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
begin{eqnarray*}
A_{xy} = sum_{k=0}^{infty} v^{k+1} ~_k p_{xy} q_{x+k:y+k}
end{eqnarray*}
where (_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
begin{eqnarray*}
A_{xy}^{IND} = sum_{k=0}^{infty} v^{k+1} ~_k p_x ~_k p_y (1- p_{x+k} times p_{y+k}).
end{eqnarray*}
From 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
begin{eqnarray*}
q_x^{revised} = (1-c) times q_x^{base}.
end{eqnarray*}
This 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.)
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.