The traditional “first to die” insurance pays 1 at the end of the year of death of either \\(x\\) or \\(y\\). The expected present value of this insurance is
\n\\begin{eqnarray*}
\nA_{xy} = \\sum_{k=0}^{\\infty} v^{k+1} ~_k p_{xy} q_{x+k:y+k}
\n\\end{eqnarray*}
\nwhere \\(_k p_{xy}\\) is the probability that both \\(x\\) and \\(y\\) survive \\(k\\) years and \\(q_{x+k:y+k}\\) is the probability that at least one of \\(x+k\\) and \\(y+k\\) dies within the year. To evaluate this, it is customary to begin by assuming independent lives, so that
\n\\begin{eqnarray*}
\nA_{xy}^{IND} = \\sum_{k=0}^{\\infty} v^{k+1} ~_k p_x ~_k p_y (1- p_{x+k} \\times p_{y+k}).
\n\\end{eqnarray*}
\nFrom the formula for \\(A_{xy}^{IND}\\), it is easy to see that as interest increases, the expected present value decreases. What about mortality? Let us think of \\(x\\) as a male life and \\(y\\) as a female life. Consider a mortality adjustment to male lives of the form
\n\\begin{eqnarray*}
\nq_x^{revised} = (1-c) \\times q_x^{base}.
\n\\end{eqnarray*}
\nThis is a coarse adjustment but will give us a flavor as to what happens to insurance prices as mortality increases or decreases. Viewers should verify that \\(A_{xy}^{IND}\\) increases as the adjustment coefficient \\(c\\) decreases. (Although a separate exercise, at \\(c \\approx 0.35\\), male mortality is roughly equivalent to female mortality for our Indonesian data.)<\/p>\n
The dynamic graph shows values of \\(A_{xy}^{IND}\\) plotted against interest rate \\(i\\) and age, where for plotting purposes we use a common age \\(x=y\\). The graph shows the dynamic effect over \\(c\\) ranging from -0.5 to 0.5. This graph shows that the expected present value \\(A_{xy}^{IND}\\) decreases as either \\(i\\) or \\(c\\) increases. Moreover, it also gives a feel for the amount of the increase. Based on the magnitude of these increases, one interpretation is that changes in interest rates have a greater effect than changes in mortality on the expected present value.<\/p>\n