Formula sheets

Kolmogorov forward $\;\dfrac{d}{dt}{}_tp_x^{ij}=\sum_{k\ne j}\left({}_tp_x^{ik}\mu^{kj}_{x+t}-{}_tp_x^{ij}\mu^{jk}_{x+t}\right)$

Thiele's differential equation $\;\dfrac{d}{dt}{}_tV^{(i)}=\delta_t\,{}_tV^{(i)}+P^{(i)}_t-\sum_{j\ne i}\mu^{ij}_{x+t}\left(b^{(ij)}_t+{}_tV^{(j)}-{}_tV^{(i)}\right)$

Projected unit credit accrued liability $\;AL_t=\text{accrued benefit at projected final salary}\times{}_{r-x}p_x\,v^{r-x}\,\ddot a_r$

Profit signature $\;\Pi_t={}_{t-1}p_x\cdot Pr_t$, NPV $=\sum_t\Pi_t v_r^t$ at hurdle rate $r$

Lee–Carter $\;\ln m_{x,t}=\alpha_x+\beta_x\kappa_t+\varepsilon_{x,t}$