We can now apply these same basic principles to account value formulas.
Use (AV) for account value instead of AS, use (FA) for face amount instead of benefit, and ignore lapsation. From the asset share formula,
begin{eqnarray*}
(~_k AV + G_k – e_k)(1+i_k^c) &= &
q_{[x]+k} left(FA_{k+1} + E_{k+1}right) + p_{[x]+k} ~_{k+1} AV \
&=& q_{[x]+k} left(FA_{k+1} + E_{k+1} – ~_{k+1} AV right) +~_{k+1}
AV
end{eqnarray*}
With only one decrement (death), we have dropped the cause notation. Further, use (i_k^c) for the interest credited. Now, we will define the Cost of Insurance to be
begin{eqnarray*}
CoI_{k} =v_q q_{[x]+k} left(FA_{k+1} + E_{k+1} – ~_{k+1} AV
right)
end{eqnarray*}
where (v_q) is a discount factor. In the prior example, we used 5% for both the interest credited and the CoI discount factor.
With discount factor (v_q), the Cost of Insurance is
begin{eqnarray*}CoI_{k} =v_q q_{[x]+k} left(FA_{k+1} + E_{k+1} – ~_{k+1} AV
right) .
end{eqnarray*}
With this, we define the account value
begin{eqnarray*}
~_{k+1} AV = (~_k AV + G_k – e_k -CoI_k )(1+i_k^c)
end{eqnarray*}
For Type B UL, the additional death benefit is
begin{eqnarray*} ADB_{k+1} = FA_{k+1} – ~_{k+1} AV = text{constant}
end{eqnarray*}
so the recursion as presented is easy to calculate. Not so for Type A.
We can use account values (with a possibly different interest rate) to determine the profit during the year (at time (k+1))
begin{eqnarray*}
Pr_{k+1} &= & left( _k AV + G_k -e_kright) (1+ i_k) –
q_{[x]+k}^{(d)} left(FA_{k+1} + E_{k+1}right) \
&~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV – p_{[x]+k}^{(tau)} ~_{k+1} AV .
end{eqnarray*}