Copula Models of Economic Capital for Insurance Companies

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1 Copula Models of Economic Capital for Insurance Companies By Jessica Mohr and Thomas Vlasak Advisor: Arkady Shemyakin 1. Summary of Problem Financial and economic variables have proven notoriously difficult to forecast, and a perfect example of the necessity of accurate predictions is the financial crisis in We worked towards modeling one such financial variable, economic capital, particularly for life insurance companies. Economic capital is the amount of capital that a firm, usually in financial services, needs to ensure that the company stays solvent given its risk profile [1]. Economic capital implies a deeper examination of correlation and distribution of risks and assets, as opposed to the more formulaic risked based capital [1]. The novelty in our approach is the application of copula models, or a multivariate probability distribution used to describe fine dependence between random variables, particularly in the low interest environment insurance companies currently face. While we worked mostly with the asset variables this summer, we have also begun to expand to include liability variables as well. 2. Data Selection We began by gathering data on the liabilities side. As we were focused on life insurance companies, we collected data on mortality and morbidity. For mortality, we were directed from the Society of Actuaries (SOA) website to the Human Mortality Database. The Human Mortality Database contains detailed mortality data from 1933 to 2015 and measures mortality in terms of births, deaths, population size, exposure-to-risk, death rates, life tables and life expectancy [2]. We are not yet sure which measure of mortality will be the most useful for our purposes. For morbidity, the Centers for Disease Control and Prevention (CDC) has an excellent database sorting by cause of injury, demographic, and geographic information [3]. Lapse rate data, which shows the rate at which customers fail to pay their premiums, was collected from a 2013 SOA study [4]. However, due to the formatting of the data its usefulness in our modeling is still unclear as it is reported as a survival rate broken down by the age of the policy. Future

2 researchers may be interested in exploring lapse rates further as we suspect lapse rates may be correlated with macroeconomic indicators. In layman s terms, if the economy tanks we suspect that one of the first monthly expenses cut would be a vested life insurance policy. On the assets side, we logged on to the UST Bloomberg Terminal and retrieved over ninety fixed income indices, or weighted averages of real world bonds that Bloomberg felt was representative of the overall market [5]. All of the indices were initially worth one hundred dollars and began on January 1 st, We retrieved the daily closing price up until July 7 th, The indices were sorted by region, government vs private, sector, yield, and combinations thereof. 3. Dealing with Autocorrelation This index of US Treasury Bonds is representative of what we were dealing with. The upward trend, while desirable for the overall economy, is not something we are interested in for our final model. We are only interested in the change in the asset s price that is dependent on other markets or liability variables. The time series also appears somewhat autocorrelated, meaning the value of one day is influenced by the value of the previous day.

3 We controlled for these factors using ARIMA modeling. For the US Treasury Index, we ended up with a model with one autoregressive term (ϕ), one difference, and a constant (μ). Giving us the following equation: Y t -Y t-1 = μ + ϕ 1 (Y t-1 Y t-2 ) + E Solving for the residuals (E) we are left with the following plot. 4. Model Selection We had too many variables to feasibly make well-fitting ARIMA models by hand, so we utilized the auto.arima function in R which selects models based on their AIC values. This type of automated model selection favors less complex models with a high goodness of fit. We suspected some of the automated models were overfit, as they contained multiple autoregressive and moving average terms, which is more lags than one would intuitively suspect for a financial variable. To confirm R was producing decent models we selected five indices that loosely represented the assets portfolio of a life insurance company as determined by a report issued by the National Association of Insurance Commissioners (NAIC), and compared the models created automatically in R to models constructed by hand using Minitab

4 [6]. The indices selected were, Bloomberg US Corporate (BUSC), Bloomberg Global High Yield Corporate (BHYC), Bloomberg US Emerging Markets (BEM), Bloomberg Sovereign French (BFRA), and Bloomberg USD Investment Grade Corporate Financial Sector (BUSCFI). The models we constructed were similar to the automated models, except the models created by hand typically utilized fewer terms. Upon comparing the residual plots however, we noticed no systematic differences between the automated and hand-made models, leading us to conclude that the automated models were acceptable. 5. Assigning Tentative Marginal Distributions The next step is to assign marginal distributions to the residuals. We started by simply fitting normal distributions over a histogram of the residuals. Our representative example is below. We noted first that the normal curve is a poor fit. This intuition was later supported when we used an automated Shapiro-Wilk test on all of our residuals, and none tested normal at any standard alpha. The second take away we took from this test is the apparent asymmetry in the tails. This asymmetry seemed to hold in every marginal distribution we inspected, and thus the next marginal

5 distribution family we want to try is a T-distribution with lower degrees of freedom and a skew parameter, as this distribution could more accurately model both the observed fat tails and left skew. As a final note, when we inspected scatterplots of two, or even three variables, the left-skewed tails are discernable, as can be seen in the scatterplots below. Scatterplot of BUSCRES vs BEMRES BUSCRES BEMRES 0 1 2

6 3D Scatterplot of BFRARES vs BEMRES vs BUSCRES 1 BFRARES BUSCRES BEMRES 6. Description of Copulas A copula is a multivariate probability distribution for which the distribution is a function of separate marginal distributions. The equation is as follows: P(X < x 0, Y < y 0 ) = C[F(x 0 ),G(y 0 )] This model rooted in Sklar's Theorem, which states, "Let H be a joint distribution function with margins F and G. Then there exists a copula C such that for all x,y H(x,y) = C(F(x),G(y) If F and G are continuous, then C is unique. The role of Sklar s theorem is that not just every copula function with marginal distributions as arguments is a valid bivariate distribution. It states that every valid bivariate distribution can be represented as a copula of its marginals" [7]. We will likely use either a T-Copula or an Archimedean Copula. 7. Future Directions Ultimately, we hope to create a practical model which includes both assets and liabilities that describes their dependency with a high degree of accuracy. This would allow insurance

7 companies to apply their own data in order to know their risk profile, particularly in a lowinterest environment. References [1] Staff, Investopedia. Economic Capital. Investopedia, InterActive Core, 1 Oct. 2015, Accessed 28 Aug [2] Shkolnikov, Vladimir, and Magali Barbieri. HMD Main Menu. Human Mortality Database, Max Planck Institute for Demographic Research, Accessed 28 Aug [3] Injury Prevention & Control. Centers for Disease Control and Prevention, Centers for Disease Control and Prevention, 20 Apr. 2017, Accessed 29 Aug [4] US Individual Life Persistency. Society of Actuaries, Society of Actuaries, LIMRA, Dec. 2012, rate. Accessed 29 Aug [5] Bloomberg. (2017) Bloomberg Professional. [Online]. Available from Bloomberg Terminal (Accessed: 7 July 2017). [6] Year-End 2013 Insurance Industry Investment Portfolio Asset Mixes. NAIC Capital Markets Special Report, National Association of Insurance Commissioners, 6 May 2014, Accessed 29 Aug [7] Shemyakin, Arkady, and Alexander Kniazev. Introduction to Bayesian estimation and copula models of dependence. Hoboken, NJ, Wiley, 2017.

8 Appendix-Numerical Results of the 5 Selected Indices Bloomberg US Corporate Bloomberg Global High Yield Corporate Bloomberg US Emerging Markets Bloomberg Sovereign French Bloomberg USD Investment Grade Corporate Financial Sector BUSC BHYC BEM BFRA BUSCFI > BUSCmodel 1. ARIMA Models (in R) Series: Bloom$`BUSC Index` ARIMA(0,1,0) with drift Coefficients: drift s.e sigma^2 estimated as : log likelihood=-685 AIC= AICc= BIC= > BHYCmodel Series: Bloom$`BHYC Index` ARIMA(1,1,1) with drift Coefficients: ar1 ma1 drift s.e sigma^2 estimated as : log likelihood= AIC=859 AICc= BIC= > BEMmodel Series: Bloom$`BEM Index` ARIMA(0,1,3) with drift Coefficients: ma1 ma2 ma3 drift s.e sigma^2 estimated as : log likelihood= AIC= AICc= BIC= > BFRAmodel Series: Bloom$`BFRA Index` ARIMA(0,1,0) with drift

9 Coefficients: drift s.e sigma^2 estimated as : log likelihood= AIC= AICc= BIC= > BUSCFImodel Series: Bloom$`BUSCFI Index` ARIMA(1,1,4) with drift Coefficients: ar1 ma1 ma2 ma3 ma4 drift s.e sigma^2 estimated as : log likelihood= AIC= AICc= BIC= Correlation P-Value 2. Pearson Correlation Matrix of the Residuals BUSCres 1.00 BUSCres BHYCres BEMres BFRAres BUSCFIres BHYCres BEMres 0.37 BFRAres 0.54 BUSCFIres