Another market consistency thread :)

[MindFull]

Hi All,

I've done a lot of reading on the forum as well as externally trying to bog down this concept. (Special thanks to Mark and Mugono's threads that helped tremendously). I think I'm almost settled but I wanted to clarify my understanding.

So one way the market consistent approach was explained (IMO) is the concept of the risks that the liability is supposed to be exposed to. So for an insurance liability, that would include lapse, mortality, expenses (?) but it would not include market risk because from the p/h's perspective, they are paying for mitigation agst mort, lapse (non hedgeable). Said another way, since $100 of a bond and $100 of a stock both provide $100, the liability value won't change based on what asset is chosen so there's no market related risk. So both are looked at the same way. As such, market risk should not be included in the discount rate. Is this correct?

The second approach is the one from the notes where it speaks to a replicating portfolio concept. I took CT8 many MANY years ago so I'm a bit rusty but basically I accept the replicating portfolio concept and that risk-free rates are req'd in the use of a market consistent valuation. So if a risk free ZCB can replicate an annuity liability then they should both have the same value. I guess my hang up is why use a risk-free ZCB? We could have also used a risky ZCB to replicate the liab. CFs but then the liab would not use the risk-free rate (it would use the rate associated with the risky ZCB). Is it because of the concept of requiring risk-free rates for a market consistent valuation? I feel my logic is becoming a bit circular so some help would be appreciated.

Thanks much!

 

[CapitalActuary]

Some of the what you wrote sounds sort of right but I think it's a bit muddled. I'll try to help, but this is partly for my own benefit as I haven't worked in life insurance or pensions so I'm curious if I understand this 'market consistent' business myself.

Let's take a simple example with an insurance liability, where we pay out $100 in 1 year with a 5% chance (upon death, say) and the risk-free rate is 25% and cannot change. The expected value of the cash-flow is $5, and discounted 1 year this is a $4 expected present value. Note the cash-flow is stochastic, but there isn't any market risk, just the 5% chance of death (mortality risk). The market price for this liability would be higher than $4, because nobody is going to take on this liability for just $4 - they'd want some compensation for the mortality risk and also to make a profit. Similarly, a customer is willing to pay more than $4 for a policy like this because of their risk aversion and willingness to pay for this.

Assume we can observe the market price of this liability and it is $4.17. This would mean the market implied risky discount rate was 20% (the solution to 4.17*(1+r)=5). The difference of 5pp between this and the risk-free rate of 25% is to account for the mortality risk.

In practice though, one cannot observe the market price for this liability, hence we can't imply the risky discount rate either. In order to come up with a market consistent value we either: 1) come up with a risky discount rate 'r' that takes account of mortality risk, and use this to work out the market consistent value, or 2) come up with an adjustment to the $4 expected present value which gets us directly to a market consistent value.

E.g. in order to come up with $4.17 as a market consistent value, we'd need to come up with a risky discount rate of 20% using method 1, or we'd need to calculate our adjustment as +$0.17 using method 2. Method 2 is generally what is done in practice, e.g. in Solvency II this adjustment is called the risk margin or in other places it's called the "cost of residual non-hedgeable risks".

Coming back to what you wrote: "since $100 of a bond and $100 of a stock both provide $100, the liability value won't change based on what asset is chosen". I don't think the latter is implied by the former (I'm also not sure exactly what you mean by the former). I'd just say that valuing the liability here is something independent of which assets you hold.

In your second paragraph, you say that 'risk-free rates are required in market consistent valuation'. This isn't true, as the example above hopefully illustrates. The market consistent value of $4.17 we could arrive at by discounting at a risky discount rate of 20%. Or alternatively by discounting at the risk-free rate of 25% then applying the 'adjustment' of +$0.17 to get to $4.17. Simply discounting at the risk-free rate would have given $4 which is below the market consistent value. In a market consistent valuation you should discount cash-flows at rates which reflect their riskiness, or otherwise discount at risk-free then apply an adjustment to your answer.

 

[MindFull]

Some of the what you wrote sounds sort of right but I think it's a bit muddled. I'll try to help, but this is partly for my own benefit as I haven't worked in life insurance or pensions so I'm curious if I understand this 'market consistent' business myself.

Let's take a simple example with an insurance liability, where we pay out $100 in 1 year with a 5% chance (upon death, say) and the risk-free rate is 25% and cannot change. The expected value of the cash-flow is $5, and discounted 1 year this is a $4 expected present value. Note the cash-flow is stochastic, but there isn't any market risk, just the 5% chance of death (mortality risk). The market price for this liability would be higher than $4, because nobody is going to take on this liability for just $4 - they'd want some compensation for the mortality risk and also to make a profit. Similarly, a customer is willing to pay more than $4 for a policy like this because of their risk aversion and willingness to pay for this.

Assume we can observe the market price of this liability and it is $4.17. This would mean the market implied risky discount rate was 20% (the solution to 4.17*(1+r)=5). The difference of 5pp between this and the risk-free rate of 25% is to account for the mortality risk.

In practice though, one cannot observe the market price for this liability, hence we can't imply the risky discount rate either. In order to come up with a market consistent value we either: 1) come up with a risky discount rate 'r' that takes account of mortality risk, and use this to work out the market consistent value, or 2) come up with an adjustment to the $4 expected present value which gets us directly to a market consistent value.

E.g. in order to come up with $4.17 as a market consistent value, we'd need to come up with a risky discount rate of 20% using method 1, or we'd need to calculate our adjustment as +$0.17 using method 2. Method 2 is generally what is done in practice, e.g. in Solvency II this adjustment is called the risk margin or in other places it's called the "cost of residual non-hedgeable risks".

Coming back to what you wrote: "since $100 of a bond and $100 of a stock both provide $100, the liability value won't change based on what asset is chosen". I don't think the latter is implied by the former (I'm also not sure exactly what you mean by the former). I'd just say that valuing the liability here is something independent of which assets you hold.

In your second paragraph, you say that 'risk-free rates are required in market consistent valuation'. This isn't true, as the example above hopefully illustrates. The market consistent value of $4.17 we could arrive at by discounting at a risky discount rate of 20%. Or alternatively by discounting at the risk-free rate of 25% then applying the 'adjustment' of +$0.17 to get to $4.17. Simply discounting at the risk-free rate would have given $4 which is below the market consistent value. In a market consistent valuation you should discount cash-flows at rates which reflect their riskiness, or otherwise discount at risk-free then apply an adjustment to your answer.

Hiya,
So my statement about the 100 bond/100 stock was (based on my understanding) supposed to point to the fact that choosing a riskier asset to back the liability (and hence causing exposure to market risk) should not change the value of the liability so hence the market risk should not apply to the liability. I think your eg is a great one and I understand the risk margin concept. My issue is coming from the use of the risk free rate in the market consistent framework (which I believe is mandatory), which I am assuming is coming from the BS/option replicating strategy logic (based on the numerous threads I've read). I just wanted to check if there's something else I'm missing with regards to the explanation.

Thanks!

 

[MindFull]

Hiya,
So my statement about the 100 bond/100 stock was (based on my understanding) supposed to point to the fact that choosing a riskier asset to back the liability (and hence causing exposure to market risk) should not change the value of the liability so hence the market risk should not apply to the liability. I think your eg is a great one and I understand the risk margin concept. My issue is coming from the use of the risk free rate in the market consistent framework (which I believe is mandatory), which I am assuming is coming from the BS/option replicating strategy logic (based on the numerous threads I've read). I just wanted to check if there's something else I'm missing with regards to the explanation.

Thanks!

Some of the what you wrote sounds sort of right but I think it's a bit muddled. I'll try to help, but this is partly for my own benefit as I haven't worked in life insurance or pensions so I'm curious if I understand this 'market consistent' business myself.

Let's take a simple example with an insurance liability, where we pay out $100 in 1 year with a 5% chance (upon death, say) and the risk-free rate is 25% and cannot change. The expected value of the cash-flow is $5, and discounted 1 year this is a $4 expected present value. Note the cash-flow is stochastic, but there isn't any market risk, just the 5% chance of death (mortality risk). The market price for this liability would be higher than $4, because nobody is going to take on this liability for just $4 - they'd want some compensation for the mortality risk and also to make a profit. Similarly, a customer is willing to pay more than $4 for a policy like this because of their risk aversion and willingness to pay for this.

Assume we can observe the market price of this liability and it is $4.17. This would mean the market implied risky discount rate was 20% (the solution to 4.17*(1+r)=5). The difference of 5pp between this and the risk-free rate of 25% is to account for the mortality risk.

In practice though, one cannot observe the market price for this liability, hence we can't imply the risky discount rate either. In order to come up with a market consistent value we either: 1) come up with a risky discount rate 'r' that takes account of mortality risk, and use this to work out the market consistent value, or 2) come up with an adjustment to the $4 expected present value which gets us directly to a market consistent value.

E.g. in order to come up with $4.17 as a market consistent value, we'd need to come up with a risky discount rate of 20% using method 1, or we'd need to calculate our adjustment as +$0.17 using method 2. Method 2 is generally what is done in practice, e.g. in Solvency II this adjustment is called the risk margin or in other places it's called the "cost of residual non-hedgeable risks".

Coming back to what you wrote: "since $100 of a bond and $100 of a stock both provide $100, the liability value won't change based on what asset is chosen". I don't think the latter is implied by the former (I'm also not sure exactly what you mean by the former). I'd just say that valuing the liability here is something independent of which assets you hold.

In your second paragraph, you say that 'risk-free rates are required in market consistent valuation'. This isn't true, as the example above hopefully illustrates. The market consistent value of $4.17 we could arrive at by discounting at a risky discount rate of 20%. Or alternatively by discounting at the risk-free rate of 25% then applying the 'adjustment' of +$0.17 to get to $4.17. Simply discounting at the risk-free rate would have given $4 which is below the market consistent value. In a market consistent valuation you should discount cash-flows at rates which reflect their riskiness, or otherwise discount at risk-free then apply an adjustment to your answer.

So I've done some more reading and it seems you are correct in the sense that you can use a risky rate but using a risk free rate would be easier since you'd have to calculate a different risky rate for every asset/derivative combo?
Apologies to everyone for this unnecessarily long thread.

PS. Mods, can we delete threads? *Hides face*

 

[CapitalActuary]

Each cash-flow needs to be discounted using an appropriate risky rate, otherwise (if you discount everything at risk-free) you need to make risk-margin-like adjustments to account for the fact you didn't use an appropriate rate.

 

[MindFull]

Each cash-flow needs to be discounted using an appropriate risky rate, otherwise (if you discount everything at risk-free) you need to make risk-margin-like adjustments to account for the fact you didn't use an appropriate rate.

While reading this again, I realised that maybe one of my issues is how I view the liability and the assets that back the liability. So in your eg, you said that the liability doesn't have any market risk but in my head, the risk free rate is being used because the assets backing the liability use a risk free rate. I can see how the insurance liability won't be exposed to market risk but the assets backing the liability will be.
Is that incorrect?

 

[CapitalActuary]

In my example I tried to keep things simple by saying the risk-free rate would not change, so there will be absolutely zero market risk for the insurance liability.

The asset side of the balance sheet is the same in theory. E.g. if you hold a bond, then the market consistent value of that bond is either the coupons and principal discounted at a risky discount rate, or discounted at a risk-free rate with an adjustment made to account for this. But of course, you don’t need to go through this process for a bond because you can just look up what the market price is, which by definition is market consistent. You can’t do the same for the insurance liabilities.

 

[MindFull]

Hi All,

I've done a lot of reading on the forum as well as externally trying to bog down this concept. (Special thanks to Mark and Mugono's threads that helped tremendously). I think I'm almost settled but I wanted to clarify my understanding.

So one way the market consistent approach was explained (IMO) is the concept of the risks that the liability is supposed to be exposed to. So for an insurance liability, that would include lapse, mortality, expenses (?) but it would not include market risk because from the p/h's perspective, they are paying for mitigation agst mort, lapse (non hedgeable). Said another way, since $100 of a bond and $100 of a stock both provide $100, the liability value won't change based on what asset is chosen so there's no market related risk. So both are looked at the same way. As such, market risk should not be included in the discount rate. Is this correct?

The second approach is the one from the notes where it speaks to a replicating portfolio concept. I took CT8 many MANY years ago so I'm a bit rusty but basically I accept the replicating portfolio concept and that risk-free rates are req'd in the use of a market consistent valuation. So if a risk free ZCB can replicate an annuity liability then they should both have the same value. I guess my hang up is why use a risk-free ZCB? We could have also used a risky ZCB to replicate the liab. CFs but then the liab would not use the risk-free rate (it would use the rate associated with the risky ZCB). Is it because of the concept of requiring risk-free rates for a market consistent valuation? I feel my logic is becoming a bit circular so some help would be appreciated.

Thanks much!

I know this thread has seen quite a lot of action but I'd love it if a tutor could help clarify some of the issues I'm having. Specifically whether the liability is exposed to market risk (if so, then the risk is hedged using an asset that replicates its CFs exactly?), and also the the meaning behind the eg I've seen Mark use (a bond worth 100 and a share worth 100 are worth the same so that shouldn't impact the liability CF that they are backing?)

Thank you!

 

[Em Francis]

I know this thread has seen quite a lot of action but I'd love it if a tutor could help clarify some of the issues I'm having. Specifically whether the liability is exposed to market risk (if so, then the risk is hedged using an asset that replicates its CFs exactly?), and also the the meaning behind the eg I've seen Mark use (a bond worth 100 and a share worth 100 are worth the same so that shouldn't impact the liability CF that they are backing?)

Thank you!

When using a market-consistent approach to value the liability, it doesn't matter what is actually backing the liability. There are two main market-consistent approaches, if using a replicating approach you would take the market value of the assets which matches the liability cashflows, so I think in your example above this will be the zero coupon bond. In a risk-neutral valuation you discount your liability cashflows by the risk-free rate, ie the yield on a government bond or a swap (adjusted for credit risk). If the liability contains non-market risk, such as mortality and withdrawals, then an additional amount needs to be held above the best estimate assumptions, eg the risk margin.
Does this help?

 

[MindFull]

When using a market-consistent approach to value the liability, it doesn't matter what is actually backing the liability. There are two main market-consistent approaches, if using a replicating approach you would take the market value of the assets which matches the liability cashflows, so I think in your example above this will be the zero coupon bond. In a risk-neutral valuation you discount your liability cashflows by the risk-free rate, ie the yield on a government bond or a swap (adjusted for credit risk). If the liability contains non-market risk, such as mortality and withdrawals, then an additional amount needs to be held above the best estimate assumptions, eg the risk margin.
Does this help?

Good Day Em,
Thanks so much for replying. This does help quite a bit though I think I'm still unclear about the background details. Still, I think I know enough to get me through (fingers crossed)!

P.S. Does your statement mean then that the liability cashflows (things like Term/WL) aren't exposed to market risk (just non-insurance risk)?

 

[Em Francis]

Good Day Em,
Thanks so much for replying. This does help quite a bit though I think I'm still unclear about the background details. Still, I think I know enough to get me through (fingers crossed)!

P.S. Does your statement mean then that the liability cashflows (things like Term/WL) aren't exposed to market risk (just non-insurance risk)?

Remember when valuing the liability, you wouldn't take the market value of those backing assets but when valuing the assets you would. And in answer to your second question, you will need to break down your products by expected liability cashflows. So yes, as the main assumptions for term assurance products will be mortality, they are mainly exposed to non-market risk. Another product such as an endowment could be exposed to both market and non-market risk.