Keeping the number of risk
neutral scenarios as low as
possible
without impeding on the quality of the valuation
Life insurance products are often characterized by embedded options or guarantees that are
contractually promised to the policy holder. The value of these options or guarantees depends heavily
on the prevailing economic conditions. Think for instance of the level of the interest rate in case of
profit-sharing options, or the return on investments in case of Unit-Linked products with a return
guarantee. The value of these embedded options or guarantees is an integral part of the market value
of insurance liabilities and a correct valuation is therefore essential.
The valuation of embedded options or guarantees in insurance liabilities is important for life insurers for
reporting purposes and internal steering. And it has also gained much more attention because of
Solvency II, the European wide regulatory regime for insurance companies, which has officially come into
force at the beginning of this year. Due to the inherent uncertainty in the future pay-off structure of
insurance contracts containing these embedded options or guarantees, risk-neutral valuation techniques
need to be used to determine their current value.
Risk neutral valuation
The most common and widely used technique for valuing embedded options or guarantees is Monte
Carlo simulation. Using Monte Carlo simulation for valuation purposes is in general based on risk neutral
scenarios. In a risk neutral world every individual is indifferent with respect to risk and therefore expects
to make a return equal to the risk-free rate for all types of investments. The premise of a risk neutral
world simplifies the valuation of options or guarantees: the (future) option cash flows can be determined
within the Monte Carlo experiment and subsequently be discounted along the path of the short-term
risk-free rate to determine the current value of the underlying embedded or guarantee. Please also note
that the risk neutral valuation provides the correct valuation in every world (thus also the risk averse
‘real’ world and not only the risk neutral world).
Contrary to the well-known Black-Scholes formula for put and call options on the asset side of the
insurer’s balance sheet, reliable analytical formulas can generally not be derived for embedded options
or guarantees within insurance liabilities due to their inherent complexity. Therefore, the Monte Carlo
simulation method (based on risk neutral scenarios) is in practice the preferred method for most
insurance companies. Although the method shows enormous flexibility in its ability to correctly value all
types of embedded options and guarantees, computation times can become quite long when the number
of scenarios used for the valuation is large. In order to reduce the computation time needed for the
valuation, insurers are always looking for ways to reduce the time needed without harming the quality of
the actual valuation and keeping the prerequisites – also from a regulatory point of view – in mind.
Keeping the number of risk neutral scenarios as low as possible
without impeding on the quality of the valuation
1
No-arbitrage condition in risk neutral scenario set
A risk neutral scenario (valuation) set should always fulfil a number of requirements (which have also
been identified and prescribed by EIOPA1 for Solvency II purposes). One of the main requirements that
have to be fulfilled is that the risk neutral scenario set should be arbitrage free. Technically speaking, a
risk neutral scenario set should contain the so-called martingale property in order to be arbitrage free.
To illustrate what the martingale property is and how this property could be tested in the risk neutral
scenarios, consider the following example: The present value of €1 invested now in any asset class for any
period of time should be equal to €1 (not taking into account other cash flows) regardless of the development of the future scenarios that are considered in the valuation of the future cash flows corresponding
to this asset. The martingale property is highly sensitive to the volatility in the risk neutral scenarios.
Therefore, typically a large number of scenarios should be used to fulfil the martingale property. The
stricter the test (which essentially means tightening the confidence band for the martingale property to
be accepted), the more scenarios are required.
Enforcing the martingale property
In order to facilitate this to insurers, Ortec Finance has constructed a specialized technique to perfectly
fulfill the martingale property in any risk neutral scenario set irrespective of the number of scenarios
used for valuation. This technique is based on thorough academic research and reduces the statistical
error involved in the risk neutral valuation set. Essentially, the technique is comparable to a moment
matching technique. It avoids more pragmatic and less robust methods such as (for instance) optimizing
the random seed. Further advantages of the method are that martingale conditions are easily fulfilled for
small scenario sets and computation times for valuation processes improve.
The valuation of embedded options is an involved topic for an insurer. Insurers often sell multiple
insurance products with different characteristics for each product, so that characteristics cannot just be
aggregated into a small number of groups for reporting and valuation purposes. In practice a relative
large number of product or policyholder characteristics should be simulated separately for valuation
purposes. The larger the number of individual product or policyholder groups, the higher the
computation times for obtaining the valuation will be. High computation times are highly disliked by
insurers, because it slows down the risk management and reporting process. Especially for (regulatory)
reporting the current deadlines are tight and will only become even tighter going forward.
The computation time increases even further in case the number of risk neutral scenarios used for
valuation increases. Therefore, insurers would like to keep the number of scenarios that are needed for
a correct valuation of the embedded options or guarantees as low as possible, but with the requirement
that (amongst others) the martingale property is fulfilled. Otherwise one does not have a qualified risk
neutral scenario set and cannot guarantee the quality of the valuation. By employing the specialized
technique of Ortec Finance one can be sure with regards to the correctness of the valuation even at
low(er) scenario numbers.
1
See EIOPA publication “Guidelines on the valuation of technical provisions”; sections 55 to 60;
https://eiopa.europa.eu/Publications/Guidelines/TP_Final_document_EN.pdf
Keeping the number of risk neutral scenarios as low as possible
without impeding on the quality of the valuation
2
Figure: Check on the martingale property for an equity series. In this test the discounted equity index should be
equal to the initial index (=1). As can be seen, the higher the number of scenarios the better the replication of the
martingale property; i.e. the blue line in the graph lies closest to 1 for each time period. For small scenario sets the
errors increase however. By using the proprietary approach of Ortec Finance, we are able to generate risk neutral
scenario sets that contain the martingale property even at low scenario numbers as can be seen by the development of the green line.
Ortec Finance’s risk neutral ESG
The specialized technique for enforcing the martingale property within the risk neutral scenario sets is
part of the proprietary risk neutral ESG of Ortec Finance. In the risk neutral ESG one has the flexibility to
generate (stress) scenario sets and a wide range of analysis tools are available for validation. Amongst
others, validation of the martingale property is available for all relevant variables. In this way the ESG
user easily gains insight into the quality of the risk neutral scenario set used for valuation purposes.
Keeping the number of risk neutral scenarios as low as possible
without impeding on the quality of the valuation
3
Rotterdam
Ortec Finance bv
Boompjes 40
3011 XB Rotterdam
The Netherlands
Tel. +31 (0)10 700 50 00
Amsterdam London
Ortec Finance bv
Naritaweg 51
1043 BP Amsterdam
The Netherlands
Tel. +31 (0)20 700 97 00
Ortec Finance Ltd
Suite 9.10, City Tower
40 Basinghall Street
London,EC2V 5DE
United Kingdom
Tel. +44 (0)20 3770 5780
Pfäffikon
Ortec Finance AG
Poststrasse 4
8808 Pfäffikon SZ
Switzerland
Tel. +41 (0)55 410 38 38
www.ortec-finance.com
Toronto
Ortec Finance Canada Inc
250 University Avenue #200
Toronto, ON M5H 3E5
Canada
Tel. +1 416 736 4955