Interest Rate Theory Cheat Sheet

Spot and Forward Rates

• Spot rate $s_t$: $(1 + s_t)^t$ = price of $t$-year zero-coupon bond
• Forward rate $f_{t,t+1}$: $(1 + s_{t+1})^{t+1} = (1 + s_t)^t \cdot (1 + f_{t,t+1})$
• Bootstrapping: Extract spot rates sequentially from par bond prices

Duration Formulas

• Macaulay Duration: $D = (1/P) \sum t \cdot v^t \cdot CF_t$
• Modified Duration: $D_{\text{mod}} = D / (1 + y)$
• Dollar Duration: $DD = D_{\text{mod}} \cdot P$
• Effective Duration: $D_{\text{eff}} = (P_- - P_+) / (2 \cdot P_0 \cdot \Delta y)$

Convexity

• Macaulay Convexity: $C_{\text{mac}} = (1/P) \sum t(t+1) \cdot v^{t+2} \cdot CF_t$
• Modified Convexity: $C_{\text{mod}} = C_{\text{mac}} / (1+y)^2$
• Price approximation: $\Delta P / P \approx -D_{\text{mod}} \cdot \Delta y + 0.5 \cdot C_{\text{mod}} \cdot (\Delta y)^2$

Immunization

• Redington conditions: (1) $PV(A) = PV(L)$, (2) $D(A) = D(L)$, (3) $C(A) \gt C(L)$
• Full immunization: Asset cash flows bracket liability timing with matched PV and duration
• Cash flow matching: Dedicated portfolio with exact cash flow alignment