# Simulations and Volatility problem

#### JosephK

##### New Member
Hi all

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

#### Link

##### Ninja say what!?!
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.
What you're doing sounds interesting to me. Would you mind elaborating more on what you're trying to do? I might be able to help.

#### JosephK

##### New Member
More detail

Hi Link

Sorry for the delayed response.

To give you a bit more background:

My company insures the "volatility cost" (i.e. the cost of fewer people dying than expected) under a portfolio of annuitants. Typically, we would take this portfolio, run simulations and determine a cost. This is all good and well if you only need to do it once. The problem though is that the portfolios are not closed, and the annuity amounts increase over time (we don't know in advance at what rate). To avoid having to rerun the simulations every time an additional annuitant joins the portfolio, I want to generate factors for:

1. The additional expected payments (in other words the threshold above which the volatility cost kicks in). This part is easy
2. The additional volatility cost added by the life, which will be dependent on age (i.e. survival probability) and annuity size, as well as the existing size of the portfolio and its spread of annuity sizes.