I have an interesting statistical problem that I am trying to wrap my head around. I'll explain by way of any example:

Let say I have group of 3 people currently receiving annuity payments, for one year only (to simply) as follows:

Age Annuity Probability of survival Expected payment

67 $1000 0.9 900

72 $800 0.8 640

74 $1200 0.75 900

(E[payment] = prob survival * annuity amount)

So the total expected payment is 900 + 640 + 900 = 2 440

Ok, so now I generate a few thousand mote-carlo simulations. In some simulations, people will die and the total amount will be less than 2 440. In other simulations no one will die and the total payment will be 3 000.

I am not interested in cases where the payments are below expected, only where above.

There will be a lot of volatility here since there are only 3 people. (It takes only 1 person dying to ensure the actual payment is below expected)

(bear with me, I'm getting to what I want to ask)

So, lets say I added an additional 100 people to the group. This will reduces the volatility significantly, meaning the relative cost of total payments exceeding the expected amount will come done, and it will keep coming down until we have infinitely many people in the group, at which time the volatility should be zero.

So firstly, more people = lower volatility

Now, lets also look at the spread of annuity payments. If you start with one person who earns $1000 and you add another person who also earns $1000 the volatility drops a lot. If however, you were to add someone who earns $1 instead the volatility drops very little even though you've doubled the number of lives.

So volatility cost = some function of annuity size and number of people. (it will also depend on the survival probability, but I don't think that needs to be allowed for here)

What I am looking to achieve here is to determine aged-based factors for the cost of the volatility. To be more specific, in stead for taking an actual group of annuitants and running monte-carlo simulations on them to determine the cost, what I want to do is to pre-generate the volatility cost that would be added by a adding a specific life, with a specific probability of dying to a group of a certain size with a certain spread of annuity amounts. The sum of these individual volatility costs would give the total volatility cost.

I've attached a spreadsheet showing the volatility cost for different group sizes as generated assuming that the spread of of annuity payments is uniform.

How do I deal with the effect of variably distributed annuity payments?

Thanks