Financial Mathematics Formulas Cheat Sheet

Interest Rate Conversions

• Effective rate: i = (1 + i^(m)/m)^m - 1
• Force of interest: delta = ln(1 + i)
• Discount rate: d = i/(1+i) = 1 - v
• Present value factor: v = 1/(1+i) = e^(-delta)

Annuity Formulas

Type	Present Value	Accumulated Value
Annuity-Immediate	a(n|i) = (1 - v^n) / i	s(n|i) = ((1+i)^n - 1) / i
Annuity-Due	a-double-dot(n|i) = (1 - v^n) / d	s-double-dot(n|i) = ((1+i)^n - 1) / d
Perpetuity-Immediate	1/i	N/A
Perpetuity-Due	1/d	N/A
Increasing Annuity	(Ia)(n|i) = (a-double-dot(n|i) - nv^n) / i	(Is)(n|i) = (s-double-dot(n|i) - n) / i

Bond Formulas

• Price: P = Fr * a(n|i) + C * v^n
• Premium/Discount: P - C = (Fr - Ci) * a(n|i)
• Makeham: P = K + (g/i)(C - K) where K = C * v^n

Duration and Convexity

• Macaulay Duration: D = sum(t * PV(CF_t)) / Price
• Modified Duration: D_mod = D / (1 + i)
• Convexity: C = sum(t(t+1) * PV(CF_t)) / ((1+i)^2 * Price)
• Price change: dP/P approx -D_mod * di + 0.5 * C * (di)^2

Immunization Conditions (Redington)

• PV(assets) = PV(liabilities)
• Duration(assets) = Duration(liabilities)
• Convexity(assets) > Convexity(liabilities)