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
begin{eqnarray*}
CoI_{k}^f =v_q q_{[x]+k} left(FA_{k+1} – ~_{k+1} AV ^f right)
end{eqnarray*}
and the associated account value
begin{eqnarray*}
~_{k+1} AV^f = (~_k AV + G_k – e_k -CoI_k ^f )(1+i_k^c)
end{eqnarray*}
Recall 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
begin{eqnarray*}
CoI_{k}^c &=&v_q q_{[x]+k} (gamma_{k+1} -1) times ~_{k+1} AV ^c
end{eqnarray*}
and the associated account value
begin{eqnarray*}
~_{k+1} AV^c = (~_k AV + G_k – e_k -CoI_k ^c )(1+i_k^c)
end{eqnarray*}
The account value is thus
begin{eqnarray*}
~_{k+1} AV = min left(~_{k+1} AV^f , ~_{k+1} AV ^c right) .
end{eqnarray*}
With
begin{eqnarray*}
~_{k+1} AV^f &=& (~_k AV + G_k – e_k -CoI_k ^f )(1+i_k^c)\
~_{k+1} AV^c &=& (~_k AV + G_k – e_k -CoI_k ^c )(1+i_k^c)
end{eqnarray*}
we can write this recursively as
begin{eqnarray*}
~_{k+1} AV = (~_k AV + G_k – e_k -CoI_k )(1+i_k^c) ,
end{eqnarray*}
where
begin{eqnarray*}
CoI_k = max left(CoI_k ^f , CoI_k ^c right)
end{eqnarray*}
To use the recursive formula for the account value, we need to calculate the cost of insurance. To this end, we start with
begin{eqnarray*}
CoI_{k}^f &=&v_q q_{[x]+k} left(FA_{k+1} – ~_{k+1} AV ^f right) \
~_{k+1} AV^f &=& (~_k AV + G_k – e_k -CoI_k ^f )(1+i_k^c)
end{eqnarray*}
We can write
begin{eqnarray*}
CoI_{k}^f = v_q q_{[x]+k} left(FA_{k+1} – (~_k AV + G_k – e_k
-CoI_k ^f )(1+i_k^c) right)
end{eqnarray*}
so
begin{eqnarray*}
CoI_{k}^f (1-v_q q_{[x]+k}(1+i_k^c)) = v_q q_{[x]+k} left(FA_{k+1}
– (~_k AV + G_k – e_k )(1+i_k^c) right)
end{eqnarray*}
which yields
begin{eqnarray*}
CoI_{k}^f &=&frac{v_q q_{[x]+k} left(FA_{k+1} – (~_k AV + G_k –
e_k )(1+i_k^c) right)}{1-v_q q_{[x]+k}(1+i_k^c)}
end{eqnarray*}
Similarly, using
begin{eqnarray*}
CoI_{k}^c &=& v_q q_{[x]+k} (gamma_{k+1} -1) times ~_{k+1} AV ^c \
~_{k+1} AV^c &=& (~_k AV + G_k – e_k -CoI_k ^c )(1+i_k^c)
end{eqnarray*}
We can write
begin{eqnarray*}
CoI_{k}^c &=& v_q q_{[x]+k} (gamma_{k+1} -1) (~_k AV + G_k – e_k
-CoI_k ^c )(1+i_k^c)
end{eqnarray*}
so
begin{eqnarray*}
&&CoI_{k}^c (1+v_q q_{[x]+k} (gamma_{k+1} -1)(1+i_k^c)) \
&=& v_q q_{[x]+k} (gamma_{k+1} -1) (~_k AV + G_k – e_k )(1+i_k^c)
end{eqnarray*}
which yields
begin{eqnarray*}
CoI_{k}^c &=&frac{v_q q_{[x]+k} (gamma_{k+1} -1) (~_k AV + G_k –
e_k )(1+i_k^c)}{1+v_q q_{[x]+k}(gamma_{k+1} -1)(1+i_k^c)}
end{eqnarray*}