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Interest rate risk - Exam FM formula sheet

Every Exam FM formula in interest rate risk on one printable sheet, condensed from the 5 reference pages that teach them.

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Interest rate risk

Macaulay duration

Definition: $D_{Mac} = \dfrac{\sum_t t\,C_t v^{t}}{\sum_t C_t v^{t}}$
Zero-coupon bond: $D_{Mac} = n$
Level annuity: $D_{Mac} = \dfrac{(Ia)_{\overline{n}|}}{a_{\overline{n}|}}$
Perpetuity: $D_{Mac} = \dfrac{1+i}{i}$

Modified duration

Definition: $D_{Mod} = -\dfrac{P'(i)}{P(i)} = \dfrac{D_{Mac}}{1+i}$
First-order estimate: $P(i+\Delta i) \approx P(i)\left(1 - D_{Mod}\,\Delta i\right)$
Percentage change: $\dfrac{\Delta P}{P} \approx -D_{Mod}\,\Delta i$

Convexity

Modified convexity: $C_{Mod} = \dfrac{P''(i)}{P(i)} = \dfrac{\sum_t t(t+1)C_t v^{t+2}}{P}$
Macaulay convexity: $C_{Mac} = \dfrac{\sum_t t^{2} C_t v^{t}}{P}$
Second-order estimate: $P(i+\Delta i) \approx P\left(1 - D_{Mod}\Delta i + \tfrac{1}{2}C_{Mod}(\Delta i)^{2}\right)$

Immunisation

Redington condition 1: $PV_A = PV_L$
Redington condition 2: $D_A = D_L$ equivalently P'_A = P'_L
Redington condition 3: $C_A > C_L$ assets strictly more convex
Full immunisation: $\text{one asset each side of the liability, PV and duration matched}$

Swaps

Swap rate: $R = \dfrac{1 - P(n)}{\sum_{t=1}^{n} P(t)}, \qquad P(t) = (1+s_t)^{-t}$
Equivalently: $R = \dfrac{\sum_{t} P(t) f_{t-1,t}}{\sum_t P(t)}$ a PV-weighted average of the forward rates
Market value later: $V = (R_{\text{new}} - R_{\text{old}})\sum_{t} P(t)$

Interest rate risk - Exam FM formula sheet. Formulas condensed from the free reference pages on this site. Not affiliated with or endorsed by the Society of Actuaries.