2. Joint Life and Last-Survivor Annuities and Insurances - Continuous

We are now in a position to express the expected present value (EPV) of several life contingent risks using multiple state as well as traditional notation. It is convenient to begin with the continuous case when we know the exact moment of payment.

A joint life insurance is a life insurance which pays a benefit of 1 immediately upon the first to die of $x$ and $y$. It is also known as a \textit{first to die insurance}. In this case, the random present value of future benefits is $v^{T(xy)} = \exp(-\delta T(xy))$ which has expected present value
\begin{eqnarray*}
\bar{A}_{xy} =
\int_0^{\infty}
e^{-\delta t} ~_t p_x ~_t p_x \left(\mu_{x+t} + \mu_{y+t} \right) ~ dt .
\end{eqnarray*}

Similarly, the EPV of last-survivor insurance, with pays a benefit of 1 immediately upon the second death, is
\begin{eqnarray*}
\bar{A}_{\overline{xy}} =
\int_0^{\infty}
e^{-\delta t} \left( ~_t q_y ~_t p_x \mu_{x+t} + ~_t q_x ~_t p_y \mu_{y+t} \right)~ dt .
\end{eqnarray*}

A joint life annuity is an annuity payable until the first death of $x$ and $y$. That is, there is a continuous payment at the rate of 1 per year while both are alive. In this case, the EPV is
\begin{eqnarray*}
\bar{a}_{xy} =
\int_0^{\infty} e^{-\delta t} ~_t p_{xy} ~ dt.
\end{eqnarray*}

The random variable underpinning the joint life annuity is $\bar{a}_{\overline{T(xy)|}} = \frac{1-\exp(-\delta T(xy))}{\delta}$. So, as with single life functions, we have
\begin{eqnarray*}
\text{joint life annuity rv} = \frac{1 - \text{joint life insurance rv}}{\delta}
\end{eqnarray*}
Taking expectations of each side yields
\begin{eqnarray*}
\bar{a}_{xy} = \frac{1 - \bar{A}_{xy}}{\delta},
\end{eqnarray*}
a familiar relationship

A last survivor annuity is an annuity payable where there is a continuous payment at the rate of 1 per year while at least one of $x$ and $y$ is alive. In this case, the EPV is
\begin{eqnarray*}
\bar{a}_{\overline{xy}} = \bar{a}_x + \bar{a}_y - \bar{a}_{xy}
\end{eqnarray*}
and can be related to last-survivor insurance through the expression
\begin{eqnarray*}
\bar{a}_{\overline{xy}} = \frac{1 - \bar{A}_{\overline{xy}} }{\delta}.
\end{eqnarray*}