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