Exam FM Formula Sheet: Every Formula You Need to Memorize
A comprehensive formula sheet covering every key formula for SOA Exam FM.
Interest Rate Fundamentals
Discount factor: v = 1/(1+i). Discount rate: d = i/(1+i) = iv = 1 - v. Relationship: (1+i)(1-d) = 1. Force of interest: delta = ln(1+i). Converting back: i = e^delta - 1. d = 1 - e^(-delta). Nominal rates: (1 + i^(m)/m)^m = 1 + i for interest compounded m times per year. (1 - d^(m)/m)^m = 1 - d for discount compounded m times per year.
Accumulation functions: compound interest a(t) = (1+i)^t. Simple interest a(t) = 1 + it. Continuous compounding a(t) = e^(delta*t). Variable force of interest: a(t) = exp(integral from 0 to t of delta(s) ds). Present value function: a^(-1)(t) = 1/a(t) = v^t for compound interest.
Annuity Formulas
Annuity-immediate (payments at end of period): a-angle-n = (1 - v^n)/i. s-angle-n = ((1+i)^n - 1)/i. Annuity-due (payments at beginning of period): a-double-dot-n = (1 - v^n)/d. s-double-dot-n = ((1+i)^n - 1)/d. Key relationships: a-double-dot-n = (1+i) * a-angle-n. s-double-dot-n = (1+i) * s-angle-n. a-angle-n = 1 - v^n follows from i * a-angle-n = 1 - v^n.
Perpetuities: a-angle-infinity = 1/i. a-double-dot-infinity = 1/d. Deferred annuity: m|a-angle-n = v^m * a-angle-n. Continuous annuity: a-bar-angle-n = (1 - v^n)/delta. Payable m-thly: a-angle-n^(m) = (1 - v^n)/i^(m).
Increasing annuity-immediate: (Ia)-angle-n = (a-double-dot-n - n*v^n)/i. Decreasing annuity-immediate: (Da)-angle-n = (n - a-angle-n)/i. Geometric annuity: PV = (1 - ((1+g)/(1+i))^n)/(i - g) when i is not equal to g. When i = g, PV = n/(1+i). Increasing perpetuity-immediate: (Ia)-angle-infinity = 1/(i*d) = (1+i)/i^2.
Loan Amortization
Level payment: R = L / a-angle-n. Outstanding balance (prospective): B_t = R * a-angle-(n-t). Outstanding balance (retrospective): B_t = L*(1+i)^t - R*s-angle-t. Interest in period t: I_t = i * B_{t-1}. Principal in period t: P_t = R - I_t = R * v^(n-t+1). Principal amounts form a geometric sequence: P_t = P_1 * (1+i)^(t-1).
Total interest paid over the life of the loan: n*R - L = n*R - R*a-angle-n = R*(n - a-angle-n). Sum of principal payments equals L. Sinking fund method: interest payment = L*i, sinking fund deposit = L/s-angle-n(j), total outlay = L*i + L/s-angle-n(j). When i = j, total outlay = L/a-angle-n(i), matching the amortization payment.
Bond Pricing and Amortization
Basic formula: P = Fr * a-angle-n + C * v^n, where F = face value, r = coupon rate, C = redemption value, n = periods. Premium/discount formula: P = C + (Fr - Ci) * a-angle-n = C + C(g - i) * a-angle-n, where g = Fr/C is the modified coupon rate. Makeham formula: P = K + (g/i)*(C - K), where K = C*v^n.
Book value: B_t = Fr * a-angle-(n-t) + C * v^(n-t). Premium amortization in period t: P_t = C(g - i) * v^(n-t+1). These amounts form a geometric sequence increasing by factor (1+i). Bonds between coupon dates: Dirty Price = B_t * (1+i)^f, Clean Price = Dirty Price - Fr*f, where f is the fraction of the coupon period elapsed.
Callable bonds: if bought at premium (g > i), assume earliest call. If bought at discount (g < i), assume latest call. When call prices vary, check each possible call date and use the minimum price.
Term Structure, Duration, and Immunization
Spot rates: PV of $1 at time t is (1 + s_t)^(-t). Forward rates: (1 + f(t,t+1)) = (1 + s_{t+1})^(t+1) / (1 + s_t)^t. Bond price with term structure: P = sum of CF_t / (1 + s_t)^t.
Macaulay duration: D = sum(t * PV(CF_t)) / P. Modified duration: D* = D/(1+i). Convexity: C = sum(t*(t+1) * PV(CF_t)) / (P*(1+i)^2). Price change approximation: dP/P is approximately -D* * di + (1/2) * C * (di)^2.
Redington immunization conditions: (1) PV(assets) = PV(liabilities), (2) D(assets) = D(liabilities), (3) C(assets) > C(liabilities). Full immunization of a single liability: asset cash flows must occur both before and after the liability date, with PV and duration matching.
Derivatives and Financial Instruments
Forward price (no income): F = S_0 * (1+i)^T. With continuous dividends at rate delta: F = S_0 * e^((r-delta)*T). With discrete dividends: F = (S_0 - PV(dividends)) * (1+i)^T. Put-call parity: C(K,T) - P(K,T) = S_0 * e^(-delta*T) - K * e^(-r*T). For non-dividend-paying stock: C - P = S_0 - K*v^T.
Swap rate: R = (1 - v^n) / sum(v^t for t = 1 to n). The swap rate makes the present value of fixed payments equal to the present value of floating payments. Interest rate swap: fixed-rate payer pays R and receives the floating rate each period.
Investment Analysis and Returns
Net present value: NPV = sum of CF_t / (1+i)^t for t = 0, 1, ..., n. Internal rate of return: the rate i where NPV = 0. Dollar-weighted rate of return: solve B_0*(1+i)^T + sum C_t*(1+i)^(T-t_j) = B_T. Time-weighted rate of return: (1+i_TW) = product of (1+i_k) for each sub-period, where i_k is the holding period return between consecutive cash flows.
Gordon growth model: P = D_1/(r - g), where D_1 is the next dividend, r is the required return, and g is the constant growth rate. Fisher equation: (1+i) = (1+r)*(1+pi), relating nominal rate i, real rate r, and inflation pi. Short selling profit: S_0 - S_T - dividends + interest on proceeds and margin.