SOA Exam FM: Financial Mathematics for Actuaries Part 1

Explore accumulation functions by comparing a constant one, a 7% compound-interest growth, and a variable investment path, and define the effective rate of interest i(t) and the relation a(t+1)=a(t)(1+i(t)).

Expand the accumulation function and derive the amount function A(t) from little a(t), showing how I(t) relates to A(t) and a(t) for actuaries.

Explore simple interest, its non-compounding accumulation, and the step-function form of a(t); derive the effective rate and solve a $77 at 8% vs $70 at 10% example for when balances equal.

Explore accumulation functions across multiple periods, deriving values with compounding, calculating period-specific effective rates, and expressing A_t as a product of (1+i_j) to handle variable year rates.

Explore compound interest and apply the accumulation formula (1+i)^t. Solve two account scenarios using logs and a quadratic to find k approximately 0.030437.

Master present value calculations and the discount factor v to determine today’s amount needed for payments. Learn with examples: a 3-year, 75,000 goal at 5% and a two-year 10,000 annuity.

Learn to solve unknown time and unknown interest rate problems using algebra or a financial calculator. Apply examples with 1000 at 5% ending at 1276.2815625 and doubling 100 at 7%.

Explore the effective rate of discount, its relation to the effective rate of interest, and memorize key formulas d = I/(1+I) and I = d/(1-d) with practical examples.

Explore nominal rates of interest and how to compute their effective and annual equivalents for various compounding frequencies, including monthly, quarterly, and semiannual examples.

Compute annual effective rates from nominal rates compounded quarterly and semiannually, set up and solve equations of value, and determine future values for mixed-rate scenarios.

Explain nominal rates of discount and their link to discount rates D, using a quarterly compounded example to find present value of 950 for 1000 due in one year.

Learn nominal rates of discount with quarterly and semiannual compounding, solving the equation of value to compute d, j, and k and the five-year accumulation of fund c.

Explore the constant force of interest, the continuous-compounding framework where delta equals a'(t)/a(t) and e to the delta t gives accumulation, with present value e to the negative delta t.

Explore non-constant force of interest, derive delta t = a'(t)/a(t) and the accumulated value via e^(integral delta t dt), and compute present value and effective rates using a(t) = 3t^2+4t+2.

Explore non-constant forces of interest through multiple examples, deriving accumulation factors from variable delta t, solving equations of value with nominal discounts and force functions.

Explore Taylor series expansions and sigma notation for 1/(1−r), 1/(1+r), e^r, and ln(1+r), and learn i, delta, and d relationships, including e^δ = 1+i and d = 1−e^{−δ}.

Explore the time value of money and equations of value under positive interest rates. Solve deposits, loans, and unknown cash flows through examples with varying compounding and scenarios.

Learn spot rates from time zero to time t, compute present values, and understand yield curves, forward rates, and normal versus inverted term structures.

Learn how forward rates are derived from spot rates, including the forward rate from time t1 to t2 and the deferral notation, with practical examples and common pitfalls.