Financial Mathematics
Rates and forces
Convert fluently among effective interest i, nominal i^(m), discount rate d, and force of interest δ, using (1+i) = (1 + i^(m)/m)^m = e^δ and v = 1/(1+i). Note that d = i/(1+i) = 1 − v, a relation that shortcuts many annuity problems.
Annuities and loans
For a level annuity-immediate, a_n = (1 − v^n)/i; the annuity-due is ä_n = (1+i)a_n = (1 − v^n)/d. A loan's outstanding balance equals the present value of remaining payments, and the sinking-fund method must agree with amortization in total cost. Always confirm when the first payment occurs.
Bonds and duration
Price equals the present value of coupons plus redemption. Whether a bond trades at a premium or discount is determined by coupon versus yield. Macaulay duration is D = Σ t·PV(CF_t)/Price; modified duration is D/(1+i); convexity captures curvature. Redington immunization matches present value, duration, and convexity across assets and liabilities, and holds only for small, parallel yield shifts.
Worked example
Ten annual payments of 1,000 beginning immediately, with i = 5%. Here d = 0.05/1.05 = 0.047619 and v^10 = 0.613913, so
PV = 1000 × (1 − 0.613913)/0.047619 = 8,107.82.
Traps
Off-by-one-period errors between due and immediate annuities, misuse of a nominal rate as if it were effective, and quoting Macaulay duration when modified duration is required are the classic slips. Internal rate of return can also be non-unique when cash flows alternate in sign.