investment return under market consistent approach

[Svensson]

Hi

My understanding is that -
1. modern valuations (e.g. MCEV, Solvency) seek to adopt a market consistent approach.
2. under a market consistent approach, when we set the investment return assumption for insurance liabilities, the expected investment return can be set as the risk-free rate (aka risk-neutral calibration approach).
3. under a market consistent approach, we use the risk-free rate as the discount rate.
4. in other words, the investment return equals the discount rate.

I do not fully understand what are the justifications for that. I can think of two main justifications:
1. a theoretical justification: if we will use the "real" expected return, which is probably higher than the risk-free, then we will have an extra risk. The extra return and the extra risk would be offsetting each other, and we would end up with the same value as if we will use the risk-free.
2. a mathematical justification: it can be shown that we must use risk-free in order to be arbitrage-free, or in order to successfully replicate some market value, or something like that. if this is a valid justification, than my question is: in a risk-neutral valuation we indeed use risk-free, but we also use an adjusted probability measure (i.e. compatible probabilities). who says that the best estimate probabilities are the compatible probabilities?

I would appreciate clarifications. the basic question is why we set the investment return as the risk-free discount rate.
thanks

 

[Mark Willder]

Hi

My understanding is that -
1. modern valuations (e.g. MCEV, Solvency) seek to adopt a market consistent approach.
2. under a market consistent approach, when we set the investment return assumption for insurance liabilities, the expected investment return can be set as the risk-free rate (aka risk-neutral calibration approach).
3. under a market consistent approach, we use the risk-free rate as the discount rate.
4. in other words, the investment return equals the discount rate.

I do not fully understand what are the justifications for that. I can think of two main justifications:
1. a theoretical justification: if we will use the "real" expected return, which is probably higher than the risk-free, then we will have an extra risk. The extra return and the extra risk would be offsetting each other, and we would end up with the same value as if we will use the risk-free.
2. a mathematical justification: it can be shown that we must use risk-free in order to be arbitrage-free, or in order to successfully replicate some market value, or something like that. if this is a valid justification, than my question is: in a risk-neutral valuation we indeed use risk-free, but we also use an adjusted probability measure (i.e. compatible probabilities). who says that the best estimate probabilities are the compatible probabilities?

I would appreciate clarifications. the basic question is why we set the investment return as the risk-free discount rate.
thanks

Hi Svensson

I would agree with everything you have written above. Thanks for writing this out so clearly.

Subject CM2 demonstrates that under the assumption of no arbitrage, we have a formula based on the risk-free rate. This is shown in various situations, eg binomial trees, Black-Scholes, the 5 step method.

Subject CM2 also shows us that we should adjust the probabilities to use the risk neutral probability Q as you've said. These are not the best estimate or real world probabilities. Pricing using the measure Q is how the market allows for risk.

All of the above applies when we look at cashflows that we can hedge/replicate with available assets. We need a deep and liquid market for these assets to do this. So this will generally only apply to market assumptions, such as the interest rate or inflation.

However, we don't have a deep and liquid market for other assumptions such as mortality or lapses. So for these we use the best estimate assumption and recognise that we are not using the probability Q and so have not made suitable market-consistent allowance for risk. This is why we add a risk margin when valuing these aspects (eg the notes show how to use a cost of capital approach to calculate a market-consistent value for mortality etc).

Best wishes

Mark