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 * (1 + f_(t,t+1))
• Bootstrapping: Extract spot rates sequentially from par bond prices

Duration Formulas

• Macaulay Duration: D = (1/P) sum t * v^t * CF_t
• Modified Duration: D_mod = D / (1 + y)
• Dollar Duration: DD = D_mod * P
• Effective Duration: D_eff = (P_- - P_+) / (2 * P_0 * Delta_y)

Convexity

• Macaulay Convexity: C_mac = (1/P) sum t(t+1) v^(t+2) CF_t
• Modified Convexity: C_mod = C_mac / (1+y)^2
• Price approximation: Delta P / P approx -D_mod * Delta y + 0.5 * C_mod * (Delta y)^2

Immunization

• Redington conditions: (1) PV(A) = PV(L), (2) D(A) = D(L), (3) C(A) > 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