SII TP - market consistency

[Mateusz]

Market consistent valuation of SII technical provisions is sometimes explained with references to results of the theory of options/derivatives pricing, but
personally I struggle a bit to understand to what extent this theory is applicable to insurance liabilities. Chapter 18 gave me some insights, but I'm not sure if they're correct.

When we want to price an option which depends on the value of some underlying asset, we take the expected value of the payoff it generates, with respect to a martingale measure, and discount it using a risk free rate. This is based on the idea of replication.

When we move to insurance liabilities, we no longer deal with the market risk only. In most cases the emerging cash flows either depend only on insurance risks (mortality, lapse etc.) or on both insurance and market risks (e.g. unit-linked or WP policies).

In the first case, for SII we need to project future cash flows using real-world probabilities, and discount using risk free rates of interest. Then add the risk margin on top. Does this approach basically assume that these projected cash flows are treated as FIXED for the replication argument to be applied? (ie find an asset to match the CFs in all possible states, but because our cash flows are treated as fixed, we are basically limited to an asset that's guaranteed to earn the risk free rate). And that the market value of the non-market risks is reflected seperately in the risk margin?

And in the second case (i'm assuming a simple case with no options/gtees), to project benefits or charges directly linked to the value of some underlying assets, we'd additionally use a fact that in a risk-neutral valuation risky assets earn the risk free rate (as the expected return). So the unit fund or asset shares need to be projected using the risk free rates of return.

I think the key point I'd like to confirm here is whether there's little (or nothing) that can be borrowed from the theory of risk-neutral valuation to deal with the insurance risks (ie reflect in the price possible deviations from the best estimates). And that this is why the problem of placing a MCV on insurance liabilities is split into two parts: BEL and RM?

Thanks,
Mateusz

 

[Mark Willder]

Market consistent valuation of SII technical provisions is sometimes explained with references to results of the theory of options/derivatives pricing, but
personally I struggle a bit to understand to what extent this theory is applicable to insurance liabilities. Chapter 18 gave me some insights, but I'm not sure if they're correct.

When we want to price an option which depends on the value of some underlying asset, we take the expected value of the payoff it generates, with respect to a martingale measure, and discount it using a risk free rate. This is based on the idea of replication.

When we move to insurance liabilities, we no longer deal with the market risk only. In most cases the emerging cash flows either depend only on insurance risks (mortality, lapse etc.) or on both insurance and market risks (e.g. unit-linked or WP policies).

In the first case, for SII we need to project future cash flows using real-world probabilities, and discount using risk free rates of interest. Then add the risk margin on top. Does this approach basically assume that these projected cash flows are treated as FIXED for the replication argument to be applied? (ie find an asset to match the CFs in all possible states, but because our cash flows are treated as fixed, we are basically limited to an asset that's guaranteed to earn the risk free rate). And that the market value of the non-market risks is reflected seperately in the risk margin?

And in the second case (i'm assuming a simple case with no options/gtees), to project benefits or charges directly linked to the value of some underlying assets, we'd additionally use a fact that in a risk-neutral valuation risky assets earn the risk free rate (as the expected return). So the unit fund or asset shares need to be projected using the risk free rates of return.

I think the key point I'd like to confirm here is whether there's little (or nothing) that can be borrowed from the theory of risk-neutral valuation to deal with the insurance risks (ie reflect in the price possible deviations from the best estimates). And that this is why the problem of placing a MCV on insurance liabilities is split into two parts: BEL and RM?

Thanks,
Mateusz

Hi Mateusz

Thank you for the question. I think your logic above all makes sense.

A first point to make is that Solvency II is not strictly covered by SP2. So the exact form of Solvency II only gets covered in SA2. Instead SP2 covers the general principles behind market-consistent valuations of liabilities. Of course Solvency II is an example of a market-consistent valuation, so it might help to think about Solvency II if that is familiar from your work. But it's worth avoiding using Solvency II names for things in SP2, so we have an illiquidity premium in SP2 rather than a matching adjustment, and a solvency capital requirement in SP2 is not necessarily calculated in the way that the SCR is in Solvency II.

Yes, you're right about the best estimates though. We're going to use best estimate assumptions for all the parameters that can't be observed in the market (the course says that this applies when there is no deep and liquid market from which to derive market values directly). That means we're assuming that mortality and lapses are fixed.

Given those best estimate assumptions we can then use our financial economic theory to say that their value must be the expected present value using the risk-free rate and the equivalent martingale measure.

Yes, consistent with this the unit fund and asset shares are assumed to grow at the risk-free rate, just as we assume that shares grow at a risk-free rate when pricing derivatives. But just like pricing derivatives, any financial options (eg a maturity guarantee on a unit-linked policy) will use the volatility of the actual assets held.

But you are right to query the best estimate assumption for mortality etc. According to the above, we've perfectly replicated our best estimate liabilities with matching assets and valued these appropriately. This gives us the best estimate liabilities of Solvency II. But no-one would accept these liabilities for this price, as there's a 50/50 chance of making a loss if our demographic assumptions are worse than expected. This is where the risk margin comes in. We'd only accept these liabilities if we would expect to earn a suitable return on the capital held against the risk of adverse experience. This leads us to the cost of capital approach in SP2, which we call the risk margin in Solvency II. So yes you're right, this bit has nothing to do with risk neural valuation.

I hope that helps to confirm your thoughts.

Thanks for asking the question. Hopefully the answer is helpful to other students too.

Best wishes

Mark

 

[mugono]

This is a great question and response from Mark; so much so that I felt compelled to contribute . I have added some complimentary comments to support / add to Mark's response below (in the hope that you and others find it helpful).

Looking at the reinsurance market: Reinsurance premiums will typically apply a loading to the best estimate cost that the insurer will need to pay a reinsurer to provide coverage. For example, in a longevity swap the fixed leg is set at outset and will be based on the best estimate cost + a loading. I am not aware of any reinsurer who is not in the business of (trying to) making money within their stated risk appetite / tolerance. This supports the idea that a market consistent value of liabilities is more than the best estimate cost.

Insurance means insurance: Discounting the resulting insurance cash flows at the risk-free rate preserves the principle that insurers are in the business of writing insurance (and no other) business. Applying a discount rate that is higher than risk-free introduces non-insurance risk. Investors can / should invest in another business if they want exposure to non-insurance risk. SII largely preserves this principle although there are a couple of exceptions (MA and VA). Be careful here: the final landing place of (any) regulation will include a number of lenses, e.g. technical, political, legal etc; i.e. it is not entirely decided on technical grounds.

 

[studier]

Hi Mateusz

But you are right to query the best estimate assumption for mortality etc. According to the above, we've perfectly replicated our best estimate liabilities with matching assets and valued these appropriately. This gives us the best estimate liabilities of Solvency II. But no-one would accept these liabilities for this price, as there's a 50/50 chance of making a loss if our demographic assumptions are worse than expected. This is where the risk margin comes in. We'd only accept these liabilities if we would expect to earn a suitable return on the capital held against the risk of adverse experience. This leads us to the cost of capital approach in SP2, which we call the risk margin in Solvency II. So yes you're right, this bit has nothing to do with risk neural valuation.

I hope that helps to confirm your thoughts.

Thanks for asking the question. Hopefully the answer is helpful to other students too.

Best wishes

Mark

I was under the impression that the risk margin is there to allow for a risk-neutral valuation. Adverse probabilities in the risk-neutral world will be higher than the real world, eg if you look at the probability of default (PD) implied by credit risky asset prices when using the riskfree rate to discount, the PD is higher than observed (real-world) PDs.

As you are discountign at the risk free rate to value S2 liabilties, we should be using risk-neutral mortality/longevity/lapse rates but obviously we don't have a liquid market to obsevrve them. Therefore we use best-estimate and the risk-margin is effectively an adjustment to the real-world probabilities to make them risk-neutral.

We see a similar thing in the fundamental spread calculation in the matching adjustment. It uses historic (real world) PDs, but discount cahflows at the riskfree rate. The cost of downgrade element is an approimxation to convert PDs from real world to risk neutral.

Note I haven't actually seen this written fomarlly but is how I have understood it.

 

[Mark Willder]

The fundamental spread is a concept from Solvency II that is not covered in the SP2 course. So I'll leave discussion of that for another time.

We need to be careful when discussing risk-neutral valuations in SP2. Note that a risk-neutral valuation is only the same thing as a market-consistent valuation in limited circumstances. SP2 covers market-consistent valuations.

A risk neutral valuation requires us to have a deep and liquid market in the assets that we need in our replicating portfolio. As there is no deep and liquid market for mortality, then a risk neutral valuation for insurance contracts is a non-starter.

So instead we're looking at a market-consistent valuation. This would price the cashflows in a consistent way with what the market would be prepared to accept.

As a starting point we use best estimate assumptions for things like mortality. If we assume that the cashflows are now fixed at their best estimates then we can calculate their value using risk-neutral techniques as we could replicate these cashflows in the market. If there was no variability around these best estimates then the risk neutral valuation and the market-consistent valuation would be the same.

However there is variability, so this risk-neutral valuation of the best estimate cashflows will be unacceptable to the market. Mugono rightly says that reinsurers want to make a profit. Insurers want a profit that compensates them for the risk taken on and the capital needed to cover the risk too. Similarly studier is right that the market wouldn't accept a best estimate of default rates to value corporate bonds. Fortunately corporate bonds often have a deep and liquid market, so we can actually observe what price the market places on the cashflows. But sadly we don't have a market price for our insurance cashflows.

So we need to adjust our best estimate to get a market price and this is where the concept of the risk margin / cost of capital approach comes in.

Best wishes

Mark

 

[Mateusz]

Thank you Mark for excellent responses and mugono for your contributions! Very helpful indeed.

We'd only accept these liabilities if we would expect to earn a suitable return on the capital held against the risk of adverse experience. This leads us to the cost of capital approach in SP2, which we call the risk margin in Solvency II

In theory, I imagine one could come up with alternatives to the cost of capital approach? Not that I'm looking for one, but I'm wondering if there is something special about this approach that makes it the default choice in calculating a loading for adverse deviations in a market consistent price?

Discounting the resulting insurance cash flows at the risk-free rate preserves the principle that insurers are in the business of writing insurance (and no other) business. Applying a discount rate that is higher than risk-free introduces non-insurance risk.

Could you elaborate a little more? It's an interesting point, but I'm not sure I've understood it. Are you referring here to the risk of not earning this higher rate of return on the backing assets?

We need to be careful when discussing risk-neutral valuations in SP2. Note that a risk-neutral valuation is only the same thing as a market-consistent valuation in limited circumstances. SP2 covers market-consistent valuations.

A risk neutral valuation requires us to have a deep and liquid market in the assets that we need in our replicating portfolio. As there is no deep and liquid market for mortality, then a risk neutral valuation for insurance contracts is a non-starter.

So instead we're looking at a market-consistent valuation. This would price the cashflows in a consistent way with what the market would be prepared to accept.

I think this is an important distinction and one I was hoping to find in this thread.

Thanks,
Mateusz

 

[studier]

The fundamental spread is a concept from Solvency II that is not covered in the SP2 course. So I'll leave discussion of that for another time.

We need to be careful when discussing risk-neutral valuations in SP2. Note that a risk-neutral valuation is only the same thing as a market-consistent valuation in limited circumstances. SP2 covers market-consistent valuations.

A risk neutral valuation requires us to have a deep and liquid market in the assets that we need in our replicating portfolio. As there is no deep and liquid market for mortality, then a risk neutral valuation for insurance contracts is a non-starter.


Best wishes

Mark

Mateusz: ignore this post, it's a very techincal point to Mark that is beyond SP2 (I don;t even think it's covered in SP6).

Mark: I don;t believe that a replicating portfolio is required for risk-neutral pricing. Risk-neutral pricing requires the absence of arbitrage*.

The first fundamental theorem of asset pricing states: A market** is arbitrage free if and only if there exsits an equivalent martignale measure.

The second fundamental theorem of asset pricing states: The equivalent martingale measure is unique if and only if the market is complete.

https://en.wikipedia.org/wiki/Fundamental_theorem_of_asset_pricing

Therefore, if we assume the market is arbitrage free, we can can use risk neutral pricing. The question then becomes what is the risk-neutral measure (RNM)/market price of risk.

1. If the market is complete, the RNM is unique and the maths tell you it.

2.It the market is not complete, the RNM is not unique. If there is a deep and liquid market, we can infer the market price of risk/RNM from market prices.

3. If there isn;t a deep and liquid market, then you use expert judgement to decide the market price of risk and hence the RNM.


*this is true in a discrete model. For a continuos model we require a similar concept called "no free lunch with vanishing risk"
** with discrete prices!

 

[Mark Willder]

Mateusz: ignore this post, it's a very techincal point to Mark that is beyond SP2 (I don;t even think it's covered in SP6).

Mark: I don;t believe that a replicating portfolio is required for risk-neutral pricing. Risk-neutral pricing requires the absence of arbitrage*.

The first fundamental theorem of asset pricing states: A market** is arbitrage free if and only if there exsits an equivalent martignale measure.

The second fundamental theorem of asset pricing states: The equivalent martingale measure is unique if and only if the market is complete.

https://en.wikipedia.org/wiki/Fundamental_theorem_of_asset_pricing

Therefore, if we assume the market is arbitrage free, we can can use risk neutral pricing. The question then becomes what is the risk-neutral measure (RNM)/market price of risk.

1. If the market is complete, the RNM is unique and the maths tell you it.

2.It the market is not complete, the RNM is not unique. If there is a deep and liquid market, we can infer the market price of risk/RNM from market prices.

3. If there isn;t a deep and liquid market, then you use expert judgement to decide the market price of risk and hence the RNM.


*this is true in a discrete model. For a continuos model we require a similar concept called "no free lunch with vanishing risk"
** with discrete prices!

Yes, this goes well beyond the course. I'll stick with the general concepts we covered in CM2.

For CM2 we'd see it like this. If the market is complete then we can find the assets that replicate or hedge the payoffs. If the replicating portfolio does exactly the same as our payoff then it must have the same price as the payoff to avoid an arbitrage. We can express this in different ways, but I think that is sufficient to understand the ideas in SP2 and SA2.

Best wishes

Mark

 

[mugono]

Could you elaborate a little more? It's an interesting point, but I'm not sure I've understood it. Are you referring here to the risk of not earning this higher rate of return on the backing assets?

Thanks,
Mateusz

Yes, sure. One way to think about it is to consider what applying a discount rate higher than the risk-free rate means for the purpose of valuing insurance liabilities. The concept of giving (partial) credit within the liability discount rate for part or all of the spread on a credit risky asset [implied by your second question] introduces risk that is arguably unrelated to the underlying insurance liability.

An insurance customer pays a premium for insurance coverage, e.g. death, morbidity etc. From their point of view the insurance policy is risk-free. Illiquidity premiums, credit downgrades, defaults etc are not risks that a customer is exposed to. The component of the spread in excess of the risk-free rate exists to compensate for risk; and its inclusion arguably breaks the 'insurance means insurance' principle that I describe in the earlier post.

 

[Mark Willder]

Thank you Mark for excellent responses and mugono for your contributions! Very helpful indeed.


In theory, I imagine one could come up with alternatives to the cost of capital approach? Not that I'm looking for one, but I'm wondering if there is something special about this approach that makes it the default choice in calculating a loading for adverse deviations in a market consistent price?


Could you elaborate a little more? It's an interesting point, but I'm not sure I've understood it. Are you referring here to the risk of not earning this higher rate of return on the backing assets?



I think this is an important distinction and one I was hoping to find in this thread.

Thanks,
Mateusz

Yes, you're right. The cost of capital approach is not the only way to allow for the non-hedgeable risks in our market-consistent valuation. The course also mentions the possibility of applying a margin directly to the assumptions, eg using a higher than best estimate mortality rate for term assurances. However, the cost of capital approach gets particular prominence in the course because it is what is used in Solvency II as the risk margin. It fits quite nicely with what insurers do with risk, as holding capital is what we do. Shareholders supply the capital and so require a return on it. I imagine it doesn't work so well in non-insurance settings, but I've not looked into alternative ways to do it.

Mugono makes an interesting point and it reminds me of some interesting stories I heard when market-consistent embedded values were first being introduced (these don' come into the syllabus until SA2). A simple story might help. What is worth more: £100 of risky assets or £100 of risk-free bonds? Well a financial economist would say that they were both worth the same - £100. Extending this argument, the cost of a without-profits annuity shouldn't be impacted by what we invest in. Pricing in a market-consistent way would mean using the risk-free rate as mugono says.

Now (and this bits controversial) despite what I've just said, many people (including some actuaries) believe that £100 of risky assets are worth more than £100 of risk-free bonds. The risky assets have a higher expected return, so if we invest in risky assets to back our annuities we'll have more profit and a higher embedded value (using a traditional non-market-consistent approach). The financial economist would say that's cheating - sure, the risky assets have a higher expected return, but they also have more risk, and the two cancel out meaning that they are still worth £100 now.

In fairness, both points of view have some merit. It's the job of investment managers to find assets that they think provide superior risk-adjusted returns. Mugono's point is that if you think the £100 of risky assets are better, then you should be an investment manager rather than an insurer. For example, the insurance regulator gets very concerned about the risk, and so generally pushes up the capital requirements for non-matching strategies.

I should leave it there, but it's an interesting area that has provoked enormous debate in the actuarial profession. For example, you can find lots of papers by Andrew Smith that would advocate a more market-consistent approach.

Best wishes

Mark

 

[Mateusz]

Thank you Mark, these are all very helpful insights!

My last question in this thread would be the following. Market consistent value of insurance liabilities is interpreted/defined as the amount that another insurer would require to take over the obligations. But when we're valuing a portfolio of liabilities which is in a 'net asset' position (ie negative liability), then we'd actually need to think about a price that a third party would be willing to pay? And in this case discounting best estimate profits at risk free rates and then deducing a risk margin would be very similar (conceptually) to a MCEV calculation?

I know it's probably not that common to have a portfolio of insurance contracts with a significant negative BE liability at a valuation date other than close to inception, but I think profitable term assurance policies would be one example.

Thank you
Mateusz

 

[Mark Willder]

Hi Mateusz

Yes, that's right. An MCEV calculation is very similar to a market-consistent valuation for solvency purposes. Yes, regular premium term assurances usually have a negative best estimate liability near the start of the contract.

Best wishes

Mark