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