General Discrete Policy. It is important to monitor the actual experience of a block of business. To this end, we use carats, or “hats”, to denote actual values. We use
* (hat{i}_k) is the actual rate of return from invested assets,
* (hat{e}_k) is the actual annual expenses per contract,
* at death, (hat{q}_{[x]+k}^{(d)}) is the realized fraction of deaths,
* at withdrawal, (hat{q}_{[x]+k}^{(w)}) is the realized fraction of withdrawals, and
* at survival to the end of the year, (hat{p}_{[x]+k}^{(tau)} = 1 – (hat{q}_{[x]+k}^{(d)}+hat{q}_{[x]+k}^{(w)})) is realized fraction that survived.
With these quantities, the profit during the year (at time (k+1)) is
begin{eqnarray*}
Profit_{k+1} &= &
left( _k AS + G_k -hat{e}_kright) (1+ hat{i}_k) – hat{q}_{[x]+k}^{(d)} left(b_{k+1} + E_{k+1}right) \
&~~~~~~~~-& hat{q}_{[x]+k}^{(w)} ~_{k+1} CV – hat{p}_{[x]+k}^{(tau)} ~_{k+1} AS \
&= &left( _k AS + G_k -hat{e}_kright) (1+ hat{i}_k) – hat{q}_{[x]+k}^{(d)} left(b_{k+1} + E_{k+1} – ~_{k+1} ASright) \
&~~~~~~~~-& hat{q}_{[x]+k}^{(w)} (~_{k+1} CV -~_{k+1} AS) – ~_{k+1} AS
end{eqnarray*}
For this illustration, we have that assumed premiums (G_k) and settlement expenses (E_k) are the same as actual values. We also assume that ( _k AS) represents the terminal (year-end) company obligation (that does not depend on experience). Recall the recursive asset share calculation
begin{eqnarray*}
(~_k AS + G_k – e_k)(1+i_k) &=& q_{[x]+k}^{(d)} left(b_{k+1} + E_{k+1} – ~_{k+1} AS right) \
&~~~~~~~~+& q_{[x]+k}^{(w)} ( _{k+1} CV – ~_{k+1} AS) + ~_{k+1} AS
end{eqnarray*}
Substituting for (_{k+1} AS), we may write the profit during the year is
begin{eqnarray*}
Profit_{k+1} &= &
left( _k AS + G_k -hat{e}_kright) (1+ hat{i}_k) –
hat{q}_{[x]+k}^{(d)} left(b_{k+1} + E_{k+1} – ~_{k+1} ASright) \
&~~~~~~~~-& hat{q}_{[x]+k}^{(w)} (~_{k+1} CV -~_{k+1} AS) – ~_{k+1} AS \
&= & left( _k AS + G_k right) (hat{i}_k – i_k) \
&~~~~~~~~+& e_k (1+ hat{i}_k) – hat{e}_k (1+ i_k) \
&~~~~~~~~+& left(b_{k+1} + E_{k+1} – ~_{k+1} ASright) (q_{[x]+k}^{(d)} – hat{q}_{[x]+k}^{(d)} ) \
&~~~~~~~~+& left( _{k+1} CV – ~_{k+1} ASright) (q_{[x]+k}^{(w)} – hat{q}_{[x]+k}^{(w)} ) .
end{eqnarray*}
This is one way to decompose the profit into identifiable components, known as the analysis of surplus. Here, we interpret these portions of the profit:
* (left( _k AS + G_k right) (hat{i}_k – i_k)) – due to favorable investment experience,
* (e_k (1+ hat{i}_k) – hat{e}_k (1+ i_k)) – due to favorable expense experience,
* (left(b_{k+1} + E_{k+1} – ~_{k+1} ASright) (q_{[x]+k}^{(d)} – hat{q}_{[x]+k}^{(d)} )) – due to favorable mortality experience, and
* (left( _{k+1} CV – ~_{k+1} ASright) (q_{[x]+k}^{(w)} – hat{q}_{[x]+k}^{(w)} )) – due to favorable withdrawal experience.