Multiple Life Models: Joint Life and Last Survivor

Joint Life Status

The joint life status (xy) fails at the first death. The survival function is t_p_{xy} = t_p_x * t_p_y (assuming independence). The force of mortality is mu_{xy} = mu_x + mu_y. Insurance and annuity values use these survival probabilities: A_{xy} is whole life insurance on joint life status, and a-ddot_{xy} is the annuity payable while both survive. The relationship A_{xy} = 1 minus d * a-ddot_{xy} still holds. Joint life annuities are used in pension benefits that cease at the first death of a couple.

Last Survivor Status

The last survivor status fails at the second death. Using the identity T_{last survivor} = T_x + T_y minus T_{joint life}, all last survivor values derive from single and joint life values: a-ddot_{last survivor} = a-ddot_x + a-ddot_y minus a-ddot_{xy}. Similarly, A_{last survivor} = A_x + A_y minus A_{xy}. Contingent insurance A_{xy}^1 pays on the death of (x) only if (x) dies before (y). The relationship A_{xy} = A_{xy}^1 + A_{xy}^{-1} splits joint life insurance by order of death. Exam LTAM tests these identities extensively.