1. The Need for Stochastic Models for Insurers

Modelling has long been a critical component of life insurance companies, and this historically arose due to the complexity and long dated nature of the liabilities.

One can go all the way back to the development of a with-profits product, first suggested by James Dodson in 1756, in a document entitled “First Lectures on Insurance”. A key feature was the development of a whole-of-life assurance policy, when only short-term (1–2 year) assurance was sold. The summary features included a description of a with-profit system “where participating and non-participating policies were simultaneously provided” and “The surplus on non-participating business would be distributed equitably amongst participating policyholders, in proportion to how much profit their own policies had generated for the corporation by virtue of their longevity”. The product as described above was later implemented by Equitable Life, which subsequently led the development of Life Assurance within United Kingdom.

The with-profits product also emerged as a key product within the United Kingdom, and along with it the role of an actuary to calculate and model the distributions from the surplus of the with-profits fund. The sophistication in modelling increased over time as the with-profits product emerged, with huge strides in the understanding of mortality, following the founding of the Institute of Actuaries in 1848 (Faculty of Actuaries in 1856). A detailed exposition of these developments can be found in the 8 volume History of Insurance (Jenkins & Yoneyauma, Reference Jenkins and Yoneyauma2000).

Most of the models at this time still had a single view of the future, either best-estimate or with various shades of prudence for the purposes of pricing, valuation and reserving. This huge volume of actuarial thought developed at a time when the main investments of insurance companies were fixed interest, loans and securities, which provided relatively low yields at a time when long-term inflation was virtually nil (19th century), and focussed very much on deterministic models. The conventional actuarial concept of a single fixed rate of interest was considered reasonably appropriate in these circumstances. However, over the 30 years from 1920 to 1950, British life offices started increasingly investing in equity and property, with the equity allocations increasing from 2% in 1920 to 21% by 1952 (Turnbull, Reference Turnbull2017) which increased the importance and complexity of the asset model.

These were the immortal words that that were brought to Robert Oppenheimer’s mind as he saw the detonation of the first ever atomic bomb in 1945 (Wikipedia, n.d.). 1945 was also the year that the first electronic computer was built at the University of Pennsylvania in Philadelphia by John Mauchly and Presper Eckert. Electronic Numerical Integrator And Computer (ENIAC) was an impressive machine, it had 18,000 vacuum tubes with 500,000 solder joints (Metropolis, Reference Metropolis1987).

ENIAC brought the renaissance of an old mathematical technique known at the time as statistical sampling. Statistical sampling had fallen out of favour because of the length and tediousness of the calculations.

While recovering from an illness in 1946 Stanislaw Ulam questioned the probability of winning a game of Canfield Solitaire (Wikipedia, n.d.). After spending a lot of time trying to come up with an abstract solution to the problem, he hit upon the idea of using sampling methods which could be used with the aid of fast computers (Eckhardt, Reference Eckhardt1987).

During the review of ENIAC in spring of 1946 at Los Alamos, Stanislaw Ulam discussed the idea of using Statistical sampling with John von Neumann to solve the problem of neutron diffusion in fissionable material. This piqued Neumann’s interest and in 1947 he sent a handwritten note to Robert Richtmyer outlining a possible approach to the method for solving the problem as well as a rudimentary algorithm (Neumann & Richtmyer, Reference Neumann and Richtmyer1947). This was most likely the first ever written implementation of the statistical sampling technique.

It was during this time that Nicholas Metropolis suggested the name Monte Carlo Method – a suggestion not unrelated to the fact that Stanislaw Ulam had an uncle who would borrow money from relatives because he “just had to go to Monte Carlo” (Metropolis, Reference Metropolis1987).

Metropolis and Ulam published their seminal paper on the Monte Carlo method in 1949 (Metropolis & Ulam, Reference Metropolis and Ulam1949). The method was implemented successfully for use in development of the Hydrogen Bomb and subsequently went on to be used in a wide array of fields including finance.

One of the core aspects of the Monte Carlo method is the use of Random Numbers to see the evolution of the decision paths. For the first implementation of the Monte Carlo method, the pseudo random number generator used was von Neumann’s “middle-square digits” (Metropolis, Reference Metropolis1987). The generator produced uniform random deviates which were then transformed into non-uniform deviates based on a transformation function.

Like Monte Carlo methods, random number generation has evolved substantially since their use in 1949 and continues to be developed at a rapid pace within the field of mathematical finance as well as outside.

A paper by the Maturity Guarantee Working Party (MGWP) (Benjamin et al., Reference Benjamin, Ford, Gillespie, Hager, Loades, Rowe, Ryan, Smith and Wilkie1980) summarises the need for stochastic models quite eloquently:

Thus, there was an increasing awareness of the uncertainty in future solvency driven by the inherent uncertainty in the capital markets, as indicated in Figure 1.

Figure 1 An example simulation of a single path

The need for stochastic models for a life insurer with path-dependent liabilities and a global asset allocation then provides some challenges, which we aim to summarise in the next section.

2. The Challenge Inherent in Stochastic Asset Models (SAMs)

In the context of history, whilst stochastic asset modelling has come relatively late to the party of complex actuarial modelling, it is perhaps the most challenging aspect in some respects. In addition to the pure inherent uncertainty of future capital market outcomes, there are a number of other factors which compound the complexity, as we see in Figure 2.

Figure 2 Asset allocation for an average pension fund (United Kingdom)

3. Evolution of Asset Models

Having taken as input a large range of literature on Actuarial history, conducted a range of interviews with developers and users of economic scenario generators (ESGs), and pored through a range of Actuarial and Academic papers, journals and books, it appears that there were a number of stages in the evolution of ESGs, as shown in Figure 3.

Figure 3 Evolution of economic scenario generators

This led to broadly four stages of ESGs, and we shall explore each in further detail in the next four sections:

• ■ Random Walk Models

• ■ Time Series Models

• ■ Market Consistent (MC) and Arbitrage-Free Models

• ■ Value at Risk (VaR) Models

Whilst the VaR Models are in practice less complex relative to the other ESGs and there is a strong distinction between the two; in general, ESGs are generally time series models that consider the evolution of an economic quantity over time, whereas VaR models focus more on a statistical distribution of outcomes over a shorter time period – usually up to 1 year in the insurance industry. Nevertheless, we have included them both in the scope of this paper as their history is to some extent interlinked.

4. Random Walk Models

In the warm summer months of 1905, Karl Pearson was perplexed by the problem of the “random walk”, as illustrated in Figure 4. He appealed to the readers of Nature for a solution as the problem was – as it still is – “of considerable interest”. The random walk, also known as the drunkard’s walk, is central to probability theory and still occupies the mathematical mind today. Among Pearson’s respondents was Lord Rayleigh, who had already solved a more general form of this problem in 1880, in the context of sound waves in heterogeneous materials. With the assistance of Lord Rayleigh, Pearson concluded that “the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!” (Nature, 2006).

Figure 4 Random Walks

In Reference Bachelier1900 Louis Bachelier published his truly remarkable doctoral thesis, La Théorie de la Spéculation (Theory of Speculation), on random walk or Brownian motion. Bachelier is credited with being the first person to apply the concept of random walk to modelling stock price dynamics and calculating option prices. It was his pioneering research that eventually led to development of the efficient market hypothesis (Rycroft & Bazant, Reference Rycroft and Bazant2005).

It was Norbert Weiner who provided the rigorous mathematical construction of Brownian motion and proceeded to describe many of its properties: it is continuous everywhere but nowhere differentiable. It is self-similar in law, i.e. if one zooms in or zooms out on a Brownian motion, it is still a Brownian motion. It is one of the best known Lévy processes (càdlàg stochastic processes, i.e. right continuous left limit process with stationary independent increments) and it is also a martingale.

It was in 1951 in the Memoirs of the American Mathematical Society; Volume No. 4, a paper titled “On Stochastic Differential Equation” was published by Kiyoshi Ito (Reference Ito1951). This seminal work first introduced the concept of “Ito’s Lemma” which went on to become a fundamental construct in financial mathematics.

Around the same time Harry Markowitz (Reference Markowitz1952) published his paper “Portfolio Selection” which was the first influential work on mathematical finance which captured attention outside academia. Markowitz (Reference Markowitz1952), the paper, together with his book “Portfolio Selection: Efficient Diversification of Investments” (Markowitz, Reference Markowitz1959), laid the groundwork for what is today referred to as MPT, “modern portfolio theory”.

Prior to Markowitz’s work, investors formed portfolios by evaluating the risks and returns of individual stocks. Hence, this leads to construction of portfolios of securities with the same risk and return characteristics. However, Markowitz argued that investors should hold portfolios based on their overall risk-return characteristics by showing how to compute the mean return and variance for a given portfolio. Markowitz introduced the concept of the efficient frontier which is a graphical illustration of the set of portfolios yielding the highest level of expected return at different levels of risk. These concepts also opened the gate for James Tobin’s super-efficient portfolio and the capital market line and William Sharpe’s formalisation of the capital asset pricing model (CAPM) (Rycroft & Bazant, Reference Rycroft and Bazant2005).

This leads to a theoretically optimal asset mix where one can define a utility function, and subsequently choose portfolio weights to maximise expected utility. In practice, utility functions contain unknown risk aversion parameters, so we use theory to explain the relationship between asset mix and risk aversion and identify “efficient” portfolios that are best for a given investor’s risk preferences.

There were several advantages to random walk models; they captured the key moments of risk, return and correlation of different asset classes. They were also practical and intuitive, with a small number of intuitive parameters and easily connected to the efficient markets hypothesis. Additionally, the common collection and reporting of historical statistics made them very amenable to calibrate to historical data.

A further development in financial economics came in 1973, when two papers in two different journals had a major impact on the world of modern finance. This biggest impact to the Actuarial profession was not until much later, as we went on a journey through time series models, and the Wilkie model in particular.

5. Time Series Models

5.2. Actuarial implementation of time series models

In November 1984 A. D. Wilkie presented to the Faculty of Actuaries his paper “A Stochastic Investment Model for Actuarial Use”. In his introduction to the model, Wilkie describes the advantage of a stochastic model – allowing the user to move beyond considering the average outcome and look at the range of outcomes. In the language of the previous section, the Wilkie model is essentially an ARMA modelFootnote ¹ but the relationships between the modelled variables does add some complications.

The model described in the 1984 paper applied many of the ideas of time series modelling and forecasting to produce a SAM. Wilkie had been a member of the 1980 MGWP which developed a reserving model for (unit linked) maturity guarantees. The MGWP modelled only equity prices, but did adopt a time series approach based on historic values of the De Zoete equity index. The MGWP modelled the equity price as the ratio of dividend pay-out and the dividend yield. The dividend pay-out was modelled as a random walk with non-zero mean with the dividend yield fluctuating around a fixed mean. The 1984 paper adopted a similar approach for equities, but extended it to other asset classes by adding inflation and a long-term fixed interest yield.

The 1984 paper adopted a “cascade” structure to link the modelled investment variables, with inflation as the entry point. In the 1970s and early 1980s inflation fluctuations, as shown in Figure 5, had a large influence on asset prices.

Figure 5 UK inflation from 1950 (Source: ONS)

Professor Wilkie wrote:

“The substantial fluctuations that have been observed in the rate of inflation, the prices of ordinary shares and the rate of interest on fixed interest securities lead one to wish to consider more carefully likely possible future fluctuations in these variables. The actuary should not only be interested in the average return that may be achieved on investments, but in the range of possible returns. Unless he does this, he cannot know to what extent any single figure he chooses is sufficiently much “on the safe side”. A consideration also of the possible fluctuations in investment experience may be of value in considering alternative strategies for investment policy, bonus declaration, etc”.

“The classical model used by financial economists to describe the stochastic movement of ordinary share prices has been that of a random walk. The MGWP showed that, over a long time period, a model based on dividends and dividend yields was more appropriate”.

It is interesting to observe that Professor Wilkie chose a specific time series formulation with two primary and two secondary variables over a fuller multi-variate structure. Wilkie clearly states in the paper that he is considering a model suitable for actuarial use – over the long term – and distinguishes an actuarial model from a model that an economist or statistician might build to prepare short term forecasts.Footnote ² Wilkie’s specific time series formulation is discussed in a separate note (itself 30 pages long). The formulation reflects the relationships Wilkie found in data from UK time series (1919–1982) and that were considered appropriate for the intended use of the model – with the process adopted being transparent. Whilst the formulation is more parsimonious than a full multi-variate structure, it is far from simple – e.g. the Consol’s yield is a third-order autoregressive process.

The 1984 paper describes the model in full but the cascade structure linking the investment variable is described in Figure 6.

Figure 6 Cascade structure of Wilkie model (Source: Wilkie, Reference Wilkie1984)

Wilkie extended the model presented in the 1984 paper in a 1995 paper. In the period between the two papers the actuarial profession had reviewed the model’s structure (Geoghegan, Reference Geoghegan, Clarkson, Feldman, Green, Kitts, Lavecky, Ross, Smith and Toutounchi1992), with assistance from Wilkie, and the 1995 update (Wilkie, Reference Wilkie1995) addressed some of the challenges and also extended the asset coverage of the model.

6. MC and Arbitrage-Free Models

It was in 1973 that two papers in two different journals had the biggest impact on the world of modern finance. Fischer Black and Myron Scholes (Reference Black and Scholes1973) published the paper “The Pricing of Options and Corporate Liabilities” in the Journal of Political Economy and Robert Merton (Reference Merton1973) published the paper “On the pricing of corporate debt: the risk structure of interest rates” in the in Journal of Finance. These papers introduced methodologies for valuation of financial instruments, in particular one of the most famous equations in finance, the “Black-Scholes” Partial Differential Equation (PDE)

(1) $${{\partial V} \over {\partial t}}{\plus}{1 \over 2}\sigma ^{2} S^{2} {{\partial ^{2} V} \over {\partial S^{2} }}{\plus}rS{{\partial V} \over {\partial S}}{\minus}rV{\equals}0$$

where $V$ is the option value, $S$ the underlying asset, $\sigma $ the volatility and $r$ the risk-free interest rate.

It was during the 1970s that another breakthrough on the industry side was the foundation of the Chicago Board Options Exchange (CBOE). CBOE became the first marketplace for trading listed options. The market was quick to adopt these models and by 1975, almost all traders were valuing and hedging option portfolios using the Black–Scholes model built in their hand calculators. From a tiny market trading only 16 option contracts in 1973, the derivatives market has grown enormously in notional amount to trillions of dollars (Akyıldırım & Soner, Reference Akyıldırım and Soner2014).

The work of Black and Scholes led to an explosion in research in mathematical finance. To recount all the discoveries since 1973 in the world of mathematical finance would be an attempt in futility. Initially simple models (1 factor) but more sophisticated models followed (better able to capture initial yield curve and an implied volatility surface)

• ■ Examples for interest rates include: Vasicek, Hull White, Cox–Ingersoll–Ross, LIBOR market model.

• ■ Examples for equities include Geometric Brownian Motion, Geometric Brownian Motion with drift, and extensions to include term structure, volatility surface (see Figure 7), stochastic volatility as well as jumps.

Figure 7 Example of a volatility surface

The work of Black and Scholes was picked up by Academic actuaries relatively quickly, with one of the early papers by Brennan & Schwartz (Reference Brennan and Schwartz1976) from the University of British Columbia in Vancouver, dated December 1975.

However, there was a significant lag before widespread use within the actuarial community specifically within the United Kingdom, partly due to the then alien concept of no-arbitrage and dynamic hedging concepts, and partly because a probabilistic model solving for a specific ruin probability as suggested by Wilkie at least solved the stochastic nature of the issue (see previous chapter), together with the report by the (unit linked) maturity guarantee working party and the subsequent widespread use and adoption of the Wilkie model.

Having said that, it was actually a paper by Professor Wilkie in 1987 on “An Option Pricing Approach to Bonus Policy” (Wilkie, Reference Wilkie1987), that drew the link between option pricing and with profit policies and introduced the idea of option pricing and financial economics to a number of UK Actuaries.

Into the early and mid-1990s, there was increasing awareness and subsequent references to, and discussion about, Financial Economics. For example, in March 1993, there was a Sessional meeting and debate on “This house believes that the contribution of actuaries to investment could be enhanced by the work of financial economics”. This was one of a series of debates on the subject in the mid-90s, both at the Institute and Faculty of Actuaries. The topic of financial economics was also formally noted in a wide ranging paper on “The Future of the Profession”, commissioned by the Councils of both the Institute and Faculty of Actuaries in December 1994. The study was presented by Nowell et al. to the Faculty of Actuaries in November 1995 and the Institute of Actuaries in January 1996 (Nowell, 1996).

In particular, it highlighted a requirement for additional skills to achieve “a better understanding of investment vehicles and financial economics”. The paper also highlighted a need for structurally incorporating the new skillsets in terms of education and Continuous Profession Development (CPD):

• ■ financial economics;

• ■ understanding of investment instruments;

• ■ financial condition reporting (e.g. risk of ruin);

• ■ a broad-based understanding of risk and finance; and

• ■ the ability to apply these, together with the more traditional skills, to a wide variety of problems in an innovative way.

On a more practical scale, there was a detailed paper by A. D. Smith (March, Reference Smith1996) on “How Actuaries can use Financial Economics”. The aim of the paper is neatly summarised in the introductory section:

The paper goes on to provide a series of case studies, on the concepts of risk neutrality, risk discount rates, arbitrage and their applications to option pricing and asset-liability models and methodology. The first concluding remark was also quite insightful into the novelty of financial economics at the time:

There were also a number of other papers on even more specific applications of financial theory, and one that particularly comes to mind was by Exley et al. (Reference Exley, Metha and Smith1997) on the financial theory of defined benefit pension schemes, presented to the Institute of Actuaries in April 1997.

This focussed on the application of MC valuation techniques to the valuations of pensions schemes, and had a profound impact on the subsequent management and investment strategy of pension schemes.

Meanwhile, there were also some regulatory developments that further enshrined the concept of a MC valuation for an insurer’s balance sheet.

• ■ 1997 – the International Accounting Standards Board started working on a new accounting standard for insurance.

• ■ 1999 – The Faculty and Institute Life Board established a working party – the “Fair Value Working Party”. This presented a paper to the Institute and Faculty in November 2001.

• ■ 2002, 2003 – Consultations published by UK’s FSA.

• ■ 2003 – FSA published the Integrated Prudential Sourcebook, requiring a MC valuation for any firm with more the £500m in with profit liabilities.

• ■ This would be soon followed by the Individual Capital Assessment (ICA), which required stressing the newly created MC balance sheet to 99.5th percentile 1-year VaR, which is described more fully in the following chapter.

7. VaR Models

In the early years of the millennium, following a notable regulatory failure, there was a desire to improve the approach to regulatory supervision of UK Life Insurers and particularly with profits companies. Given that one of the key issues was a desire to monitor guarantees and options which existed within life insurance contracts, the use of a MC valuation of liabilities and assets, which was being considered as an accounting standard, formed a natural starting point. However, the previous regulatory framework, which presumed the existence of implicit liability margins, would not sit well with a margin-free MC view of liabilities. Hence a new approach was needed.

Given that banks had developed “ ${\rm VaR}$ ” as a mechanism for determining capital adequacy that could be applied to realistic valuations, this appeared a reasonable starting point for insurers. However, the nature of banking and insurance and consequently their supervisory regimes are significantly different so consideration had to be given to the implications of these differences.

In its simplest form ${\rm VaR}$ requires the identification of the amount of capital required to ensure that in the event of a stress of defined probability the stressed assets held remain sufficient to cover the stressed liabilities. If there is only one risk factor then this concept is relatively simple to model once the distribution of out-turns of the risk are known. However, for even a simple Life Insurer there would usually be multiple risks relating to liability risks, asset risks and valuation basis stresses. For a multi-line Life Insurer, the numbers of risks can be numbered in hundreds. Many of these risks will not be independent and it is normal practice to make allowance for the correlations, typically using copulas. In these cases, there will be a surface of possible combinations of stresses each of which may be considered possible with the defined probability level. The particular stress of this level which produces the largest capital stress is sometimes referred to as the critical stress and is the stress which defines the required capital.

${\rm VaR}$ modelling was a requirement of the ICA regime introduced in the UK ~ 2004. A similar requirement is now required under Solvency II. In both cases the defined probability level is a year on year stress with probability of $0.5\,\%\,$ . This is sometimes referred to as a 1 in 200 years event.

The methodology used here is effectively an extension of the MC methodologies except that the result is typically far more sensitive to the tail of the distribution rather than being averaged over the whole distribution. Also, it is common practice to derive the probability distribution from historic data rather than option prices. In theory the model could be calibrated from option prices but at the levels of risks considered many of the prices used would be far out of the money or deeply in the money and therefore the level of trading would be such that the markets would be far from considered “deep and liquid”. Also markets tend to be preoccupied with specific current threats, which may typically be considered as more likely than required for this form of calibration rather than a long term view of the possible yet extreme events.

One particular feature of this form of modelling which is shared with the MC models, but in this case is far more significant is that the use of correlations applied to a large number of single stresses means there tends not to be a coherent view of the scenarios to which the stressed events correspond and therefore it is difficult to apply expert economic judgement to the plausibility of the model. Further available data for many of the risks involved may be limited, or considered limited, by the lack of relevance of early data. This means that the observed stress events will be limited exaggerated versions of what has actually happened in that period, which may lead to excessive stresses but reduced exposure to those events which have not happened. Consider, e.g. a calibration of interest rate stresses from 10 years of data taken from a period of consistently falling interest rates. One may expect this to overstate the risk of further fall and understate the risk of sharp rises.

Deriving correlations from historic data can also be subject to similar data issues but these can be magnified if it is considered that in stress correlations would be different to those experienced in non-stressed circumstances.

A number of practical and theoretical issues arise with these models, many of which are more significant because of the paucity of data issue referred to above. Fitting a probability distribution to data which does not truly derive from a known distribution potentially leads to high levels of models as indicated by the very different distributions that are derived by trying to fit historic data to a variety of plausible statistical distributions. For example, the Extreme Events working party published a numbers of papers on this topic from 2007–2013 Frankland et al. (Reference Frankland, Holtham, Jakhria, Kingdom, Lockwood, Smith and Varnell2011, Reference Frankland, Smith, Wilkins, Varnell, Holtham, Biffis, Eshun and Dullaway2009, Reference Frankland, Eshun, Hewitt, Jakhria, Jarvis, Rowe, Smith, Sharp, Sharpe and Wilkins2013) and continues to research in the area.

A notable criticism of ${\rm VaR}$ models has been the failure to reflect contagion events, as experienced in the 2008 global financial crisis, when sound financial assets are sold to raise cash in order to meet solvency requirements where other assets have collapsed in value. ${\rm VaR}$ models are typically flat with respect to time, implicitly assuming a constant distribution of asset volatility over time. Other questions that remain open is how one reconciles the implied MC distribution with extreme 1-year VaR stresses derived from historical experience, as well as acknowledgement of the reality that most standard commercial risk packages, by definition, consider VaR on an ex-post basis.

8. Summary of Past and Current uses of ESGs

The historical and present uses for which an ESG is commonly needed, based on interviews with experienced industry practitioners, are as follows:

• ■ Strategic Asset allocation.

• ■ General long-term investment strategy; assessment of investment strategies against investor required return and appetite for risk.

• ■ Business decisions and management actions based on solvency in future scenarios/probabilities of ruin from an ALM.

• ■ Product design, prior to launching products.

• ■ MC Pricing of products once the products have been designed.

• ■ Financial planning for individuals and institutions.

• ■ Risk Management and hedging, to control the volatility of the balance sheet.

• ■ Economic Capital requirements: determining the quantity of assets to ensure liabilities are met with a sufficiently high probability to meet the company’s risk appetite.

• ■ Regulatory Capital requirements: determining the quantity of assets to ensure liabilities are met with a sufficiently high probability for regulatory solvency.

• ■ Valuation of complex liabilities or assets, particularly when these contain embedded options or guarantees.

Of the different uses, the last two have largely arisen owing to changes in regulation over the last decade and a half; starting with the Prudential Sourcebook for Insurers (Financial Conduct Authority, 2003), which required every insurance company with more than £500m of with-profit liabilities to use stochastic valuation techniques, and reinforced by Pillar II ICA and its successor Solvency II, which requires a VaR calculation.

Amongst the interviewees, it was widely felt that knowledge of ESGs is important for making decisions thereupon. Given the preponderance of interviews with ESG developers, it is not surprising that the interviewees felt this was an important point. The user must have enough knowledge to understand the limitations of a model and to make sure it is appropriate for the task in hand. This is a lower degree of understanding than a full understanding of the mathematical properties or being able to recreate the model from scratch.

Model calibration (as well as model design) and the challenge that model developers face in communicating the limitations of a model were felt to be important factors in ensuring successful use of an ESG. A number of interviewees pointed to over reliance on model calibrations, whilst not pointing to any specific instances of this occurring in practice. Communication of the data underlying calibrations as well as calibration goals and limitations were additional themes mentioned in the interviews.

There was unanimous agreement that general awareness of ESGs, both amongst users and management, has improved over time. Additionally, there was a general feeling that ESGs should be published in peer reviewed journals. ESG developers interviewed however highlighted that the implementation was important and the low-level implementation of a particular academic model could make a difference to its behaviour and yet the model could still be described as the XYZ academic (and peer reviewed) model. One interviewee commented that the use of non-peer review models placed more responsibility on the model user.

9. Factors Influencing the Evolution of ESGs

The evolution of ESGs occurred over quite a long time period, circa the past 40–50 years, and the evolution can be attributed to a number of factors:

Technical Criteria

• ■ Goodness of fit to historical data.

• ■ Ability to forecast outside the sample used for calibration, also called “back-testing”.

• ■ Desirable statistical properties of estimated parameters, such as unbiasedness, consistency and efficiency.

• ■ Accuracy in replicating the observed prices of traded financial instruments such as other options and derivatives.

Technical Criteria are frequently listed in Published Papers.

Social Criteria

• ■ Ease of design, coding and parameter estimation.

• ■ Commercial timescale and budget constraints.

• ■ Auditable model output that can be justified to non-specialists in intuitive terms.

• ■ Compatibility of model output and input data fields with available inputs, and with requirements of model users.

• ■ Ability to control model output.

• ■ Meeting regulatory approval e.g. use of ESG in Internal Model Approval Process.

The existence of Social Criteria, such as the above, is inevitable. The existence of Social Criteria is less widely recognised (than the existence of Technical Criteria) and is very rarely discussed in Published Papers.

Technology criteria:

• ■ Developments in computing power.

• ■ Use of Monte Carlo methods, and simulation.

• ■ Black Scholes and option pricing theory.

Regulatory criteria/insurance industry-wide events:

• ■ Equitable Life.

• ■ Realistic Balance Sheet (and moving away from statutory Pillar 1 Peak 1 reporting).

• ■ ICA and Pillar II.

• ■ Solvency II for European wide adaptation of the above.

Most interviewees have suggested regulatory led changes were the largest factor driving ESG evolution historically, e.g. the move to MC valuations led in the United Kingdom by the FSA. Some changes have been driven by unforeseen changes in the economic environment, e.g. negative interest rates arising out of the 2008 global financial crisis, some user-led by shortcomings in ESGs and some have been developer-led, e.g. the Wilkie model introducing stochastic modelling for the reserving of guarantees.

In general ESG developers wished to improve their models themselves on finding shortcomings. This has been driven partly by a desire to lead the market and by extra research and knowledge that has become available over time. User and regulator dissatisfaction has been less of a driver. Designer dissatisfaction at elements in their ESGs has been very specific to the particular ESG in question.

Historically users found the most pertinent limitations of ESGs to be: a lack of flexibility in models; inability to model some of the real-world behaviour observed, e.g. jumps in equities, negative interest rates; and a lack of calibration to observed market behaviour. Most of the issues have been addressed over time, mainly through a combination of enhanced knowledge and available technology.

The rest of this section discusses the observations that were made by interviewees on the historic development of ESGs from random walk models, followed by time series models, then MC and arbitrage-free models and most recently the 1-year ${\rm VaR}$ models in use for economic capital and related purposes at the present time.

There was a broad consensus that time series models (such as the Wilkie model) supplanted random walk models owing to more accurate behaviour relative to the real-world and time series modelling was more realistic and useful to actuaries controlling for actuarial risk. Additionally, there were concerns over the lack of mean reversion in random walk models although a number of interviewees mentioned that the Wilkie model exhibited too strong mean reversion properties. However, since the development of time series models, each new type of ESG has not generally replaced the previous type. Time series, market-consistent and 1y ${\rm VaR}$ models are all still in use but for their own particular purposes.

There was a widespread view amongst the interviewees that MC ESGs were being developed in the late 1990s before realistic balance sheet reporting was introduced in 2003, whilst acknowledging that the FSA changes led to demand for MC-based ESGs. The general consensus at the time was a wariness of the new technique and a focus on perceived inconsistencies within MC valuation techniques, but has come to be seen in a more positive light over time with more objective values and the focus on path dependent modelling.

MC models of the early 2000s were designed for different purposes and required an arbitrage-free framework for all assets which reproduces market pricing to address valuation requirements. The time series model never had this in mind as it is a framework to extend time series modelling of asset prices in the real world. Hence, they were addressing a different problem so cannot be strictly compared. It should be noted however that, whilst the spotlight and focus both in insurance companies and external literature has been on MC models, the Real-World models have continued to co-exist alongside in the background.

The development of ${\rm VaR}$ stimulated another phase of ESG development, although they co-existed with MC models, but used for capital calculations rather than replacing MC ESGs. Presently, the insurance industry has standardised around the 1-year ${\rm VaR}$ technique, e.g. the 1 in 200 probability of ruin approach under Solvency II. This regulation-led development has put the focus on 1-year ${\rm VaR}$ techniques rather than multi-year ${\rm VaR}$ models. The interviewees widely expressed the view that there is no point in doing anything more than absolutely necessary in order to gain regulatory approval, further enshrining the 1-year ${\rm VaR}$ current approach. Interviewees suggested multi-period ${\rm VaR}$ models are difficult to run and interpret, and any insight into the long term is missed as a result.

Figure 8 shows the perceived balance between empirical data-driven models and Economic Theory-driven models historically over time.

Figure 8 Balance between Empirical data-driven models and Economic Theory. VaR=Value at Risk

10. The Way Forward

This section has been updated for some comments from the wider community at the sessional meeting at Staple Inn on 22 February 2018.

We have some salient observations from the research, discussions and interviews that we have collated over the course of this exercise. Whilst the forces driving change and development of ESGs have changed over time, the need to understand the capital markets and their interactions for investment strategy and business decision making have continued to play a key role. This was the case for the random walk and time series (Wilkie) models. For some years thereafter, the developmental effort was driven by regulations and was focused on more specific applications of MC valuation and 1-year VaR distribution for the ICAS and Solvency II regimes.

This begs the question, what will determine future model developments? There we believe that whilst a natural curiosity and desire to better understand and model the capital markets will lead to a level of steadily increasing sophistication, there may be exogenous factors that lead the development efforts. These could likely include

• ■ Insurance Regulations – Once again insurance regulations are likely to play a key role in the medium term, this time with the multi-year approach required by the Own Risk and Solvency Assessment (ORSA). This is likely to lead to a renewed focus on longer-term real-world modelling within the insurance industry. For companies that previously relied on external providers, a renewed interest and enthusiasm in the use of in-house capital market views for business and strategy decision making. This will also create a demand in insurance companies for the specialist expertise required for capital markets research and modelling, at the same time as the MiFID II regulations and transparency on commission result in an excess supply from the sell-side (asset management and investment bank research). This may result in an influx of capital markets modelling ideas from the asset management industry.

• ■ Greater Transparency to customer: Impact of other wider regulations requiring transparency to the customer on the distribution and profile of portfolio outcomes. This was also extensively discussed at the sessional meeting and in particular the need to use capital markets and portfolio modelling for both individuals and institutions.

• ■ Financial planning and education: At a higher level, it may also require greater awareness of capital markets and financial planning in the actuarial education system, with a particular focus on lifetime and retirement financial planning.

Most of these changes are likely to provide an opportunity to re-define models for the better, whilst some changes such as the ORSA may focus development on a specific aspect for a period of time.

There are other changes which are more difficult to foresee, e.g. the allowance for the evolution of monetary policy, given we have only had a decade of data under the “unconventional” Quantitative Easing programmes deployed by central banks globally, and zero year under any Quantitative Tightening Scenarios. Together with other longer-term trends in capital markets, such as the globalisation of inflation, this would likely lead to a very different starting point for creating an ESG. This was nicely articulated by Stuart Jarvis at the Staple Inn sessional meeting, where he explained that if he were to build the Wilkie model today, he would be using gross domestic product (GDP) growth rather than inflation as the key variable for a causal capital markets model (CMM).

For completeness, it is also worth mentioning technology. At a basic level, the improvement in cloud computing and big data allowing greater granularity of modelling (although to be a good model, the complexity would still need to be traded off against parsimony based on the Occam’s razor principle). At a philosophical level, there may be a disruptive threat in the form of a machine learning algorithm creating a better brute-force ESG. The only saving grace for budding investment professionals is that machine learning algorithms are constrained by Father Time, in that they can only learn as fast as the markets operate in real time (which is still much faster than any human …)

We should also consider learnings from different countries where the use of ESGs has been very different from the United Kingdom. The working party has carried out some initial work on this, but any further research, contacts and interviewees would be much appreciated. Key international developments include work by Mary Hardy in Canada to develop and extend the concept of regimes and regime switching models. Other countries seem to have put some resource to centralised knowledge on ESGs, ranging from United States, where the Society of Actuaries has built a minimal “actuarial profession” ESG that is widely used by the smaller insurers, to Switzerland where the regulator has invested heavily in ESG expertise. Europe has so far focussed resource on European wide regulations and standards which put some loose constraints on consistent ESGs, but without focussed development on economic scenarios with the exception of the 1-year ${\rm VaR}$ stresses for different asset classes under the “standard model”.

On a final note, the use of the ESG acronym for ESG is itself under threat! This is largely due to the growing importance of the alternative expansion – Environmental, Social and Governance. Perhaps we should consider a different acronym such as CMM, or SAM?