Formula sheets

Annuity-immediate and due $a_{\overline{n}|}=\frac{1-v^n}{i},\qquad \ddot a_{\overline{n}|}=\frac{1-v^n}{d}=(1+i)\,a_{\overline{n}|}$

Nominal to effective $\;\left(1+\tfrac{i^{(m)}}{m}\right)^m=1+i=e^{\delta}=\left(1-\tfrac{d^{(m)}}{m}\right)^{-m}$

Bond price (face $F$, coupon $r$, redemption $C$, yield $i$): $P=Fr\,a_{\overline{n}|i}+Cv^n$

Outstanding balance (prospective) after $k$ payments of $R$: $B_k=R\,a_{\overline{n-k}|}$

Macaulay and modified duration $D_{mac}=\frac{\sum t\,v^t CF_t}{\sum v^t CF_t},\qquad D_{mod}=\frac{D_{mac}}{1+i},\qquad \frac{\Delta P}{P}\approx -D_{mod}\,\Delta i+\tfrac12 C\,(\Delta i)^2$

Redington immunization: $PV_A=PV_L$, $D_A=D_L$, $C_A>C_L$.