Volatile Times
D.A. van den As
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: D.A. van den As
Student nr: 10004690
Email: duco.vdas@gmail.com
Date: September 1, 2015
Supervisor: prof. dr. ir. M.H. Vellekoop Second reader: dr. L.J. van Gastel
The purpose of this report is to measure the performance of Solvency II’s adjustments to the interest rate term structure. More specifically we analyze the effectiveness of the Volatility Adjustment in reducing balance sheet volatility caused by Solvency II’s risk-based approach.
This is achieved with a prospective analysis where we simulate an arbitrary balance sheet 1 year into the future using a calibrated one-factor Hull-White model; and subsequently an analysis of hedging portfolios in retrospect where we use solvency ratios to measure the performance of the Volatility Adjustment over the past 8 years.
The simulations show that regardless of the correlation structure or duration match the insurer’s own funds is destabilized when the Credit Risk Adjustment or Volatility Adjust-ment is applied to the interest rate term structure. The impact of the VA is however much lower when an insurer has lower asset duration than liability duration, making the VA more interesting for these insurers. Furthermore we find that standard duration matching is not adequate in the Hull-White model and therefore we use a more sophisticated dura-tion matching technique. It turns out that the choice of fitted swapdura-tion maturities in the calibration procedure is highly important for the simulation results.
The analysis of hedging portfolios in retrospect shows that solvency ratios increase dra-matically if the Volatility Adjustment is used in volatile times. This is caused by both lower Solvency Capital Requirements as well as relative high own funds. The most stable solvency ratios are obtained with hedging portfolios similar to the VA reference portfolio.
Keywords Solvency II, volatility adjustment, credit risk adjustment, Hull-White, Vasicek,
1 Introduction 1
2 Solvency II 4
2.1 Quantitative requirements . . . 4
2.2 Governance and risk management . . . 5
2.3 Disclosure and transparency . . . 5
2.4 Implementation . . . 6
2.5 Chapter two summary . . . 6
3 Modelling Solvency II’s interest rate term structure 8 3.1 Basic interest rate term structure . . . 8
3.2 Credit Risk Adjustment . . . 9
3.3 Volatility Adjustment . . . 11
3.3.1 Volatility Adjustment calculation . . . 11
3.3.2 Currency Volatility Adjustment . . . 12
3.3.3 Country Volatility Adjustment . . . 13
3.3.4 Reference portfolios . . . 16
3.3.5 Risk correction . . . 20
3.3.6 Analysis of Volatility Adjustment . . . 22
3.3.7 A discontinuous Volatility Adjustment . . . 23
3.4 Extrapolation of interest rate term structures . . . 25
3.4.1 EIOPA’s June 2015 Technical specifications . . . 27
3.5 Chapter three summary . . . 29
4 Model Outline 31 4.1 Asset and liability characteristics . . . 31
4.1.1 Asset portfolio . . . 31
4.1.2 Liability portfolio . . . 33
4.1.3 Liability interest rate sensitivity . . . 34
4.2 Balance sheet stability . . . 36
4.2.1 A model for simulations . . . 36
4.2.2 Hull-White model . . . 37
4.2.3 Credit Risk Adjustment and Volatility Adjustment simulations . . . 39
4.2.4 Correlation structure . . . 40
4.2.5 Asset liability duration match . . . 42
4.3 Hedging portfolios in retrospect . . . 46
4.3.1 Duration match . . . 46
4.3.2 Performance measure . . . 48
5.1.1 Credit Risk Adjustment and Volatility Adjustment simulations . . . 50
5.1.2 Asset & Liability distributions . . . 52
5.1.3 Own funds distribution . . . 55
5.1.4 Quality of simulations . . . 58
5.2 Hedging portfolios in retrospect . . . 60
5.2.1 Hedging and liability portfolios over time . . . 60
5.2.2 Solvency ratios . . . 62
6 Conclusion 65
A Modelling Solvency II’s interest rate term structure 70
1
Introduction
With the introduction of Solvency II scheduled for January 2016 insurers are faced with new challenges concerning the approach towards volatility in liability valuations. The risk-based approach of Solvency II should ensure on one hand a better reflection of the current status in the financial markets but on the other hand may introduce a more volatile balance sheet because the present value of liabilities must be marked to market. This report will accomplish an insight into the effects that interest rate term structure adjustments have on the stability of the balance sheet of a Dutch insurer.
Solvency II was adopted by the Council of the European Union and the European Parliament in November 2009. Four years later, in December 2013, the Council adopted a directive setting the application date of Solvency II at January 2016. The delay was due to complexity and long lasting debates between the insurance industry and the Trilogue parties (European Parliament, European Commission and the Council of the European Union) about details in the calculation of the interest rate term structure for liability valuation and the resulting balance sheet volatility. There were concerns that the risk-based approach of Solvency II could lead to artificial volatility and pro-cyclical investment behaviour without the introduction of some measures to address the valuation of long-term insurance products.
After five quantitative impact studies, the European Insurance and Occupational Pen-sions Authority (EIOPA) provided the European Commission in 2013 with technical find-ings in a Long-Term Guarantee Assessment. This led to the adoption of Omnibus II in April 2014 which supplemented Solvency II and included several measures to address the special properties of long-term insurance products. More specifically a Volatility Adjustment (VA) was introduced to counteract the destabilizing effect of the mark to market valuation of liabilities. The VA is a permanent positive or negative adjustment to the interest rate term structure, based on the spread of pre-specified reference portfolios. This adjustment should result in a decrease of balance sheet volatility, should prevent pro-cyclical investment be-haviour and protect against short-term volatility.
To this date no academic literature seems available about the impact of this specific interest rate term structure adjustment on the balance sheet of insurers. Moreover no historical impact analysis of this adjustment seems available, which is most likely due to the fact that the first technical specifications were only released in February 2015 and that insurers will tend to hold the limited amount of information in-house.
Long-term insurance products are of increasing interest for insurers due to the charac-teristics of these products. Relative to banks, insurers have liabilities with higher maturities on their balance sheets. The illiquid character of liabilities and the absence of short-term deposits make it possible for insurers to invest using a long-term strategy. Mortgage loans are an example of an asset to finance long-term insurance products. In the past years the total amount of mortgage loans on the balance sheet of Dutch insurers increased to over
41 billion (DNB, 2015) at the end of 2013.
The VA introduced in Omnibus II is important, because short-term market volatility can affect the balance sheet of insurers dramatically. Failure to recognize the special prop-erties of long-term insurance products will lead to an increase in the required regulatory capital and thus to higher prices for policyholders or lower dividends to shareholders.
Interest rate term structure adjustments play a key role in insurers asset-liability man-agement (ALM). The objective of ALM is to choose the optimal investment portfolio con-sisting of a speculative portfolio that exploits risk premiums and a hedging portfolio that immunizes their liability structure. In this way they protect themselves against changes in the interest rate term structure. A report by (Stanley and Wyman, 2012, p. 9) found that small adjustments to the interest rate term structure can lead to significantly improved sol-vency ratios. These solsol-vency ratios are an indicator for the extent to which the insurer can meet her future obligations. An insurer looking to stabilize her solvency ratio and thereby protecting herself against market volatility will thus find difficulties when adjustments to the interest rate term structure occur.
In this report we analyse the impact of Solvency II’s interest rate term structure ad-justments on the stability of an insurers balance sheet. We pay special attention to the VA which has as primary goal to reduce balance sheet volatility. We analyze the effectiveness of the VA in reducing balance sheet volatility by first taking a prospective outlook and then an analysis of hedging portfolios in retrospect. Therefore we must first find a suitable simulation model and use this model to project an arbitrary balance sheet 1 year ahead. After this is done, we evaluate the performance of the VA in retrospect by constructing different hedging portfolios and pretend the VA was in effect before the financial crisis in 2008.
We expect the interest rate term structure adjustments to have some effect on the stability of an insurers balance sheet. This effect could be undermined by the different calculation methodologies of the various interest rate term structure adjustments. Where 1 adjustment is determined over a previous’ years data, the other is determined using present data based on the spread of pre-specified reference portfolios. The design of these adjustments could cause the interest rate term structure including adjustments to move in unwanted directions in volatile times.
For our analyses in retrospect we need multiple time series; firstly the interest rate swap rates to determine the interest rate term structure; secondly data is needed for the interest rate term structure adjustments. This data includes 3-month Euribor, 3-month Overnight Index Swap and government and non-government yields. EIOPA exactly specified (EIOPA, 2015b) the data sources for the calculation of the interest rate term structure adjustments. Details of this data can be found in Appendix table A.1.
Inevitably we are forced to make several assumptions in this report. In the first stage of our report where the calculation of the interest rate term structure adjustments is given, we assume the information released by EIOPA to be correct and to be roughly applicable to past years of market volatility. Regarding these weights and risk corrections, we are then forced to alter these weights and risk corrections marginally due to an insufficient amount
of available data. In the second stage, where we analyze hedging portfolios in retrospect, we assume that duration matching is adequate to show the impact of the interest rate term structure adjustments.
This report is set up as follows. In chapter 1 we provide an overview of the properties and goals of Solvency II as well as possible implementation challenges. In chapter 2 we show the calculation of Solvency II’s interest rate term structure including adjustments. In chapter 3 we outline the models we use in our simulations and analysis of hedging portfolios in retrospect. Then in chapter 4 we present the results and finally we present our conclusion regarding the effectiveness of adjustments to the interest rate term structure.
2
Solvency II
Solvency II is the successor of Solvency I and will come into force in January 2016. Solvency I stems from 1973 and introduced regulatory capital that insurers are required to hold for unforeseen circumstances. After several additions to Solvency I it became apparent that EU solvency rules should be reviewed thoroughly (European Commission, 2007, p. 1). This led to the adoption of the final version of Solvency II in April 2014.
The main objectives of Solvency II are to harmonize the regulatory frameworks for insurers across the European Union; to ensure financial health of insurers in volatile times; and to protect policyholders and the financial stability as a whole. Three instruments aim to accomplish these objectives. The first instrument is to specify minimum amounts of capital that insurers are required to hold in order to survive volatile times. The second instrument is to ensure a solid governance and risk management system. The third and final instrument is to ensure harmonized disclosure standards and transparency. In that way the public is better informed and insurers can better anticipate and handle difficult situations.
The structure of this framework is to establish an early-warning mechanism which enables supervisors to intervene before things start to go wrong. The framework requires insurers to take all relevant risks into account which will be reflected in the required regulatory capital and eventually in the design of their products. If things do go wrong, the supervisor has enough time to address the situation before insolvency materializes.
Solvency II is not automatically binding in all countries in the European Union but needs to be implemented according to the national constitutions. In the last section of this chapter we elaborate on possible challenges that may arise when Solvency II is implemented in all European Union countries. But first we focus on the instruments to achieve Solvency II’s goals, where our main source of input is a publication by the European Commission (2007) on Solvency II’s structure.
2.1
Quantitative requirements
The first instrument focuses on the capital requirement that an insurer is required to hold in order to survive volatile times. The insurer will first have to calculate the total amount needed to cover expected future claims from policyholders. This amount is called the technical provision and should equal the amount another insurer would pay in order to take over the insurer’s obligations to policyholders. In addition to the technical provision the insurer should have sufficient funds to cover the required regulatory capital to meet obligations with 99.5% certainty in 1 year’s time. This required regulatory capital can either be calculated using the standard formula of Solvency II or using an internal model that is verified by the supervisor. In later chapters we will refer to the technical provisions as the present value of liabilities and handle the required regulatory capital as an additional
add-on on top of the liabilities.
The standard model has a modular structure in which risks are divided in modules and sub-modules. One problem in the standard model is the correlation structure between (sub-)modules used to generate one total capital requirement. Kousaris (2011) argues that the covariance matrix used to aggregate the SCR is not sufficient and that two major un-derlying assumptions are flawed. The first flawed assumption is that a multivariate normal distribution is used and second that linearity between losses and risks is assumed. EIOPA recognizes that non-linearity between losses exist in market risks such as interest rate risk and that probability distributions are most like not normally distributed (EIOPA, 2010, p. 8). They cope with these downsides by choosing the correlation parameters such that they approximate 99.5% Value at Risk. Though these assumptions simplify calculations, important information like tail dependence is not fully taken into account. The standard model underestimates the capital needed by a large amount if two risks simultaneously coincide (Kousaris, 2011).
Insurers with advanced risk management systems can use internal models to calculate, and possibly reduce, their regulatory capital requirement. With an internal model they can possibly align the management of their solvency position more closely with their own view on required regulatory capital. A report by (Stanley and Wyman, 2012, p. 47) however states that: ”in some cases supervisors may insist on assumptions in internal models that lead to higher capital charges than the standard formula”. This may cause insurers to alter their internal model away from the most accurate reflection of their business model.
2.2
Governance and risk management
The second instrument is to ensure a solid governance and risk management system. These components are mainly implemented and assessed using the Own Risk and Solvency As-sessment (ORSA). The main goal of an ORSA is to consider the impact of risks that are not explicitly considered in the standard model and possibly improve shortcomings in the standard model. An ORSA should identify deviations of the insurer’s risk profile from the assumptions underlying the standard model and add a prospective focus to Solvency II. An important remark about the ORSA is that it is not designed as an additional capital requirement. A capital requirement resulting from an ORSA is a possibility for the supervi-sor if capital requirements of the ORSA and standard model do not coincide. In summary the ORSA includes the possible risks, the consequences for the financial position and ways to avert these risks.
2.3
Disclosure and transparency
The third and final instrument is structured to ensure harmonized disclosure standards and transparency. Solvency II requires insurers to report information to a far greater extent than currently is the case. Information under Solvency II is published in four components,
the first component is the reporting of the solvency and financial position, the second component the regulatory report including quantitative information, the third component consists of a report containing pre-defined scenarios and the influence on the financial position and the fourth component contains the policy concerning the disclosure of infor-mation and reporting to the regulator. Important inforinfor-mation about the financial health of the insurer is reported to the outside world, which should increase stability of insurers and competition between insurers.
2.4
Implementation
Solvency II is not automatically binding in all countries in the European Union and needs to be implemented according to the national constitutions. In this final section we look at possible problems that may arise in the implementation phase in each country in the European Union. First we discuss the implementation in European law and then the im-plementation in national law.
The implementation of Solvency II in European law is divided in three stages. The first stage called ”Level 1”, is the directive that sets out the basic principles and deadlines for a European Union-wide implementation. The second stage called ”Level 2”, is the stage where the European Commission defines the specific technical requirements and implements them in European Law. This process was delayed several times due to 5 quantitative impact studies. Based on these studies, the advice from EIOPA and feedback from the industry, the European Commission adopted the final Level 2 measures in October 2014. The third and final stage called ”Level 3”, contains technical standards and guidance developed by EIOPA and national regulators to supplement Level 1 and Level 2 legislation.
After the three levels of Solvency II are implemented in European law they possibly need implementation in some European countries. Whether or not additional implementation is necessary depends on the national constitution, which implies that European countries can change elements or procedures of Solvency II. Important about the Volatility Adjustment is that some European countries require prior regulatory approval. This means the Volatility Adjustment cannot be applied before the regulator consents. For example, the regulator in the United Kingdom and Germany will need prior approval, whereas the Netherlands, France and Italy will not require regulatory approval. These differences affect the level-playing field that the European Union has set as one of her primary goals. Two identical insurers, where one is situated in the Netherlands and one in the United Kingdom can have different capital requirements due to the prior regulatory approval process.
2.5
Chapter two summary
In this chapter we have seen the goals of Solvency II as well as the instruments to achieve these goals. The goals are achieved through three instruments; capital requirements calcu-lated using a standard or internal model; solid governance and risk management systems
mainly through the ORSA; and finally harmonized disclosure and transparency standards to increase stability and competition between insurers. We mentioned that the standard model may underestimate the capital needed by a large amount if two risks simultaneously coincide.
The structure of Solvency II is to establish an early-warning mechanism which enables supervisors to intervene before things start to go wrong. Lastly, we have seen that national implementation can cause problems in the level-playing field between insurers in different European countries due to the fact that some countries require prior approval before the Volatility Adjustment can be applied. In the next chapter we determine Solvency II’s interest rate term structure including adjustments.
3
Modelling Solvency II’s interest rate term
structure
Solvency II’s interest rate term structure consists of the basic interest rate term structure calculated from swap rates and two adjustments. Solvency II introduced the Credit Risk Adjustment (CRA) to adjust interest rate swaps for their credit risk and Omnibus II introduced the Volatility Adjustment (VA) to stabilize balance sheet volatility and to prevent pro-cyclical investment behaviour.
The goal of this chapter is to calculate Solvency II’s interest rate term structure, which we need to determine the impact of these adjustments on the stability of an insurers balance sheet. The main source of input for this chapter is EIOPA’s technical specification (EIOPA, 2015b). We assume that these technical specifications are applicable to past years of market volatility so that we can reliably replicate both adjustments over the past 8 years.
This chapter is set up as follows. First we calculate the basic interest rate term structure from interest rate swaps in section 3.1, then we calculate and subtract the CRA in section 3.2 and add the VA in section 3.3 to construct Solvency II’s interest rate term structure.
3.1
Basic interest rate term structure
The basic interest rate term structure is the interest rate term structure without adjust-ments and it is constructed using Euro interest rate swap rates. This swap rate is the fixed rate that a party must pay to the counter party, when the floating Euro Interbank Offered Rate (Euribor) is received. These swap rates must be derived from deep, liquid and transparent (DLT) markets to ensure reliability. Deep refers to a large quantity of transactions that can take place without significantly affecting the price. Liquid means that instruments can be converted by buying or selling without significantly affecting the price and transparent means current trade and price information is readily available to the public. According to EIOPA, swap rates meet the DLT criteria up to 20 years for the Euro. The highest maturity where DLT swap markets are available is called the last liquid point (LLP). Because DLT swap markets are not available in all countries, other methods are introduced by EIOPA to derive the basic interest rate term structure for such coun-tries. We focus on a Dutch insurer, therefore details on the calculation of the interest rate term structure for non-Euro area insurers will only be discussed briefly. When DLT swap markets are not available, government bonds that meet the DLT criteria must be used instead. If DLT government bonds are also not available, the interest rate term structure must be agreed upon by the insurer and national supervisor in that respective country. At the moment this last situation is not present in the European Union but there are some countries that have DLT markets for government bonds and not for swaps such as Croatia, Iceland and Poland.
EIOPA defined the LLP at 20 years for the Euro, which implies that an extrapolation method must be used to calculate the basic interest rate term structure beyond the LLP. This extrapolation is carried out by EIOPA using the Smith and Wilson (2001) extrapola-tion method. However, before we can use the Smith-Wilson extrapolaextrapola-tion method we need to subtract the CRA and add the VA to the basic interest rate term structure. Therefore we will first explain the calculation method of the CRA in the next section.
3.2
Credit Risk Adjustment
The interest rate term structure in Solvency II is calculated using interest rate swaps for Euro area countries. Interest rate swaps are agreements between two parties to exchange a periodic fixed interest payment and a periodic floating interest payment based on an interest rate over a term to maturity. The interest rate payments are exchanged for a specified period based on a notional amount.
This implies that some credit risk is present because credit risk is the risk that a borrower will fail to repay a certain loan. In interest rate swap agreements, the credit risk in the agreement itself is limited because this credit risk is symmetrical and also secured with collateral. The majority of credit risk in interest rate swaps is therefore present in the underlying 3-month Euro interbank offered rate (Euribor). This is the interest rate that banks charge each other for lending funds. The higher the risk that these banks become insolvent, the higher the interest that borrowers will require for lending capital. This implies that credit risk is not absent in interest rate swaps and therefore the CRA is appropriate. The CRA in Solvency II is applied as a parallel downward shift of the interest rate term structure. The CRA is defined as half of the average daily spread over 1 year (255 days1), between 3-month Euribor and Overnight Indexed Swap (OIS) with the same maturity. After half of the average spread over 1 year is determined this average is capped between 10 and 35 basis points (bps):
CRA = min(max(50% · (Euribor − OIS), 0.1), 0.35) bps. (3.1)
The CRA is capped to ensure a sufficient decrease when markets exhibit low volatility and on the other hand maximizes the adjustment when volatility in credit spreads is present.
The CRA is computed using DLT swap markets when these are available. When DLT swap markets are not available and the basic interest rate term structure is determined using DLT government bond markets, the CRA equals the CRA for the Euro. Because there is no European country in which the DLT market requirement is not met, we will not discuss the situation where both DLT swap and DLT government bond markets are not available.
Based on the data in Appendix table A.1 and the above specifications we computed the historical development of the CRA and show the results in figure 3.1.
Figure 3.1: CRA for the Euro in basis points to be subtracted from the basic interest rate term structure to incorporate a realistic assessment of expected losses, unexpected credit risk or any other risk. Sample from April 2004 to June 2015.
In the historical development of the CRA we can make some interesting observations:
• The effect of the backwards based calculation of the CRA is clearly visible. When the Euribor over OIS spread is high, the CRA adjusts with a delay because it is determined as a moving average over the past year.
• When spreads are low the CRA equals its lower bound of 10 basis points. When spreads increase the CRA increases, and when spread are very high the CRA reaches its upper bound of 35 basis points. In order for the CRA to increase, a prolonged pe-riod of high spreads is necessary. The same observation occurs when spreads decrease, the CRA needs time to adjust to lower levels even when spreads already returned to normal levels.
The observations we made point out that the CRA increases when volatility in credit spreads increases. This causes the interest rate term structure including CRA to decrease in volatile times and assigns a higher present value to liabilities during a crisis. This continues to have an effect even after the volatility of the crisis diminishes due to the prolonged low spread needed to let the CRA decrease. A report by the European Commission (2007, p. 7) described pro-cyclicality as: ’rules that unnecessarily amplify swings in underlying economic cycles or contribute to excessive market movements’. Based on this definition we find that the CRA indeed causes pro-cyclicality.
The CRA can now be subtracted from the basic interest rate term structure. With the credit risk adjusted interest rate term structure known, we can move on to the VA.
3.3
Volatility Adjustment
The Volatility Adjustment (VA) was introduced in Omnibus II (2014) after EIOPA pro-vided the European Commission in 2013 with the technical findings on the Long-Term Guarantee Assessment (2013). In this Long-Term Guarantee Assessment EIOPA gathered several key points that the insurance industry brought forward as important deficiencies in the then foreseen regulation. CRO Forum (2014) stated that:
’It is of great importance to shelter insurance undertakings against undue short-term mar-ket volatility and pro-cyclicality. Undervaluing the VA by 10 basis points could have an impact of at least 12 billion Euro on surplus capital resources in the whole of Europe’. A large move away from long-term insurance products to shorter term assets was expected, because in their opinion it would become too expensive for insurance companies to value long-term insurance products in volatile markets. This was also expected to result in a move away from insurers offering long-term insurance products, because these products would not be viable any more due to the excessive capital charge that had to be incurred on policyholders.
Because insurance companies were seen as a stabilizer of systemic risk (EIOPA, 2013, p. 41), the limiting role of insurance companies could result in more systemic risk thereby increasing the risk of a collapse of the financial system as a whole. By increasing the stability of an insurer’s balance sheet, the insurer may be less inclined to sell assets in volatile times, preventing contagion effects across financial markets.
These issues where discussed in the Long-Term Guarantee Assessment that EIOPA sent to the Trilogue parties. The Trilogue parties recognized the need for a solution and eventually adopted Omnibus II which included the VA to counteract balance sheet volatility and to prevent pro-cyclical investment behaviour.
3.3.1
Volatility Adjustment calculation
The VA is rounded in basis points and added as a parallel upwards or downwards shift to all maturities until the last liquid point (LLP). The VA consists of two components, a component specific for a currency and a component specific for a country. The country component is included to represent the volatility more precisely per country.
There are two important facts to consider when applying the VA. The first fact is that both components should only apply to a liability portfolio in a specific currency or country. The insurer obtains a liability portfolio with subportfolios that have a distinct VA for the currencies and countries in that subportfolio.The second fact is that the decision of the insurer to use the VA, must be applied consistently such that the insurer cannot only choose to apply the VA in volatile times when they find application most suited (European Council and the European Parliament, 2014, Art. 51).
Both the currency and country VA are determined using risk adjusted spreads of pre-specified government and non-government bond reference portfolios. The spreads of all reference portfolios are adjusted with a risk correction to take into account a realistic
assessment of expected losses, unexpected credit risk or any other risk. In figure 3.2 an overview of the VA is given.
Volatility Adjustment Currency Volatility Adjustment Country Volatility Adjustment Non-government bonds currency Government bonds currency Non-government bonds country Government bonds country Weighted reference portfolio Risk Correction Weighted reference portfolio Risk Correction Weighted reference portfolio Risk Correction Weighted reference portfolio Risk Correction
Figure 3.2: Overview of VA calculation.
Due to the complexity of the VA calculation, we start of with the general methodology of both the currency and country specific VA and postpone the reference portfolio spreads and risk corrections to section 3.3.4 and 3.3.5 respectively.
3.3.2
Currency Volatility Adjustment
In this section we begin with the general methodology of the currency VA. The currency VA is defined as 65% of the spread between the yield of a reference portfolio, containing currency specific assets over the interest rate and a risk correction term (European Council and the European Parliament, Art. 77d):
V Acrncy = 0.65 · SRC−crncy. (3.2)
A visual representation of the currency VA is given in figure 3.3.
S = Currency portfolio spread over interest rate (1.00%) RC = Risk correction
(0.25%)
S-RC = Risk adjusted currency spread (S-RC = 0.75%)
Currency VA =
65% of risk adjusted currency spread (65% * (S-RC) = 0.4875%)
The currency specific reference portfolio spread consists of two components. The first component is the currency spread on government bonds and the second component is the currency spread of non-government bonds. These components are weighted using weights given by EIOPA (Appendix table A.4). The weights are obtained by a ratio between the respective aggregated market values. For example, the weight for government bonds is obtained by dividing the aggregated market value of government bonds by the aggregated market value of government bonds and other assets like equity and property. If a change in market value occurs, EIOPA will publish these changed weights at least on quarterly basis (European Council and the European Parliament, 2014, Art. 77e(1c)).
We analyze the VA for a Dutch insurer and therefore we use the weights for the Euro in the formulas for the currency VA:
ScrncyEuro = 0.387 · max(SgovEuro; 0) + 0.482 · max(Snon−govEuro ; 0). (3.3)
The currency specific reference portfolio spread cannot be negative and the non-government bond weight is larger than the government bond weight. After the spread SEuro
crncy from the
reference portfolio is calculated, a risk correction (RC) is deducted to incorporate credit risk. This risk correction represents a realistic assessment of expected losses, unexpected credit risk or any other risk and consists of two components that are weighted using the same weights:
RCcrncyEuro = 0.387 · RCgovEuro+ 0.482 · RCnon−govEuro . (3.4)
The first component of the risk correction corresponds to the risk correction for the gov-ernment bonds spread and the second component corresponds to the risk correction for the non-government bonds spread. The calculation of the RC is postponed to section 3.3.5. After the spread ScrncyEuro and risk correction RC are determined, the risk adjusted currency spread can be calculated using:
SRC−crncyEuro = ScrncyEuro− RCEuro
crncy. (3.5)
The risk corrected currency spread can be negative if RC > Scrncy, which is the case
when markets are excessively optimistic. The reference portfolio spread will then be low and when combined with a positive RC this will results in a negative VA. The resulting currency VA equals:
V AEurocrncy = 0.65 · SRC−crncyEuro . (3.6) The currency VA is then added to the interest rate term structure before extrapolation and results in an increase if the risk corrected currency spread is positive.
3.3.3
Country Volatility Adjustment
The second component of the VA is a country specific VA and is included to describe the volatility more precisely per country. The currency reference portfolios can be seen as a more general version of the country reference portfolios because the country VA is only
added if additional criteria are met. The risk corrected country spread must exceed the risk corrected currency spread more than twice and the country specific spread must at least be 100 basis points:
V ARC−country = 0.65 · max(SRC−country− 2 · SRC−crncy; 0), (3.7)
if SRC−country > 100 basis points.
In this section we discuss the general methodology of the country VA. The calculation of the reference portfolio spread and risk correction is again postponed to section 3.3.4 and 3.3.5 respectively. The structure of the country VA is showed in figure 3.4.
S = Country portfolio spread over interest rate (2.50%) Amount by which country
portfolio spread exceeds twice the currency portfolio spread (0.5%)
Country VA = 65% of the amount by which the country portfolio spread exceeds twice the currency portfolio spread (0.65*0.5% = 0.325%)
Currency portfolio spread over interest rate (1.00%)
Figure 3.4: Country VA, equal to 65% of the difference between the country spread and twice the currency spread. The country spread must also be higher than 100 basis points.
When the risk corrected country spread is at least twice the risk corrected currency spread and the risk corrected country spread is lower than 100 basis points, the total VA equals the currency VA:
V Atotal = 0.65 · SRC−crncy. (3.8)
This means there is a discontinuity point when using the VA and this can cause great difficulties when hedging the interest rate term structure. A very small increase could lead to a substantial increase in the total VA and to a shift in the interest rate term structure because of the country VA. For example: if the risk corrected currency spread equals 40 basis points, and the risk corrected country spread equals 99 basis points, the country VA is zero. If this risk corrected country spread increases to 101 basis points, the country VA equals:
0.65 · (101 − 2 · 40) = 13.65 basis points. (3.9)
The interest rate term structure will increase with 40 basis points (0.65 · (101 − 40)) instead of 26 basis points (0.65 · 40), while the country VA only increases 2 basis points.
The risk corrected country spread is determined in the same way as the risk corrected currency spread SRC−crncy. This means that we calculate the risk corrected country spread
(Ap-pendix table A.4) and obtain:
ScountryN L = 0.411 · max(SgovN L; 0) + 0.485 · max(Snon−govN L ; 0). (3.10)
The weights given by EIOPA for the country VA differ from the weights for the currency VA, because both the government and non-government spread weight are higher. This indicates that both government and non-government bonds constitute a larger value relative to the total value of assets in the Netherlands (section 3.3.2). Again we postpone the calculation of the spread of the government and non-government bonds reference portfolio to section 3.3.4. The risk correction term for the country VA is determined in the same way as the risk correction term for the currency VA and can be calculated using:
RCcountryN L = 0.411 · RCgovN L+ 0.485 · RCnon−govN L . (3.11)
The resulting risk corrected country spread then equals:
SRC−countryN L = ScountryN L − RCN L
country. (3.12)
Now we have defined both the risk corrected currency and country spreads, the total VA can be seen in figure 3.5 and calculated using:
V Atotal = 0.65 · (SRC−crncyEuro + max(S N L
RC−country− 2 · S Euro
RC−crncy; 0)), (3.13)
if SRC−country > 100 basis points.
Country VA (0.325%)
Currency VA (0.4875%) Total VA (0.8125%)
Figure 3.5: VA, consisting of both the currency component and the country component.
The total VA is then added to the interest rate term structure after the CRA is sub-tracted as a parallel shift until the LLP.
After the LLP, EIOPA uses the Smith-Wilson extrapolation method to calculate the interest rate term structure for long maturities. In the next section we first describe the reference portfolios and risk correction term as specified by EIOPA.
3.3.4
Reference portfolios
The VA is an adjustment to the interest rate term structure, which is based on the spread of pre-specified reference portfolios. These reference portfolio spreads are input for equa-tions (3.3) and (3.10), that we used to calculate the currency and country VA for the Netherlands.
The reference portfolio weights will be updated annually, based on data submitted at the previous year-end. So for example, the reference portfolio weights for 2016 are based on the year-end 2014 submission (EIOPA, 2015b, Art. 198). This lag creates stability in the VA reference portfolios because a sudden shift in market conditions will not immediately change the weights in the reference portfolios.
Both currency and country VA consist of a portfolio based on government and non-government bonds. This implies we need a Euro currency non-government and non-non-government portfolio and a Dutch country government and non-government portfolio. In this report we assume the non-government portfolios of the Euro and the Netherlands to be the same, because of insufficient country specific non-government bond data:
Snon−govEuro = Snon−govN L . (3.14)
In the next sections we will first analyze the government portfolios and then the non-government portfolio.
Government portfolios
In this section the government portfolios for the Euro currency and the Dutch country VA are analyzed. These portfolios consist of different countries weighted according to the market value, duration and credit quality of the respective country. Following the technical specifications (EIOPA, 2015b) we obtain the Euro currency government portfolio and the Dutch country government portfolio in table 3.1.
These weights differ slightly from the weights specified by EIOPA because insufficient data was available for Luxembourg, Malta, Slovenia and Croatia. The full specifications can be found in Appendix table A.2. Important considerations for omitting these countries are the relative small weights in both the Euro currency and Dutch country government portfolio. The contribution of the countries omitted is added to comparable countries ac-cording to the credit rating of each particular country. A more visual representation of the government reference portfolios used can be found in figure 3.6.
EIOPA states in her technical specifications that a CRA must be determined for each respective currency. Both government portfolios use a non-Euro country which implies ad-ditional CRA calculations for Poland because the currency in Poland is not the Euro. In Poland, no DLT swap markets are available which suggests that the government bond mar-ket must be used instead. However, when only government bond marmar-kets are available, the CRA for that currency is the same as the CRA for the Euro currency as we have explained in section 3.2. Therefore the government reference portfolio spreads can be calculated using
Currency (Euro) portfolio Country (NL) portfolio Country Rating Weight (%) Duration (years) Weight (%) Duration (years)
Austria AA+ 4.40 9.7 9.0 12.8
Belgium AA 10.0 8.3 5.0 12.1
Czech Rep. A+ 0.1 5.8 -
-Finland AAA 0.8 8.8 2.0 12.0
France AA 31.8 9.6 10.0 17.1
Germany (+LU) AAA 11.7 9.5 33.0 13.4
Ireland (+MT) A- 1.3 5.5 - -Italy BBB 25.2 7.3 1.0 10.8 Netherlands AAA 4.2 9.6 38.0 10.6 Poland A- 0.4 7.4 1.0 10.6 Portugal BB+ 1.1 4.3 - -Slovakia A 0.6 7.2 - -Spain (+SI/HRK) BBB 8.4 9.3 1.0 10.9
Sum / Average duration 100.0 8.51 100.0 12.5
Table 3.1: Government reference portfolios for the Euro and the Netherlands used in this report. Ratings are obtained by comparing and choosing the second best of the Standard and Poors, Moody’s and Fitch ratings obtained on 15 April 2015.
Austria: 4.4% Belgium: 10% France: 31.8% Germany: 11.7% Italy: 25.2% Netherlands: 4.2% Spain: 8.4% Other: 4.3%
(a) Euro currency government portfolio
Austria: 9% Belgium: 5% France: 10% Germany: 33% Netherlands: 38% Other: 5%
(b) NL country government portfolio Figure 3.6: Government reference portfolios used in this report. For visibility all countries under
4 percent are grouped in category ’other’. In the reference portfolios used in this report these will be included (table 3.1).
only the CRA for the Euro.
The government portfolios for the Euro and the Netherlands are now computed by multiplying the weights from table 3.1 with the yields of the respective countries. The maturity of the yields is determined by the average duration in the reference portfolios. We use a maturity of 8 years for the Euro government portfolio and a maturity of 12 years for the Dutch government portfolio. We rounded these durations to the lower integer because more liquidity is available in our data in those maturities. The development over time of these weighted government portfolios is shown in figure 3.7.
01−2007 05−2008 10−2009 03−2011 08−2012 01−2014 06−2015 0% 1% 2% 3% 4% 5% 6% Date Yield
8Y Weighted Euro Currency Portfolio 12Y Weighted NL Country Portfolio
Figure 3.7: Development of Euro currency and Dutch country government portfolios. They fol-low similar paths until late 2008 after which some deviation occurred that disap-peared after August 2013.
With the weighted government portfolio yields we are able to determine the government currency spreads for the Euro (SEuro
gov ) and the Netherlands (SgovN L). EIOPA determines the
government currency spread using the internal rate of return (IRR) for non-periodic cash flows. This method calculates the return from an investment and is defined as the rate that makes the net present value of all cash flows equal to zero taking into account the exact number of days in between two cash flow dates. For example, if the yield of the weighted 8 year government bond portfolio is 4%, the PV equals:
P V = −1 + (1 + 0.04)
8
(1 + r)D = 0, (3.15)
where D is the number of days expressed in years between the cash flow dates and r the rate that makes these cash flows zero. This method is repeated for the interest rate term structure including the CRA. The government currency spreads for the Euro (SEuro
gov ) and
the Netherlands (SN L
gov) are then obtained by subtracting the 8 or 12 year interest rate term
structure IRR, from the IRR of the government portfolios.
In the next section we will analyze the non-government portfolios that are used to calculate the government portfolio spread for both the Euro and the Netherlands.
Non-government portfolios
In this report we assume that the non-government bond portfolio for the Euro is equal to the non-government bond portfolio for the Netherlands (equation (3.14)). This is because of the lack of country specific non-government bond data. We use the weights that EIOPA specified in the technical specification according to the market value, duration and credit quality of the respective non-government bond. The non-government portfolio consists of
bonds of financials and non-financials based on their respective credit rating, see table 3.2.
Non-government portfolio Asset class Rating Weight (%) Duration (years)
Financial AAA 17.6 6.4 Financial AA 11.8 5.9 Financial A 23.5 4.6 Financial BBB 12.9 4.3 Non-financial AAA 1.2 6.3 Non-financial AA 5.9 5.5 Non-financial A 12.9 5.9 Non-financial BBB 14.1 5.2 Sum / Average duration 100.0 5.4
Table 3.2: Specifications of the non-government reference portfolios for the Euro and the Nether-lands used in this report. Weights are adjusted because some country specific non-government bond data is not available.
These weights differ slightly from the weights specified by EIOPA because non-government data was not available for some countries. Full specifications of the data can be found in Appendix table A.3. We proportionally increase the weights of the non-government port-folio to include the 15% of the data that is not accounted for. The portport-folio weights for the asset classes in table 3.2 are therefore normalized to include the missing 15% of data. Regardless of whether or not this assumption is plausible, it is at this moment not possible to construct a more realistic portfolio using the information given by EIOPA. A visual representation of the non-government reference portfolios used can be found in figure 3.8.
Fin AAA: 17.6% Fin AA: 11.8% Fin A: 23.5% Fin BBB: 12.9% Non−fin AAA: 1.2% Non−fin AA: 5.9% Non−fin A: 12.9% Non−fin BBB: 14.1%
Figure 3.8: Non-government reference portfolios used in this report.
The yields for non-government portfolios for the Euro and the Netherlands are now computed by multiplying the weights from table 3.2 with the yields of the respective bonds. The maturity of the yields is determined by the average duration in the reference portfolio. We therefore use a maturity of 5 years for the non-government portfolio. The development over time of the yield for this weighted non-government portfolio is shown in figure 3.9.
01−2007 05−2008 10−2009 03−2011 08−2012 01−2014 06−2015 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Date Yield
5Y Weighted Non−government Portfolio
Figure 3.9: Historical development of the weighted non-government bond portfolio. A large peak at the end of 2008 is clearly visible after which the reference portfolio yields decline rapidly until early 2011. After a relative small peak, they declined until now.
Now that the 5 year weighted non-government portfolio yields are computed, we de-termine the non-government portfolio spread using the IRR method described earlier. The IRR of the 5 year basic interest rate term structure including the CRA is subtracted from the IRR of the weighted non-government portfolio yield to obtain the non-government currency and country spread for the Euro (Snon−govEuro ) and the Netherlands (Snon−govN L ).
3.3.5
Risk correction
So far we have determined both the government and non-government portfolio spreads. In this section we compute the risk adjusted currency and country spread (equation (3.5) and (3.12)), by subtracting the risk corrections from the government and non-government portfolios spreads found earlier. The RC consists of two components, a government RC and a non-government RC that are weighted using weights given by EIOPA (Appendix table A.4). The RC for government bonds depends on the percentage exposure to member states or non-member states:
RCgov =
30% · LTAS for exposure to member state government bonds 35% · LTAS for exposure to non-member state government bonds.
(3.16)
The Long-Term Average Spread (LTAS) is the average spread over the past 30 years between the currency or country interest rate term structure and the basic interest rate term structure with CRA. As the name would suggest, this is a measure for the average amount by which the country interest rate term structure exceeds the risk-free interest rate term structure.
probability of default (PD) plus the cost of downgrade (CoD):
RCnon−gov = max(35% · LTAS, PD + CoD). (3.17)
The PD and CoD are not included in the government RC because Recital 22 of the Dele-gated Regulation (European Commission, 2015) states that: ’where no reliable credit spread can be derived from default statistics, as in the case of exposure to sovereign debt, the risk correction for the VA should be equal to the portion of the LTAS set out in Article 77c of Directive 2009/138/EC’.
Fortunately, the LTAS, PD and CoD are provided by EIOPA in the technical specifi-cations. The LTAS is given for government and non-government bonds in each currency, country and maturity.
The government RC is equal to the correct maturity LTAS times 30% for member state government bonds and 35% to non-member state government bonds. The Euro currency portfolio with 8 year duration therefore has the 8 year LTAS rates and the Dutch portfolio with 12 year duration has the Dutch 12 year LTAS rates. The non-government RC also consists of the PD and CoD. They are determined in a similar way to the government RC, that is, by finding the correct country or currency and maturity. Following these steps we compute the RC for the government portfolio of the Euro and the Netherlands as well as the RC for the non-government portfolio that is the same for both countries:
Government Non-government Euro RC (%) NL RC (%) RC (%) 30% LTAS 0.1133 0.0096 -35% LTAS - - 0.4674 PD - - 0.1067 CoD - - 0.0824 RC 0.1133 0.0096 0.4674
Table 3.3: Risk corrections for the Euro and the Netherlands.
These RCs are not directly subtracted from the currency, country or the non-government reference portfolio spread, but are determined by the spread between the IRR of the re-spective reference portfolio yield and the IRR of the rere-spective reference portfolio yield minus the RC found above:
RCnon−govEuro = IRRnon-gov reference portfolio− IRRnon-gov reference portfolio−RCnon−gov, (3.18)
and is repeated for the Euro currency RC (RCgovEuro) and Dutch country RC (RCgovN L). Now we have computed the RC for the Euro currency and the Dutch country portfolio, we can determine the VA for a Dutch insurer. We subtract the RC from the weighted Euro currency and Dutch country portfolio spreads and obtain the risk corrected spreads. The VA is then computed using equation (3.13). In the next section we analyze the VA for a
Dutch insurer and comment on its development over the past 8 years.
3.3.6
Analysis of Volatility Adjustment
In this section we analyze the VA by looking at the historical development over the past 8 years. The VA for a Dutch insurer over time is shown in figure 3.10 and consists solely of the currency VA because the conditions for the additional country VA as described in section 3.3.3 are not met. For the country VA to come into action, the country spread must be higher than 100 basis points and the country spread must at least be twice as high as the currency spread. The VA we see in figure 3.10 is therefore the Euro currency VA.
01−2007 05−2008 10−2009 03−2011 08−2012 01−2014 06−2015 −0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% Date Adjustment VA Netherlands CRA
Figure 3.10: Historical development of the VA from January 2007 to June 2015 for a Dutch insurer. The VA equals the Euro currency VA because the additional requirements for the Dutch country VA are not met.
When we look at the historical development of the VA we can make the following interesting observations:
• In the beginning of our sample the VA is negative, which is possible when the RC is larger than the currency portfolio spread. This implies that the decrease of the interest rate term structure by the CRA is reinforced by the VA when the financial markets are relatively calm.
• In 2009 the VA increased to more than 200 basis points and to 150 basis points in early 2012. This will most definitely result in extreme decreases of the present value of liabilities in volatile times. In the rest of our sample one can see that the VA almost always exceeds the CRA resulting in a higher interest rate term structure, and therefore lower present value of liabilities when the VA is used in volatile times. • The VA is quicker to react to volatility of the financial markets than the CRA. This is also logical because the CRA is a backwards based adjustment and the VA is an adjustment calculated in real time.
3.3.7
A discontinuous Volatility Adjustment
The VA calculated in the above example equals the Euro currency VA. In our earlier discussion about the VA in section 3.10 we noticed the country VA to have a possible discontinuity point, because the condition of a country spread of at least 100 basis points is necessary for the country VA to come into effect. Because we want to analyze this behaviour we pick a country which shows more volatility. The country that we choose for additional analysis is Greece, because of the financial crisis in 2007 and the sovereign debt crisis in the years after. In order to calculate the VA for Greece we obtain the Greek reference portfolio weights in table 3.4.
Country (GR) portfolio Country Rating Weight (%) Duration (years)
Austria AA+ 1.0 4.9 Belgium AA 4.0 3.6 France AA 19.0 5.6 Germany AAA 10.0 4.2 Greece CCC+ 22.0 6.2 Ireland A- 2.0 6.8 Italy (+RON) BBB 14.0 7.2
Netherlands (+LU) AAA 11.0 6.1
Poland A- 2.0 6.1
Portugal BB+ 5.0 5.7
Slovakia A 1.0 8.9
Spain BBB 6.0 6.3
USD AAA 3.0 8.1
Sum and Average duration 100.0 6
Table 3.4: Government reference portfolio for Greece used in this report. The ratings are ob-tained by comparing and choosing the second best of the Standard and Poors, Moody’s and Fitch ratings obtained on 15 April 2015.
France: 19% Germany: 10% Greece: 22% Italy: 14% Netherlands: 11% Poland: 5% Spain: 6% Other: 13%
Figure 3.11: Greek government reference portfolio used in this report. For clarity all countries under 4 percent are omitted. In the reference portfolios used in this report these will be included (table 3.4).
Because insufficient data was available for Romania and Luxembourg, the weights differ slightly from the official weights given by EIOPA. The full specifications of the Greek country portfolio can be found in Appendix table A.2. An important consideration for omitting these countries is the relative small weight in the country government portfolio which can justify omitting these countries. The contribution of the countries omitted is added to a comparable country according to credit rating. A visual representation of the government reference portfolios is shown in figure 3.11.
The risk correction needed to calculate the risk corrected country spread for Greece, makes use of the Euro non-government portfolio RC and a RC for Greece individually. So the non-government portfolio satisfies:
RCnon−govGR = RCnon−govEuro . (3.19)
The RC for Greek government bonds equals: 1.6669 and is a lot higher than the Dutch RC. With this RC we have computed the development of the VA for Greece in figure 3.12.
01−2007 05−2008 10−2009 03−2011 08−2012 01−2014 06−2015 −0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Date Adjustment VA Greece VA Netherlands CRA
Figure 3.12: Historical development from January 2007 to June 2015 of the VA for Greece and the Netherlands where the latter equals the currency VA.
The conditions for the country VA for Greece to come into effect are satisfied, which leads to the following interesting observations:
• The development of the VA for Greece followed roughly the same development as the VA for the Netherlands until 2011. The VA for Greece is negative at the beginning of our sample resulting in an even further increase in the present value of liabilities.
• In March 2011 the country VA for Greece comes into effect and the two deviate from each other. The VA for Greece almost hits 350 basis points which would result in an enormous upwards shift of the interest rate term structure resulting in an enormous decrease in the present value of liabilities.
• At the beginning of 2012 the VA for Greece starts to decrease and approaches the VA of the Netherlands.
The effect of the discontinuity of the country VA is not as pronounced as we would have expected to have seen. Though some small peaks are visible in April 2011 and May 2015, discontinuities as a result of the country VA resulting in a sudden shift of the interest rate term structure are absent.
We have seen that the VA shows some peculiar behaviour in the last years. The VA can cause an even further decrease of the interest rate term structure when volatility is low causing the present value of liabilities to increase further. Important is that if the VA is applied by an insurer in a currency with a low RC (equation (3.17)), this causes the VA to be relatively high when volatility in markets is low. Therefore it will be more attractive for an insurer in certain currencies with low RC’s to apply the VA. This is also exactly the purpose of the VA, to promote investments in low-risk countries with low RC’s resulting in a higher benefit from using the VA.
3.4
Extrapolation of interest rate term structures
At this stage we constructed the basic interest rate term structure with CRA and VA until the LLP. This is the final point where EIOPA deems the DLT market conditions to be satisfied. After the LLP the interest rate term structure is extrapolated using the Smith and Wilson (2001) extrapolation method. In order to obtain a better understanding of the steps we took so far, we provide an overview in figure 3.13. Here the steps that are needed to calculate Solvency II’s interest rate term structure can be found.
Risk adjusted interest rate Deduct CRA Remove data beyond LLP Swap rates Apply VA Extrapolate to UFR Add VA until LLP Extrapolate to UFR with VA until LLP Discount using interest rate with VA Discount using interest rate without VA yes no
Figure 3.13: Interest rate term structure according to EIOPA (2015b).
The Smith-Wilson extrapolation method uses the idea of an ultimate constant level to which the forward rate converges which is called the Ultimate Forward Rate (UFR). The UFR is based on estimates of expected inflation and the long-term average of short-term
real rates. The UFR must be stable and may only change when long-term expectations about inflation or long-term average of short-term real rates change. For the Euro area, the UFR equals 4.2 percent and this consists of 2% expected inflation and 2.2% long-term average of short-term real rates. With this UFR the interest rate term structure can be extrapolated beyond the LLP.
The method that EIOPA developed to determine the LLP is rather interesting. It is called the residual volume criterion and first calculates the outstanding volume for all maturities of both government and non-government bonds and then calculates the maturity (x) where the cumulative sum of volume is 6% smaller than the total value of outstanding bonds at all maturities:
LLP = inf(x ∈ R :XVolumebonds(x) ≥ 0.94Volumetotal) (3.20)
For the Euro the total volume of government and non-government bonds is determined by summing the bonds in all Euro countries and this results in a LLP of 20 years. For currencies other then the Euro, the LLP is also chosen according to this DLT market assessment. Why EIOPA uses 6% is not specified in the technical specifications, nor is it based on empirical evidence. This point is chosen because supposedly the DLT market criteria after the LLP is not met. However EIOPA persists in using a LLP of 20 years for the Euro which in our opinion strengthens the idea that political incentives and the ”pure” mathematical calculation of the interest rate term structure are intertwined.
After the LLP is determined for the Euro we can compute the convergence point. This is the point where the forward rate must equal the UFR that is chosen in the Smith-Wilson extrapolation method. The forward rate is the future rate between two consecutive times and is defined as:
fk : yearly rate between times t = k ≥ 0 and t = k + 1. (3.21)
The convergence point is determined as max(LLP + 40, 60) and for the Euro we obtain a convergence point of 60 years. This means that after 60 years the forward rate must have converged to the UFR that was chosen. In Sweden for example a LLP of 10 years exists which does not reduce the convergence point. The swap rates in the United Kingdom are liquid until 50 years and this implies a convergence point of 90 years.
Now we have the LLP and the convergence point, the rate at which the forward rate converges to the UFR must be chosen. In EIOPA’s technical specifications the speed of convergence (α) is chosen using a goal seek approach. The difference (τ ) between the forward rate and UFR must be smaller than 1 basis point at the convergence point:
|UFR − forward rateconvergence point| < 0.0001. (3.22)
The α that we use to calculate this UFR must be calibrated to 6-digit precision and the minimum value for α is 0.05, which is to ensure timely convergence. The UFR parameters for the Euro are summarized in table 3.5.
Parameter EIOPA Ultimate Forward Rate 4.2% Last Liquid Point 20 years Convergence Point 60 years Convergence Precision τ 1 basis point Convergence Precision α 6-digit precision Minimum α value 0.05
Table 3.5: EIOPA’s UFR parameters used in interest rate term structure extrapolation.
Now all the details of the interest rate term structure calculation are known, we are able to calculate the interest rate term structure including adjustments that is used to determine the present value of liabilities in Solvency II:
0 10 20 30 40 50 60 70 80 90 100 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Maturity (year) Forward rate Swap rate Swap rate incl. CRA Swap rate incl. VA and CRA Extrapolated
UFR
LLP
Figure 3.14: Interest rate term structure as of 1 June 2015 according to the technical spec-ifications of EIOPA. Credit Risk Adjustment equals 10 basis points, Volatility Adjustment equals 18 basis points and α equals 0.131503.
In this figure the interest rate including CRA and VA is showed. The forward rate is nowhere near the UFR of 4.2% at the LLP. The α is determined using the goal seek approach of EIOPA and decreases when the VA is added to the basic interest rate term structure with CRA because of the already higher level of the forward rate.
3.4.1
EIOPA’s June 2015 Technical specifications
In this section we compare our results with EIOPA’s latest June technical specifications (EIOPA, 2015c). At an advanced phase of writing this report, new technical specifications were published by EIOPA. In previous sections we showed the calculation methodology of Solvency II’s interest rate term structure and in this section we compare our results against EIOPA’s monthly published adjustments and the June technical specifications.
An interesting new feature is a simplification to the data used to calculate the Euro currency VA. The government reference portfolio used for the Euro currency VA may be calculated using the European Central Bank (ECB) ALL curve instead of the weighted portfolios. In figure 3.15 we compare the ECB ALL curve with specified maturity of 8.7 years with the reference portfolio we used in the calculation of the Euro currency VA. The ECB ALL curve consists of a weighted average of all Euro area government bonds. Exact weights are not published by the ECB, only some selection criteria (ECB, 2015).
On the left side the full development over our data sample is shown and the increments are plotted against each other on the right:
01−2007 05−2008 10−2009 03−2011 08−2012 01−2014 06−2015 0% 1% 2% 3% 4% 5% 6% Date Yield
ECB ALL curve VA Government portfolio
(a) ECB ALL - VA government portfolio
−0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 ECB(t+1)−ECB(t) VAport(t+1)−VAport(t) (b) Increment plot
Figure 3.15: Comparison between ECB ALL 8.7 year curve with government reference portfolio for the Euro currency VA. Correlation between the two is 0.9957.
We find a very high correlation between the two. When we look at the development on the left we see that they almost coincide except that the ECB ALL curve is marginally higher from 2009 onwards. The assumption that we made regarding constant weights in the reference portfolios is therefore a valid assumption given the fact that the ECB ALL curve coincides with the VA government portfolio that we calculated. To show the extend that our calculations differ from EIOPA’s published information or from the ECB ALL curve, we provide the following table:
This report EIOPA ECB curve Month CRA VA CRA VA CRA VA December 2014 10 11 10 17 10 13 January 2015 10 9 10 14 10 10 February 2015 10 3 10 9 10 5 March 2015 10 7 10 11 10 9 April 2015 10 10 10 15 10 12 May 2015 10 12 10 18 10 12
Table 3.6: Comparison of CRA and VA between the government reference portfolio used in this report, the ECB ALL curve and EIOPA’s monthly reported values. Adjustments are in basis points.
We find that the CRA is correct in all cases which is not surprising because the CRA is calculated as a weighted average over 1 year and therefore less fault sensitive. However, the VA shows a deviation of 5 basis points on average. The ECB ALL curve is closer to EIOPA’s value with an average of 4 basis points below. In July 2015 EIOPA released the Matlab programming code (EIOPA, 2015a) that can be used to calculate the interest rate term structures. Therefore we were able to find the cause of these deviations.
To compute the reference portfolios, we used the average duration to calculate the reference portfolio yield. We rounded the average duration towards its lower integer because insufficient data was available at the exact duration that was given. Furthermore differences occur because EIOPA calculates the VA values on exactly the last day of the month. This implies that interpolation is needed between Friday rates and Monday rates. Since we have calculated the VA on the last working day of the month, this causes a small difference with EIOPA’s given values. Furthermore the CRA is calculated by taking the average over 365 days, which implies that interpolation in weekends is necessary. We have taken the average over 255 working days which may cause deviations from EIOPA’s given values.
Most unclear was EIOPA’s method to derive the portfolio spreads. After the Matlab programming code was released we found that portfolio spreads are calculated using the Internal Rate of Return (IRR) for non-periodic cash flows. This method is used by EIOPA with the actual amount of days between two cash flow dates to incorporate leap years. This improved the values for the VA significantly. Furthermore the calculation method for the risk corrections became clearer. The RC is calculated by subtracting the RC from the portfolio yield and calculating the spread between the IRR over the reference portfolio yield minus the IRR over the reference portfolio yield minus the RC (section 3.3.5). Before we came across these differences we used the pure yield to calculate the portfolio spread over the interest rate term structure and applied the RC as a parallel shift.
3.5
Chapter three summary
In this chapter we have determined the interest rate term structure including adjustments used in Solvency II. The interest rate term structure adjustments were analyzed in detail and we commented on its behaviour in the period from January 2007 to June 2015.
The CRA can cause pro-cyclical investment behaviour, because the CRA is slow to react since it is calculated as an average over the previous year. It takes time for the CRA to decrease when market volatility already decreased, resulting in a downward shift of the interest rate term structure when this is not necessary.
Furthermore the VA can have an emphasizing effect on the increase of the present value of liabilities when portfolio spreads are very low. This is because the risk correction can be larger than the risk corrected currency spread resulting in a negative VA. Therefore, if portfolio spreads are low, the VA promotes investments in countries with low RC’s because then the VA is relatively highest. When market volatility is present, the VA increases rapidly, which results in a higher interest rate term structure and lower present value of liabilities.
Because we found that the VA for the Netherlands equals the currency VA, we ana-lyzed the country VA for Greece and found that the country VA influences the VA quite dramatically. At first sight we noted possible discontinuity problems with the country VA, but this did not have a large impact on the Greek VA.
We assumed the weights of the reference portfolios to have remained the same over the time, which we found is a good assumption because the government portfolio coincides with the ECB ALL curve that was suggested by EIOPA as simplification. The reference portfolio weights will be updated annually, based on data submitted at the previous year-end. So, for example, the reference portfolio weights for 2016 are based on the year-end 2014 submission. This lag creates stability in the VA reference portfolios because market volatility will not directly result in changes in the reference portfolio weights.
In the next chapter we outline the model that we use in our prospective analysis and analysis of hedging portfolios in retrospect.
