For a joint life annuity on lives \\(x\\) and \\(y\\) with payments of 1 per year in advance, the EPV can be written recursively as:
\n\\begin{eqnarray*}
\n\\ddot{a}_{xy} = 1 + v p_{xy} ~\\ddot{a}_{x+1:y+1}.
\n\\end{eqnarray*}<\/p>\n
The EPV is
\n\\begin{eqnarray*}
\n\\ddot{a}_{xy} = \\sum_{k=0}^{\\infty} v^k ~_k p_{xy} = \\ddot{a}_{xy}^{00}.
\n\\end{eqnarray*}
\nBy conditioning (or using the Chapman-Kolmogorov equations), we may write
\n\\begin{eqnarray*}
\n~_{k+1} p_{xy} = ~_{k+1} p_{xy}^{00} = p_{xy}^{00} ~_k p_{x+1:y+1}^{00} = p_{xy} ~_k p_{x+1:y+1} .
\n\\end{eqnarray*}
\nThus, using \\(k=j+1\\), we have
\n\\begin{eqnarray*}
\n\\ddot{a}_{xy} &=& 1 + \\sum_{k=1}^{\\infty} v^k ~_k p_{xy} = 1 + \\sum_{j=0}^{\\infty} v^{j+1} ~_{j+1} p_{xy} \\\\
\n&=& 1 + v p_{xy} \\sum_{j=0}^{\\infty} v^j ~_j p_{x+1:y+1} =1 + v p_{xy} \\ddot{a}_{x+1:y+1}.
\n\\end{eqnarray*}\n<\/p><\/div>\n<\/div>\n
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