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The Actuary The magazine of the Institute & Faculty of Actuaries
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Investment: The best route?

In the pursuit of an internal model for Solvency II, it is not surprising that insurers tend to spend a significant amount of time focused upon the underlying methodology. It is only natural for actuaries to gravitate toward their modelling comfort zone, but the pursuit of a perfect model can lead to a seemingly endless research phase, potentially 
leading to significant delays to the project.

The quest for perfection is a laudable goal, but reaching milestones along the journey 
is preferred to inaction caused by results 
that are unable to foster 100% certainty.

The selection of a liability approximation method for internal model purposes should be driven not only by the suitability to the products but also the envisioned applications, with consideration of how risk will be embedded into decision making. 
The ambitions and requirements of the insurer may shift over time and, as the actuaries gain experience with the method, so it is important to ensure that the model is able to evolve with these adjustments.

Liability approximation methods, such as replicating portfolios, curve/formula fitting and least squares Monte Carlo simulation, have emerged to address the limitations of existing actuarial systems, most notably calculation speed. Each method has its own advantages and limitations, but it is worth keeping in mind that estimations can only be as good as the underlying measurements it is trying to emulate. Any flaws embedded in the assumptions of the actuarial software will be magnified by proxy measurement error. Some insurers have recognised this possibility and have considered enhancing or 
replacing their existing core actuarial models.

A considerable amount of 
published material about the various techniques is available from the report ‘Portfolio replication is not only about capital’ http://bit.ly/mVYj2g, but the main approaches adopted by European insurance groups for Solvency II internal models have been dominated by replicating portfolios and curve-fitting techniques.

Curve fitting
Curve fitting applies a multi-dimensional surface to a number of pre-calculated values under user-defined joint stresses. The joint stresses are not limited to just economic risk drivers, so all material risks (such as mortality, credit spreads, interest rates) can be incorporated up front. Curve fitting can be implemented using a parametric approach that aims to fit a linear or 
non-linear functional form to the entire surface. Non-parametric curve fitting involves localised regression at each plot point and doesn’t assume any global functional form.

Curve fitting is very well suited to the time zero liability valuation, but its use for business planning and projection beyond one year is less reliable. Its main advantage over portfolio replication is that the risk drivers are effectively interconnected and as such, the approach is less reliant on copulas for aggregation of the different risk types.

Replicating portfolios
The replicating portfolio liability approximation approach endeavours to emulate a liability under a wide range of market conditions using financial instruments that can be valued in a market consistent fashion. For a specified set of fitting scenarios (typically a combination of real world, risk neutral and shock scenarios), the liability cashflows are provided. Cashflows for the specified universe of replicating instruments are generated using the same set of fitting scenarios. The replication involves the parameterisation of an optimisation process to minimise the absolute or squared error under all scenarios and time steps.

Replicating portfolios require a 
strong intuition and understanding of 
both the liability and the instruments being selected to replicate the product. 
The selection of fitting scenarios can also have an important impact on the ability 
to get a good replication. Other criticisms of the technique involve over-fitting and the temptation to cosmetically alter the result, although these are addressed through awareness and experience. 
The main drawback identified is the necessity to aggregate the replicating portfolio’s market risks with non-market risks using a copula method. The selection and justification of an appropriate copula (Gaussian, Student’s t, Frank, Gumbel, and so on) is non-trivial.

Replicating portfolios have varied applications beyond valuation, to the management of options and guarantees, product design and pricing, hedge analysis, asset liability management, risk appetite, business planning and ‘what-if’ analysis. The iterative process provides the actuary with a better appreciation for the behaviour of the assets but also the liabilities, and it can actually foster improved communication and greater risk awareness across the organisation since the ALM team and asset managers will also be able to benefit by using the common language of replicating portfolios.

Identify the goal and work backwards
The adoption of proxy modelling techniques for internal modelling purposes continues to gather momentum, yet there continues to be uncertainty about which technique is suitable for the business and whether benefits anticipated prior to implementation are likely to be realised. In selecting an appropriate proxy model, there needs to be comfort with the methodology at various levels of the organisation. Prioritising the various business applications sought after by senior management will assist the selection of a model that not only efficiently calculates required capital, but can also help ensure that the business is being run in a more risk-aware fashion.

Andrew Waters

Andrew Waters is vice president of insurance for
QuIC, part of the Markit Group