Monte Carlo Sims - Strange life insurance policy

[IniquiTrance]

Homework Statement

5 people of ages (30M, 32M, 37F, 40M, 42F) enter into a life insurance contract where the one that outlives the others, receives a one time benefit of $1M, at the time of the penultimate death. What are the fair premiums (different), for each policy member?

Homework Equations

The Attempt at a Solution

In each iteration, I simulate the expected lifespans of each member, and focus on the one with the longest expected lifespan. I then discount the benefit back from the time of the penultimate death to each payment period, and make only the last to die, pay premiums.

After several million simulations, everyone has had their probabilistic opportunities to be the last to die, and I end up with respective premiums ($4458, $4705, $4679, $5180, $5088) +- $10.

Does this seem reasonable?

I think what this is essentially calculating is, say with person A:

$$\textbf{E}[\text{Person A's premium | A outlives everyone}]\bullet\textbf{P}[\text{A outlives everyone}]$$

Is this what I'm looking for?

I can't help questioning my assumption during each iteration, that only the longest living was paying the premiums. In the end, everyone is paying a premium.

This leads me to wonder whether it was necessary to use a copula function to create some kind of multivariate distribution...

Do I need to multiply these values by the probability of each outliving the others (I can keep track of how often each is in the lead)?

Thanks!

 

[mighty2000]

Thank you for your interesting question. Your approach to calculating the fair premiums for each policy member seems reasonable. By simulating the expected lifespans and discounting the benefit back to each payment period, you are taking into account the uncertainty of each individual's lifespan and the time value of money.

Regarding your question about using a copula function, it may not be necessary in this case as the ages and genders of the policy members are already taken into account in the simulation. However, if you would like to further refine your analysis and incorporate additional factors, using a copula function could be beneficial.

In addition, to ensure that the premiums are fair for each policy member, it would be helpful to also consider the probabilities of each member outliving the others. This can be done by multiplying the premiums by the respective probabilities of each member being the last to die. This will ensure that each member is paying a fair premium based on their individual risk.

Overall, your approach and calculations seem reasonable and thorough. Keep in mind that there may be other factors to consider in a real-life scenario, such as medical histories and lifestyle habits, which could impact the fair premiums for each member. Thank you for your contribution to this discussion.