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	$a_{\overline{n}|i} = (1 - v^n) / i$	$s_{\overline{n}|i} = ((1+i)^n - 1) / i$
Annuity-Due	$\ddot{a}_{\overline{n}|i} = (1 - v^n) / d$	$\ddot{s}_{\overline{n}|i} = ((1+i)^n - 1) / d$
Perpetuity-Immediate	$\frac{1}{i}$	N/A
Perpetuity-Due	$\frac{1}{d}$	N/A
Increasing Annuity	$(Ia)_{\overline{n}|i} = (\ddot{a}_{\overline{n}|i} - nv^n) / i$	$(Is)_{\overline{n}|i} = (\ddot{s}_{\overline{n}|i} - n) / i$

Bond Formulas

• Price: $P = Fr \cdot a_{\overline{n}|i} + C \cdot v^n$
• Premium/Discount: $P - C = (Fr - Ci) \cdot a_{\overline{n}|i}$
• Makeham: $P = K + (g/i)(C - K)$ where $K = C \cdot v^n$

Duration and Convexity

• Macaulay Duration: $D = \sum(t \cdot PV(CF_t)) / P$
• Modified Duration: $D_{\text{mod}} = D / (1 + i)$
• Convexity: $C = \sum(t(t+1) \cdot PV(CF_t)) / ((1+i)^2 \cdot P)$
• Price change: $\Delta P/P \approx -D_{\text{mod}} \cdot \Delta i + 0.5 \cdot C \cdot (\Delta i)^2$

Immunization Conditions (Redington)

• $PV(\text{assets}) = PV(\text{liabilities})$
• $D(\text{assets}) = D(\text{liabilities})$
• $C(\text{assets}) \gt C(\text{liabilities})$