Financial Mathematics
Financial Mathematics tests the time-value toolkit actuaries apply to reserves, premiums and asset-liability work: accumulation and discounting, annuity and bond valuation, the term structure of interest rates, and interest-rate risk measures.
Core rules
- Force of interest δ(t) links continuously to accumulation: A(t) = A(0)exp(∫δ). Nominal rates convertible m times per year satisfy (1+i) = (1+i^(m)/m)^m.
- Annuity-immediate versus annuity-due differ by a factor of (1+i); perpetuity values are limiting cases.
- Spot rates price cash flows individually: P = Σ C_t/(1+s_t)^t. Forward rates follow from no arbitrage: (1+s_2)² = (1+s_1)(1+f_{1,2}).
- Macaulay duration is the PV-weighted mean term; modified duration D/(1+y) gives the percentage price change per unit yield change. Convexity corrects for curvature.
- Redington immunisation requires equal PV, equal duration, and assets more convex than liabilities.
Worked example
A 2-year bond pays an 8% annual coupon on face 100. With s₁ = 5% and s₂ = 6%, price = 8/1.05 + 108/1.06² = 7.619 + 96.119 = 103.738. The implied forward rate is 1.06²/1.05 − 1 = 7.0095%. Duration weights are 0.0734 at t=1 and 0.9266 at t=2, giving D = 1.927 years and modified duration 1.927/1.06 = 1.818.
Traps
- Mixing nominal and effective rates, or annual and monthly periods, in duration and annuity formulas.
- Applying a single yield where the term structure is given.
- Multiple IRRs when cash flows change sign more than once; use NPV profiles.
- Quoting convexity in years² but applying it without the ½(Δy)² factor.
- Immunisation is instantaneous: duration drifts as time and yields move.