Hi, I am solving q23.4 of chapter 23. Not able to understand the steps of the solution given after drawing the life tables. Could anyone please help me in understanding them logically?
Hi Siddharth P0 is the standard premium paid by everyone at the start of the contract. It is equal to the value of the benefits. Policyholders could die in the first year with probability q_50. Or they could die in the second year, which means they must survive the first year, ie p_50 * q_51. P1 is the premium paid by those that take up the option. This is paid at time 1, so the probability of claiming is just q_51. All of the above calculations are calculated using the standard premium basis, ie normal mortality, as this is the deal with policies with options. However the premiums calculated above will not be enough because in reality there will be anti-selection amongst the option takers. We calculate the additional premium required to offer the mortality option by calculating the actual cost of the benefits less the value of the premiums we've calculated above. To calculate the actual benefits we still use q_50 for the first year as everyone is still assumed to have normal mortality. However for the second year we split the policyholders in two: 70% do not take up the option and claim with standard mortality q_51 (0.7 * p_50 * q_51 = 0.7 * 0.997492 * 0.002809 = 0.00196137) and 30% who do take the option and claim with heavier mortality q' (0.3 * p_50 * q'_51 = 0.3 * 0.997492 * 0.005618 = 0.0016812). Hopefully the calculations make a bit more sense now that I've added in the extra working for the value of the benefits. Best wishes Mark
