copula models of economic capital for insurance …...companies. economic capital is “the amount...
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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 2008. 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
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 1st, 2010. We retrieved the daily closing price up until July 7th,
2017. 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.
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:
Yt -Yt-1 = μ + ϕ1 (Yt-1 – Yt-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
[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
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.
210-1-2-3
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
BEMRES
BU
SC
RES
Scatterplot of BUSCRES vs 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 < x0, Y < y0) = C[F(x0),G(y0)]
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
-2-1
0
-2
-1
0
-2
1
0
2
1
SERARFB
SERMEB
SERCSUB
D Scatterplot of BFRARES vs BEMRES vs BUSCRES3
companies to apply their own data in order to know their risk profile, particularly in a low-
interest environment.
References
[1] Staff, Investopedia. “Economic Capital.” Investopedia, InterActive Core, 1 Oct. 2015,
www.investopedia.com/terms/e/economic-capital.asp. Accessed 28 Aug. 2017.
[2] Shkolnikov, Vladimir, and Magali Barbieri. “HMD Main Menu.” Human Mortality
Database, Max Planck Institute for Demographic Research, www.mortality.org/. Accessed
28 Aug. 2017.
[3] “Injury Prevention & Control.” Centers for Disease Control and Prevention, Centers for
Disease Control and Prevention, 20 Apr. 2017, www.cdc.gov/injury/wisqars/nonfatal.html.
Accessed 29 Aug. 2017.
[4] “US Individual Life Persistency.” Society of Actuaries, Society of Actuaries, LIMRA, Dec.
2012, www.soa.org/Search.aspx?q=lapse rate. Accessed 29 Aug. 2017.
[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,
www.naic.org/capital_markets_archive/140506.htm. Accessed 29 Aug. 2017.
[7] Shemyakin, Arkady, and Alexander Kniazev. Introduction to Bayesian estimation and
copula models of dependence. Hoboken, NJ, Wiley, 2017.
Appendix-Numerical Results of the 5 Selected Indices
Bloomberg US Corporate BUSC
Bloomberg Global High Yield Corporate BHYC
Bloomberg US Emerging Markets BEM
Bloomberg Sovereign French BFRA
Bloomberg USD Investment Grade Corporate Financial Sector BUSCFI
1. ARIMA Models (in R)
> BUSCmodel Series: Bloom$`BUSC Index` ARIMA(0,1,0) with drift Coefficients: drift 0.0247 s.e. 0.0078 sigma^2 estimated as 0.1178: log likelihood=-685 AIC=1374.01 AICc=1374.02 BIC=1385.17 > BHYCmodel Series: Bloom$`BHYC Index` ARIMA(1,1,1) with drift Coefficients: ar1 ma1 drift 0.5734 -0.2503 0.0352 s.e. 0.0492 0.0582 0.0119 sigma^2 estimated as 0.09049: log likelihood=-425.5 AIC=859 AICc=859.02 BIC=881.32 > BEMmodel Series: Bloom$`BEM Index` ARIMA(0,1,3) with drift Coefficients: ma1 ma2 ma3 drift 0.3205 0.2080 0.0920 0.0320 s.e. 0.0225 0.0236 0.0229 0.0107 sigma^2 estimated as 0.08585: log likelihood=-373.44 AIC=756.88 AICc=756.91 BIC=784.78 > BFRAmodel Series: Bloom$`BFRA Index` ARIMA(0,1,0) with drift
Coefficients: drift 0.0197 s.e. 0.0079 sigma^2 estimated as 0.1236: log likelihood=-732.09 AIC=1468.19 AICc=1468.19 BIC=1479.35 > BUSCFImodel Series: Bloom$`BUSCFI Index` ARIMA(1,1,4) with drift Coefficients: ar1 ma1 ma2 ma3 ma4 drift -0.6742 0.6775 0.0403 0.0718 0.0909 0.0248 s.e. 0.1234 0.1240 0.0272 0.0278 0.0233 0.0066 sigma^2 estimated as 0.06854: log likelihood=-151.8 AIC=317.61 AICc=317.67 BIC=356.67
2. Pearson Correlation Matrix of the Residuals
Correlation P-Value
BUSCres BHYCres BEMres BFRAres BUSCFIres
BUSCres 1.00 0.0000
0.00 0.8262
0.37 0.0000
0.54 0.0000
0.96 0.0000
BHYCres 0.00 0.8262
1.00 0.0000
0.63 0.0000
-0.06 0.0050
0.09 0.0001
BEMres 0.37 0.0000
0.63 0.0000
1.00 0.0000
0.23 0.0000
0.40 0.0000
BFRAres 0.54 0.0000
-0.06 0.0050
0.23 0.0000
1.00 0.0000
0.52 0.0000
BUSCFIres 0.96 0.0000
0.09 0.0001
0.40 0.0000
0.52 0.0000
1.00 0.0000