Dr Alexander van Haastrecht explains how the accuracy of market-consistent valuations can be improved using replicating portfolio control variates
Market consistent valuation of insurance liabilities is the most important cornerstone of Solvency II, market consistent embedded value (MCEV) and the international financial reporting standard IFRS 4, Phase 2. An accurate modelling of the insurance liability is of utmost importance and, for this reason, simulation methods have become the standard method for valuing insurance liabilities.
Monte Carlo simulation allows for maximum flexibility in the modelling of embedded options and dynamic policyholder behaviour in local cashflow models. The caveat of the simulation methods lies in the fact that accuracy is highly dependent on the number of scenarios used. The Monte Carlo estimate is, in theory, guaranteed to converge to its true value if we let the underlying number of simulations go to infinity. However, in practice, only a limited number of scenarios can be used and there can be a significant uncertainty surrounding the (not yet) converged liability estimate.
In order to obtain a negligible simulation error, enabling robust hedging and P&L reporting, it is not uncommon for more than 50,000 scenarios to be used for complex products. Many traditional actuarial systems cannot cope with such a high number of scenarios. Another way to improve the accuracy of Monte Carlo valuations is to employ variance reduction techniques.
These techniques improve the simulation accuracy without requiring further scenarios to be calculated.
This article demonstrates how the use of replicating portfolio control variates can dramatically improve the simulation accuracy. A real-life case study demonstrates that using 1,000 scenarios with a replicating portfolio as control variate can give a simulation as accurate as over 3m ordinary simulations would provide.
Reducing simulation noise
Control variates can improve the simulation efficiency by correcting for known simulation errors. For instance, in valuing insurance products that incorporate a high percentage of fixed cashflows, we can compare the simulated value of a portfolio of zero-coupon bonds with the theoretical values based on the yield curve. These differences can be used to correct for the simulation errors incorporated in the underlying insurance product, as shown in Figure 1.
The left box in this example represents a standard Monte Carlo simulation that contains simulation noise for the entire product. Using control variates, simulation noise is eliminated for large parts of the product, such as profit-sharing and fixed cashflows. Only a small simulation noise remains for the product features that cannot be replicated using standard financial instruments.
Effectively, the control variate reduces the need to simulate known product features that can be replicated using standard financial instruments and valued exactly from no-arbitrage or using a closed-form formula. Consequently, only a small portion of the underlying insurance remains subject to simulation noise and the valuation accuracy is improved significantly
The control variate technique
On calculating the market consistent value, L, of an insurance product, the basic Monte Carlo estimator is frequently employed. This estimator averages the discounted product payoffs over all simulation paths, hence providing the following liability estimate:
Although the standard estimator is an unbiased estimator that eventually converges to the true liability value, it is often not very efficient. This means that, typically, a relatively high number of simulations is required to obtain accurate estimates.
Suppose, however, that on each sample path we also calculate another portfolio of related financial instruments along with the underlying insurance liability from which the theoretical value is known in closed-form, such as zero-coupon bonds, swaptions and/or equity options. We can then compare the simulated prices of these instruments with their theoretical values, and use these differences to correct for the simulation noise of the liability value of interest.
The control variate estimator for the insurance product liability value L is given as:
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This takes into account to what degree simulation errors in the control are correlated to similar simulation noise in the liability estimates.
The control variate estimator will always lead to a lower variance than the ordinary estimator and consequently leads to more accurate simulation estimates. That is, the overall improvement is equal to:
where p2 denotes the squared correlation between the control portfolio and underlying liability. Hence, the effectiveness of the control variate technique directly depends on the extent to which the financial instruments in the control portfolio can be attributed to the total variability of the underlying insurance product: the higher the correlation, the higher the effectiveness of the control variate adjustment.
A European example
In Figure 2, we consider the effectiveness of replicating portfolio control variates for a large European traditional insurance portfolio with profit sharing linked to fixed-income investment gains. The use of zero-coupon bonds and swaptions are tested as control variates, for which the scatterplots of the resulting replications are shown.
As expected, the full replicating portfolio is able to explain more of the variability, especially for large losses of the underlying cashflow model. Consequently, the full replication will be more effective as control variate for the underlying insurance liability.
The realised improvements for the zero-coupon bond and full replicating portfolio control variate (CV) portfolios are provided in Table 1.
The use of the simplified control variate consisting of a portfolio of zero-coupon bonds for the fixed cashflows already leads to a simulation uncertainty of 13 basis points and hence reduces the simulation noise by a factor of more than 20. The use of the full replicating portfolio further improves the results, with a simulation noise of 5 basis points. As a consequence, using 1,000 scenarios in combination with the replicating portfolio as control variate gives an estimation as accurate as using more than 3.5m normal simulations.
Summary
Obtaining accurate market-consistent valuation is of great importance for the valuation, risk management and reporting of insurance liabilities. This article demonstrates that the use of replicating portfolio control variates can significantly improve the simulation accuracy of market-consistent liability valuations by removing the uncertainty within parts of the product that can accurately be replicated by a closed-form theoretical valuation.
The level of improvement will be dependent on the extent to which the replication portfolio is able to explain the variability of the true underlying insurance product. The higher the replication portfolio R2 and so the more accurately it reflects the underlying product, the higher the overall variance reduction will be. A moderate R2 of 85% provides an accuracy improvement of a factor of 7; an R2 of 99.9% gives an improvement factor of 1,000.
For the real-life insurance portfolio case study, improvements of over a factor of 3,000 could be observed. Depending on availability of a good-quality replicating portfolio, similar improvements can be attained for other insurance portfolios.
References
F. Chan. Smart modelling: using control variates to boost performance, Insurance matters Asia Pacific, 2009.
P. Jackel. Monte Carlo Methods in Finance. Wiley Finance, 2002.
P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer Verlag, 2003.
S. Morrison. Replicating Portfolios for Economic Capital: Replication or Approximation?,
Barrie and Hibbert Working Paper, 2008.
A, Ng. Replicating Portfolios, SOA Life Spring Meeting Presentation, 2009.
D. Schrager, Replicating Portfolios for Insurance Liabilities, Aenorm, 2008.
The author would also like to thank Paul Elenbaas of ING Insurance & Investment Management EurAsia's market-consistent methodology team
