CM1 Actuarial Mathematics: Theory of Interest and Contingencies
The Actuarial Mathematics domain of CM1 tests two linked skills: valuing deterministic cash flows through the theory of interest, and valuing payments that depend on survival or other decrements through contingencies. Most marks come from setting up a correct equation of value and solving it, so precision about timing and basis matters more than formula recall.
Deterministic cash flows
- v = 1/(1+i), d = i/(1+i), δ = ln(1+i); ä_n = (1−v^n)/d, a_n = (1−v^n)/i, s_n = a_n(1+i)^n.
- Equation of value: choose one comparison date; earlier amounts accumulate, later ones discount. Solve for an unknown rate by interpolation between two trial rates.
- Bond price = PV of coupons + PV of redemption; dirty price = clean price + accrued interest.
- Macaulay duration D = Σ t·PV_t / Σ PV_t; modified duration = D/(1+i); convexity adds curvature. Redington immunisation needs matched present value, matched duration, and asset convexity above liability convexity.
Contingencies
- A_x = Σ v^(k+1)·ₖp_x·q_(x+k); ä_x = Σ v^k·ₖp_x; the identity A_x = 1 − d·ä_x links the two.
- Equivalence principle: EPV(premiums) = EPV(benefits) + EPV(expenses).
Worked example
With ä_50 = 15.00 at i = 4% and a whole life assurance of 1,000 payable at the end of the year of death: d = 0.04/1.04 = 0.038462, so A_50 = 1 − 0.038462 × 15.00 = 0.42308. Then P × 15.00 = 1,000 × 0.42308, giving P = 28.21 a year.
Traps that cost marks
- Paying a benefit at the moment of death but discounting with i rather than δ.
- Mixing ä and a, or d and i, in the assurance–annuity identity.
- Prospective and retrospective reserves diverge once the basis changes mid-term.
- Ignoring select mortality in the years immediately after underwriting.