Exam FM Formula Sheet: Every Formula You Need to Memorize

Interest Rate Relationships

v = 1/(1+i). d = i/(1+i) = iv = 1-v. (1+i)(1-d) = 1. delta = ln(1+i). i = e^delta - 1. d = 1 - e^(-delta). (1+i^(m)/m)^m = 1+i. (1-d^(m)/m)^m = 1-d. Accumulation: a(t) = (1+i)^t for compound, a(t) = 1+it for simple, a(t) = e^(delta*t) for continuous. Variable force: a(t) = exp(integral_0^t delta(s) ds).

Annuity-immediate: a-angle-n = (1-v^n)/i. s-angle-n = ((1+i)^n - 1)/i. Annuity-due: a-double-dot-n = (1-v^n)/d. s-double-dot-n = ((1+i)^n - 1)/d. Relationships: a-double-dot-n = (1+i)*a-angle-n. s-double-dot-n = (1+i)*s-angle-n. Perpetuity: a-angle-infinity = 1/i. a-double-dot-infinity = 1/d.

Loans, Bonds, and Annuity Variations

Level payment: R = L/a-angle-n. Outstanding balance: B_t = R*a-angle-(n-t) (prospective) = L*(1+i)^t - R*s-angle-t (retrospective). Interest: I_t = i*B_{t-1}. Principal: P_t = R - I_t = R*v^(n-t+1). Increasing annuity: (Ia)-angle-n = (a-double-dot-n - n*v^n)/i. Decreasing: (Da)-angle-n = (n - a-angle-n)/i. Geometric: PV = (1-((1+g)/(1+i))^n)/(i-g).

Bond price: P = Fr*a-angle-n + C*v^n = C + (Fr - Ci)*a-angle-n. Book value: B_t = Fr*a-angle-(n-t) + C*v^(n-t). Makeham: P = K + (g/i)*(C-K). Sinking fund deposit: D = L/s-angle-n(j).

Duration, Derivatives, and Investment

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: dP/P approximately equals -D*di + (1/2)*C*(di)^2. Immunization: PV_A = PV_L, D_A = D_L, C_A > C_L. Forward price: F = S_0*(1+i)^T (no dividends). With dividends: F = (S_0 - PV(div))*(1+i)^T. Put-call parity: C - P = S_0 - K*v^T. Swap rate: R = (1-v^n)/sum(v^t). Fisher: (1+i) = (1+r)(1+pi). DWRR: solve B_0*(1+i)^T + sum C_t*(1+i)^(T-t) = B_T. Gordon model: P = D_1/(r-g). NPV = sum CF_t/(1+i)^t. IRR: rate where NPV = 0.