# Use of asset volatility in unit linked BEL calculations

Discussion in 'SA2' started by gruhaa, Apr 4, 2018.

1. Hi

Can anyone please help me in understanding what is the use of volatility of assets returns assumption in unit linked BEL calculation?
We assume that for all the unit funds growth rate assumption is risk free rate(dont understand how different unit funds have the same growth rate? ) and, in case we have maturity guarantees, then why we need stochastic model when our growth is risk free with no uncertainty or do we simulate risk free rates in future?
Also, i have a tax related question, if for BLAGAB, there is no tax on 'I minus E', then company dont get the tax relief on expenses. For company, the trading profit calculated net of expenses hence it take account of relief expenses provides, then what relief it is talking about?

Thanks

2. ### Lindsay SmithermanActEd TutorStaff Member

Solvency II requires use of a risk-neutral market-consistent valuation approach, basically meaning that the expected return on all assets is the risk-free rate and a risk-free discount rate should be used. (It might be worth reviewing your CT8 notes as a reminder of the risk-neutral valuation approach?).

If there is a maturity guarantee within the UL product (or, indeed, any financial guarantee) then it has to be valued stochastically, not deterministically, otherwise the time value of that guarantee will not be captured in the BEL.

The investment returns would be simulated stochastically for each type of asset. These simulations will be calibrated so that the expected return on each asset (averaged across the simulations) is the risk-free rate, irrespective of asset type. However, the variation of simulated returns around that expected rate will depend on the asset type, and specifically on the volatility assumption used for each. So, for example, there would be a higher volatility assumption for returns (around the risk-free rate) for equities than there would be for cash.

3. ### Lindsay SmithermanActEd TutorStaff Member

Apologies but I don't think that I fully understand your question. Would you mind rephrasing it (ideally under a separate thread with a title linked to Tax, so that other students can find it more easily with the search tool)?

Are you asking about when a proprietary company is XSE, so that it cannot gain immediate tax relief on all of its expenses in the "I-E" calculation, as it has to carry forward some as XSE (defined as the excess of profit over {I-E})? it would then get relief on this carried forward XSE if it has sufficient I in the next tax period, although it may find itself stuck as XSE and therefore unable to gain this relief at any point.

However, if I have misunderstood your point do let us know.

4. Hi Lindsay,

I think he/she means that if a company is not XSE, it can be said that:

a) it will not get relief on expenses;
b) it will be taxed on trading profits.

But the formula for trading profits is broadly P + I - E - C - increase in reserves etc. Since we are deducting E in the calculation of trading profits, how then can we say the company will not get relief on expenses?

5. ### Lindsay SmithermanActEd TutorStaff Member

If a company is not XSE then it will be XSI. In which case I-E > minimum profits, so it will be taxed on I-E, not profit, so it will receive tax relief on E (the taxable amount of I-E is reduced by E).

I'm sorry but still not quite sure that I am understanding the question - apologies!

6. ### Lindsay SmithermanActEd TutorStaff Member

Let me try this instead. I think we need to think about companies that are XSE, not companies that are not XSE. (But apologies again if I have misunderstood)

BLAGAB business should be taxed on an "I-E" basis. However, HMRC state that there is a minimum level of taxation that has to be paid, which is the corporation tax rate x minimum profit. This puts life insurance companies in line with any type of company.

Let's think about the situation where I=300, E=175, minimum profit=150. This would mean that the company is XSE, since minimum profit > I-E. The taxable amount is the higher of the two, which means that the company is being taxed on 150 (the minimum profit test "bites") rather than on I-E of 125. The company has therefore paid tax on 25 more than it should have. This 25 is carried forward as XSE.

Bearing in mind that the company should be paying tax on I-E (because this is the BLAGAB fund), it has effectively only been allowed to count E=150 in its I-E tax calculations. It has been taxed on a total of 150, and if E is restricted to 150 then I-E = 300-150=150 is achieved. So the BLAGAB tax calculation has only counted 150 of E rather than the 175 of E that it has actually incurred. The company has not obtained relief (on the I-E basis) on 25 of the E, so this amount is carried forward. And the company should be able to offset that 25 against I in the next tax year, unless it is stuck in an XSE position.

Although, as you point out, the profit calculation deducts E, it also adds in any release of reserve. The reserves would allow for one fewer year of expenses, so this would offset (except to the degree to which actual is not equal to the valuation basis).

Has that hit what you were concerned about?

7. Thanks Mandla for elaborating this further.
What I meant with this was if the government says that for BLAGAB busiess, there will be no tax on Investment return and the invested assets will roll up with gross return and pass on to customer taxfree. To give you a reference, see Q&A Bank part 1, ques 1.4.

8. ### Lindsay SmithermanActEd TutorStaff Member

Ah, thank you - the question reference is very helpful. This is a hypothetical situation in which the government is proposing (as you say) that a new type of unit-linked policy can be sold which is not taxed on "I-E" anymore: "I" is therefore accrued on a gross basis and it is no longer possible to gain tax relief on the "E". This is not particularly good news for the company, because the benefit of having gross (ie non-taxed) "I" is mostly passed on to the policyholder, and the company can no longer gain tax relief on the "E".

If I have understood correctly, I think that your point is that if it is taxed on trading profits instead of "I-E" then there is still relief on "E" because the profit calculation includes "-E". However, as I indicated above, the profit calculation also adds in any release of reserve. The reserves would allow for one fewer year of expenses, so this release of expense reserve would offset against the -E (except to the degree to which actual is not equal to the valuation basis). If taxed on trading profits, the company is basically just being taxed on profit margins loaded into the business, as they emerge.

Hope that helps. Thanks for the clarification on the source of the query.

9. absolutely it helped. thanks a lot.

I also have one more question on chapter 8, I couldnt understand very well, when the fund is contracting and company has unrealised gains, in what situation discounting of taxation is required and in what situation discounting wont be required? the paragraph written on core read(in unbold font) is not very clearly setting out the difference between the situations

10.
11. In that case, to project the unit fund, we actually consider the investment return from asset as Rfr plus volatility?

12. ### Lindsay SmithermanActEd TutorStaff Member

Yes - the simulated distribution of investment returns on each asset type will have a mean of the rfr and volatility around that mean according to the inherent volatility of returns on that asset type.

13. ### Lindsay SmithermanActEd TutorStaff Member

Hi - I have started a new thread to answer this, so that other students can find it more easily by title:

https://www.acted.co.uk/forums/inde...wance-in-unit-pricing-contracting-fund.14900/

14. thanks lindsay.

Just to give it a firm understanding, i have a follow up question that, if we were using real world expectation calibration, then we have used long term expected return of that particular asset, say expected average rate is 5% and therefore standard deviation around this average return of 5% is say 1%. Thus under real world expectation, 1% is the volatility.
But in risk neutral, the average return would be risk free rate, say 1% but should the volatility would be higher in this case because the return is calibrated to risk free for example say 4%. So when we simulate the return in future for the underlying asset, we would get almost same simulated return under both the scenarios say around in the range 3%-5%?

Am I getting this right?

15. ### Lindsay SmithermanActEd TutorStaff Member

You wouldn't get simulated returns falling so closely in the same range for this example. The simulated returns under the risk-neutral calibration will have the 1% risk-free rate as their mean, with variability either side of that expected rate. If you were simulating returns under a real-world calibration they would have a mean of 5%, with variability either side of that rate.

You are, however, correct that there will be different volatility measures used for each calibration. The risk-neutral volatilities are typically derived from option prices: the volatilities required are those which equate the theoretical price (under a risk-neutral valuation approach, ie with risk-free drift) with traded prices. This is unlikely to be exactly the same as a "real world" volatility - although they should be of broadly the same order (under both measures, more risky assets will have higher volatilities than less risky assets).

Bear in mind that the two calibrations are done using different probability measures (risk-neutral vs real world). There would also be different discount rates applied.

Risk-neutral calibration of ESGs is a fairly complex technical area, which you wouldn't be expected to know the details of for SA2 - beyond the basics of what they are trying to achieve. If you are interested in exploring further for personal reasons, there will be technical papers available on the internet which go into more detail.

16. thanks Lindsay, you explained really well.

I have a last question around nested stochastic problem mentioned in chapter 15 page 19
It is written as-We want to project what think will happen to assets and liability in the future-this requires real-world calibration eg investment return interest rate etc and this is called outer model.
However, to calculate the liability at any particular time may also require a stochastic simulation eg stochastic model to calculate the cost or option which is calibrates to be market consistent(risk neutral) called inner model.

I have two ques in this content:
1. why outer model is real world calibration and inner model is risk neutral? shouldnt it be consistent? what difference it makes?

2. If we want to calculate CoG, we need future investments return, interest rates and other economic assumption which we already derived in outer model then why cannt we use that if if there was consistency and there will be no nested stochastic problem?

17. ### Lindsay SmithermanActEd TutorStaff Member

Imagine that a UK company (which has guarantees within its products) wants to answer the question "What is the probability that we will be insolvent in five years' time?"

They first need to project forward their assets and policy information for five years using a real-world stochastic model (the outer model). This model needs to be real-world because they want to get a realistic view of what the value of assets could be at that future point in time. It needs to be stochastic because they want to use it to determine a probability; let's say that they have performed 10,000 simulations.

For each of these 10,000 simulated outer model outcomes, the company also needs to work out what the liabilities and solvency capital requirements are in five years' time, in order to be able to complete its assessment of whether it is solvent or not at that point. It will have to do this on the Solvency II basis, which is a risk-neutral market-consistent valuation approach. So, for each of these 10,000 simulated outcomes, it will then have to perform another stochastic calculation (because it has guarantees) but now on a risk-neutral calibration (because that is what Solvency II requires). This is the inner model.

To make things more complicated, the company should also be testing whether it goes insolvent at any point during the five years under any of its 10,000 real-world simulations, not just at that single date in five years' time. So it should, in theory, be doing these inner model risk-neutral simulations at regular intervals during the five year real-world projection period. Hence we have the complexity of nested stochastic calculations, on the two different calibrations.

Once the risk-neutral Solvency II valuations have been performed, the company can count how many of the 10,000 real-world simulations (from the outer model) result in it going insolvent, and this gives the probability of insolvency. It needed to use a real-world calibration for the outer model in order to obtain a real-world probability of insolvency. (A risk-neutral model would give a risk-neutral probability, which is an artificial construct that does not have real-world meaning.)

A useful rule of thumb is that real-world calibration is normally used if we are interested in understanding what the future looks like (since based on realistic expectations of the future), and a risk-neutral calibration tends only to be used when we are interested in valuing something ie calculations involving projecting forward and then discounting (since based on replicating market prices).

Does that help?

Aladinsane and Chinj like this.
18. Hi,

I'm just trying to get my head around this. So to confirm, when calculating the COG as part of the BEL under SII you would always use a market consistent valuation? What would be the stochastic variable in this case? Normally I would have thought investment returns would be but under market consistent valuation they are risk free.

Thanks,

Max

19. Hi Lindsay,

Just a couple of additional questions:
• When calculating the SCR, assuming you have guarantees embedded in some of your products would you need to re-run stochastic calculations for each stress of the SCR (so all market, insurance etc stresses)? So assuming 5000 runs say for each stress (including base scenario) this would mean needing to do tens of thousands of runs?
• Would the above process including the nested stochastic loops equally apply to projecting the SCR when calculating the RM and so is the reason simplified drivers are used instead?
• Under the risk neutral calibration all assets are expected to earn risk free rate. If we use a VA can we increase the risk free in respect of the VA?
Thanks again,

Max

20. Yes you would use a MC valuation and the stochastic variable would be investment returns. I think to Lindsay's point a few posts up, the expected return is risk free but the volatility depends on the asset. If you back the gtee with equities then it will have a different cost of gtee than if you backed it with gilts, even if both grow at risk free, because of the different volatility around risk free of these assets. Volatility is key in a risk neutral valuation basically.

Lindsay Smitherman likes this.
21. Thanks Aladinsane, that makes sense now!