I recently attended a conference where the following question came up: when you calibrate an Economic Scenario Generator for computing Solvency II technical provisions, which risk-free interest do you calibrate it to? The risk-free rate as defined by Solvency II, or the one observed in the market? That turns out to be a tricky question…

#### Regulation says: do both

The Solvency II regulation is pretty clear about this. Article 22(3) states that your economic scenario generator must be arbitrage free and must reproduce the observed market prices. If the observed market prices involve quotes for EUR OIS swaps (which most consider close enough to risk-free) all EUR denominated assets should yield this interest rate on average, for the model to be arbitrage free.

There is a third requirement, however, which is that the ESG must be consistent with the regulatory risk-free rate – including volatility adjustment, matching adjustment, and transitional on the risk-free rate. Put differently, all EUR denominated assets should yield the *regulatory* risk-free rate on average, for the model to be arbitrage free.

That is kind of hard. Hence the cake metaphor.

#### Why are we having this discussion?

Before diving into solution mode, a short historical digression.

When Solvency II was put into place, there was a lot of fuss about discounting rates. Naturally, because the discounting rate of future obligations can mean life or death for an insurance company with long dated obligations. The purist choice for a discounting rate, would be to take the one observed in the market. If this was done, there would be no need to have this discussion, because there would only be one risk-free rate to calibrate to. There were, however, some good reasons not to go with this approach

- Over time, there is no good candidate in the market for a risk-free rate. There is a tendency to use a particular rate as risk free for some time, only to discover that it is not risk free.
- Insurance liabilities are often way longer than the market is willing to hedge, so there is nothing to calibrate to in the long end.
- Using the purist approach would have missed some really good opportunities for serving special interests by playing tricks with the risk-free rate.

The current situation is, that you can end up discounting on several different rates for each currency in the same model: with and without volatility adjustment and potentially with several different matching adjustments, and they are all supposed to be considered risk free.

#### Wiggling yourself out of the maze

In some cases, the contradiction can be swept under some simplifying assumptions. If the cash flow whose value you have to estimate is independent of future interest rates, it is sufficient to discount the expected cash flow on the relevant initial interest rate, and you really do not need ESG. Everyone can all live with a practical solution in spite of theoretical difficulties.

When interest rates can influence the cash flow, and you are in ESG territory, you cannot really close your eyes to the contradiction. There are certainly ways to go about this, but there can be many different ways, with different consequences. Questions you should ask yourself are:

- If you keep a EUR in a risk-free account to cover a future benefit payment (discounted on the regulatory risk-free rate), how does it earn a higher interest than if it covers a financial market payment (discounted on the market risk-free rate)?
- How do the spreads between the different risk-free rates develop? Is market data even a relevant guide to this?
- How do you keep the complexity down, here?
- If you have an ESG vendor, do they provide answers to the above questions, or do you have to bolt that on yourself?

The answers to these questions can be quite fundamental to, how you value and hedge options and guarantees. And answering them requires a mix of traditional quant skills and actuarial experience.

Drop me a line if you need help with this.