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.

5 topics, 17 formula lines. Every line is condensed from the Exam FM formula pages, which stay free to read in full. Print this page to get the sheet on paper; the site chrome drops away.

Interest rate risk

Macaulay duration

Definition: $D_{M a c} = \frac{\sum _{t} t C _{t} v ^{t}}{\sum _{t} C _{t} v ^{t}}$
Zero-coupon bond: $D_{M a c} = n$
Level annuity: $D_{M a c} = \frac{( I a ) _{\overline{n} ∣}}{a _{\overline{n} ∣}}$
Perpetuity: $D_{M a c} = \frac{1 + i}{i}$

Modified duration

Definition: $D_{M o d} = - \frac{P ^{'} ( i )}{P ( i )} = \frac{D _{M a c}}{1 + i}$
First-order estimate: $P (i + Δ i) \approx P (i) (1 - D_{M o d} Δ i)$
Percentage change: $\frac{Δ P}{P} \approx - D_{M o d} Δ i$

Convexity

Modified convexity: $C_{M o d} = \frac{P ^{''} ( i )}{P ( i )} = \frac{\sum _{t} t ( t + 1 ) C _{t} v ^{t + 2}}{P}$
Macaulay convexity: $C_{M a c} = \frac{\sum _{t} t ^{2} C _{t} v ^{t}}{P}$
Second-order estimate: $P (i + Δ i) \approx P (1 - D_{M o d} Δ i + \frac{1}{2} C_{M o d} (Δ i)^{2})$

Immunisation

Redington condition 1: $P V_{A} = P V_{L}$
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: $one asset each side of the liability, PV and duration matched$

Swaps

Swap rate: $R = \frac{1 - P ( n )}{\sum _{t = 1}^{n} P ( t )}, P (t) = (1 + s_{t})^{- t}$
Equivalently: $R = \frac{\sum _{t} P ( t ) f _{t - 1, t}}{\sum _{t} P ( t )}$ a PV-weighted average of the forward rates
Market value later: $V = (R_{new} - R_{old}) \sum_{t} P (t)$

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.

Interest rate risk - Exam FM formula sheet

What is on this sheet

Interest rate risk

Interest rate risk

Macaulay duration

Modified duration

Convexity

Immunisation

Swaps