insurance products with profitsharing
Milou Teeuwen
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: Milou Teeuwen Student nr: 10691405
Email: milouteeuwen@gmail.com Date: January 14, 2016
Supervisor: Prof. Dr. ir. M.H. Vellekoop Second reader: Prof.Dr. Roger Laeven Supervisor Deloitte: Nick van Pelt
Abstract
Since the recent financial crisis Counterparty Credit Risk (CCR) must be taken into account in the fair valuation of derivative contracts. This means that adjustments must be made to the valuation of a derivative. The adjustment that captures the credit risk of the counterparty is the Credit Valuation Adjustment (CVA), and the one that captures an entity’s own credit risk is the Debit Valuation Adjustment (DVA). The new derivative value then equals the original value minus CVA plus DVA.
Insurance companies do not yet take into account CVA and DVA in the pricing of their liabilities. However, if insurers hedge the interest rate risk induced by the liabilities they can become involved in derivative transactions that are subject to a CVA charge. Due to the impact of CVA/DVA the actual asset value may be different, which could induce a mismatch between assets and liabilities. This thesis studies the impact of this potential mismatch for a case study involving a life insurance contract with a profitsharing feature.
In this case it can be concluded that if CVA and DVA are taken into account, the value of the own funds decreases due to a decrease in asset values. This results in shortages on the balance sheets, from which it can be concluded that the impact of CVA and DVA on the balance sheet of insurers is significant.
Keywords Credit Valuation Adjustment (CVA), Debit Valuation Adjustment (DVA), Bi-lateral Credit Valuation Adjustment (BCVA), Counterparty Credit Risk (CCR), Valuation, Insurance liabilities, profitsharing
Contents
Preface viii
1 Introduction and overview 1
1.1 Fair derivative valuation and the Credit Valuation Adjustment . . . 1
1.2 Insurance portfolios and CVA . . . 1
1.3 Thesis outline . . . 2
1.4 Scope . . . 2
2 Background information 3 2.1 Counterparty Credit Risk . . . 3
2.2 CVA in context . . . 4
2.3 Bilateral credit valuation adjustment . . . 4
3 Literature review 6 3.1 Embedded options . . . 6
3.2 Credit risk and CVA . . . 7
3.2.1 Loss given default . . . 7
3.2.2 Expected exposure . . . 8
3.2.3 Default probability . . . 9
4 Methodology 10 4.1 Selection of liabilities and assets . . . 10
4.1.1 Selection of the liability portfolio . . . 10
4.1.2 Selection of the asset portfolio . . . 11
4.2 Valuation of liabilities and assets . . . 13
4.2.1 Valuation of the liabilities . . . 13
4.2.2 Valuation of the assets . . . 13
4.3 Valuation models . . . 14
4.3.1 Short rate models . . . 14
4.3.2 CVA and DVA . . . 16
4.4 Chosen contract and model parameters . . . 19
4.4.1 The insurance contract . . . 19
4.4.2 The asset portfolio . . . 19
4.4.3 The Hull White One-Factor model . . . 19
5 Results 21 5.1 Value of the liabilities . . . 21
5.2 Value of the asset portfolio . . . 22
5.2.1 Risk-free value of the hedge portfolio . . . 22
5.2.2 CVA and DVA value of the swap contract . . . 23
5.3 Combined results . . . 25
5.3.1 Risk-free valuation . . . 25
5.3.2 Asset valuation including CVA/DVA . . . 26
6 Results for the hedge portfolio containing a cap 28
6.1 Risk-free value of the assets . . . 28
6.2 CVA value of the cap contract and DVA value of the profitsharing feature 28 6.3 Combined results . . . 28
7 Sensitivity analysis and stress test 31 7.1 Sensitivity analysis . . . 31
7.1.1 Impact of a change in the mean-reversion parameter . . . 31
7.1.2 Impact of a change in the volatility parameter . . . 32
7.1.3 Impact of a shift in the Euribor and EONIA curve . . . 32
7.2 Stress test . . . 34
7.2.1 The stress scenario . . . 34
7.2.2 Results of the stress test . . . 35
8 Conclusion and discussion 37 8.1 Conclusion . . . 37
8.1.1 Recommendations . . . 38
8.2 Discussion . . . 38
Bibliography 39
List of Figures
1.1 Hypothetical balance sheet mismatch between assets and liabilities. . . . 2
2.1 Global OTC derivatives market by outstanding notional amounts. . . . 3 2.2 Euribor/OIS-spread. . . 4
4.1 Schematic representation of the methodology. . . 10 4.2 Possible cashflow structure of the life insurance policy for the policyholder. 11 4.3 Possible cashflows of the swap contract and the profitsharing feature. . . 12 4.4 The EONIA discount curve observed at 30-09-2015. . . 19 4.5 The Euribor curve observed at 30-09-2015. . . 20
5.1 Simulated short rates r(t) (a) and constructed interest rate paths R(t, t + 0.5) (b). . . 21 5.2 One simulation of the 6-month Euribor rate R(t,t+0.5) (a) and the
cor-responding cashflows stemming from the profitsharing feature (b). . . . 22 5.3 The risk-neutral distribution of the net present values of the swap contract. 23 5.4 Hazard rates for Aegon, ABN Amro, Nationale Nederlanden and Santander. 24 5.5 The exposure profile for the payer swap per simulation (a) and exposure
profiles for EPE, ENE and EE (b). . . 24 5.6 Balance sheets without CVA and DVA for the 100% (a) and 80% (b)
hedge scenarios. . . 25 5.7 Balance sheets for Aegon versus ABN Amro (a) and for Nationale
Ned-erlanden (NN) versus Santander (b) in the 100% hedge scenario. . . 26 5.8 Balance sheets for Aegon versus ABN Amro (a) and for Nationale
Ned-erlanden (NN) versus Santander (b) in the 80% hedge scenario. . . 26
6.1 The exposure profile for the cap contract and profitsharing feature per simulation (a) and the average exposure profile (b). . . 29 6.2 Balance sheets for Aegon versus ABN Amro for the hedge portfolio that
contains a cap (a) and for the hedge portfolio that contains a swap (b). 29 6.3 Balance sheets for Nationale Nederlanden versus Santander for the hedge
portfolio that contains a cap (a) and for the hedge portfolio that contains a swap (b). . . 30
7.1 The average interest rate paths for shocks in a, σ and the Euribor curve based on the averages over 20.000 simulations for R(t, t + 0.5). . . 34 7.2 The 6-month Euribor rate (R(t, t + 0.5)) over the period 02-01-2002 until
08-07-2015. . . 35
3.1 Credit conversion factors for specific contract types and times to maturity taken from the Basel II accord (Basel Committee on Banking Supervision,
2005). . . 8
4.1 CDS Spreads (in basispoints) of selected insurance companies with re-spect to different maturities. . . 17 4.2 CDS Spreads (in basispoints) of potential counterparties with respect to
different maturities. . . 17
5.1 Parameter assumptions and results of the liability valuation. . . 22 5.2 The impact of CVA/DVA on the own funds and on the total value of the
balance sheet. . . 27 5.3 Required markup values for a perfect risk-free hedge and a perfect hedge
that includes CVA/DVA for Aegon versus ABN Amro. . . 27 5.4 Required markup values for a perfect risk-free hedge and a perfect hedge
that includes CVA/DVA for Nationale Nederlanden versus Santander. . 27
6.1 Difference between the BCVA value for the swap contract and CVA value for the cap contract. . . 30
7.1 Results of calculations based on a=0.03 (base case) and a=0.06 (test case). The differences between the base and test case are shown additionally. 31 7.2 Results of calculations based on σ=0.008 (base case) and σ=0.016 (test
case). The differences between the base and test case are shown additionally. 32 7.3 Results of calculations based on the Euribor and EONIA curve at
30-09-2015 (base case) and the same curves plus 50 basispoints (test case). The differences between the base and test case are shown additionally. . . 33 7.4 Results of calculations based on the Euribor and EONIA curve at
30-09-2015 (base case) and the same curves minus 50 basispoints (test case). The differences between the base and test case are shown additionally. . 33 7.5 CDS Spreads (in basispoints) of selected counterparties with respect to
different maturities in 2011 and the multiplication factor between credit spread data of 2015 and 2011. . . 35 7.6 Results of the stress test, in which the third column displays the absolute
differences between the base and stress case and in which the fourth column displays the multiplication factor between the base and stress case. 36
Preface
This thesis is written in order to complete the master Actuarial Science and Mathemati-cal Finance at the University of Amsterdam. Writing this thesis was done in combination with an internship at Deloitte Financial Risk Management. I am very thankful for all the guidance and support I have received from everyone. First I would like to thank my supervisor of the University of Amsterdam, professor Vellekoop, for the support he gave during the past four months. He always asked critical questions and gave very helpful suggestions during our meetings. Next I would like to thank Nick van Pelt, who was my supervisor at Deloitte. He took the time to read my thesis almost every week and gave very valuable comments. Also a special thanks to the rest of the Deloitte FRM department. Everybody was very interested in my research and willing to help if needed.
Introduction and overview
1.1
Fair derivative valuation and the Credit Valuation
Ad-justment
Pre-crisis valuations of OTC-derivatives were performed without considering credit ad-justments, which means that derivatives were valuated as if counterparty credit risk (CCR) was negligible. CCR is the risk that the counterparty in a transaction defaults during the trading period and does not make required payments. The recent financial crisis brought new light on this approach, as it became clear that even large counter-parties are not default-free.
For this reason IFRS 131prescribes that derivatives must be priced at fair value and defines this value as the exit price, i.e. ‘the price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date’ (International Valuation Standards Council, 2013). It also prescribes that counterparty credit risk, as well as an entity’s own credit risk, should be reflected in the fair value of derivatives. The required adjustment stemming from credit risk of the counterparty is the credit valuation adjustment (CVA), while the one adjusting for an entity’s own credit risk is the debit valuation adjustment (DVA). When valued together they are referred to as the bilateral credit valuation adjustment (BCVA)
(Franz´en and Sj¨oholm,2014).
1.2
Insurance portfolios and CVA
This thesis studies the impact of CVA and DVA on the balance sheet of insurance com-panies, by answering the following research question:
“What is the impact of CVA/DVA on the balance sheet of an insurance company that contains insurance products with a profitsharing feature and how can insurers deal with this impact?”
To answer this question a stylised balance sheet is studied. On the liability side the balance sheet of an insurance company contains the values of the insurance products. The value of an insurance product is equal to the sum of present values of future cashflows. In this thesis a specific insurance product is studied, namely a life insurance product that contains a profitsharing feature. Such a product induces certain risks, such as mortality and interest rate risk. The focus here will be on interest rate risk and other risks are left out of scope.
In order to hedge the interest rate risk the insurer must select assets that replicate the cashflows of the liabilities. Certain assets might be derivatives that are subject to
1IFRS stands for the International Financial Reporting Standard
2 Milou Teeuwen — The impact of the credit valuation adjustment
a CVA charge. Due to this the actual value of the insurer’s asset portfolio can change to a higher or lower value. This could create a mismatch between the liabilities and the assets, as is shown in figure1.1.
Figure 1.1: Hypothetical balance sheet mismatch between assets and liabilities.
In a scenario where CVA is higher than DVA, the actual value of the hedge portfolio will be lower than the original value of the same portfolio. In the opposite scenario the actual value will be higher. This has a direct impact on the own funds of an insurance company. It is good to know the exact impact of CVA and DVA on the balance sheet, so that appropriate measures can be taken. A possible measure might be to increase or decrease the premium of the insurance contracts.
1.3
Thesis outline
In the next chapter background information on counterparty credit risk and on CVA is provided. Chapter 3 reviews existing literature on insurance policies with embedded options and on CVA calculations. Chapter4describes the used methodology and results are discussed in chapter 5 and 6. Then chapter 7 shows the results of a sensitivity analysis and a stress test, which means that this chapter shows how results change if certain underlying parameters are changed. Finally a conclusion is drawn in chapter8.
1.4
Scope
Next to CVA and DVA there are a lot of other valuation adjustments, for example the Funding Valuation Adjustment (FVA), the Collateral Valuation Adjustment (CollVA) and the capital Valuation Adjustment (KVA). The analysis of the impact of these other VA’s will not be within the scope of this thesis. Furthermore it is assumed that deriva-tive contracts between the insurer and specified counterparties are not supported by collateral.
Background information
First this chapter gives background information on counterparty credit risk. Next the different contexts in which CVA plays a role are pointed out. Finally the Bilateral Credit Valuation Adjustment is discussed. This is the value that accounts for both CVA and DVA.
2.1
Counterparty Credit Risk
Ghamami (2013) defines CCR as ‘the risk that a counterparty in a transaction will
default prior to expiration of a trade and will not, therefore, make the payments required by the contract’. Counterparty credit risk can be seen as a combination of market risk and credit risk. Market risk originates from movements in underlying variables, such as interest rates. This risk determines the exposure one has towards his counterparties. Credit risk is the risk a party defaults, so credit risk defines the credit quality of a certain counterparty. CVA prices counterparty credit risk and allows for distinguishing between the two described components (Gregory,2012).
Counterparty credit risk is particularly relevant in the OTC-derivatives market. OTC-derivatives are derivatives that are traded ‘over-the-counter’. This means that these are bilateral contracts, i.e. contracts between two specific parties, without any protection. Trading in OTC-derivatives has grown significantly in the past years, as is shown in figure 2.1. This rapid increase can be explained by the fact that OTC-derivatives offer customisation, so that contracts better meet the specific needs of clients. Derivatives that are traded on an exchange do not offer these customisations.
Figure 2.1: Global OTC derivatives market by outstanding notional amounts.
4 Milou Teeuwen — The impact of the credit valuation adjustment
Before the financial crisis, many institutions considered counterparty credit risk to be negligible in trades with large high-quality rated financial institutions. A concept related to this is the spread between the Euro swap curves and the Euro overnight curve, as this spread can be seen as an indicator for interbank credit risk. Before the financial crisis spreads between these rates were around 1 basispoint, but at the time Lehman Brothers went bankrupt these spreads increased significantly as is shown in figure2.2. This rapid increase in interbank credit risk brought new focus on counterparty credit risk and on pricing and valuation of derivatives.
Figure 2.2: Euribor/OIS-spread.
2.2
CVA in context
This thesis focuses on CVA in the context of pricing and accounting. In this context CVA is the adjustment to the valuation of a certain derivative in order to account for counterparty credit risk.
CVA can also be viewed in a regulatory context. From a regulation point of view CVA is the amount of capital that is needed to cover future losses due to defaulting counterparties and thus the amount that financial institutions must set aside. It is a 99.5% VaR measure that the regulatory capital covers future losses due to the credit spread volatility of counterparties (Franz´en and Sj¨oholm,2014). This is also the context in which the Solvency II framework considers counterparty credit risk (Gatzert and
Martin,2012).
2.3
Bilateral credit valuation adjustment
Derivative contracts differ from simple loan contracts, because in derivative contracts there are two parties that can default instead of only the lender. When only one party in a transaction has a risk of defaulting, while the other is default free, we speak of a Unilateral Credit Valuation Adjustment (UCVA) which is equal to CVA.
In a bilateral contract both parties that enter the derivative contract bring their own risk of defaulting prior to the contract maturity. This needs to be represented in the valuation, in order to obtain a fair price. When an entity also takes into account its
own credit risk in valuing derivatives, next to counterparty credit risk, we speak of a debit valuation adjustment (DVA). The DVA ‘represents the CVA that a counterparty would be expected to hold against its exposure to the entity’ (International Valuation
Standards Council, 2013). Hence if we denote a calculation made by the investor with
the subscript i and one made by the counterparty with the subscript c, we must have that CVAi = DVAc. In this case the CVA calculated by the investor is equal to the
DVA calculated by the counterparty (Facchini, 2013). The Bilateral Credit Valuation Adjustment (BCVA) is then equal to the difference between CVA and DVA. Bo and
Capponi(2014) explain that the BCVA can be seen as the market price of credit risk, as
the BCVA is equal to the difference between ‘the price of a portfolio transaction, traded between two counterparties assumed default fee, and the price of the same portfolio where the default risk of both counterparties is accounted for’.
The BCVA calculation can be both non-contingent and contingent (International
Valuation Standards Council, 2013). In a non-contingent BCVA calculation both the
CVA and DVA are calculated separately and are then added together. In a contingent BCVA calculation the CVA and DVA are calculated simultaneously, while assuming dependence between the credit risk of both parties. In both cases it holds that BCVA = CVA - DVA (European Banking Authority, 2015). In this thesis the non-contingent approach is used.
Chapter 3
Literature review
This thesis focuses on the impact of the credit valuation adjustment in the context of life insurance contracts containing a profitsharing feature. This chapter first reviews the existing literature on embedded options and then the literature on counterparty credit risk and the credit valuation adjustment.
3.1
Embedded options
A profitsharing feature is a specific type of option that can be part of an insurance contract. Options that are added to plain insurance contracts are called embedded op-tions. Insurance contracts including embedded options have been extensively researched, for example by Briys and De Varenne (1997), Grosen and Jørgensen (2002) and Xu,
Ren, and Zheng(2009). More specificallyBouwknegt and Pelsser(2002) studied the fair
valuation of insurance contracts with a profitsharing feature based on a typical Dutch insurance policy.
In the paper ofBriys and De Varenne(1997) a valuation model is derived that prices life insurance policies by taking into account stochastic interest rates, default risk and profitsharing. Their model is based on contingent claim theory and models the insurance liability as a set of options. For a with-profits life insurance policy three scenarios can then be distinguished, namely bankruptcy of the insurance company, a scenario in which the insurance company can only make the guaranteed payments and a scenario in which the insurance company can pay out bonuses related to the with-profit option. For these three scenarios the payoffs for the policyholder suggests that the liabilities have features that are similar with contingent claims. By using the option pricing framework market values for the liabilities can then be computed.
In their paper Grosen and Jørgensen (2002) describe that the bankruptcy of many insurance companies between 1980 and 1990 can be explained by three reasons: misman-agement of interest rate guarantees, mismanmisman-agement of credit risk and application of poor accounting principles. Insurers ignored the value of and premiums for guarantees, resulting in decreasing safety margins when interest rates dropped significantly. This led to the understanding that these guarantees induce credit risk. Regulators addressed this issue by putting a maximum on the guaranteed interest rate, but a more correct route to address this issue is to determine a fair market value and hedge of the guarantees. Based on this line of argumentation, this paper describes a contingent claim model used for modelling insurance liabilities including various types of embedded options based on the model of Briys and De Varenne(1997) (The BV model). In addition to the BV model they incorporate regulatory intervention resulting in a reduction of insolvency risk.
Xu et al. (2009) focus on with-profit life insurance policies in China. Their model
is also based on contingent claim theory and uses Monte Carlo techniques to calculate values for different types of with-profit embedded option types. They discuss the impact
of different bonus strategies and different scenarios for the risk-free interest rate on the fair value of the insurance policy. They find that the value of the with-profit option cannot be ignored in pricing insurance policies and that different bonus strategies and risk-free interest rates have a high impact on the value of the liability. They also find that the impact of these factors is specifically high for an annual bonus option.
The paper ofBouwknegt and Pelsser(2002) illustrates how the market value of a life insurance policy that contains a profitsharing feature can be determined by making use of replicating portfolios. They describe how the market value of fixed deterministic cash flows can be obtained from the current interest rate term structures, and that the market value of the profitsharing feature can be determined by making use of interest rate derivatives. Furthermore, Zaglauer and Bauer (2008) show how German participating life insurance policies can be valuated using Monte Carlo methods. They describe a way to work with stochastic interest rates, instead of assuming deterministic or constant rates.
3.2
Credit risk and CVA
The impact of credit risk associated with bond investments in insurers’ portfolios in-cluding life insurance policies with cliquet-style guarantees is examined by Gatzert and
Martin(2014). They focus on credit risk related to bond investments and find that it is
relevant for the risk assessment to take into account credit risk for these investments. In the Netherlands these cliquet-style guarantees are less common, so therefore this thesis will not focus on this embedded option type. Despite this, their finding that credit risk is relevant in investments of insurers emphasises the importance of the research question of this thesis.
The credit valuation adjustment explicitly prices counterparty credit risk and is de-fined as ‘the difference between the risk-free portfolio value and the true portfolio value that takes into account the possibility of institution and counterparty default’ (Brigo,
2012). The literature discusses several ways to price CVA, as there is no specific method prescribed by IFRS. The only requirement is that valuation methods make maximal use of observable inputs (European Banking Authority,2015). For example,Liu(2015) describes how to model CVA based on least-square Monte Carlo methods. His method first simulates the time of default and all cash flows that occur when a trading position is entered. Furthermore the nested least-square Monte Carlo method is used to calculate the future market value of the specific derivative contract. Then CVA can be calculated based on the default probabilities and the future market values. Nested simulations refer to the method in which two scenario sets are used: outer scenarios for the change in the variables during the first time period and inner scenarios for the evaluation of the expectation. Outer scenarios are then based on the real world and inner scenarios on the risk-neutral world. This method is however very time-consuming and computation time can be reduced by using a least-square Monte Carlo approach by which the inner scenarios are calculated with a regression.
For calculating CVA four components are of relevance, namely the discount factors, the loss given default (LGD), the expected exposure (EE) and the default probability (DP). All elements can be calculated separately, and afterwards they can be combined to obtain the CVA value. The rest of this section discusses literature on calculating the loss given default, the expected exposure and the default probabilities.
3.2.1 Loss given default
For modelling the LGD, Qi and Zhao (2011) compare six different methods in terms of model fit and predictive accuracy. Four of the studied methods are parametric, namely OLS regression, fractional response regression, inverse Gaussian regression and
8 Milou Teeuwen — The impact of the credit valuation adjustment
inverse Gaussian regression with beta-transformation. The other two methods are non-parametric and these are the regression tree method and the neural network method. They find that the non-parametric methods have a better model fit and predictive ac-curacy than the parametric methods. The neural network method provides no direct way of showing the complex underlying non-linear relationships of the model, so they find that the regression tree method can be seen as the best of the studied methods. As IFRS favours the use of observable inputs, most financial institutions use recovery rates that are implied by the market to calculate the LGD, which is equal to one minus the recovery rate (European Banking Authority, 2015). According to Moody’s, the LGD implied by the market for financial institutions is approximately 60% (see appendix A).
3.2.2 Expected exposure
To calculate the exposure, a variety of methods is described in the literature. The most simple way of calculating the exposure is described by the ‘current exposure method’. In this method the future exposure is computed of three components: the current exposure, the potential future exposure and the posted collateral. The future exposure is then approximated by adding the potential future exposure to and subtracting the posted collateral from the current exposure (Franz´en and Sj¨oholm,2014). The potential future exposure is equal to the multiplication of the notional of the contract with the credit conversion factor. The appropriate conversion factor can be obtained from table 3.1.
Contract type < 1 year 1-5 years > 5 years Interest rates 0.0% 0.5% 1.5% FX and gold 1.0% 5.0% 7.5%
Equities 6.0% 8.0% 10.0%
Precious metals except gold 7.0% 7.0% 8.0% Other commodities 10% 12.0% 15.0%
Table 3.1: Credit conversion factors for specific contract types and times to maturity taken from the Basel II accord (Basel Committee on Banking Supervision,2005).
More accurate approaches to calculate the exposure are referred to as semi-analytical methods. These methods are more sophisticated than the add-on approach and do not involve time-consuming Monte Carlo simulations. However, these approaches still volve some approximations; calculations depend on simplified assumptions about in-volved risk factors and the methods mostly ignore netting effects (Gregory,2012). Net-ting effects refer to the effect of netNet-ting agreements, i.e. agreements that can be made to net exposures of multiple contracts that are closed with one specific counterparty.
The most sophisticated way to model the expected exposure profile is by making use of simulations. Then the expected exposure profile can be split into a positive part for CVA calculations and a negative part for DVA calculations (International
Valua-tion Standards Council,2013). To determine the exposure profile,Antonov and Brecher
(2012) andDe Graaf, Feng, Kandhai, and Oosterlee (2014) describe that Monte Carlo simulations can be used. In the first mentioned article, a so-called ‘thin-out’ procedure is used to efficiently optimise the calculation of the exposure in large vanilla swap port-folios. Their method ‘thins out’ the portfolio payments by approximating the portfolio by a single payment stream. The second article describes three methods to calculate the exposure profile by calculating option values for simulated Monte Carlo scenarios. All methods described in this article contain two elements. The first is a forward sweep to generate scenarios for the future and the second is a backward sweep to calculate exposures along the asset paths that are generated by the forward sweep.
3.2.3 Default probability
The default probability can be based on historical data, but market implied default probabilities are preferred (European Banking Authority, 2015). Determining default probabilities on historical data is mostly based on transition matrices displaying prob-abilities of moving from one credit rating to another (Tennant, Emery, and Cantor,
2008). An example of such a table can be found in appendix B.
Market implied default probabilities are based on credit spreads or suitable proxies
(International Valuation Standards Council,2013). Deriving default probabilities from
credit spreads is a risk-neutral approach and within this approach there are more ways to obtain a credit spread. Credit spreads can for example be obtained from premiums of credit default swaps (CDS), from traded spreads of asset swaps and from bond prices
(Gregory, 2012). Using premiums of credit default swaps is the most direct way of
obtaining credit spreads. In a credit default swap the buyer pays a fee to buy protection for a certain amount of debt for a specified reference company, so the premium of the CDS represents the credit spread of the specified reference company. The appropriate method to use depends on the counterparty, as for instance for smaller counterparties no credit default swaps exist and proxy methods must be used.
A way to model the default probability based on credit spreads is described byCao,
Zhou, and Chi (2012). They describe that the default probability can be calculated
from the credit spread and loss given default by DP = CS/(CS + LGD). Next to this,Gregory(2012) argues that conditional default probabilities must be used, i.e. the default probability at a certain time is conditional on the fact that default has not yet occurred in a previous year. Here default is assumed to be driven by a Poisson process. Mathematically the cumulative default probability at a future time t is then given by DP (t) = 1 − exp(−ht). Here h is the hazard rate of default which can be approximated by the credit spread divided by the LGD (h = CS/LGD). This formula applies when the hazard rate is assumed to be constant.
This thesis studies the CVA in an insurance context and will contribute to the existing literature by examining the possible mismatch that arises on the balance sheet of insurers due to the inclusion of credit valuation adjustments in hedge portfolios.
Chapter 4
Methodology
To study the impact of CVA/DVA on the balance sheet of insurance companies a certain portfolio consisting of insurance products on the liability side and a specified hedge portfolio on the asset side needs to be selected. Then both the assets and liabilities must be modelled and valued. The liabilities will be valued without including counterparty credit risk and for the assets also CVA and DVA will be taken into account. Finally the values of the assets and liabilities will be compared to study the impact of CVA/DVA. A schematic representation of the described steps is given in figure4.1. All steps will be discussed in more detail in the remainder of this chapter.
Figure 4.1: Schematic representation of the methodology.
4.1
Selection of liabilities and assets
4.1.1 Selection of the liability portfolioThe studied insurance contract is a life insurance policy that contains a profitsharing feature. The plain contract pays out a certain amount of money X(1 + r)T when the
insured is still alive at the payout date, time T (in years from today, i.e. time zero). The policy is financed by a premium payment equal to X. For example if the interest basis r is equal to 2% and the premium payment is equal to e500.000, the guaranteed amount is equal to e742.974 at time T = 20. In this thesis mortality risk is not taken into account, i.e. the assumption is made that the policyholder will always be alive at time T .
Next to the plain contract there is a profitsharing feature in place to compensate policyholders for the fact that it might be possible to make a better return in the market than the contract return r. The profitsharing feature pays out the excess returns over r
on a (fictional) investment in a sudden redeemable loan contract with a maturity date equal to T . The amount invested in this loan is equal to the paid premium X and the interest rate on the loan at time t is the 6-month Euribor rate. Throughout this thesis the 6-month Euribor rate is denoted by R(t, t + 0.5), hence by an interest rate that for each time t, has a maturity equal to t+0.5. The excess returns, i.e. returns larger than r, are paid out in cash to the policyholder at the end of each year. A problem arises when returns are lower than r, since no buffers can be built up to compensate for negative values of the profitsharing feature. For this reason a markup of 1% is in place, which reduces the payoffs of the profitsharing feature to the excess returns over (r + 1%). The payout amounts at time t = 1, 2, .., T are thus equal to max[R(t, t + 0.5) − r − 1%, 0]X, with R(t, t + 0.5) the 6-month Euribor rate at time t.
The cashflow structure of the complete insurance contract can be described by figure
4.2, in which green arrows represent positive cashflows and red arrows represent negative cashflows for the policyholder.
Figure 4.2: Possible cashflow structure of the life insurance policy for the policyholder.
4.1.2 Selection of the asset portfolio
The insurance company now has an obligation to the policyholder and this induces certain risks for the insurer. One of these risks is the interest rate risk, since the value of the liability changes if interest rates change. As described previously a problem arises when interest rates drop below the interest basis r, since the insurer must yearly make a minimal return equal to r. To limit exposure to interest rate risk the insurer can invest in a hedge portfolio, consisting of assets that (partly) replicate the cashflows of the insurance contract.
The cashflow of the plain insurance contract can be replicated with the payoff from a zero coupon bond. If the insurer invests in a zero coupon bond with a notional equal to X and an interest rate equal to r the payoff from this bond exactly matches the final payment that the insurer must make to the policyholder. In this thesis it is assumed that the insurance company will buy the zero coupon bond with these specifications from the Dutch government.
The derivative that perfectly matches the cashflows of the profitsharing feature is a cap consisting of caplets for each payout date. A caplet has a payoff equal to max(Rt−
K, 0)X, with Rt the reference interest rate, K the strike and X the notional. If Rt
represents the 6-month Euribor rate R(t, t + 0.5) and if the strike is chosen to be equal to r + 1% the payoffs of the caplets are exactly equal to the payoffs of the profitsharing feature.
12 Milou Teeuwen — The impact of the credit valuation adjustment
Even though a series of caplets seems to be the best derivative to hedge the prof-itsharing feature, this derivative is not a common hedge instrument for insurers. More often, insurers use swaps for constructing a hedge portfolio. Another downside of hedg-ing with caplets is that a caplet is a unilateral derivative contract, which means that on the asset side only CVA and no DVA plays a role. Finally hedging with caplets is also more expensive in terms of premiums. These premiums are left out of scope in this thesis, but it must be pointed out that expensive premiums might be a reason not to choose a cap as the preferred hedge instrument for the profitsharing feature. For these reasons a swap is chosen to be the hedge instrument for the profitsharing feature in the main part of this thesis. To give a complete overview results for a hedge portfolio containing a cap are shown additionally in chapter6.
Entering a payer swap generates payoffs at the payout dates equal to (Rt− K)X
with Rt the reference interest rate, K the fixed rate and X the notional. Here payout
dates are equal to t = 1, ..., T for both the fixed and floating leg. If Rtis chosen equal to
R(t, t + 0.5) and if K is chosen to be equal to r + 1% the positive payoffs from this swap contract are exactly equal to the payoffs of the profitsharing feature. However, this swap contract can also produce negative cashflows in case the floating rate is lower than the fixed rate. For this reason K is chosen to be equal to r, so that the markup of 1% can be used to compensate for years in which the swap payoffs are negative. To visualise this specified hedge a possible cashflow structure for the swap contract together with a payoff structure for the profitsharing feature is given in figure 4.3.
0 2 4 6 8 10 12 14 16 18 20 Time in years -10 -5 0 5 10 15 20 Payoff (x1000 Euro) Swap contract Profitsharing
Figure 4.3: Possible cashflows of the swap contract and the profitsharing feature.
In this thesis two scenarios for the hedge portfolio are considered, namely a 100% and an 80% hedge portfolio. Here a 100% hedge portfolio means that all cashflows stemming from the insurance liability are hedged, while in an 80% hedge portfolio the profitsharing feature is only hedged for 80%. The 20% that is not hedged is then kept in cash. In the 80% hedge scenario the hedge products are the same, only the notional of the swap contract changes to 0.8X. The value of the zero coupon bond remains the same as in the 100% hedge scenario. By analysing these two scenarios it is possible to see whether a linear decrease of the hedge percentage results in a linear decrease of the CVA/DVA impact.
4.2
Valuation of liabilities and assets
4.2.1 Valuation of the liabilitiesThe value of the insurance liability is equal to the net present value of future cashflows of the liability. The described insurance policy contains a linear component (the fixed payout value) and a non-linear component (the payouts from the profitsharing feature). For determining the present value of the linear component a market discount curve is needed. In this thesis the market curve used for discounting the fixed payout value is the EONIA curve. EONIA stands for the Euro Overnight Index Average and the EONIA curve can be seen as an approximation of the risk-free interest curve. The present value of the plain insurance contract can then be determined by:
P V = X(1 + r)TP (0, T )EONIA, (4.1)
with X the value of the notional, r the fixed interest basis defined in the insurance contract, T the maturity of the contract and P (0, T )EONIA the EONIA discount factor
used for discounting a payment at time T back to time t = 0.
In order to determine the value of the non-linear components of the described policy, scenarios for R(t, t + 0.5) are required. These scenarios are obtained by making use of a short rate model. Details on short rate models are provided in section 4.3.1. Once scenarios for future interest rates have been derived, it is possible to determine the value of the profitsharing feature for each scenario by discounting all future cashflows of the profitsharing feature. Cashflows take place each year at times t = 1, 2.., T . For determining the present value of the profitsharing feature the following formula will be used: P V = T X t=1 t ∆t−1 Y i=0 P (ti, ti+ ∆t) max[R(t, t + 0.5) − r − 1%, 0]X . (4.2)
In this formula P V is the present value of the future cashflows, R(t, t + 0.5) is the 6-month Euribor rate at time t, r is the interest basis described in the contract and P (ti, ti + ∆t) is the discount factor used for discounting a payment at time ti + ∆t
back to time ti. The termQ
t ∆t−1
i=0 P (ti, ti+ ∆t) is equal to P (0, t), but presented here in
terms of monthly simulations. The value of the profitsharing feature is then equal to the average present value over all the simulations. The total value of the liability is equal to the value of the plain insurance contract plus the value of the profitsharing feature.
Under IFRS 9 an insurer should separate some embedded derivatives in insurance contracts from the host contract and measure them at fair value (International
Ac-counting Standards Board, 2004). This means that in principal the profitsharing
fea-ture should be measured at fair value, meaning that for this feafea-ture also DVA plays a roll. For the main part of this thesis this is not taken into account. However, chapter 6
shows what happens if DVA is included in the value of the profitsharing feature. Since the profitsharing feature has the same cashflow structure as the described cap, valuing DVA for the profitsharing feature is equal to valuing CVA for the cap. How this is done is described in section 4.3.2.
4.2.2 Valuation of the assets
Valuation of the asset portfolio is split in risk-free valuation and valuation of CVA and DVA. Here risk-free valuation means determining the value of the assets without considering counterparty credit risk. The risk-free value of the assets is then equal to the sum of present values of future cashflows stemming from the assets. The asset portfolio studied in this thesis consists of a zero coupon bond and a payer swap. Additionally a hedge consisting of a zero coupon bond and a cap is studied and results of this hedge
14 Milou Teeuwen — The impact of the credit valuation adjustment
portfolio are shown in chapter 6. The risk-free value of the zero coupon bond is equal to the present value of X(1 + r)T. As described in section4.1.2the cashflows of the cap
perfectly replicate the cashflows of the profitsharing feature, which means that also the risk-free value of the cap is equal to the risk-free value of the profitsharing feature.
For the swap contract the payoffs depend on the interest rate. To determine scenarios for these rates a short rate model is used. Details on short rate models are provided in section 4.3.1. Once simulated scenarios for the interest rate have been obtained, the present value of the payer swap can be determined per simulation by discounting all payoffs. Payoffs take place at each time t = 1, 2, .., T and are equal to the multiplication of the difference between the fixed and floating rate with the notional. For calculating the present value of the payer swap the following formula is used per simulation:
P V = T X t=1 t ∆t−1 Y i=0 P (ti, ti+ ∆t) (R(t, t + 0.5) − r)X . (4.3)
The value of the payer swap is then equal to the average present value over all the simulations.
Next to the risk-free value of the assets, values for CVA and DVA are determined for all derivatives except for the bond. For the zero coupon bond no CVA is calculated, as it is not likely that the Dutch government will default. For the valuation of CVA the following formula is used:
CV A = LGD
T
X
i=1
DF (ti)EP E(ti)DP (ti−1, ti). (4.4)
With:
• LGD the Loss Given Default, i.e. the amount that the creditor cannot recover as a percentage of the total outstanding amount;
• DF (ti) the Discount Factor, i.e. the relevant risk-free discount factor based on the EONIA curve and denoted by P (0, ti);
• DP (ti−1, ti) the Default Probability, i.e. the probability that the counterparty
can-not make the required payments because of a default between times ti−1 and ti;
• EP E(ti) the Expected Positive Exposure, i.e. the payments and unrealised gains
that an entity expects to receive from it’s counterparty.
To calculate the DVA value the same formula as for CVA applies, only the Expected Positive Exposure is replaced by the Expected Negative Exposure, i.e. the payments and unrealised losses that an entity expects to pay to its counterparty.
As can be seen from formula (4.4), four parameters are of interest. The discount fac-tors are obtained from the EONIA curve. How the other three parameters are calculated is described in section 4.3.2.
4.3
Valuation models
This section first describes interest rate models. Next the relevant components of the CVA/DVA formula are discussed.
4.3.1 Short rate models
Numerous models for the short rate are described by Hull(2012). These models can be divided in equilibrium models and no-arbitrage models. Hull explains that equilibrium
models do not exactly fit term structures that are encountered in practice. Oppositely, no-arbitrage models are specifically designed to exactly fit today’s term structure of interest rates. Due to this fact a no-arbitrage model will be used in this thesis.
The first model named by Hull (2012) is the Ho-Lee model described by Ho and Lee (1986), which models the short rate according to dr = θ(t)dt + σdW (t). Here θ is a time-dependent function that determines in which direction the short rate moves at time t and σ is the short rate’s standard deviation. The variable W (t) follows a Wiener process. The main advantage of this model lies in the analytic tractability, but a serious disadvantage is that this model does not include mean-reversion.
Another model for modelling the short rate is the Hull White One-Factor model, described by Hull and White (1990). This model describes the short rate process by dr = [θ(t) − ar]dt + σdW (t). This model adds mean-reversion at rate a to the Ho-Lee model and keeps the same amount of analytic tractability. Furthermore, interest rates can become negative in this model. Most literature discusses this as a disadvantage of the model, but nowadays negative interest rates are in fact observed in the market. For this reason it is no longer a disadvantage of the model.
There also is a Hull White Two-Factor model that provides a richer pattern of term structure movements and volatilities, but is less commonly used. This model adds a disturbance term and describes the short rate process by dr = [θ(t) + u − ar]dt + σ1 dW (t)1. Here the variable u is initially zero and is described by du = −bu dt +
σ2dW (t)2, with b a mean-reversion rate lower than rate a.
A model that does not allow negative interest rates is the Black-Karasinski model described byBlack and Karasinski(1991), in which the short rate is modelled by d ln r = [θ(t) − a ln r]dt + σdW (t). A main disadvantage of this model is that it has less analytic tractability than the other described models. It is for example not possible to value bonds in terms of the short rate r by making use of direct formulas.
The model that will be used in this thesis is the Hull White One-Factor model, due to the analytic tractability and mean-reversion component. This model is more extensively described below.
The Hull White One-Factor Model
The Hull White One-Factor model describes the short rate process under a risk neutral measure by:
dr(t) = [θ(t) − ar(t)]dt + σdW (t), (4.5)
with r(t) the short rate at time t, θ(t) the function to fit the initial term structure, a the constant mean-reversion rate, σ the constant short rate volatility and W (t) a Wiener process. The short rate r(t) must not be confused with the fixed interest basis r defined by the insurance contract. Based on (Hull and White, 1990) the solution of equation (4.5) is equal to: r(t) = r(0)e−at+ Z t 0 θ(t)e−a(t−τ )dτ + σ Z t 0 e−a(t−τ )dW (τ ). (4.6)
Furthermore, the function θ(t) can be calculated from the initial term structure by:
θ(t) = Ft(0, t) + aF (0, t) +
σ2
2a(1 − e
−2at), (4.7)
with F (0, t) the instantaneous forward rate with maturity t observed at time 0, equal to −δtδ ln P (0, t). The subscript t denotes a partial derivative with respect to t. In this formula the last term is usually negligible. Using this expression for θ(t) equation (4.6) can be rewritten, resulting in a simplified expression for r(t):
r(t) = α(t) + σ Z t
0
16 Milou Teeuwen — The impact of the credit valuation adjustment with α(t) = F (0, t) +σ 2 2a(1 − e −2at). (4.9)
Once the short rate is modelled, it is possible to construct interest rate paths based on the obtained short rates. For each simulation for the short rate zero coupon bond prices at time t, i.e. discount factors, can be constructed by:
P (t, T ) = A(t, T )e−B(t,T )r(t) (4.10)
with A(t, T ) and B(t, T ) equal to:
A(t, T ) = P (0, t) P (0, T )expB(t, T )F (0, T ) − σ2 4a(1 − e −2at )B(t, T )2 B(t, T ) = 1 a1 − e −a(T −t). (4.11)
Then the yield at time t with maturity T is given by:
R(t, T ) = − ln(P (t, T ))
T − t . (4.12)
4.3.2 CVA and DVA
As described by formula (4.4) CVA and DVA are built up from the discount factors, the loss given default, the expected exposure and the probability of default. In this thesis the EONIA curve is used for discounting at time zero. This section describes the other three components.
Loss given default
The loss given default (LGD) is the percentage of the exposure that is lost in case the counterparty defaults. It is therefore equal to one minus the recovery rate. The Basel
Committee on Banking Supervision (2015) explains that for the LGD there are three
scenarios when a counterparty defaults: 1. full recovery, 2. sale of assets and collateral and 3. in-between scenario’s. The LGD is specific for each trade and determination involves information about the counterparty type, legally enforceable terms and held collateral. Typically the LGD is based on assumed industry levels. For most derivative contracts a LGD of 60% is assumed, based on the average found by Moody’s for the financial institutions industry (see appendix A). This is also the percentage used in the valuations made in this thesis.
Default probability
Earlier a historical approach was used to determine default probabilities. An approach that is based on market observable factors is however preferred. Therefore the default probability will be based on credit spreads. How to obtain the credit spread depends on the counterparty, as is described in section 3.2.3. In this thesis the credit spread is based on Credit Default Swaps and only counterparties for which there is a CDS spread available are taken into account.
Default is assumed to be driven by a Poisson process, which leads to the follow-ing formula for the cumulative default probability before a future period u, assumfollow-ing constant hazard rates:
Here h is the hazard rate which can be approximated by CDS(u)/LGD. The instan-taneous default probability then becomes equal to the derivative of F (u), given by:
dF (u)
du = h · exp[−hu]. (4.14) The marginal default probability is the probability that default occurs between two specified time points. This probability is given by:
DP (t1, t2) = F (t2) − F (t1). (4.15)
The survival probability S(u) is the probability that default does not occur. This prob-ability is equal to 1 − F (u).
To determine default probabilities a specific insurance company must be studied. Furthermore a counterparty must be specified. In this thesis all Dutch insurance com-panies for which there is a CDS spread available are taken into account. A selection of counterparties is also made. For the swap contract the counterparty will be a bank. Table 4.1 and table 4.2 give an overview of quoted CDS spreads for the selected in-surance companies and potential swap counterparties respectively. All CDS spreads are obtained from Bloomberg and based on observations from 30-09-2015.
Insurance company 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y Aegon 35 44.2 8.7 76 92.1 104 124 141.4 Nationale Nederlanden 26.5 31.5 51.4 60.6 76.9 87 105.8 118.9
Table 4.1: CDS Spreads (in basispoints) of selected insurance companies with respect to different maturities. Counterparty 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y ABN Amro 30.1 32.7 44.5 62.9 82.6 87.4 96.5 105.6 Barclays 41.4 42.4 52.8 59.6 70.9 80.2 101.5 116.6 BNP Paribas 42.6 56 64 68 81 88.4 108 122.9 Citigroup 30.2 38.5 53.1 71.1 84.3 102.9 130.3 147.7 Credit Agricole 47.5 55 64 68 81 89.6 110.5 125.4 Credit Suisse 47.8 56.2 64.2 70 83.3 95.2 116 134 Deutsche Bank 46.1 57.1 71 82.8 95 107.2 127.2 139.7 Goldman Sachs 38.2 44.6 53.4 68.6 86.8 104.7 130.8 150.7 HSBC 43.2 53.4 60.1 76.5 85.6 95.8 114.8 129.2 ING Group 38 42.5 50.1 59 73.6 82.7 104.3 119.1 JP Morgan 30 38.3 51.6 63.6 75.5 91 113.8 134.1 Rabobank 38.1 40.6 48.6 55 66.1 76 90 106 RBS 44.5 50.8 54.9 71.9 84 94.3 115 130 Santander 104.5 92.4 118 127.2 144 159.6 187.5 203.8 SNS Bank 408.6 471 454.6 439.3 462.8 457.5 455 462.5 UBS AG 35.3 40.8 46.4 51 62.1 71.9 88 101
Table 4.2: CDS Spreads (in basispoints) of potential counterparties with respect to different maturities.
In this thesis calculations will be based on two different swap contracts, namely on a contract between Aegon and ABN Amro and on a contract between Nationale Nederlanden and Santander. In this way one contract is studied in which the insurer has a higher default probability than the counterparty and one contract is studied in which this situation is reversed. Here only the default probabilities will be based on real
18 Milou Teeuwen — The impact of the credit valuation adjustment
insurance companies and counterparties. The selected insurance product and specified hedge products are fictional and are not based on the actual balance sheets of both Aegon and Nationale Nederlanden.
Expected exposure
The expected exposure refers to the market value of the derivative during the contract period. The expected exposure at time t thus equals the average amount that is lost when the counterparty defaults at time t.
How to determine the expected exposure depends on the type of contract. As de-scribed in section4.1.2the chosen hedge instruments are a zero coupon bond combined with a payer swap or a cap. For the zero coupon bond no CVA and thus no expected exposure is calculated, as it is not likely that the Dutch government will default.
For a swap contract the expected exposure is determined from the interest rate sim-ulations. It thus uses the scenarios from the Hull White model. The expected exposure for each time t = 0, ∆t, .., T is then equal to the present value of the swap contract at time t:
Etj = P V (Swap)tj for j = 0, 1, .., N, with N = T · ∆t
= R(tj, tj + 0.5) − K∆tX + N X i=j+1 i−1 Y k=j P (tk, tk+ ∆t) · (R(ti, ti+ 0.5) − K)∆tX . (4.16)
with P (t, t + ∆t) the relevant discount factors, R(t, t + 0.5) the floating rate equal to the 6-month Euribor rate at time t, K the fixed interest rate and X the notional. The values of R(t, t + 0.5) are based on the simulations for the interest rate. The values for P (t, t + ∆t) are also based on the simulations, so that for each simulated interest rate path adjusted discount factors are used. For mathematical convenience it is assumed that the cashflows of the swap occur at each time t = 0, ∆t, .., N and are equal to ∆t times the yearly cashflows.
As described previously, the expected positive exposure is used for calculating CVA and for calculating DVA the expected negative exposure is used. These are calculated by:
EP Etj = max(Etj, 0) and
EN Etj = min(Etj, 0).
(4.17)
For the cap the expected exposure is also determined from the interest rate simu-lations and thus based on the output of the Hull White model. The expected exposure for each time t = 0, ∆t, .., T is then equal to the present value of the cap at time t:
Etj = P V (cap)tj for j = 0, 1, .., N, with N = T · ∆t
= max(R(tj, tj + 0.5) − K, 0)∆tX + N X i=j+1 i−1 Y k=j P (tk, tk+ ∆t) · max(R(ti, ti+ 0.5) − K, 0)∆tX . (4.18)
Due to the maximum function the expected exposure can only be positive. This exposure profile can be used for calculating CVA in the cap contract with the bank and for calculating DVA in the profitsharing contract with the policyholder.
4.4
Chosen contract and model parameters
4.4.1 The insurance contractIn order to price the insurance policy some assumptions about the policy must be made. It is assumed that the policyholder is 45 years old at t = 0 and that time T is equal to 20 years. Time t = 0 is chosen to be the 30th of September 2015.
The plain insurance contract pays out a value of X(1+r)T. In this thesis the premium X is chosen equal toe500.000 and the fixed interest basis r is chosen equal to 1%. Also a discount curve is needed to determine the present value of the final payment. Here the interest rate used for discounting is the EONIA as the EONIA curve can be seen as an approximation of the risk-free interest curve. On t = 0 the EONIA discount curve is equal to the one displayed in figure4.4. Based on this curve the value of P (0, T ) for T = 20 is equal to 0.764706. 0 2 4 6 8 10 12 14 16 18 20 Time in years 0.75 0.8 0.85 0.9 0.95 1 1.05
Eonia discount factor
Figure 4.4: The EONIA discount curve observed at 30-09-2015.
4.4.2 The asset portfolio
The asset portfolio that is studied in the main part of this thesis consists of a zero coupon bond and a payer swap. For the zero coupon bond the same parameters as for the plain insurance contract are used. Furthermore it is assumed that the zero coupon bond is bought from the Dutch government, so no CVA will be calculated for the bond as the default probability of the Dutch government is assumed to be zero. For the swap contract the fixed rate, the floating interest rate and the notional must be specified. Here the fixed rate is equal to r, which is equal to 1%. The floating rate is based on the 6-month Euribor rate R(t, t + 0.5) and simulations for this rate are obtained by making use of the Hull White One-Factor model. Finally the notional is equal to the premium payment X, which is equal to e500.000. Additionally a hedge portfolio consisting of a zero coupon bond and a cap is studied. For the cap the same parameters as for the profitsharing feature apply.
4.4.3 The Hull White One-Factor model
The short rates are based on the Euribor curve at 30-09-2015, which is displayed in figure
4.5. This means that also the values for the profitsharing feature, the swap contract and the cap are based on this curve. The short rate is simulated 20.000 times, over a horizon of 20 years and by taking 12 time steps per year. Furthermore α and σ are chosen to
20 Milou Teeuwen — The impact of the credit valuation adjustment
be equal to 0.03 and 0.008 respectively, based on a calibration on market data. Needed interest rate maturities for determining P (t, T ) and R(t, T ) are maturities of one month and half a year, as the maturity of one month is needed to determine a discount factor for each time step and the maturity of half a year is needed to determine scenarios for the 6-month Euribor rate.
0 5 10 15 20 25 30 35 40 45 50 Time in years 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016
Euribor interest rate
Results
First this chapter discusses the results of the risk-free valuation of the liabilities. Next results for both the risk-free and CVA/DVA valuation of the assets are presented. Finally the combined results are discussed.
5.1
Value of the liabilities
As described in section4.2 simulated short rates are needed to obtain the value of the insurance product. These rates are obtained using the Hull White One-Factor model and an example of the output is displayed in figure 5.1a. The simulated short rates are then used to construct interest rates for different maturities at all time points between now (t = 0) and maturity. These rates make it possible to determine the value of the products that depend on R(t, t + 0.5). Simulations of the 6-month Euribor rate between t = 0 and T = 20 are displayed in figure 5.1b. Figures of the short rate r(t) and the 6-month Euribor rates R(t, t + 0.5) are very much alike. This is due to the relatively short maturity of the Euribor rates, namely a maturity of six months.
(a) Simulated short rates (b) Constructed interest rate paths
Figure 5.1: Simulated short rates r(t) (a) and constructed interest rate paths R(t, t + 0.5) (b).
An example of a simulation of the 6-month Euribor rate is shown in figure 5.2a
and the corresponding payoff structure for the profitsharing feature is displayed in fig-ure 5.2b. Figure 5.2b shows the payoffs on a yearly basis, since the payoffs from the profitsharing feature are paid out yearly.
Given the simulations of the of the interest rate and the corresponding payoff schemes for the profitsharing feature, it is possible to calculate the value of the profitsharing fea-ture for each simulated interest rate path with formula (4.2). Based on the assumptions described in section4.4.1, the value of the profitsharing feature is equal toe62.418. The
22 Milou Teeuwen — The impact of the credit valuation adjustment 0 50 100 150 200 Time in months -1 0 1 2 3 4 5 6 Rate (%) R(t,t+0.5) r+1%
(a) Simulation of the interest rate curve
0 2 4 6 8 10 12 14 16 18 20 Time in years 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Payoff (Euro)
(b) Payoffs of the profitsharing feature
Figure 5.2: One simulation of the 6-month Euribor rate R(t,t+0.5) (a) and the corre-sponding cashflows stemming from the profitsharing feature (b).
value of the plain contract with r = 1% is determined with formula (4.1) and equal to e466.407. This results in a total value of the liability equal to e528.825. The parameter assumptions and results are summarised in table5.1.
Item Value Number of simulations 20.000 Notional X e500.000 interest basis r 1% Maturity time T 20 Discount factor P (0, T ) 0.764706 Payoff value plain contract e610.095 Value profitsharing feature e62.418 Value plain contract e466.407 Total present liability value e528.825
Table 5.1: Parameter assumptions and results of the liability valuation.
5.2
Value of the asset portfolio
5.2.1 Risk-free value of the hedge portfolioA hedge of 100%
As described in section 4.1.2 the 100% hedge portfolio consists of a zero coupon bond and a payer swap. The payoff and present value of the zero coupon bond match exactly with the payoff and present value of the plain insurance contract and are equal to e610.095 and e466.407 respectively.
The swap value is determined from the interest rate simulations and figure5.3gives an overview of the risk-neutral distribution of net present values of the payer swap based on 20.000 interest rate simulations. For a fixed rate of 1% and a notional of e500.000 the value of the payer swap is equal to e49.788. Since the value of the profitsharing feature is equal toe62.418, a hedge consisting of a zero coupon bond and a payer swap generates a shortage of e12.630.
-800 -600 -400 -200 0 200 400 Swap net present value (x1000 Euro)
0 200 400 600 800 1000 1200 1400
Figure 5.3: The risk-neutral distribution of the net present values of the swap contract.
A hedge of 80%
In this scenario, only 80% of the profitsharing feature is hedged. This means that the notional of the swap contract changes from e500.000,- to e400.000,-. The value of the zero coupon bond remains the same as in the 100% hedge scenario. Due to this the total value of the hedge portfolio becomes equal to (0.8 · 49.788) + 466.407 = e506.237. A value of 0.2 · 49.788 =e9.958 is then kept in cash. Considering this scenario the same shortage is generated as in the 100% hedge portfolio.
5.2.2 CVA and DVA value of the swap contract
For the swap contract this section discusses the results of the CVA/DVA calculations for the scenarios Aegon versus ABN Amro and Nationale Nederlanden versus Santander. Insight is given in what happens if the counterparty has a lower default probability than the insurer (Aegon versus ABN Amro) and what happens if this situation is reversed (Nationale Nederlanden versus Santander). As described in section 4.3.2for both cases the EONIA curve is used for discounting and the LGD is taken to be equal to 60%.
The first step involves determining the default probabilities of both the insurance company and the counterparty. How to calculate these probabilities is described in section4.3.2. As described in this section a loss given default, a time horizon and credit spreads are needed. Since credit spreads are not available for each maturity the data is interpolated so that hazard rates can be obtained for all maturities between half a year and ten years. For maturities shorter than half a year, the value for the credit spread with a maturity of half a year is used. For maturities larger than 10 years, the value for the credit spread with a maturity of ten years is used. This results in hazard rates as displayed in figure 5.4 for Aegon, ABN Amro, Nationale Nederlanden and Santander. For the CVA calculations marginal default probabilities are used, which are calculated by formula (4.15).
Next the exposures must be simulated. The exposure profile displays the present value of future cash flows from the payer swap at each point in time. How to deter-mine these values is described in section 4.3.2. The positive exposures can be used for determining the CVA value and negative exposures for determining the DVA value. Ex-posure outcomes for the swap contract are given in figure 5.5, in which the left figure displays the exposures per simulation and the right figure displays the exposure profiles for the expected exposure (EE), the expected positive exposure (EPE) and the expected
24 Milou Teeuwen — The impact of the credit valuation adjustment 0 50 100 150 200 Time in months 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Hazard rate (%) (a) Aegon 0 50 100 150 200 Time in months 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Hazard rate (%) (b) ABN Amro 0 50 100 150 200 Time in months 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Hazard rate (%) (c) Nationale Nederlanden 0 50 100 150 200 Time in months 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 Hazard rate (%) (d) Santander
Figure 5.4: Hazard rates for Aegon, ABN Amro, Nationale Nederlanden and San-tander.
negative exposure (ENE). It can be noted that the expected exposure at time zero is not the same as the present value of the swap contract. This is due to the fact that for calculating the exposures the assumption is made that cashflows for the swap occur every month and are equal to 1/12 times the yearly cashflows.
(a) Exposure per simulation
0 50 100 150 200 Time in months -40 -20 0 20 40 60 80
Average exposure (x1000 Euro)
ENE EPE EE
(b) Expected Exposure Profiles
Figure 5.5: The exposure profile for the payer swap per simulation (a) and exposure profiles for EPE, ENE and EE (b).
Finally CVA, DVA and BCVA can be calculated. To determine the CVA value formula (4.4) is used. The DVA value is determined using the same formula, but by
taking the expected negative exposure instead of the expected positive exposure. As described previously the BCVA value is equal to CVA minus DVA. The new derivative value is then equal to the risk-free value minus the BCVA value.
To determine CVA and DVA in the 80% hedge portfolio, the only thing that changes is the notional of the swap contract. Results for this hedge portfolio are also given in this section.
Swap for Aegon versus ABN Amro
Here the impact of CVA/DVA is presented for the swap contract between Aegon and ABN Amro, in which Aegon pays a fixed rate of 1% and in which ABN Amro pays the 6-month Euribor rate. Over 20.000 simulations, the value of CVA is equal to e11.566, the value of DVA is equal to e4.185 and the value of BCVA then becomes e7.381. In the 80% hedge scenario CVA equals e9.253 and DVA equals e3.348. This results in a BCVA value equal to e5.905. As 0.8 · 7.381 = 5.905, it is shown that a linear decrease in the hedge percentage results in a linear decrease in the absolute CVA impact, .
Swap for Nationale Nederlanden versus Santander
Here the impact of CVA/DVA is presented for the swap contract between Nationale Nederlanden and Santander, in which Nationale Nederlanden pays a fixed rate of 1% and in which Santander pays the 6-month Euribor rate. Over 20.000 simulations, the value of CVA is equal to e19.904, the value of DVA is equal to e3.606 and the value of BCVA then becomes e16.298. In the 80% hedge scenario CVA equals e15.923 and DVA equals e2.885. This results in a BCVA value equal to e13.038.
5.3
Combined results
5.3.1 Risk-free valuationThe results that are described in the section that discusses the risk-free valuation (5.2.1) are shown in the balance sheets displayed in figure 5.6a and 5.6b for a hedge of 100% and a hedge of 80% respectively. Due to low interest rates the present value of the swap contract is lower than the present value of the profitsharing feature, resulting in a shortage of e12.630.
(a) Balance sheet 100% hedge (b) Balance sheet 80% hedge
Figure 5.6: Balance sheets without CVA and DVA for the 100% (a) and 80% (b) hedge scenarios.
26 Milou Teeuwen — The impact of the credit valuation adjustment
5.3.2 Asset valuation including CVA/DVA
Summing the results described in section 5.2.2 results in the balance sheets displayed in figure5.7for the 100% hedge scenarios and in figure5.8for the 80% hedge scenarios. The value of the BCVA must be subtracted from the risk-free asset value, so that a negative BCVA value results in an increase in the asset value and vice versa.
(a) Aegon versus ABN Amro (b) NN versus Santander
Figure 5.7: Balance sheets for Aegon versus ABN Amro (a) and for Nationale Neder-landen (NN) versus Santander (b) in the 100% hedge scenario.
(a) Aegon versus ABN Amro (b) NN versus Santander
Figure 5.8: Balance sheets for Aegon versus ABN Amro (a) and for Nationale Neder-landen (NN) versus Santander (b) in the 80% hedge scenario.
From these balance sheets it becomes clear that a decrease of 20% in the hedge percentage results in a decrease of 20% in the BCVA value. There is however a difference in relevant impact. This is visible when looking at the own funds or the total balance sheet value. An overview of the total impact of both the 100% and 80% hedge scenarios on the own funds and on the total balance sheet value is given in table 5.2.
To study the impact in terms of the markup on the profitsharing feature it is de-termined which markup value results in a perfect risk-free hedge and which markup value results in a perfect hedge that includes CVA/DVA. These results are shown in table5.3for Aegon versus ABN Amro and in table5.4for Nationale Nederlanden versus Santander. Both tables apply to the 100% hedge scenario.