We now modify our definition of the Cost of Insurance to accommodate Type A policies and the “corridor factor.” To simplify matters, drop settlement expenses and define the unmodified version to be
\n\\begin{eqnarray*}
\nCoI_{k}^f =v_q q_{[x]+k} \\left(FA_{k+1} – ~_{k+1} AV ^f \\right)
\n\\end{eqnarray*}
\nand the associated account value
\n\\begin{eqnarray*}
\n~_{k+1} AV^f = (~_k AV + G_k – e_k -CoI_k ^f )(1+i_k^c)
\n\\end{eqnarray*}
\nRecall that we can think of the corridor factor as \\(\\frac{\\text{AV + ADB}}{\\text{AV}}=\\frac{\\text{ADB}}{\\text{AV}}+1\\). So, now suppose that the corridor factor \\(\\gamma_{k+1}\\) is (exogenously) given. Define a modified cost of insurance
\n\\begin{eqnarray*}
\nCoI_{k}^c &=&v_q q_{[x]+k} (\\gamma_{k+1} -1) \\times ~_{k+1} AV ^c
\n\\end{eqnarray*}
\nand the associated account value
\n\\begin{eqnarray*}
\n~_{k+1} AV^c = (~_k AV + G_k – e_k -CoI_k ^c )(1+i_k^c)
\n\\end{eqnarray*}<\/p>\n
The account value is thus
\n\\begin{eqnarray*}
\n~_{k+1} AV = \\min \\left(~_{k+1} AV^f , ~_{k+1} AV ^c \\right) .
\n\\end{eqnarray*}
\nWith
\n\\begin{eqnarray*}
\n~_{k+1} AV^f &=& (~_k AV + G_k – e_k -CoI_k ^f )(1+i_k^c)\\\\
\n~_{k+1} AV^c &=& (~_k AV + G_k – e_k -CoI_k ^c )(1+i_k^c)
\n\\end{eqnarray*}
\nwe can write this recursively as
\n\\begin{eqnarray*}
\n~_{k+1} AV = (~_k AV + G_k – e_k -CoI_k )(1+i_k^c) ,
\n\\end{eqnarray*}
\nwhere
\n\\begin{eqnarray*}
\nCoI_k = \\max \\left(CoI_k ^f , CoI_k ^c \\right)
\n\\end{eqnarray*}<\/p>\n
To use the recursive formula for the account value, we need to calculate the cost of insurance. To this end, we start with
\n\\begin{eqnarray*}
\nCoI_{k}^f &=&v_q q_{[x]+k} \\left(FA_{k+1} – ~_{k+1} AV ^f \\right) \\\\
\n~_{k+1} AV^f &=& (~_k AV + G_k – e_k -CoI_k ^f )(1+i_k^c)
\n\\end{eqnarray*}
\nWe can write
\n\\begin{eqnarray*}
\nCoI_{k}^f = v_q q_{[x]+k} \\left(FA_{k+1} – (~_k AV + G_k – e_k
\n-CoI_k ^f )(1+i_k^c) \\right)
\n\\end{eqnarray*}
\nso
\n\\begin{eqnarray*}
\nCoI_{k}^f (1-v_q q_{[x]+k}(1+i_k^c)) = v_q q_{[x]+k} \\left(FA_{k+1}
\n– (~_k AV + G_k – e_k )(1+i_k^c) \\right)
\n\\end{eqnarray*}
\nwhich yields
\n\\begin{eqnarray*}
\nCoI_{k}^f &=&\\frac{v_q q_{[x]+k} \\left(FA_{k+1} – (~_k AV + G_k –
\ne_k )(1+i_k^c) \\right)}{1-v_q q_{[x]+k}(1+i_k^c)}
\n\\end{eqnarray*}<\/p>\n
Similarly, using
\n\\begin{eqnarray*}
\nCoI_{k}^c &=& v_q q_{[x]+k} (\\gamma_{k+1} -1) \\times ~_{k+1} AV ^c \\\\
\n~_{k+1} AV^c &=& (~_k AV + G_k – e_k -CoI_k ^c )(1+i_k^c)
\n\\end{eqnarray*}
\nWe can write
\n\\begin{eqnarray*}
\nCoI_{k}^c &=& v_q q_{[x]+k} (\\gamma_{k+1} -1) (~_k AV + G_k – e_k
\n-CoI_k ^c )(1+i_k^c)
\n\\end{eqnarray*}
\nso
\n\\begin{eqnarray*}
\n&&CoI_{k}^c (1+v_q q_{[x]+k} (\\gamma_{k+1} -1)(1+i_k^c)) \\\\
\n&=& v_q q_{[x]+k} (\\gamma_{k+1} -1) (~_k AV + G_k – e_k )(1+i_k^c)
\n\\end{eqnarray*}
\nwhich yields
\n\\begin{eqnarray*}
\nCoI_{k}^c &=&\\frac{v_q q_{[x]+k} (\\gamma_{k+1} -1) (~_k AV + G_k –
\ne_k )(1+i_k^c)}{1+v_q q_{[x]+k}(\\gamma_{k+1} -1)(1+i_k^c)}
\n\\end{eqnarray*} <\/p>\n