
Joint Life expectancy
Using the Gompertz law I use the following excel formula to calculate life expectancy for a single life:
=LN(11/(EXP((ApMp)/SDp)/LN(0.5)))*SDp
Where:
Ap = Age principal life Assume 65
Mp = Mean principal – Assume 82.1
SDp = Standard deviation principal – Assume 11.4
The result is 16.1 years
If I assume the following for a 2nd life (Spouse)
As = Age spouse life Assume 60
Ms = Mean spouse – Assume 86.3
SDs = Standard deviation spouse – Assume 9.5
The result is 23.7 years
Using Newton’s method I can approximate the joint life expectancy for the couple using the following formula for the joint life probability and solving for JLE (joint life expectancy) where the joint probability = 0.50.
=(1(1(1(1EXP(EXP((ApMp)/SDp)*(1EXP(JLE/SDp))))))*(1(1(1EXP(EXP((AsMs)/SDs)*(1EXP(JLE/SDs)))))))
Result 25.8 years
I would like to know if there is a closed form method to calculate joint life expectancy using the Gompertz law, or a simpler way to approximate it. I would appreciate the answer in Excel notation.