t). For constant force δ, a(t) = e^{δt} and v(t) = e^{-δt}.

Effective rate i and discount rate d satisfy d = i/(1+i) = 1 - v, and v = 1 - d. Nominal rates: (1+i) = (1 + i^{(m)}/m)^m = (1 - d^{(m)}/m)^{-m}.

Force of interest from accumulation function: δ(t) = a'(t)/a(t) = d/dt ln a(t), so a(t) = exp(∫₀ᵗ δ(s) ds). Constant δ gives i = e^δ - 1.

Annuity-immediate: a-angle-n = (1 - v^n)/i, s-angle-n = ((1+i)^n - 1)/i. Annuity-due: ä-angle-n = (1+i)a-angle-n = (1 - v^n)/d.

Continuous annuity: ā-angle-n = (1 - v^n)/δ. Perpetuities: a-angle-∞ = 1/i (immediate), 1/d (due), 1/δ (continuous).

Increasing annuity: (Ia)-angle-n = (ä-angle-n - n·v^n)/i. Varying payments use Σ C_t v^t; split into level plus increasing components to simplify.

Equation of value: Σ inflows·v^t = Σ outflows·v^t at a chosen comparison date. Any consistent date gives the same answer; IRR is the i making net present value zero.

Real vs nominal: (1+i_real) = (1+i_nominal)/(1+inflation). Payments in geometric progression with ratio (1+g) use first payment times a-angle-n at rate j = (i-g)/(1+g).

Spot rate y_t is the yield on a zero-coupon bond maturing at t: P(0,t) = exp(-y_t·t). Forward rate f_{t,r} satisfies exp(-y_{t+r}(t+r)) = exp(-y_t·t)·exp(-f_{t,r}·r).

Instantaneous forward rate f(t) = -d/dt ln P(0,t); with continuous spot y(t), f(t) = y(t) + t·y'(t). Upward-sloping spot curve implies forward rates above spot rates.

Duration (Macaulay) D = Σ t·C_t·v^t / Σ C_t·v^t; modified duration = D/(1+i); convexity = Σ t(t+1)C_t v^t / ((1+i)^2 Σ C_t v^t).

Redemption yield solves P = Σ C_t (1+y)^{-t}; price and yield move inversely. A bond trades at par when coupon rate equals yield.

Random walk model: log return over unit period R ~ N(μ, σ²). Accumulation over n years: S_n = S_0·exp(R_1+...+R_n), with E[S_n] = S_0·exp(n(μ + σ²/2)).

Lognormal property: ln S_n ~ N(nμ, nσ²), so S_n is lognormal; Var[S_n] = S_0²·e^{2nμ+nσ²}(e^{nσ²} - 1). Percentiles use Φ^{-1}.

Brownian motion with drift: dS/S = μ dt + σ dW gives S_t = S_0·exp((μ - σ²/2)t + σW_t); the σ²/2 correction distinguishes arithmetic drift μ from geometric growth.

Term structure theories: expectations hypothesis (forwards are expected future spot rates), liquidity preference (forwards exceed expected spots due to premium), and segmentation/market segmentation.

A cashflow model specifies payment amounts, timings, and the probability distribution driving them; a present-value random variable discounts each cashflow to time 0 using the relevant discount function.

For a whole-life assurance paying 1 at the end of the year of death, PV = v^{K+1} where K is curtate future lifetime; E[PV] = A_x = Σ v^{k+1}·{k}p_x·q{x+k}.

Variance principle: Var[Z] = E[Z²] - (E[Z])². Compute E[Z²] using the same survival probabilities discounted at v² (or e^{-2δ}), giving A_x at rate (1+i)² - 1.

Actuarial present value (APV) equals expected present value under the assumed interest and mortality basis; equivalence principle sets EPV of premiums equal to EPV of benefits plus expenses.

Benefit premiums: net premium P = A_x / a_x for whole life; gross premium includes expense loadings, typically as a percentage of premium plus per-policy amounts.

Prospective reserve at time t = EPV(future benefits + expenses) - EPV(future premiums); retrospective reserve = accumulated past premiums minus accumulated past benefits, equal under the same basis.

Thiele's differential equation: d/dt V_t = δ_t·V_t + P_t - b_t - (S_t - V_t)·μ_{x+t}, where S_t is the sum at risk and μ the force of mortality.

For a portfolio, expected profit emerges from the difference between the reserving basis and the experience basis; profit signature is the sequence of expected cashflows including reserves.

Life table identities: l_{x+1} = l_x - d_x; {t}p_x = l{x+t}/l_x; {t}q_x = 1 - l{x+t}/l_x; force of mortality μ_{x+t} = -(1/l_{x+t})·dl_{x+t}/dt.

Curtate expectation of life e_x = Σ_{k≥1} kp_x; complete expectation e̊_x ≈ e_x + 1/2 under uniform distribution of deaths.

Whole-life assurance: A_x = v·q_x + v·p_x·A_{x+1}; recursive relationship allows computation from the ultimate age backward. Term assurance: A^1_{x:n} = A_x - v^n·np_x·A{x+n}.

Endowment assurance: A_{x:n} = A^1_{x:n} + v^n·np_x (pure endowment). Relationship to annuity: A_x = 1 - d·ä_x; A{x:n} = 1 - d·ä_{x:n}.

Annuity-immediate a_x = Σ v^k·_kp_x; annuity-due ä_x = (1+i)a_x. Increasing assurances use commutation functions or recursive relations with (IA)_x.

Commutation functions: D_x = v^x·l_x, C_x = v^{x+1}·d_x, N_x = Σ_{k≥0} D_{x+k}, M_x = Σ C_{x+k}. Then A_x = M_x/D_x and a_x = N_{x+1}/D_x.

Uniform distribution of deaths (UDD) assumption: _{t}q_x = t·q_x for 0 ≤ t ≤ 1, giving fractional-age approximations for benefits paid at death.

Reserves for term and endowment assurances follow the recursive relation (V + P)(1+i) = q_{x+t}·S + p_{x+t}·_{t+1}V, where S is the sum assured.

Gompertz law: μ_x = B·c^x, giving survival _{t}p_x = exp(-B·c^x·(c^t - 1)/ln c). Makeham adds a constant: μ_x = A + B·c^x.

Weibull law: μ_x = k·x^n, giving _{t}p_x = exp(-k·((x+t)^{n+1} - x^{n+1})/(n+1)).

Select mortality: rates depend on duration since selection, denoted q_{[x]+r}. Select period of h years followed by ultimate rates q_{x+r}; select rates are lower than ultimate rates for the same age.

Multiple decrement: total force μ^{(T)} = Σ μ^{(j)}. Associated single-decrement probability q_x^{(j)} relates to dependent rate via q_x^{(j)} = ∫₀¹ _tp_x^{(T)}·μ_x^{(j)}(t) dt.

Uniform distribution of decrements in the multiple-decrement table: q_x^{(j)} = a_x^{(j)}/l_x where a_x^{(j)} are independent decrements; conditional probability of decrement j given decrement occurs is a_x^{(j)}/a_x^{(T)}.

Multiple-state models: transition intensities μ^{ij}_x define the generator matrix; Kolmogorov forward equations d/dt p^{ij}(s,t) = Σ_k p^{ik}(s,t)·μ^{kj}(t) govern transition probabilities.

Two-state model (alive/dead) with constant force μ gives _{t}p_x = e^{-μt} and the expected present value of a whole-life assurance paid at the moment of death is μ/(μ+δ).

Selection effects: underwritten lives show lower early-duration mortality; temporary initial selection followed by ultimate rates, often approximated by adding a select period to standard tables.

Profit test: project premiums, benefits, expenses, interest, and reserves at the pricing basis; profit at time t = (reserve_{t-1} + premium - expenses)(1+i) - benefits - reserve_t.

Profit signature: the sequence of expected profits per policy in force at inception (profit at t times survival probability). The net present value is the sum of discounted profit signature.

Internal rate of return (IRR) solves NPV = 0; the profit test compares IRR to the hurdle rate (risk discount rate) to determine contract profitability.

Break-even year is the first year in which the cumulative present value of profits becomes positive; break-even month interpolates within that year.

Analysis of surplus decomposes the change in surplus between valuation dates into components: interest (actual vs expected), mortality (actual vs expected), expenses, lapse/withdrawal, and new business strain.

Sources of surplus: return on investments, release of margins in reserves, mortality profit, expense savings, and surrender profits. Each is computed as (actual - expected) times the exposed amount.

New business strain arises because initial expenses exceed the first premium; it reduces surplus at outset and is recovered through future profits.

Asset share is the accumulated value of premiums less benefits and expenses per policy, rolled up with the actual investment return; it informs bonus and surrender value decisions.

Gross premium reserve uses the valuation (reserving) basis, typically more prudent than the pricing basis: higher mortality, lower interest, and higher expenses.

Net premium reserve uses only net premiums (excluding expenses) but reserves for expenses implicitly through the difference between gross and net premiums.

Solvency capital requirement: capital needed so that assets exceed liabilities at a high confidence level (e.g., 99.5% one-year VaR under Solvency II).

Risk margin compensates for the cost of holding required capital; under Solvency II it is the cost of capital (6%) applied to projected SCR run-off.

Best estimate liability is the expected present value of future cashflows under realistic assumptions; technical provisions equal best estimate plus risk margin.

Reserving bases: zillmerised reserves adjust for initial expenses; modified reserves allow surplus to emerge earlier and reduce new business strain.

Stress testing and scenario analysis assess the impact of changes in interest rates, mortality, expenses, and lapses on reserves and capital.

Free assets (surplus) equal assets minus liabilities; the ratio of free assets to required capital indicates financial strength and is monitored by regulators.

Defined benefit (DB): benefit based on salary and service, e.g., n/60 × final salary per year of service. The employer bears investment and longevity risk.

Defined contribution (DC): benefit equals accumulated contributions plus investment return at retirement; the member bears investment risk and often longevity risk.

Final salary vs career average (CARE): final salary uses salary near retirement; CARE uses revalued average salary, reducing cost volatility for the employer.

Pension funding: contribution rate c satisfies EPV of contributions = EPV of benefits plus expenses; actuarial valuation uses a discount rate reflecting expected asset returns.

Accrued benefits vs projected benefits: the accrued (past service) liability uses current salary; the projected liability includes future salary growth and service.

Social security schemes: pay-as-you-go (PAYG) funds current benefits from current contributions; funded schemes build assets. PAYG depends on the dependency ratio.

Pension increases and revaluation: LPI (limited price indexation) caps inflation-linked increases; salary growth assumptions affect final salary benefits.

Actuarial modelling for pensions includes sensitivity to inflation, salary growth, longevity, and asset returns; funding level = assets / liabilities.