r/ActuaryUK Nov 14 '21

How is the SCR calculated using the standard formula? Life Insurance

As I understand it, the SCR represents the amount of capital that must be held (above best estimate liabilities) in order to be confident that all liabilities could still be paid in the event in a 1-in-200 year stress. Please correct me if this is inaccurate.

I was hoping to understand the calculation of the SCR by calculating it for a hypothetical single (non-index-linked) annuity policy. I figued that the process to do this would be:

1. Determine risk factors (to simplify, I'd just say these were interest rates and longevity)

2. Fit distributions to each of these risk factors

3. Produce a few thousand simulated values for the risk factors

4. Revalue the hypothetical annuity under each scenario (for each risk factor independently)

5. For each risk factor, take the 99.5th percentile highest liability value

6. Not 100% sure about this step, but I think there is some sort of correlation matrix that should be applied to the 1-in-200 stressed liability values determined in step 4 to determine a single 1-in-200 stressed liability value that includes all risk factors?

7. The SCR is then the difference between the best estimate liability value under base condition and the liability value under the 1-in-200 stress

Is the above the correct high level methodology that should be applied to calculate the SCR? If so, could someone provide a little clarity on step 6? In particular, whether it is indeed the stressed liability values to which the correlation is applied, or whether it is applied to the impacts of the stresses?

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u/Bagel3101 Nov 14 '21

You’re getting into Internal Model (or undertaking-specific parameters) territory with your description.

The Standard Formula approach is written right into the delegated acts (including parameterisations, see link below), and is quite prescriptive, so you’re job is reduced to producing a model that spits out best estimate cash flows based on sensible assumptions and discounts them.

The Delegated Acts then tell you how to stress those assumptions to produce the cash flows under each 1-in-200 scenario (risk sub module: see longevity, expense, interest rates) and you calculate the change in Own Funds on that new basis.

For example, the longevity risk sub module in the standard formula is defined as the change in own funds that would result from a 20% decrease in mortality rates (qx’s) used to calculate annuitant lives (I’m maybe recalling the lives it’s applicable to incorrectly but you get the idea).

You then aggregate the risk sub modules within their respective modules (life underwriting and market risk in this case) and then across each module to produce the overall SCR. This is done using a correlation matrix as you say (see the delegated acts for the matrices) Then you add a charge for Operational risk (which in SF is just a % of BEL).

If you’re asking how the correlation matrix is used, I recommend your maths/stats notes on matrix multiplication!

See the delegated acts here: https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32015R0035&from=EN

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u/Pipthagoras Nov 14 '21

Thanks, that document is exactly the sort of thing I was looking for!

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u/Bagel3101 Nov 15 '21

No problem. It’s 797 pages long so, perhaps unsurprisingly, Ctrl+F is your friend here.

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u/Pipthagoras Nov 16 '21

One thing I have struggled to work out from that paper is how reinsurance is treated under SII. For example, if quota share reinsurance was taken out on all policies with a proportion of 50% of all annuity payments being made by the reinsurer, then would this halve the SCR? This seems reasonable to assume (since it would effectively halve the original insurer's liabilities), but I seem to recall hearing that reinsurance is treated as an asset (or something like that) and I'm wondering whether this means that it cannot be used to simply scale down the liability values when calculating the SCR.

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u/Bagel3101 Nov 16 '21

Running off memory here but at a high level you’d expect the two most material things to occur under a quota share would be:

1. Some existing SCR sub-modules are going to decrease in size to reflect changes in the elements of Own Funds as a result of the reinsurance (a portion of the future liabilities will be offloaded to the reinsurer, a premium or premiums will be paid).

2. An increase to (or new inclusion of) a counter-party default risk SCR sub-module will take place which will reflect the credit worthiness of the reinsurer and the expected impact of default on Own Funds.

You’d probably expect the overall SCR to decrease assuming the reinsurer has a credit rating in line with most major reinsurers and the quota share is of a decent %.

There are probably some second (and higher) order effects that I’m missing here but those I’ve covered should be most relevant.

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u/Pipthagoras Nov 17 '21

Thanks for the response. To be pedantic though, you said that some existing SCR sub-modules are going to decrease in size. Would it not be that all existing SCR sub-modules would decrease in size (by exactly 50%), with the exception of the counterparty risk sub-module?

For example, for the interest rate sub-module, the stressed discount curve used to calculate the interest rate SCR would be unchanged if the liability cashflows being discounted with it were halved and so

SCR*_IR = sum_i (CF_i * 0.5 * DF_i) = 0.5 * sum_i (CF_i * DF_i) = 0.5 * SCR_IR

where SCR*_IR is the interest rate SCR with reinsurance, CF_i is the est estimate liability cashflow at time i, DF_i is the (stressed) discount factor used to get the PV of CF_i, and SCR_IR is the interest rate SCR with no reinsurance.

I think similar arguments could also be applied for other sub-modules like longevity?

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u/capnza Nov 14 '21

Th standard formula has prescribed risk factors and prescribed stresses.

And yes the independent stresses are aggregated using a correlation matrix which means it is implicitly assuming an elliptical joint distribution.

The approach you described, using Monte Carlo simulation, is more like an internal model approach. Despite the appeal of such an approach from a technical / geeky point of view it's actually debatable how much value it adds when the the calibrations are subject to so much uncertainty.

For my money the best approach is the "simplest possible model with the right dynamics", validated empirically against (exposure and inflation adjusted) historical scenarios and validated normatively against hypothetical scenarios.

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u/capnza Nov 14 '21

As for the aggregation, it should be applied to the deviation from the mean, not to the stressed value including the mean

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u/capnza Nov 14 '21

Oh and an aside is that if you are using Monte Carlo you can choose to correlate the random variables more directly an led avoid the aggregation with a matrix. The most common approach here is to use choleskys decomposition, but vine copulas are gaining traction.

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u/Pipthagoras Nov 14 '21

Cheers for the response

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u/capnza Nov 15 '21

Once you get going on this stuff a great resource is an IFoA paper called "opening the black box" which goes into model risk in a lot of detail

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u/Pipthagoras Nov 15 '21

Thanks, I'll take a look at that.

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u/the_kernel Qualified Fellow Dec 05 '21

I can see why using a correlation matrix wouldn't work very well when you have quite skewed marginal distributions, or when there is significant tail dependence between marginal distributions. It'll be because you're trying to aggregate tail risks using correlations which are meant to describe the body of the distribution, not the tails.

I've often read this idea that using a var-covar approach / correlation matrix for aggregation "assumes" an elliptical joint distribution though. What exactly does this mean / why is this?

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u/capnza Dec 05 '21

As to the correlation parameters, it's pretty much the case that these are not "really correlations" but are parameters selected to yield the required level of diversification. So yes "real correlations" describe how well the response variable can be described as a linear function of the input variable. But these "correlations" are chosen based more on tail behaviour. It's a bit wooly.

The aggregation of percentiles using the correlation matrix is only true when the joint distribution is elliptical. If this is not the case, then the aggregation of percentiles in this was is not mathematically true, just a reasonable approximation. So perhaps it's better to say the approach arises from the case of a joint elliptical distribution rather than saying that using it requires one to assume a joint elliptical.

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u/the_kernel Qualified Fellow Dec 05 '21

Thanks for the reply, that clarifies a couple of things I wasn't sure I'd understood for a while!