## Joint Life expectancy

Using the Gompertz law I use the following excel formula to calculate life expectancy for a single life:

=LN(1-1/(EXP((Ap-Mp)/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-(1-EXP(EXP((Ap-Mp)/SDp)*(1-EXP(JLE/SDp))))))*(1-(1-(1-EXP(EXP((As-Ms)/SDs)*(1-EXP(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.