Consider a fully continuous life insurance for (x) and (y). Suppose that
1) premiums are payable until the first death
2) a benefit of 1 is payable at the second death.
The joint mortality for (T(x)) and (T(y)) is not independent but given by
begin{eqnarray*}
~_t p_{xy} = r ~_t p_x + (1-r) ~_t p_x ~_t p_y.
end{eqnarray*}
For simplicity, assume that the mortality for (T(x)) and (T(y)) follow a common exponential distribution,
begin{eqnarray*}
~_t p_x = e^{-mu t} = ~_t p_y .
end{eqnarray*}
Show that the annual premium is (frac{mu(2mu+delta r)}{delta + (1+r)mu}.)
– Soln
+ Soln
Solution. The premium (P) is the solution of (bar{A}_{overline{xy}} = P bar{a}_{xy}). Now
begin{eqnarray*}
bar{a}_{xy} &=& int_0^{infty} e^{-delta t} ~_t p_{xy} ~ dt \
&=& int_0^{infty} e^{-delta t} left( r ~_t p_x + (1-r) ~_t p_x ~_t p_y right) ~ dt \
&=& int_0^{infty} e^{-delta t} left( r e^{-mu t} + (1-r) e^{-2mu t} right) ~ dt \
&=& frac{r}{mu+delta} + frac{1-r}{2mu+delta} = frac{delta+(1+r)mu}{(mu+delta)(2mu+delta)}
end{eqnarray*}
Further
begin{eqnarray*}
bar{A}_x &=& int_0^{infty} e^{-delta t} ~_t p_{xy} mu_{x+t}~ dt \
&=& mu int_0^{infty} e^{-delta t} e^{-mu t} ~ dt = frac{mu}{mu+delta} = bar{A}_y
end{eqnarray*}and
begin{eqnarray*}
bar{A}_{overline{xy}} = bar{A}_x + bar{A}_y – bar{A}_{xy} =2 frac{mu}{mu+delta} –
left( 1 – delta bar{a}_{xy} right) .
end{eqnarray*}
Thus
begin{eqnarray*}
P &=& frac{bar{A}_{overline{xy}}}{bar{a}_{xy}} = frac{2 frac{mu}{mu+delta} -1}{bar{a}_{xy}} + delta \
&=& frac{mu(2mu+delta r)}{delta + (1+r)mu}.
end{eqnarray*}
begin{eqnarray*}
bar{a}_{xy} &=& int_0^{infty} e^{-delta t} ~_t p_{xy} ~ dt \
&=& int_0^{infty} e^{-delta t} left( r ~_t p_x + (1-r) ~_t p_x ~_t p_y right) ~ dt \
&=& int_0^{infty} e^{-delta t} left( r e^{-mu t} + (1-r) e^{-2mu t} right) ~ dt \
&=& frac{r}{mu+delta} + frac{1-r}{2mu+delta} = frac{delta+(1+r)mu}{(mu+delta)(2mu+delta)}
end{eqnarray*}
Further
begin{eqnarray*}
bar{A}_x &=& int_0^{infty} e^{-delta t} ~_t p_{xy} mu_{x+t}~ dt \
&=& mu int_0^{infty} e^{-delta t} e^{-mu t} ~ dt = frac{mu}{mu+delta} = bar{A}_y
end{eqnarray*}and
begin{eqnarray*}
bar{A}_{overline{xy}} = bar{A}_x + bar{A}_y – bar{A}_{xy} =2 frac{mu}{mu+delta} –
left( 1 – delta bar{a}_{xy} right) .
end{eqnarray*}
Thus
begin{eqnarray*}
P &=& frac{bar{A}_{overline{xy}}}{bar{a}_{xy}} = frac{2 frac{mu}{mu+delta} -1}{bar{a}_{xy}} + delta \
&=& frac{mu(2mu+delta r)}{delta + (1+r)mu}.
end{eqnarray*}