Actuarial Mathematics
Scope of Actuarial Mathematics
This domain tests whether you can move cash flows through time correctly: effective and nominal interest, accumulation and discount, annuities, equation of value, yield rates, loan schedules, and duration. Korean exam questions often combine a simple product cash-flow pattern with a rate conversion, so the arithmetic must be clean and the timing convention explicit.
Core rules
- Effective annual rate i and nominal rate i^(m): 1+i = (1+i^(m)/m)^m.
- v = 1/(1+i), d = i/(1+i) = 1-v.
- Annuity-immediate a_n = (1-v^n)/i; annuity-due ä_n = (1+i)a_n.
- Equation of value: PV(outflows) = PV(inflows) at the chosen valuation date.
- IRR solves NPV=0; check sign changes because multiple roots are possible.
Worked example
A saver deposits KRW 1,000 at the end of each year for five years into an account crediting 5% effective. Find the accumulated value at the fifth deposit.
Use s_5 = ((1.05)^5 - 1)/0.05. (1.05)^5 = 1.2762815625, so numerator = 0.2762815625 and s_5 = 5.52563125. Accumulated value = 1,000 × 5.52563125 = KRW 5,525.63. Reasoning: the first deposit earns four years of interest, the second three, and the fifth none; the annuity formula sums those five accumulated amounts. If deposits were at the beginning of each year, multiply by 1.05 to get KRW 5,801.91.
Traps
Mixing due and immediate timing is the most common error. A nominal rate compounded monthly cannot be used directly as a monthly effective rate unless divided by 12. Do not round intermediate v factors; keep at least six decimals. When a problem asks for a loan outstanding balance, the retrospective and prospective methods must agree only if the same assumptions are used.