Study of Volatility Stochastic Processes in the Context of Solvency Forecasting for Sri Lankan Life Insurers

The main business of Life Insurers is Long Term contractual obligations with a typical lifetime of 20 - 40 years. Therefore, the Solvency metric is defined by the adequacy of capital to service the cash flow requirements arising from the said obligations. The main component inducing volatility in Capital is market sensitive Assets, such as Bonds and Equity. Bond and Equity prices in Sri Lanka are highly sensitive to macro-economic elements such as investor sentiment, political stability, policy environment, economic growth, fiscal stimulus, utility environment and in the case of Equity, societal sentiment on certain companies and industries. Therefore, if an entity is to accurately forecast the impact on solvency through asset valuation, the impact of macroeconomic variables on asset pricing must be modelled mathematically. This paper explores mathematical, actuarial and statistical concepts such as Brownian motion, Markov Processes, Derivation and Integration as well as Probability theorems such as the Probability Density Function in determining the optimum mathematical model which depicts the accurate relationship between macro-economic variables and asset pricing.


Introduction
The key determinants of a successful Service-Based Organization in Sri Lanka are Customer Relationship, Pro-activeness, Commitment and Effective Controls

Contemporary Evidence
Stochastic Process Algebras are used in Dependability Modelling where the components of the holistic system influence each other. Stochastic Volatility Modelling includes components which fluctuate over time. The model is kept together by a set of equations which are based on the historic relationships of the components and can forecast the movements of the entire system [8]. This is especially useful in Statistical Risk Management in terms of monitoring and forecasting the movement of Risk-Based Capital.

Literary Abstract
The Sri Lankan Financial Industry has had a turbulent past tainted with fraud, anti-money laundering and political manipulation which resulted in the European Commission listing the country as high financial risk. Furthermore, the era of peace bought with a 32-year-old only lasted 5 -6 years as political instability returned through scandals and unconstitutional political upheaval.
The Insurance Sector went through ups and downs in the past decade. The decline in motor vehicle registration declined the General Insurance business while process integration and automation enhanced the quality of service and margins. Product innovation has created diversification both in terms of products and distribution channels. However, majority of the market remains unpenetrated as the deep inbuilt distrust regarding the Financial Industry hampers potential. However, Enterprise Risk Management and a series of Merges and Acquisitions boosted the capabilities of the Industry through the years.
The conundrum of ageing population seems to be doing the industry a favour as it has increased the burden of dependents on a contracted labor force. Therefore retirement products, health riders and life covers have taken forefront of consumer behavior. This was only enhanced further by the COVID-19 pandemic.
The gig economy has helped improve the management of Agents. However there is a near possibility that much of the services exposed to customers will be automated and digitized through insuretech.
An ERM system is a management skeleton which must be adopted and customized as per the company's objectives and circumstances. A set of risk-centric corporate objectives are enforced, monitored and controlled through risk identification, assessment and response. A neural communication network is bound around risk-silos (sub risk segments) and reporting is executed through these networks.
A solid ERM must be able to forecast the cost of capital of doing business for

Methodology
The Research Onion methodology was adopted when producing this paper. It is an outside in method which follows a systematic approach in deciding the correct methodology for each research component [9].

Research Philosophy
The collective set of norms, beliefs and idiosyncrasies associated with the objec-

Methodological Choice
Given that this research involves a quantitative metric (Capital) which is influenced by qualitative forces (External environment), a multi-method research was conducted through mathematical, economic and business Literature where quantitative and qualitative findings and principles were incorporated.

Strategy
Strategy defines how research expectations are realized. Strategies include experiment, survey, archival research, case study, ethnography, action research, grounded theory and narrative inquiry.
The strategy used to realise research objectives of this paper is multiple case study through systematic review.

Time Horizon
This research studies past trends extensively in order to generate relationships between variables. As such, a Longitudinal Time Horizon was selected.

Techniques and Procedure
Existing literature on conceptual frameworks was collected in order to study different perspectives in terms of Capital Valuation, Risk Management methodologies and Stochastic Volatility processes. The source for this was mainly academic journal papers. In order to study past trends, publications of statistical data on historic performance in terms of economy and the insurance industry were collected. These are mainly reports and publications by Government or Regulatory bodies, mainly through the databases of CBSL, CSE and IRCSL. Newspaper articles and journal papers pertaining to the events of the insurance industry were collected in order to ascertain the nature and evolution of same.
The literature pertaining to the Insurance Industry of Sri Lanka, Capital Valuation, Risk Management methodologies and Stochastic Volatility processes were first studied in order to understand the underlying concepts and calculation bases. Historic data was then analyzed to form trends and correlations in order to form relationships between variables. The prevalent Stochastic modelling (RBC) within the country was then studied to understand if all necessary aspects are being addressed. Where necessary, the knowledge from existing studies relating to mathematics was used to improve or fill the gaps in the RBC model.

Systematic Review
Systematic Review collates and evaluates data in order to answer the specified research question while meta-analysis helps infer quantitative data with qualitative literature. The PRISMA model was adopted for this purpose [10], which details a journey through the collected data through the stages of identification, screening, eligibility testing and inclusion.
Systematic review follows seven stages, namely setting research scope, infor- Denscombe details four methods of thematic analysis, namely meta-analysis, narrative analysis, thematic synthesis and realist synthesis [11]. When compiling this paper, thematic synthesis was used to review selected articles in order to derive research themes and questions. Meta-analysis was used to critically evaluate historic literature and data in order to derive conclusions on the selected themes.
The research was then conducted in line with set objectives, both in terms of literature review and data analysis. To study the stochastic volatility processes in order to derive the algorithms defining the relationship between volatile market forces and the company's Capital.

Regulatory Risk-Based Capital
Risk-Based Capital (RBC) is the capital required to assure solvency while managing uncertainty [12]. This must be based on a market consistent valuation methodology as it shows the management the liquid assets available to absorb market shocks. Additionally, RBC has been used as the base for prudential and Solvency can be achieved by maintaining the minimum capital requirement.
The most crucial factor in capital management is determining the level of capital required in order to cover its portfolio risks in the relevant timeframe. Therefore the timely identification and quantification of such risks are mandatory in maintaining solvency alongside key performance targets. Some tools used in Capital Budgeting include game theory, real options pricing, decision trees, sensitivity analysis, scenario analysis, IRR, NPV and uncertainty absorption cash flow [13].

Risk-Adjusted Performance Measurement (RAPM)
RAPM provides a single performance measurement based on Solvency rules and is therefore a much more realistic forecast of shareholder's returns.
where expected revenue and losses are mutually exclusive [15]. In terms of Solvency 2 Regulations, it is assumed that Capital of a Bank and Insurer provides 99% and 99.5% guarantee respectively, that obligations will be met by the company. Therefore required capital here is calculated by subtracting the capital Value-at-Risk (VaR) at 99.5% confidence level. However, this could be the making or breaking point of Capital Management as this calculation requires a series of dependent and independent assumptions. If the company is over-optimistic it gives rise to the risk of insolvency in the future while over-cautiousness could create opportunity cost, foregoing probable positive cash flows.
Dacorogna suggests a series of equations with probable conditions and rationale [14]. However, given that the risk components for each company differ depending on strategy, segmentation, size and demographics, it is prudent to calculate both economic capital and required capital on the basis of recognized risks as per the company's capital structure. Hence, the company must have a clearly defined capital structure cascading down to risk components. In hindsight, this is where the Risk-Based Capital structure becomes most relevant.

Value-at-Risk (VaR)
VaR is typically the loss impact through the worst case scenario under normal or expected market conditions and can be calculated in both parametric and nonparametric methods. Non-parametric methods deviate from predictive analysis and calculate VaR through historic trends. The Historic Simulation method uses the base assumption that the considered future is similar to the recent past while

Exponential Weight Moving Average Model (EWMA)
If the probability distribution is assumed to be normal then EWMA can be used [17] to estimate σ t ; VaR at confidence level 1 − α = µ + σ t G−1(α) where G−1(α) represents the α quantile of the standard normal distribution while σ t is the standard deviation of portfolio returns.
( ) ( ) The main drawback of this method is the assumption that the dataset will follow a normal distribution which is improbable due to market volatility and uncertainty and are usually observed to follow a negatively skewed leftist distribution. Furthermore, while EWMA factors in varying and cluster volatility, it leaves out asymmetry and leverage.

Volatility Models
Volatility models seem to somewhat abate the aforementioned shortcomings.
Three widespread volatility models are GARCH family, Stochastic Volatility Model and Realised Volatility Based Model.

Autoregressive Conditional Heteroskedastic (ARCH) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
In the ARCH model introduced by Engle, conditional variance fluctuates over a given period of time as a function of historic errors thereby having a constant unconditional variable. This is mostly accurate for a single period forecast. The model provides a risk scoring mechanism based on likelihood of occurrence [18]. Bollerslev's GARCH model on the other hand generalized Engle's ARCH with corrections on the forecast period limitation and flexible lags [19].

Brownian Motion and Samuelson-Black-Scholes Model
If time is to be attached with a positivity condition where t > 0, the first formulae can be rewritten as: where ( ) t τ has replaced the usual variance σ 2 with a drift where the integral: where * τ functions as the integrated variance.
However, a customized formula must be derived if the actual proportions and relativity of the factors impacting forecasting metrics with the progression of time. This deserves a separate paper. In order to derive a sound Stochastic Volatility model, analytical option pricing, realized variance and realized power variance, OU based models and subordination must be integrated into multivariate models [21].

Solvency Capital Requirement (SCR)
An alternative approach calculated the SCR under Solvency II as a function of sub risk categories, namely equity risk and market risk [22].

Pearson's Coefficient
Pearson Coefficient (R) calculates the strength and nature of the linear relationships between two variables [23]. The coefficient can be calculated on excel using the formula = PEARSON (array 1, array 2) where the arrays represent the data sets of the two variables [24]. The result is an integer ranging from −1 to +1 and can be interpreted as follows: Weak negative It can also be plotted on a scatter diagram on excel for graphical representation. R is mainly useful when forecasting a dependent metric. In this paper, R is used to determine and confirm the macro-economic variables which impact RBC in the context of Sri Lanka in order to determine the depended and independent variables which must be subjected to the theoretical conclusion of the Literature Review. irCC is the minimum of the up-shock and down-shock scenarios of mismatch between asset and liabilities valued at present. Here, the asset and liability cash-flows are discounted factoring fixed shock scenarios and Risk Free Rates (RFR). RFR is an indicative forward rate of return which is the excess of current Treasury bond yield over current inflation rate [25]. cpCC is calculated in a similar manner by discounting cash flows of assets bearing credit risk by factoring RFR and shock scenarios.

Calculation of Risk-Based Capital
eqCC imposes risk charge factors on Listed Equity of non-related parties, Listed Equity of related parties and Unlisted Equity of non-related parties in the amount of 35%, 45% and 45% respectively. Equity is valued at current market price.
prCC and grCC bear risk charge factors of 25% and 15% respectively and is charged on its admissible exposure value.
utCC imposes risk charge factors based on the underlying instrument of the relevant Unit Trust Funds, as per Table 1.   Investors tend to make riskier investments in order to maximize returns during low risk periods. However, when the risk sentiment begins to worsen, the market produces a natural risk off phenomena where investors abandon low risk assets and flee to safe heavens as was prominent during the COVID-19 period [33]. This is evidenced by the inverse relationship between ASI and RFR as shown in Figure 2.

Relationship between Economic Output and Economic Indicators
The relationship between the Exchange Rates and Insurance Premium Growth was positive up until 2019. Given the low economic stimulus prevalent in the country observed through low credit growth, CBSL eased monetary policy by way of continuously reducing policy rates, statutory reserves and bank rates while managing the liquidity reserves through issuance and repurchase of Government Securities [34].
Growth of Insurance Premiums and Service sector is observed to have a positive relationship.
2) Relationship between Economic Output and the movement of Economic Indices The Insurance sector and its sub-sectors as well as the service sector have a strong positive correlation with GDP as observed in Table 2. Therefore, both positive  Interest rates movement and Insurance Premium growth has a strong positive correlation while Interest rates movement and ASI growth has a strong negative correlation as observed in Table 3. This is evidenced by the boost in ASI witnessed during monetary policy easing and the dips seen during high inflationary environments [35] [36] [37].
However, Interest rates movement has a weak positive correlation with GDP which indicates that interest inelastic sectors contribute massively to GDP. The exchange rate movement has low negative correlations with GDP growth and service sector growth as observed in Table 4. However, insurance premium growth has a low positive correlation indicating a low level of influence. ASI has a strong negative correlation with exchange rate movement.
The GDP growth rate directly impacts the disposable income and thereby the purchasing power which heavily influences the service sector and insurance industry. Therefore shocks on GDP through components such as Trade, Credit Growth, Remittances, Tourism, FDI's must be factored in when assessing risks to the Insurance Industry.
The positive and negative relationships interest rate growth has with insurance premium growth and ASI respectively can be used to hedge against each other. If the risk management system can predict a shortfall on the industry, resources can be deviated from business acquisition to investments thereby negating the overall impact on capital. Despite the positive and negative relationship between LKR: USD movement and Interest rates and ASI respectively, given its opposing proportions interest rates and ASI cannot be hedged against each other efficiently. The strong negative correlation as observed in Table 5 indicates that ASI and Insurance act as substitute products. The strong negative correlation with Life premium growth is prudent as Life insurance products are long term investments while the equities are mostly short term investments. Hence, the direction of both depends on the country's GDP growth as well as the risk sentiment.
It can be deduced that GDP growth and interest rates directly influence the performance of the Insurance Sector while ASI inversely influences the performance of the Insurance Industry, specifically Life Insurance. Therefore, the impact on Life Insurance Premium growth can be forecasted by multiplying the Coefficient with forecasted movement of GDP and Interest rates. Assets have higher positive correlation with Premium than Interest as observed in Table 6. This is because the Asset Base largely funded by premium cash flows in the Life Insurance Sector and is therefore funded mainly through Liabilities.

Impact of Interest Rates and Premium on Firm Assets
Available information was insufficient to assess the same for liabilities.

Derivation of Quadratic Equations for Risk Charges
It has been deduced that the main influencers of CAR are domestic economic growth, interest rates, sales and capital management, out of which economic growth and interest rates are external volatility variables. Therefore, majority of Risk Charges can be derived as an expression of these volatility elements.
If we take the customize the Brownian motion formula derived in the Literature review section; However, with the attempt to understand the basics of the model, it is prudent to build up from the most simplistic random walk formula; ( ) Taking Interest Risk Capital Charge (irCC) as the example, the following formula can be derived.
Segregate interest sensitive assets and apply the formulae separately If the influence of interest rates on bonds is relatively direct, then the constant can be derived through the regression coefficient. On the other hand, given that changes announced at the bi-monthly monetary policy review by the CBSL contributes largely to the independent movement, a qualitative metric can be used here to predict the direction of the decision taken at the upcoming monetary policy review meeting and to thereby adjust the constant. Therefore, irCC will be the summation of all interest sensitive asset categories subjected to the aforementioned methodology. eqCC can be derived in a similar yet indirect manner. Given that the Equity market responds indirectly to the changes in the interest rates, the change in ASI can be derived as an expression of the change in short term interest rates.
The formulae derived for the different interest sensitive assets can be indirect-Open Journal of Statistics ly linked to the Liability Risk Capital Charge (LRCC) computation. Given that the same interest rate forecast applies to Liabilities as well, the same independent movement can be taken while the constant has to be derived separately for each policy/rider as per the historic regression analysis.
Conversely, CRCC, COCC and RICC are dependent on the future counterpart rating. Therefore, an internal credit rating system must be developed to annually (for practical purposes) forecast any rating changes of counterparts. This must be dependent on the company's risk appetite statement as well.
OPCC depends mainly on the other risk charges and sales. Given that sales forecast is the cornerstone is financial planning, this paper assumed that there is a solid system in place for this purpose.
SVCC can hence be auto-calculated as liability metrics are being forecasted through LRCC. In order to forecast LRCC, sales, pricing strategy and market rates must all be forecasted to ascertain the liability movement. Efficiency standards and management styles must be predetermined in order to incorporate projected profits into the OPCC calculation.

2) Derivation of a Quadratic Equation for Capital
The underlying Independent variable which impacts the entirely of the Industry is Policy Decisions. Shortcomings in GDP expectations due to the decline in terms of interest rates, GDP, exchange rates and demand. Therefore, the first instance of Brownian motion application must be between Policy review expectations and the resultant change in economic metrics. These relationships much then be re-expressed as a multiple of one or two independent variables. This enables the model to express the RBC formulae through the selected independent variables thereby giving a near-accurate quadratic equation for capital projection.

Literature and Observation
The

Limitations of the Observed Solution
While the mathematics provide a sound basis of forecasting, it is entirely based on the relationship between policy decisions and resultant variable volatility.
Therefore, the most prudent methodology is to predict the first instance of policy review by way of assessing the necessity of such a measure. This gives an indication as to the magnitude of the necessity of change and the magnitude of required change. If the volatility of each variable can then be integrated into a mass economic volatility formulae, much of the resultant impacts on many industries can then be quantified using the model proposed in this paper.
However, such mass integration is yet to be initiated on an academic level in Sri Lanka. Standalone initiation itself is insufficient and impractical due to the volatile nature of the economy. Therefore, quantification of the relationship between policy decisions and macroeconomic variables must be an ongoing research project with committed veterans of the industry as well as skilled statisticians and developers.
The same is true in terms of application of the model proposed within this paper on any one Insurer. The relationships must be re-evaluated and updated constantly as per the changing trends. The country currently does not have a committed line of research in this regard.

Opportunities Arising from This Paper
The recommendations stipulated in section 6.3 can be applied for any country or

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.