With-profits BEL

A few things to think about here. Let's consider the Total BEL first.

Rather than thinking about (a) + (b) separately, I would encourage you to think about them together: the model needs to project what max{guaranteed benefit, smoothed asset share} is at the point at which a claim is being modelled, because this represents the amount of benefit payable. This then needs to be discounted back to the valuation date (and then, together with the PV of future expenses less premiums, forms the BEL).

The model that is needed has to be stochastic: you are correct that you need to project future RB rates to determine the guaranteed benefit at claim payment date. As you say, these RB rates may vary between simulations, depending on the rules used by the company for setting RB rates in practice. The projected asset share also needs to be projected stochastically. This is because you won't otherwise be allowing for scenarios under which asset values fall (or investment returns are very poor) and the asset share falls below the guaranteed benefit.

The model would then compare smoothed projected asset share and the guaranteed benefit, and the payout assumed to be made to the policyholder would be the higher of the two. (In practice models may project terminal bonus tables explicitly, but it comes down to being broadly the same thing.) This higher amount is then discounted as being the "benefits" part of the total BEL.

The calculation uses smoothed asset share rather than unsmoothed because this is what the payout will be based on. If investment returns are being smoothed upwards, the benefit liability is higher. If investment returns are being smoothed downwards, then the benefit liability is lower. In both cases, this reflects what the company expects to pay out. [I am not sure why you have said that if smoothed AS is higher than unsmoothed AS then using smoothed AS would under-estimate the liability?]

 

And now considering the BEL for guaranteed benefits only:

There is again no need to split up (a) and (b) separately, and indeed you don't need to project the asset share at all. This part of the BEL is only in relation to the guaranteed benefits, and what has accrued to date as a guarantee is a known amount, say £X. The £X just needs to be discounted back from when it is expected to be paid out.

 

In terms of thinking about the total BEL in terms of (a) and (b), as you have defined them above, it may be that you are getting the projection approach confused with the following approximation:

Total BEL (for a WP contract) is approximately equal to: current smoothed asset share + cost of guarantees (present value of the time value of the excess cost of guarantee payments over asset share). [There may also be a cost of smoothing element.]

This can be a helpful way of thinking about the total BEL for a WP contract if, for example, you are asked about how the Solvency II balance sheet might be affected by an event such as an equity market movement.

However, in terms of the model calculation itself I think that by trying to split it down into (a) and (b) you have unnecessarily complicated it and ended up confusing things.

Just think about needing to value the expected payout (and how this is defined for WP business, ie basically as the higher of guarantee and smoothed asset share) as the "benefit" part of the BEL calculation, and remember that it also needs to be split for presentational purposes into the guaranteed and discretionary components (which are not the same as your (a) and (b) split].

 

Yes - that's correct

 

Yes you are correct: terminal bonus has to be modelled either way. What I was referring to was the modelling of terminal bonus tables explicitly. Some models will be based on the projection of asset share for each policy, and the terminal bonus assumed to be given to each policy at claim is then just the excess (if any) of this projected asset share (allowing for smoothing) over the guaranteed benefit at that time. More sophisticated models generate sets of terminal bonus rate tables that would apply for each simulation and projection period. This will take into account the approach taken by the company in practice to setting terminal bonus rates, eg based on specimen policies, perhaps applying smoothing rules to the bonus rates rather than to investment returns etc.

Under the former approach, modelled benefit payout = max {smoothed asset share, guaranteed benefit}. Under the latter approach, modelled benefit payout = guaranteed benefit + modelled terminal bonus rate. The approaches are broadly equivalent; the latter allows for smoothing across policies and for the approximations that apply in reality (for practical reasons) in terms of applying the same terminal bonus rate across a group of policies, rather than having a separate rate for each individual claim.

Hope that helps.

 

Yes!

Yes possibly: but dynamic RB rates might well be linked in the model to investment returns rather than to asset shares. The modelled TB rates will depend on projected asset shares (and smoothing).

 

For a BEL calculation, both the expected investment return in the projections and the discount rate must be risk-free. (If the projection of investment returns is done stochastically, then the scenario generator would be calibrated to give the risk-free rate as the mean outcome.)

 

This thread should help:
https://www.acted.co.uk/forums/index.php?threads/2022-sep-q2-ii.19943/#post-77689