IFoA CM1 flashcards
IFoA CM1 · Build recall with a focused review session.
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- Accumulation function
- Function a(t) giving the accumulated value at time t of an investment of 1 at time 0. It satisfies a(0)=1 and is increasing for positive interest.
- Force of interest
- The instantaneous rate δ(t) = a'(t)/a(t). For constant δ, the accumulation function is e^{δt}, and the effective annual rate equals e^δ - 1.
- Present value factor
- The discount function v(t) = 1/a(t). With constant effective rate i, v^t = (1+i)^{-t}; with constant force δ, v(t) = e^{-δt}.
- Annuity-immediate formula
- a-angle-n = (1 - v^n)/i is the present value of 1 paid at the end of each of n years. Its accumulated value is s-angle-n = (1+i)^n·a-angle-n.
- Equation of value
- Equating present values of inflows and outflows at a chosen comparison date. The solution interest rate making net present value zero is the internal rate of return.
- Spot rate
- The yield y_t on a zero-coupon bond maturing at t, with price P(0,t) = exp(-y_t·t). The spot curve plots y_t against maturity t.
- Forward rate
- The rate f_{t,r} agreed now for investment from time t to t+r. It satisfies exp(-y_{t+r}(t+r)) = exp(-y_t·t)·exp(-f_{t,r}·r).
- Macaulay duration
- The weighted average time to cashflow: D = Σ t·C_t·v^t / Σ C_t·v^t. Modified duration D/(1+i) approximates the percentage price change per unit yield change.
- Lognormal asset returns
- If log returns over unit periods are independent N(μ,σ²), then ln S_n ~ N(nμ, nσ²) and S_n is lognormal. E[S_n] = S_0·exp(n(μ+σ²/2)).
- Liquidity preference theory
- Term structure theory holding that forward rates exceed expected future spot rates because investors demand a premium for holding longer-dated, less liquid bonds.
- Present-value random variable
- A random variable representing the discounted value of future cashflows whose timing and amount depend on random events such as death or survival.
- Equivalence principle
- The premium basis is set so that the expected present value of premiums equals the expected present value of benefits plus expenses at outset.
- Prospective reserve
- At duration t, the expected present value of future benefits and expenses minus the expected present value of future premiums, using the reserving 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.
- Net premium
- Premium calculated ignoring expenses, equating expected present value of benefits to that of net premiums. For whole life, P = A_x / a_x.
- Curtate expectation of life
- e_x = Σ_{k≥1} kp_x, the expected number of complete future years lived by a life aged x. Complete expectation e̊_x ≈ e_x + 0.5 under UDD.
- Commutation functions
- D_x = v^x·l_x, C_x = v^{x+1}·d_x, N_x = Σ D_{x+k}, M_x = Σ C_{x+k}. Then A_x = M_x/D_x and a_x = N_{x+1}/D_x.
- Endowment assurance identity
- A_{x:n} = A^1_{x:n} + v^n·_np_x, combining term assurance with a pure endowment. Also A_{x:n} = 1 - d·ä_{x:n}.
- Annuity-due relationship
- ä_x = (1+i)·a_x, reflecting payment at the start rather than end of each year. For n-year temporary annuities, ä_{x:n} = (1+i)·a_{x:n}.
- Reserve recursion
- (V + P)(1+i) = q_{x+t}·S + p_{x+t}·_{t+1}V for a sum assured S payable at death, linking consecutive reserves for assurance contracts.
- Gompertz law of mortality
- μ_x = B·c^x, giving _{t}p_x = exp(-B·c^x·(c^t - 1)/ln c). Mortality increases exponentially with age, with parameters B > 0 and c > 1.
- Makeham law
- μ_x = A + B·c^x, adding a constant background mortality A to the Gompertz term. It captures mortality at younger ages better than Gompertz alone.
- Select mortality
- Mortality rates depending on duration since underwriting, written q_{[x]+r}. Select rates are typically lower than ultimate rates q_{x+r} for the same attained age.
- Multiple decrement table
- A table tracking several modes of exit (death, withdrawal, retirement) from a group. Total force equals the sum of individual forces of decrement.
- Kolmogorov forward equations
- d/dt p^{ij}(s,t) = Σ_k p^{ik}(s,t)·μ^{kj}(t), governing transition probabilities between states in a multiple-state model.
- Profit test
- Projection of premiums, benefits, expenses, interest, and reserves to derive expected profits per policy, discounted at a risk discount rate.
- Profit signature
- The sequence of expected profits per policy in force at inception, obtained by multiplying each year's profit by the survival probability to that year.
- Break-even year
- The first year in which the cumulative present value of profits becomes positive, indicating when the contract recovers its initial strain.
- Analysis of surplus
- Decomposition of the change in surplus between valuations into interest, mortality, expense, lapse, and new business components, each comparing actual to expected experience.
- Gross premium reserve
- Reserve calculated on the valuation basis including future expenses and gross premiums, typically more prudent than the pricing basis assumptions.
- Risk margin
- Amount added to best estimate liabilities to compensate for the cost of holding required capital, often computed as a cost-of-capital charge on projected solvency capital.
- Defined benefit scheme
- Pension scheme where the benefit is formula-based, often salary and service related. The employer bears investment and longevity risk.
- Pay-as-you-go scheme
- Social security arrangement funding current benefits from current contributions rather than accumulated assets. Sustainability depends on the dependency ratio.
- Accrued pension liability
- Present value of benefits attributable to service already completed, based on current salary. Contrasts with projected liability including future salary growth.