# s2 and percentiles

Discussion in 'SA2' started by i-actuary, Nov 21, 2019.

1. ### i-actuaryMember

Hi,

got confused with some things.
1. the shocks at each risk are specified so to give the 99.5 percentile :
a. each subrisk or
b. in total scr ?

2. regarding the risk margin given non linearity and non separability of ind risks ends up to a percentile lower than 99.5? and what about the non headgeable scr included in risk margin calc?
thank you

2. ### Em FrancisActEd TutorStaff Member

a) The SCR is calibrated using the VaR of the basic own funds subject to a confidence level of 99.5 % over a one-year period. This calibration objective is applied to each individual risk module and sub-module. The total SCR is then calculated by combining using a correlation matrix/copula.

In theory, both hedgeable and non-hedgeable risks could be impacted, and therefore resulting in the RM being impacted.
Hope this helps.
Thanks
Em

3. ### i-actuaryMember

thank you for this.
what o dont get is:
lets suppose we have only risks.
1. the 2 shocks we apply take us to the 99.5% for each risk ?
2. if yes then by correlating the two risks we end up at 99.5 again for total scr?
or is 1 totally wrong ?

thanks

4. ### mugonoTon up Member

The aggregated SCR under 1 before diversification would give a result that is stronger than a 1 in 200 year VaR. This is because it’s unlikely that such a severe stress would occur simultaneously. 2 accounts for this; ie it ensures that the total SCR reflects the 1 in 200 level after allowing for the correlations between multiple risks.

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5. ### i-actuaryMember

thanks for this.
so based on what you are saying (and i agree) i have the following two doubts (which is what triggered my question).
1. the non market scr (the one that we use for risk margin) is at which percentile ?
2. the non mkt src for life or health at which?
3. and 3rd the rm ends up at some percentile or we dont know?

thank you

6. ### mugonoTon up Member

1. The pre-diversification non hedgeable risk stresses would be at the 99.5th confidence level. The non hedgeable risks SCR would allow for diversification. The actual percentiles (ie the ‘biting scenario’) that underlies the non hedgeable risk SCR that feeds into the risk margin would likely vary by risk. I explained this in my previous response.

2. It’s not clear what you’re asking here.

3. The risk margin is calculated as the discounted value of the sum of the non hedgeable risk SCR (at each time t) multiplied by a cost of capital rate. The percentile that underlies this SCR at each time t was explained under 1.

Last edited: Nov 28, 2019
7. ### i-actuaryMember

Hi many thanks for this.

I agree on this and I was under the same impression. However under a previous answer earlier in this post you mentioned that the pre-diversification shocks are more severe and the total scr after diversification ends up to 99.5%. So I guess I misunderstood.

What I am wondering is the following: based on theory of s2 the total scr ends up to be at 99.5% for 1 year period after all the calibrations. What I am wondering is: the non-market scr (or even the Life/health scrs in isolation) are at the 99.5% percentile too?

The risk margin as an extra capital charge corresponds to some percentile ? for example if the non-market scrs (let's exclude the op risk) correspond to 99.5% for 1 year period each, the risk margin is for example the n-year 99.5% (or something like this)?

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8. ### mugonoTon up Member

1. Each stress is calibrated (in isolation) at a 99.5% CI. If you simply add up the independently calibrated 99.5th stresses you'd get a materially stronger aggregate SCR than the 99.5th CI required.

2. The stresses are calculated at the 99.5th CI. However, after allowing for diversification the 'actual' CI for each risk that feeds into the reported SCR number will be different. For example, the implied 'biting scenario' for Risk 1 could be the 75th percentile, the implied 'biting scenario' for Risk 2 could be the 95th percentile etc (I hope you get the idea).

3. I think you're abusing the risk margin concept here . The RM = discounted value of sum of future SCR at each time t multiplied by a cost of capital rate. Each future SCR for each time t will likely have a different implied 'biting scenario' consistent with the discussion we've been having. The idea of aggregating the implied biting scenario somehow in order to obtain a "n-year implied biting scenario" isn't done and if it were, would feel a little meaningness. Others may have different views.

Last edited: Nov 28, 2019
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9. ### i-actuaryMember

Hi,thank you for your replies.
Totally agree with your points.

regarding point 2: so if we had a life company the non market scr we don't know to which percentile corresponds?
regarding point 3: The main reason for the RM abuse is that I would like to find a connection between the RM (1 year horizon) and RA (leveraging s2 RM) over the entire life time of the product.
and yes I totally agree with the biting scenario thing but for simplification reasons lets assume nothing changes

Thank you

10. ### MateuszMember

In the Standard Formula you need to calculate the 0.995-th quantile for each risk/sub-risk, and aggregate them with correlation matrix to get the total SCR at the 99.5% CI. But the reason you use 0.995-th quantiles from the distributions of standalone risks is because they show up in the formula, not because your ‘biting’ scenario is such that each risk is at exactly the 99.5-th CI. This is not true for reasons mentioned by mugono – you’d arrive at the total SCR being too high.

The correlation-based formula in the SF can be derived assuming the RV [L(1), ..., L(n)] - where L(i) are individual losses from risks i=1,2,...,n - has a multivariate normal distribution (not saying this is an explicit assumption of the SF, but practically ends up being a similar thing). Now, if you know how to calculate x such that P(L(1)+...+L(n)<=x) = 0.995 (which you do under the above set-up), how much can you tell about the individual losses x(i) in the 'biting' scenario, i.e. where the sum x(1) + … + x(n) adds up to x? I'd say not much; there's no easy answer here. In practice there are techniques to allocate the diversified capital to individual risks, and this is a topic of interest under both the SF and internal model, but not a trivial thing from a theoretical standpoint.

Last edited: Nov 28, 2019
11. ### mugonoTon up Member

Ok, it suddenly dawned on me that I’m in the weeds .

In answer to your question: I’m not entirely convinced that there is a “1 year RM horizon” unless the contract has a one year maturity; and so I think you are trying to connect apples and oranges.

Conceptually, the risk margin is an amount that you need to add to the BEL to get to the market consistent value of the liability; that a knowledgeable participate would pay for it. Technically, it isn’t a capital requirement (though I have heard actuaries describe it as such); you don’t need to arbitrarily set a confidence level or time horizon to measure it.

The cost of capital approach, is ONE (but not the only) approach for how you might go about calculating the RM. The discussion to date has focussed on the implications of one proposed approach and I think has lost / is losing sight on what the RM is and is trying to achieve.

Contracts that could be valued by replication would be regarded as the market consistent liability value and wouldn’t have a RM for eg.

Last edited: Nov 28, 2019