Bayesian modelling of financial guarantee insurance

Transcription

1 Bayesian modelling of financial guarantee insurance Anne Puustelli (presenting and corresponding author) Department of Mathematics, Statistics and Philosophy, Statistics Unit, FIN University of Tampere, Finland. Lasse Koskinen Insurance Supervisory Authority of Finland, Mikonkatu 8, P.O. Box 449, Helsinki, Finland and Helsinki School of Economics, P.O. Box 1210, FIN Helsinki, Finland and Arto Luoma Department of Mathematics, Statistics and Philosophy, Statistics Unit, FIN University of Tampere, Finland. The losses in financial guarantee insurance may be fatal during economic depression (i.e. deep recession). The number of claims and the proportion of excessive claims can be extraordinary high during that time. This indicates that economic business cycle, particularly depression periods, must be taken into account when the claim size distribution of financial guarantee insurance is modelled. Since depression is an exceptional event and usually not enough data is available on the phenomenon, it is difficult to model it with standard actuarial or credit risk models. In this research our goal is to model the claim process and to predict the premium and the required amount of risk capital needed for the claim deviation. Even though the used data is from the Finnish economy and from the financial guarantee system of the Finnish statutory earnings-related pension scheme, we suppose that it could be widely used in similar cases elsewhere. A Markov regime-switching model is used to predict the number and length of depression periods in the future. The prediction of claim amount is done by a transfer function model where the predicted business cycle is an explanatory variable. The initial risk reserve is evaluated by the use of predictive distribution. Bayesian methods are used throughout the modelling process, more specifically Gibbs sampler in the estimation of the business cycle model. Simulation results show that the required amount of risk capital is high even though depression is an infrequent phenomenon. Key Words: Business cycle, Credit risk, Depression, Gibbs sampler, Markov regime-switching model, Risk capital, Risk theory, Solvency. 1

2 2 PUUSTELLI, KOSKINEN & LUOMA 1. INTRODUCTION Financial guarantee insurance covers losses from specific financial transactions and guarantees that investors in debt instruments receive timely payment of principal and interest if there is a default. When a country experiences economic depression (i.e. deep recession) the losses in financial guarantee insurance may reach catastrophic dimensions for several years. During that time the number of claims can be extraordinary large and, what is more important, the proportion of excessive claims can be much higher than in usual periods (see e.g. Romppainen, 1998). As the future growth of the economy is uncertain, it is important to consider the level of uncertainty one can expect in the future claim process. But this kind of rare phenomenon is difficult to capture with the standard methods of risk theory or credit risk models. A mild and short downturn in the national economy increases the losses suffered by financial guarantee insurers just moderately, but catastrophic downturns in the national economy are crucial. History knows several economic depressions, which can be called catastrophic. These include the Great Depression in the 1930s, World Wars I and II, and the oil crisis in the 1970s. In recent years the Finnish experience from the beginning of the 1990s and the Asian crisis in the late 1990s are good examples. An interesting approach for analyzing the timing and effects of the Great Depression is the Regime Switching method in Coe (2002). There is no single best practice model for credit risk capital assessment (Alexander, 2005). The main approaches are structural firm value model, option theoretical approach, rating based method, macro economic model and actuarial loss model. This paper adopts the latter approach. In contrast to market risk, there has been little detailed analysis of the empirical merits of different models. On the contrary, a review of commonly used financial mathematics methods can be found e.g. in Ammann (2001). Cairns (2000) points out that uncertainty in actuarial modeling arises from three sources: uncertainty due to the stochastic nature of a given model, uncertainty in the values of the parameters in a given model, uncertainty in the model underlying what we are able to observe and determining the quantity of interest. In financial guarantee insurance the main problem seems to fall into the third category because of the complexity of the underlying risk process. Here we model the claim process of financial guarantee insurance in the economic business cycle context. This paper builds on the following three studies on the financial guarantee system of the Finnish pension scheme. Rantala and Hietikko (1988) modeled the solvency issues by means of linear models. Their main objective was to test methods for specifying bounds for the solvency capital. The linear method combined with the data not containing any fatal depression period (the article was published in 1988) - in the early 1990s Finland suffered from a depression that in many ways was much more severe than the Great Depression in the 1930s - underestimated risk. Romppainen (1996) analyzed the structure of the claim process during the depression period. Koskinen and Pukkila (2002) also applied the economic cycle model. Their simple model gives approximate results but it lacks sound statistical ground. We use modern statistical methods that offer advantages for assessing the uncertainty.

3 BAYESIAN MODELLING OF FINANCIAL GUARANTEE INSURANCE 3 From the methodological point of view we adopt the Bayesian approach recommended e.g. by Scollnik (2001). Simplified models or simplified assumptions may fail to reveal the true magnitude of the risks faced by the insurer. While undue complexity is generally undesirable, there may be situations where complexity cannot be avoided. Best et al. (1996) explain how Bayesian analysis can generally be used for realistically complex models. An example of concrete modelining is Hardy (2002) who applies Bayesian technique to a regime-switching model of the stock price process for risk management purposes. Here Bayesian methods are used throughout the modelling process, more specifically Gibbs sampler in the estimation of the business cycle model. The proposed actuarial model is used for simulating purposes in order to study the effect of the economic cycle on the needed pure premium and initial risk reserve. We apply the Markov regime-switching model to predict the number and length of depression periods in the future. The prediction of claim amount is done by a transfer function model where the predicted business cycle is an explanatory variable. More specifically, we utilize the business cycle model introduced by Hamilton (1989). In the Hamilton method all the dating decisions or, more correctly, the probability that a particular time period is in recession, are based on the observed data. The method assumes that there are two distinct states (regimes) in GNP - one for expansion and one for recession - that are governed by a Markov chain. The stochastic nature of the GNP growth depends on the prevailing state. As such the conventional dichotomy - boom and recession - business model is inadequate, because severe recession constitutes the real risk. We propose a model where the two states represent 1) the depression period state and 2) its complement state consisting of both boom and mild recession periods. The used economic data is the Finnish GNP series. The claim data is from the financial guarantee insurance system of the Finnish pension scheme. Combining the business cycle model with the transfer function model provides a new way to analyze the solvency of a financial guarantee provider with respect to claim risk. Stochastic simulations produce a wide range of possible outcomes. Given an initial buffer, in some outcomes, the insurer may seem solvent, in others it may be considered insolvent. So both qualitative and quantitative insight can be obtained on how credit risk develops. The organisation of the paper is the following. In section 2 the Finnish credit crisis in the 1990s is described. Section 3 gives the business cycle model and in section 4 the transfer function model and predictive claim amount are presented. Section 5 concludes. 2. THE FINNISH EXPERIENCE IN THE 1990S During the years Finland s GNP dropped by 12%. Naturally that period was harmful to all sectors of the economy and society as a whole. However, the injuries suffered in the insurance sector were only moderate, at least compared with the problems of the banking sector at the same time. An important exception was credit insurance related to the Finnish statutory earnings-related pension scheme of the private sector.

4 4 PUUSTELLI, KOSKINEN & LUOMA The administration of the pension scheme is decentralised to numerous insurance companies, company pension funds and industry-wide pension funds. The central body for the Finnish earnings-related pension scheme is the Finnish Centre for Pensions (FCfP). Special financial guarantee insurance was administrated by the FCfP. It was designed to be a guarantee for loans granted by pension insurance companies as well as to secure the assets of pension funds. The business was started in 1962 and continued successfully until Finland was hit by the depression in the 1990s. Then the losses took catastrophic dimensions and the financial guarantee insurance activity of the FCfP was closed. Claims paid by the FCfP are shown in Figure 1. The cost was levied to all employers involved in the mandatory scheme. Hence, pension benefits were not jeopardized. At a general level Norberg (2006) describes the risk presented to pension schemes under economic and demographic developments. Afterwards the FCfP s run-off portfolio was transferred to the new company named Garantia. A more detailed description of the case of the FCfP can be found in Romppainen (1996). In order to promote the capital supply, the FCfP had a legal obligation to grant financial guarantee insurance to company pension funds and industry-wide pension funds for which insurance was obligatory. Hence, it employed fairly liberal risk selection and tariffs. This probably had an influence on the magnitude of the losses. Hence, the data reported by Romppainen and used here can not be expected, as such, to be applicable in other environments. The risks would be smaller in conventional credit insurance, which operates solely on a commercial basis. Claim amount, Million Euros FIG. 1. Claims paid from credit insurance by the Finnish Centre for Pensions of Finland between 1980 and The lower dark part of the bar describes the final loss by August It is interesting to note that there are similar problems also in the USA at present. The corresponding US institute is Pension Benefit Guaranty Corporation (PBGC). It is a federal corporation created by the Employee Retirement Income Security Act of It currently protects the pensions of nearly 44 million American workers and retirees in 30,330 private

5 BAYESIAN MODELLING OF FINANCIAL GUARANTEE INSURANCE 5 single-employer and multiemployer defined benefit pension plans. Pension Insurance Data Book 2005 (page 31) reveals that total claims of PBGC have increased rapidly from about 100 million dollars in 2000 to 10.8 billion dollars in This increase in claims can not be explained by nation-wide depression, but it may be related to problems of special industry sectors (e.g. aviation). 3. NATIONAL ECONOMIC BUSINESS CYCLE MODEL Our first goal is to find a model by which we can forecast the growth rate of GNP. We will use the real GNP of Finland from 1860 to 2004 provided by Statistics Finland. We are particularly interested in depression periods, more precisely the number and length of them in the next say 30 years. For this purpose we will utilize the Markov regime-switching model introduced by Hamilton (1989). The original Hamilton model has two states for the business cycle: expansion and recession. In our situation, however, it is more important to detect depression, since it is the period when credit insurance will suffer its most severe losses. Therefore, we will define the states in a slightly different way in our application. Specifically, we will use a two-state regime-switching model in which the first state covers both expansion and recession periods and the second state is for depression. The Hamilton model may be expressed as y t = α 0 + α 1 s t + z t, where y t denotes the growth rate of the economy at time t, s t the state of the economy and z t a zero-mean stationary random process, independent of s t. The parameters α 0 and α 1 and the state s t are unobservable and should be estimated. We will assume that z t is an autoregressive process of order r, denoted by z t AR(r). It is defined by the equation z t = φ 1 z t 1 + φ 2 z t φ r z t r + ɛ t, where ɛ t N(0, σɛ 2 ) is an i.i.d. Gaussian error process. The growth rate at time t is calculated as y t = log(gnp t ) log(gnp t 1 ). We define the state variable s t to be 0, when the economy is in expansion or recession, and 1, when it is in depression. The transitions between the states are controlled by the first-order Markov process with transition probabilities P(s t+1 = 0 s t = 0) = p, P(s t+1 = 1 s t = 0) = 1 p, P(s t+1 = 0 s t = 1) = 1 q, P(s t+1 = 1 s t = 1) = q. Thus, the transition matrix is given by ( ) p 1 p P =. 1 q q The stationary probabilities π = (π 0, π) of the Markov chain satisfy the equations π P = π and π 1 = 1, where 1 = (1, 1).

6 6 PUUSTELLI, KOSKINEN & LUOMA The Hamilton model was originally estimated by maximising the marginal likelihood of the observed data series y t. Then the probabilities of the states were calculated conditional on these maximum likelihood estimates. The numerical evaluation was done by a kind of nonlinear version of the Kalman filter. By contrast, we will use Bayesian computation techniques throughout this paper. The Hamilton model will be estimated using the Gibbs sampler. The Gibbs sampler is a useful algorithm for simulating multivariate Markov chains (see, for example, Gelman and others, 2004). It is also called alternating conditional sampling and it is defined in terms of subvectors of θ = (θ 1, θ 2,...,θ p ), where θ is the random vector whose distribution is to be simulated. In each iteration the Gibbs sampler goes through θ 1, θ 2,...,θ p and draws values from their conditional distributions, conditional on the latest values of the other components of θ. It can be shown that this algorithm produces an ergodic Markov chain whose stationary distribution is the distribution of θ. In order to use the Gibbs sampler as an estimating algorithm, all full conditional posterior distributions of the parameters need to be evaluated. To simplify some of the expressions we will use the following notations: y = (y 1, y 2,...,y T ), y t = (y t, y t 1..., y t r+1 ), s = (s 1, s 2,..., s T ), s t = (s t, s t 1..., s t r+1 ), z = (z 1, z 2,..., z T ), z t = (z t, z t 1..., z t r+1 ), φ = (φ 1, φ 2,..., φ r ), and Z = z 0 z 1. z T 1. In the following treatment we assume y 0 = (y 0,..., y 1 r ) and s 0 = (s 0,..., s 1 r ) to be known. In fact, s 0 is not known, but we will simulate it using its stationary distribution. Using these notations the density of y, conditional on s 0, s and the parameters, can be written as p(y s 0,s, α 0, α 1, φ, σ 2 ɛ) = T t=1 ( 1 exp 1 ) 2πσ 2 ɛ 2σɛ 2 (y t α 0 α 1 s t φ z t 1 ) 2. In order to make computations easy, we chose the following prior distributions: p Beta(α p, β p ), q Beta(α q, β q ), p(φ, σɛ) 2 1 σɛ 2, p(α 0 ) 1, p(α 1 ) N(α 1 µ 0, σ0) 2 I(α 1 < 0). We obtained noninformative prior distributions for p and q by specifying as prior parameters α p = β p = α q = β q = 0.1. We also made some sensitivity analysis by using uniform priors

7 BAYESIAN MODELLING OF FINANCIAL GUARANTEE INSURANCE 7 for p and q, which corresponds to the prior parameters α p = β p = α q = β q = 1, and found that the estimation results remained essentially similar except that the posterior intervals of q became somewhat shorther and those of p longer. For α 0 and (φ, σɛ 2 ) we gave improper, noninformative prior distributions. The prior distribution of α 1 prevents it from getting a positive value. Here, N(α 1 µ 0, σ 2 0 ) denotes the Gaussian density with mean µ 0 and variance σ 2 0 and I(α 1 < 0) the indicator function obtaining the value 1, if α 1 < 0, and 0, otherwise. We specified the values µ 0 = 0.1, σ 2 0 = 0.22 as prior parameters, which results in a fairly noninformative prior distribution. In order to implement the Gibbs sampler we determined the full conditional posterior distributions of the parameters. In the following presentation we have omitted some conditioning parameters when appropriate: ( T ) T {p s 0,s} Beta [(1 s t )(1 s t 1 )] + α p, [s t (1 s t 1 )] + β p, t=1 ( T ) T {q s 0,s} Beta [s t s t 1 ] + α q, [s t 1 (1 s t )] + β q, t=1 t=1 t=1 {s t y t,y t 1,s t 1, α 0, α 1 φ, σ ɛ, p, q} ( [p 1 s t 1 (1 q) st 1 Bernoulli exp { 2α 1(y (1 p) 1 st 1 st 1 q t ) α2 1 2σ ɛ {φ y 0,y,s 0,s, α 0, α 1, σ 2 ɛ } N( (Z Z) 1 Z z, σ 2 ɛ (Z Z) 1), {σɛ 2 y 0,y,s 0,s, α 0, α 1, φ} Inv-χ 2 (T, (z Zφ) (z Zφ)/T), ) {α 0 y 0,y,s 0,s, α 1, φ, σ ɛ } N ( T t=1 y t T, σ2 ɛ T p(α 1 y 0,y,s 0,s, α 0, φ, σ ɛ ) N(α 1 ˆα 1, ˆσ 1 2 ) I(α 1 < 0), where we have denoted, } ] 1 ) + 1, t = 1,...,T, ˆα 1 = ˆσ 2 1 = T t=1 sty t σɛ 2 T t=1 st σ 2 ɛ + 1 µ σ σ 2 0 ( T ) 1 t=1 s t + 1 σ0 2, σ 2 ɛ, and y t = y t α 1 s t φ z t 1, y t = y t α 0 φ z t 1. The notation Inv-χ 2 (ν, s 2 ) means the scaled inverse-chi-square distribution, defined as νs2 χ, 2 ν where χ 2 ν is a chi-square distributed random variable with ν degrees of freedom. In Figure 2, one simulated chain, produced by the Gibbs sampler, is shown. As we can see, the chain converges rapidly to its stationary distribution and the component series of the chain

8 8 PUUSTELLI, KOSKINEN & LUOMA phi sum(st) phi q alpha p alpha sigmae Time Time FIG. 2. Iterations of the Gibbs sampler. mix well, that is, they are not too autocorrelated. The summary of the estimation results, based on three simulated chains, as well as Gelman and Rubin s diagnostics (Gelman et al. 2004) are given in Appendix 1. The values of the diagnostic are close to 1 and thus indicate good convergence. Observed growth rate of GNP Time Three predicted growth rates of GNP Time FIG. 3. The observed growth rate of GNP and three predicted series. There is an example of growth rate prediction in Figure 3. In the upper part of the figure, the observed y t is shown, that is the differenced log(gnp t ), and in the lower part of the figure,

9 BAYESIAN MODELLING OF FINANCIAL GUARANTEE INSURANCE 9 three predicted series are shown. In Figure 4, the differenced log(gnp t ) and the estimated probabilities of depression are shown. Probability of depression Growth rate of GNP Year FIG. 4. The growth rate of GNP and the probabilities of depression. 4. PREDICTION OF THE CLAIM AMOUNT In this research, our ultimate goal is to predict the pure premium and the required amount of risk capital needed for the claim deviation. The claim data is obtained from the central body of the Finnish earnings-related pension scheme and the years included in this study are The prediction of the claim distribution is done by using a transfer function model of the form x t = β 0 + β 1 x t 1 + β 2 y t + ɛ t, where x t = Φ 1 (x t ) and x t is the gross claim amount proportional to the technical provision at time t, y t the growth rate of GNP and ɛ t N(0, σ 2 ) an i.i.d. Gaussian error process. The parameters β 0, β 1 and β 2 are not known and are estimated. The series y t is predicted using the model described in the previous section. The model x t = β 0 + β 1 x t 1 + β 2 y t + ɛ t may also be expressed in the form x t β 0 = β 2 1 β 1 B y 1 t + 1 β 1 B ɛ t = β 2 (y t + β 1 y t 1 + β 2 1y t ) + ɛ t + β 1 ɛ t 1 + β 2 1ɛ t , from which one can see that x t can be obtained by applying exponentially smoothing linear filters to the input series y t and ɛ t. In our analysis we applied the probit transformation Φ 1 (.), the inverse function of the standard normal distribution function, to the variable x t. This is reasonable, since x t cannot exceed 1. The parameters β 0, β 1 and β 2 were estimated using standard Bayesian simulation for regression models. We used an otherwise uninformative prior distribution but made the

10 10 PUUSTELLI, KOSKINEN & LUOMA restriction β 1 < 1 to ensure that that the estimated model for x t is stationary. The observed proportion of claim amount to technical provision is shown in the upper part of Figure 5 and in the lower part of the figure there are three simulated predicted series. Observed claim amount / technical provision Time Three predicted claim amounts / technical provision Time FIG. 5. The observed proportion of gross claim amount to technical provision and three predicted series. The premium and the initial risk reserve is evaluated from the predictive distribution of the proportion of claim amount to technical provision. The technical provision is set to be 1 in the prediction. The premium is set to be the overall mean of all iterations through the prediction period. The balance at time t is calculated by subtracting the claim amount at time t from the cumulated premiums. The 95% and 75% values at risk are evaluated from the minimum balance of each iteration. The distribution of the minimum balance is extremely skewed, which can be explained by the rareness of depression and by the extreme losses of credit insurance once depression hits. This phenomenon can be seen from Figure 6, which shows the 95% prediction limits of the balance. The slope of the upper prediction limit is much more moderate than the slope of the lower limit. Extensive simulations (1000 iterations) were done to evaluate the pure premium, 95% and 75% values at risk for the prediction period of five years. The range of pure premium level was 1.77% 2.02% with mean 1.87%. The range of 95% value at risk was from 1.98 to 2.46 times the five-year premium with mean 2.2 and the range of 75% value at risk was from to times the five-year premium with mean CONCLUSIONS In this paper we presented an application of advanced Bayesian modelling to financial guarantee insurance. The Markov regime-switching model was used to predict the number and length of depression periods in the future. We used real GNP data to measure the economic

11 BAYESIAN MODELLING OF FINANCIAL GUARANTEE INSURANCE Time FIG. 6. The 95% prediction interval (solid lines) of balance with 20 example paths (dashed lines). growth. The claim distribution was predicted using a transfer function model where the predicted business cycle was an explanatory variable. The claim data was obtained from the central body of the Finnish earnings-related pension scheme and it includes the fatal depression period in the early 1990s. We had no remarkable convergence problems when simulating the joint posterior distribution of the parameters even though the prior distributions were noninformative or only mildly informative. For the interpretation of the results it is important to note that the risks are probably smaller in conventional companies that operate solely on a commercial basis. The simulation results can be summarized as follows. First, if the effects of economic depressions are not considered properly, there is a danger that the premiums of financial guarantee insurance will be set at a too low level. Pure premium level based on the gross claim process is assessed to be 1.7% 2.0%. In Finland the claim recoveries after the realization process of collaterals has been about 50%. Second, in order to get through a long-lasting depression period a financial insurer should have a fairly great risk reserve. The 95% value at risk for a five-year period is about 2.2 times the five-year premium. The corresponding 75% value at risk is only about 0.32 times the five-year premium. These figures illustrate the essential importance of reinsurance contracts in assessing the needed risk capital. General observations can be made on the basis of this study: In order to understand the effects of business cycle on financial insurers financial condition and better appreciate the risks, it is appropriate to extend the modelling horizon to cover a depression period;

12 12 PUUSTELLI, KOSKINEN & LUOMA A financial guarantee insurance company may benefit from incorporating responses to credit cycle movements into its risk management policy; The use of modern Bayesian methods offers significant advantages for assessing uncertainty; This study underlines the observation that a niche insurance company may need special features in its internal model. We suppose that the proposed method can also be applied to the credit risks assessment of a narrow industry sector whenever a suitable business cycle model is found. ACKNOWLEDGEMENTS The authors of this study are grateful to Yrjö Romppainen for the data he provided us, the most valuable comments and inspiring discussions. We also would like to thank Vesa Ronkainen for useful comments. REFERENCES Ammann, M. (2001): Credit risk valuation - methods, models, and applications, Springer Verlag, Heidelberg. Alexander, C. (2005): The Present and Future of Financial Risk Management, Journal of Financial economics, 3, Best, N., Spiegelhalter, D., Thomas, A., and Brayne, C. (1996): Bayesian analysis of realistically complex models, Journal of the Royal Statistical Society, Series A, 159, Cairns, A. (2000): A Discussion of Parameter and Model Uncertainty in Insurance, Insurance: Mathematics and Economics, 27, pp Coe, P. (2002): Financial crisis and the great depression - A regime switching approach, Journal of Money, Credit and Banking, 34, Gelman, A., Carlin, J. B., Stern, H. S., Rubin, D. B. (2004):Bayesian Data Analysis, Chapman & Hall/CRC, Second ed. Hamilton, J. (1989): A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, pp Hardy, M. (2002): Bayesian risk management for equity-linked insurance, Scandinavian Actuarial Journal, 3, Koskinen, L. and Pukkila, T. (2002): Risk caused by the catastrofic downturns of the national economy, 27th Transactions of the International Congress of Actuaries, Norberg, R. (2006): The Pension Crisis: its causes, possibile remedies, and the role of the regulator in Erfaringer og utfordringer Kredittilsynet , Kredittilsynet. Pension Insurance Data Book 2005:

13 BAYESIAN MODELLING OF FINANCIAL GUARANTEE INSURANCE 13 Rantala, J. and Hietikko, H. (1988): An application of time series methods to financial guarantee insurance, European Journal of Operational Research,37, Romppainen, Y. (1996): Credit insurance and the up and down turns of the national economy, ASTIN Colloquim, Copenhagen. Scollnik, D. (2001): Actuarial modelling with MCMC and Bugs, North American Actuarial Journal, 5, > summary(sims) APPENDIX 1 Iterations = 1:2500 Thinning interval = 1 Number of chains = 3 Sample size per chain = Empirical mean and standard deviation for each variable, plus standard error of the mean: Mean SD Naive SE Time-series SE alpha e e-05 alpha e e-04 phi e e-03 phi e e-03 sigmae e e-06 p e e-04 q e e-03 sum(st) e e Quantiles for each variable: 2.5% 25% 50% 75% 97.5% alpha alpha phi phi sigmae p q sum(st)

14 14 PUUSTELLI, KOSKINEN & LUOMA > gelman.diag(sims) Potential scale reduction factors: Point est. 97.5% quantile alpha alpha phi phi sigmae p q sum(st) Multivariate psrf i