Market-consistent valuation of liabilities is currently at the heart of many reporting frameworks. A widely used interpretation of the market-consistent approach is that the value of liabilities should be equal to the amount that the insurer would pay to transfer its remaining contractual obligations immediately to another entity. For certain types of insurance liabilities, this approach is relatively easy to adopt. In many other cases, however, the correct approach is less clear. In particular, insurance liabilities containing options and guarantees are typically valued using stochastic simulation precisely because a market price for the liability (or a suitable matching portfolio) cannot be observed in the market. In such cases, a suitably calibrated economic scenario generator (ESG) is likely to be used to generate a large number of asset return scenarios, which are then used within an asset-liability projection model to calculate liability values.

**Capturing all information available in the market **

A common definition of a market-consistent ESG is one which can reproduce the market prices of instruments that refl ect the nature and term of the liabilities being valued [*This is required for current UK regulatory reporting and will also be a requirement under Solvency II*”. In practice, this means that if we use the ESG output to value a particular asset, such as a risk-free bond, the price calculated should be consistent with the price observed in the market.

Although some judgment may be required to determine exactly what set of instruments best refl ects the nature and term of the liabilities, the options and guarantees embedded in most insurance contracts often share similar features with traded vanilla derivative contracts, for which market prices can easily be observed. By analysing various liability features such as the moneyness and time to maturity of the guarantees, we can usually identify an equity option (or a set of equity options) that creates similar economic exposure and whose price we will therefore aim to reproduce to demonstrate market-consistency. Given that insurers will typically write a range of policies with different guaranteed levels and terms to maturity, all likely to be backed by a diverse portfolio of assets, the range of such derivative prices is likely to be correspondingly broad [*As well as equity options, this is at least also likely to include interest rate swaptions and options on property assets*”.

While certain ESG models are sophisticated enough to capture all of this market price information, others may not have suffi cient degrees of freedom and might therefore only be able to reproduce a particular subset of such prices. As an example, let’s consider the way that the price of put options on equity indices varies with the term and strike price of the underlying option. Figure 1 (see below) shows the variation in the equity implied volatilities [*For a particular moneyness and term, there will be a one-to-one correspondence between option prices and option volatilities*” by the term and strike price of the underlying option, as at 31 December 2009, for the UK economy.

Figure 2 shows the volatility surface that is produced by an ESG model that uses a relatively simple Black-Scholes model for equity returns, in which equity volatility only varies deterministically by term but not with the level of the equity index. The model has been calibrated to the same at-the-money volatility data as shown in Figure 1, and while those volatilities (and therefore option prices) are reproduced reasonably well, the model clearly overstates the prices of in-the-money options (with a strike price above 100%), as well as signifi cantly underestimating the prices of out-of-the-money options.

By contrast, a more sophisticated equity model, such as a local volatility model in which volatility also varies by the equity index level, will be able to capture more of the variation in the market volatility surface and therefore prices a wider range of equity options more accurately. This is shown in Figure 3.

In order to try to quantify the impact of such model choice decisions, we have analysed a portfolio of with-profits contracts incorporating a set of cash guarantees applying on maturity, surrender or death. The extent to which the guarantees are in the money varies across the portfolio, and the guarantees have a term of between one and 40 years. To make the example as realistic as possible, our model allows for the management actions typically present in with-profits business, such as dynamic bonus rates, equity backing ratios and policyholder behaviour.

When using the model shown in Figure 2, the total guarantee costs on a notional liability portfolio were £29.6m, using 5,000 simulations and the end-2009 UK model calibrations. By contrast, with the model shown in Figure 3 the guarantee costs on the same liability portfolio were £32.4m, an increase of just under 10%. These results illustrate that, while both models have been calibrated to give as good a fit as possible to the same set of market prices, the difference in liability values is still significant. It is also worth noting that the average equity backing ratio in our example is only around 35%; with a higher equity backing ratio, the difference in guarantee costs would be correspondingly higher.

One approach that is often taken in cases where the ESG model is unable to capture all of the market data is to identify the average moneyness or term to maturity of the liabilities, and calibrate the ESG model to this particular point instead. However, such analysis is typically not straightforward and care is needed to ensure that the resulting liability value is not materially different to that produced by a more sophisticated model; for example, since equity option prices are a convex function of moneyness, the price of an option corresponding to an average moneyness will tend to underestimate the average guarantee costs across the whole portfolio.

Model choice should be a particular consideration in the context of Solvency II, where the requirements to demonstrate that any such simplifications in the valuation of the technical provisions are not material are particularly onerous, and the running of a more complex ESG model may be required each time such a validation is undertaken.

**Making realistic extrapolation assumptions**

While insurance options and guarantees share many similar characteristics to vanilla derivative contracts, they also typically incorporate features that generate additional economic exposures that cannot be hedged by vanilla derivatives. In particular, the management actions often associated with such policies, such as dynamic bonus rates or asset allocation policy, mean that the insurer has effectively written a complex path-dependent option that does not trade in the market, and for which a market price therefore cannot be directly observed. The same is also true when the term of the liabilities being valued is greater than the term for which market data is available.

This means there will always be an element of extrapolation from the vanilla derivative prices, used to calibrate any market-consistent ESG model, to the complex insurance options and guarantees that the ESG is being used to value. It is therefore possible for two ESG models to reproduce market prices for the same set of vanilla derivatives and still result in different liability values.

To explain why this may be the case, let’s consider again the simple example described above. Although the local volatility equity model has been extended to incorporate a wider range of market prices than the simpler Black-Scholes model, this has been done in a way that is expressly designed to reproduce that particular set of target market prices, without any particular consideration for whether the equity return dynamics generated by such an extension are realistic.

However, it will be precisely those equity return dynamics that influence not only what prices the model generates for other vanilla derivatives that are outside of the calibration sample (such as more deeply outof- the-money equity options or options with a longer term), but more importantly the values that are placed by the model on those complex options that better reflect the true nature of the liabilities being valued. While we may not be able to observe market prices for such options, by taking into account actual asset price behaviour the value placed on such an option by the model is likely to be closer to the value at which it would trade in the market and can therefore in some sense be considered more market-consistent.

To illustrate this point, we can look at a more realistic extension of the simple Black- Scholes model, such as the Bates equity model. The Bates model incorporates both a stochastic process of equity volatility, and a discrete jump process where equity values change discontinuously over very short (in practice, instantaneous) periods of time.

Like the local volatility equity model, the Bates model is sufficiently flexible to reproduce the market data shown in Figure 1 to a reasonable degree of accuracy. However, it also captures some of the equity market dynamics observed in real life that are absent from the local volatility (and Black-Scholes) models, such as volatility clustering (periods of high market volatility following each other) and increases in equity volatility during equity market falls.

As a result, the liability values generated by the two models can also be quite different. For the same with-profit liability portfolio described above, and again using end-2009 UK model calibrations, the Bates model results in guarantee costs of £36.2m, an increase of just under 12- when compared to the local volatility model. The results are summarised in Figure 4.

Model choice can have implications beyond placing a market-consistent value on a set of insurance liabilities as part of regulatory reporting. Market-consistent ESG models are also often employed as part of hedging programmes (such as delta hedging), in particular for variable annuity portfolios, where the valuation differences described above can easily translate into inappropriate hedging decisions. The appropriateness of the ESG models should therefore form part of the review of modelling within any risk management framework.

**Conclusion **

The above analysis highlights the need to use models with sufficient degrees of freedom to capture all relevant market data. Given that the ESG will usually be calibrated to the market prices of vanilla derivative instruments with less complex features than the liabilities being valued, it is also important to ensure that the model dynamics make realistic assumptions about asset price behaviour, as these are often crucial to the valuation of the complex path-dependent options and guarantees embedded in many insurance contracts.

*Adam Koursaris is currently leading Barrie & Hibbert’s research on implementation methodologies for Solvency II capital calculation. Viktor Knava is a client services manager in Barrie & Hibbert’s EMEA team and has significant experience in the development and application of stochastic models within the insurance industry*