Pricing Multi-period Insurance Risk

Preface

This Monograph equips the practicing actuary with tools and techniques to estimate, communicate, and implement profit targets for insurance contracts. Each of these activities can occur for the entire corporation, a business unit, or an individual account and different considerations apply at each level. Mildenhall and Major (2022) calls the process by which ingredient loss picks and expenses are combined with a profit provision to propose or evaluate a quoted or sold premium the “last mile” of insurance pricing. This Monograph extends PIR by integrating into the last mile the impact of time-value of money and risk from losses that emerge over multiple periods, two important features of many insurance products. Implementation is a two-way process: rates can be computed to achieve a given target or expected profits can be computed from a given premium. The term “profit target” is deliberately vague because it can be interpreted in many ways. It is defined more precisely in due course.

An implicit or explicit premium determination framework sits behind every policy sold, every strategy to expand business or re-underwrite, every rate filing, and every reinsurance purchase. They act from the micro-level problem of individual policy pricing and rate regulation all the way up to the macro-level question of strategic portfolio management. They encapsulate an insurer’s view of risk and return and are central to company management. Given this central importance, it is disappointing that no privileged framework exists—in many ways the problem is too complicated and any model must ignore parts of reality and different insurance market stakeholders have different perspectives on the “correct” answer. It is often possible, however to determine a reasonable range. This unfortunate situation stands in contrast to the single-period situation, where good answers are known, even if they are not widely known (PIR, preface).

This monograph addresses problems such as:

• Estimation or Determination of
  • An appropriate profit target for the whole corporation.
  • An appropriate profit target for a unit of insurance, based on experience or capital modeling simulation output and a corporate target.
  • The same for an account given a loss pick, expense structure, and unit target.
• Communicating
  • How profit targets are set upwards and downwards in the organization to technical and non-technical audiences in a way that is clear and actionable.
  • How profit targets are expressed.
  • How profit targets are implemented in rates and premiums to achieve desired target outcomes.
• Implement, calculate or evaluate
  • Rates from profit targets and demonstrate how they produce results that achieve the target.
  • The indicated (quoted) premium for a unit or account given a loss pick, expense structure, and profit target.
  • The expected profit that will emerge from a given price, given a loss pick, and expense structure.
  • The amount of insurance that should be offered at specified market terms.
  • The highest price at which reinsurance should be purchased given profit targets.

Target Audience

There are several potential users of this Monograph

• A corporate actuary working in strategic planning trying to
  • Estimate an overall corporate return target
  • Allocate an overall target to business unit
  • Communicate those targets to business leaders.
• A business-unit pricing actuary trying to
  • Allocate the unit’s target to a more granular level, such as a pricing tool.
  • Incorporate a given target into an account premium
  • Incorporate a profit target into filed and regulated premium rates in a manner consistent with relevant regulations and standards.
  • File rates, must pay attention to requirements of their jurisdiction.
• A line-level pricing actuary trying to
  • Determine the indicated premium for an individual account consistent with the business unit’s target
  • Determine the highest price at which reinsurance makes sense
  • Advise on what lines should be offered at specified Firm Order Terms

Approach

The Monograph adopts a divide and conquer approach to the problem, based in-part on Bühlmann’s idea of pricing “from the top-down” (Bühlmann 1985). The approach is to estimate an overall target return, split it into insurance and investment parts, and then allocate the insurance part to business units. In turn, the business units allocate their target to individual accounts and implement rates or calculate prices to achieve the targets. The approach is iterative, with feedback from the bottom to the top. The process of estimating or determining targets includes the critical step of communicating them, both upwards to senior management and downwards to underwriters and actuaries. And the process of implementing of calculating, going from profit target to premium, can be reversed to evaluate the expected profit in a given premium.

This framework does not consider investment risk, a big simplification. Investment risk is a real risk and must be part of risk a management framework. However, in many ways it is an undifferentiated background risk for pricing purposes. IFRS 17 clearly separates insurance finance costs (unwinding discount) from investment earnings, setting up a framework to evaluate investment decisions in a similar way to that we describe for insurance risk. The framework also assumes that timing risk and amount risk are separable.

A pricing model is a simplification of reality: models always involve choices about what to ignore. Common choices the actuary faces in multi-period pricing include:

• Business stand-alone or within a portfolio
• Steady-state vs. single year vs. growing business
• Full run-off vs. one-year’s emergence
• Choice of accounting standard(s)
• Choice of time period length (annual, quarterly) and organization (calendar, accident, policy year)
• Choice of investment return: risk-free, portfolio, market, prospective or embedded
• Choice of discount rates: risk-free, liquidity adjusted, risk adjusted, yield curve
• Capital structure: overall cost and tranche or component costs
• Gross vs. net steering and incorporating the market cost of reinsurance
• Models of loss emergence, distinct from payout patterns and traditional triangle-based models
• Motivation of insurer vs. pure intermediary role: management aligned with equity

These choices cannot all be made independently.

Throughout, we validate models by specifying the question they are trying to answer. With the question clear, it becomes more obvious whether the model is appropriate to the intended purpose (Actuarial Standards Board 2019). It is also instructive to stress test models with extreme parameter choices to ensure their predictions remain reasonable. Stress tests are good at teasing out theoretical weaknesses. Any model must be consistent with economic theory, market values, and the underlying cash flows. It should incorporate all information and recognize information can precede cash flows (case reserves). It must be comport to external reality, including regulatory and accounting rules. The model should acknowledge and accept uncertainty and the important role of ambiguity in diversifiable risk. The model should balance the relative uncertainties of its components (don’t look where the light is an ignore the elephants in the dark). It should be “as simple as possible but no simpler”, favor Occam’s razor, be comprehensible and explainable, accept that approximately correct is better than precisely wrong, and capture the essence of the problem.

Ratemaking vs. Costing

The distinction between costing and ratemaking is particularly important in regulated environments, and resulted in ASOP53 (Actuarial Standards Board 2017) being renamed from “Property/Casualty Ratemaking” to “Estimating Future Costs for Prospective Property/Casualty Risk Transfer and Risk Retention” during the drafting process. Lines with sufficient homogeneity and data to support price optimization rarely have highly differentiated profit targets, thus the distinction is less relevant in practice. ASOP53 introduces the concept of “the intended measure of the future cost estimate[s]” whcy may vary with but be appropriate for the intended purpose. It gives examples “the mean, the mean plus risk margin, the high or low estimate within a range of reasonably possible outcomes, and a specified percentile of the distribution of reasonably possible outcomes.” In contrast, the CAS Statement of Principles on Ratemaking (SOP) (Casualty Actuarial Society 1988) says that “A rate is an estimate of the expected value of future costs” (emphasis added). Modern accounting standards allow (IFRS 17) or require (Australian GAAP) discounted loss plus risk margin to be determined as a percentile. The SOP allows “Other business considerations are also a part of ratemaking” but does not list any. In the background section, ASOP53 allows “Throughout our history as a profession, actuarial future cost estimates have not always been the sole basis for rates and prices in risk-transfer or risk-retention transactions. For example, other important influences may include regulatory requirements and business objectives. Such other influences may support or compete with actuarial future cost estimates in deciding upon final rates and prices.” The standard’s scope “does not include items such as[nd] the balancing and interaction of potentially competing objectives related to regulation, business objectives, and actuarial cost estimates?” The Second Exposure draft of ASOP53, available on the Actuarial Standard’s Board website, included a discussion of business objectives in its scope section “This standard is limited to the estimation of future costs. While the actuary may play a key role in the company’s decisions in determining the price charged after taking into account other considerations, such as marketing goals, competition, and legal restrictions, this standard does not address the other considerations” (emphasis added). The final version of ASOP53 removed this language.

Limitations

Since “all models are wrong” there are things this Monograph does not address.

• The market cycle [why people buy insurance RMIR article]
• Investment risk
• Franchise value (value of a going concern/continuation value) and market value of insurers (quagmire quote)
• Long-duration (life) contracts where policy reserves move premium between periods
• Incorporating business considerations into pricing, such as cycle-pricing or market share considerations
• Inseparable risk and time value of money

In general, these remaining problems are very difficult, and we happily leave them as future research for other actuaries!

Case Studies, Examples, Reproducibility and Code

NN Case Studies are used throughout to illustrate the concepts.

1. Two simple one-year models: one the Cat/NonCat from PIR and another more interesting.
2. Various multi-year emergence models: \(A+B\) model, random walk model.
3. …something calibrated to industry payout patterns?

TODO: If you are a XXXX then read YYYY.

Rest of the Monograph and Reader’s Guide

The rest of the monograph is organized as follows.

1. Introduction

   1. The Problem: what are we trying to do
   2. Tyche and Norns dialog: what do we think about it out the box?
   3. Archtypal example: let’s try to solve in a simple case
   4. Why the problem is hard: why we can’t apply “out-the-box” finance
   5. Literature review: what have others done
      • Risk sharing
      • Distortion pricing
      • Multi-period pricing

2. Background

   1. Accounting (VSI)
      • Balance sheet, income statement, cash flow statement
      • Reporting, statutory, tax, and management accounting
      • Capital, equity, and debt
      • Legal requirement for equity owners (equity=management)
   2. Insurance and Insurance Accounting
      • Time periods: meaning and importance of a period = evaluations with actions; AY, CY, PY
      • One-period Ins Co. model (who are investors left ambiguous)
      • Multi-period Ins Co. model as a chaining of one-period models with a recap option; difference between new business and reserves (can lower capital adequacy for reserves)
      • Default and the equal priority rule; replacing unlimited losses with limited losses; role of insurer in limited liability
      • Insurance metrics (LPM…); target (ambiguous) vs. margin and return (McClenahan 1999), pentagon w/o time value; Ferrari with (reconcile?);
      • Stakeholders and their accounting standards
      • US GAAP and statutory (“stat”)
      • IFRS 17
      • Solvency II
      • Capital structure; debt, reinsurance, equity; cost considerations
      • Asset risk and discount rates, insureds participate in investment decisions via limited liability debate
   3. Probability
      • Densities and risk adjustments
      • Random variables, sigma algebras, and information
      • Conditional expectation
      • Adjusted probabilities and conditional expectations
   4. Finance
      • Return and time (risk once per day, week, month, year?!)
      • Liquidity risk and returns (for IFRS 17)
      • Bid and ask prices
      • General equilibrium models
      • NPV analyses
   5. Risk theory
      • Risk, diversifiable and non-diversifiable risk, ambiguity aversion; non-repeatable risks
      • Comonotonic random variables
      • Risk measures and their propertiers: TI, monotone, SA, PH, coherent, LI, comonotonic additive, convex
      • Monetary utility functions
      • Distortion functions and SRMs; Cherny quote
      • Computing a CRM and SRM: six methods
      • Layer interpretation, price as function of assets, CCoC and XTVaR capital
      • The five standard distortions; properties of standard insurance metrics; TVaR as tail risk neutral
      • BiTVaRs (diagram from BTE of types), distortions achieving a given price
      • Calibrating distortions to market pricing (Newton Raphson)
      • Risks with a given price from a given distortion
      • Bid and ask prices with an SRM
      • NA P and M
      • NA Q and average cost Q
      • Covariance interpretation of NA P
   6. Life actuarial and discounting
      • Time periods in discounting tabulations, why time starts at \(t=0\)
      • Dicounting, \(v+d=1\), \(v=1/(1+d)\), \(d=1-v\)
      • Equivalence principle premium
      • Profit signature
      • PVI / PVP
   7. Basic premium calculation principles
      • Expense loadings
      • Basic premium calculation formulae
      • Premium from the top down
        
      • Regulatory approaches, Solvency 2 cost of capital model
   8. Understanding diversification
      • Why do riskier risks pay a higher margin?
      • Transfer effects of default
      • Efron risks, non-Efron risks, and wild risks
      • Range of NA vs. stand-alone
      • Pricing as though SA
      • Four examples
   9. Capital Allocation
      • Capital allocation and insurer managment
      • NA vs equal cost allocations

3. Case studies and examples

4. Risk sharing in a one-period model and \(g\)-economies

   • Ins Co. model: identifying investors! Identifying \(g\)
   • Two approaches to \(g\): risk elicitation and observed market prices
   • \(g\)-economy unfettered and with liquidity (limited participation) and transaction expenses (limited layers); low layers map to agents
   • Spreads over \(g_{\min{}}\)
   • Importance of top layer (big, sets pricing for lower layers)
   • Importance of equity layer (residual; only non-fixed cost layer; equity=mgmt)
   • Top = bottom layers when only one layer
   • Role for brokers in structuring
   • Role for reinsurers in risk transformation and accessing risk bearers; non-comonotonic layers
   • Role for management=equity: maximize premium over cost of (other) financing
   • Approximations to average costs
   • Appropriately crediting for the value of “cheap reinsurance”

5. Discount with no emergence

   • Pricing to achieve a given profit target
   • US Stat/GAAP booking: what a clusterfuck
   • IFRS booking: what a joy, separation of insurance service revenue and finance charge, accretion of discount
   • IFRS reported discount rates and implied discount in reserves
   • Cash flow models: DCF, IRR, NPV; Cummins, Myers-Cohn
   • Robbin’s methods
   • Single policy co. vs. steady state co. vs growing co.; grow at COC for IRR=steady state
   • Maximize growth rate pricing and ergodic model (risk-return consideration)
   • Cost of regulatory capital and the adjusted NPV method (account for financing…)
   • Calibrating to a rating agency model? (Allocations…)
   • Pricing with IRFS free cash flow

6. Emergence with no discount

   • Introduction
   • Literature review for risk over time
   • Distinction between emergence and payout patterns; comparison with traditional triangle-based models, Bornhuetter-Ferguson and Cape Cod
   • P2P models: using a one-period model to price multi-period risk
   • Bernoulli time expensive results: intuitions about the value of information
   • Two-period compound models with frequency known at period 1
   • Other models of loss emergence: A+B, random walk, etc.
   • Constraint: ultimate is sum of emergence over periods, a new decompostion of risk
   • Correctly modeling casualty risk: not more risky because slow-emerging, ensure your modeling allows for the full range of undercertainty!
   • Independent sum of emergence model (different slicing of CY and AY)
   • Comparion with IFRS risk adjustment; magnitude of risk adjustment reported (vs. CSM)

7. Emergence with discount in a separable model

Acknowledgments

Thank collaborators, funding sources, etc.

Terminology and Vocab Reminders

Term	Context	Meaning
Estimate	Profit targets	compute profit targets by unit
Determine	Profit targets	same
Communicate	Profit targets	describe process, amount, and implementation
Implement	Premiums	compute premium to achieve target, usually for rates
Calculate	Premiums	same, usually for an individual risk
Evaluate	Premiums	compute profit target embedded in a premium
Validate	Models	check that the model is working as intended, compare different types of model

Definitions

Term	Definition
Actuarial reserve	Expected discounted value of future cash flows, no risk adjustment.
Actuarial value	Objective expected value with no risk margin. Discounted if appropriate. Undiscounted actuarial value for clarity.
Duration	contract duration, short-d and long-d
Emergence
Incurred	Always means ultimate incurred loss and never case incurred.
Individual risk (IR)
Market value	Risk adjusted expected value. Discounted if appropriate.
Payout pattern	long-tail (as distinct from heavy tail)
Statutory reserve	Management best estimate liability, no discount, no (explicit) risk adjustment
Surplus	Old terminology for capital
Unit

Notation

Symbol	Meaning	Reference
\(X\)	Random variable
\(\mathscr F\)	Sigma algebra

Actuarial Standards Board. (2017). Estimating Future Costs for Prospective Property/Casualty Risk Transfer and Risk Retention, (53).

Actuarial Standards Board. (2019). Modeling. Actuarial Standard of Practice No. 56, (56).

Bühlmann, H. (1985). Premium Calculation from Top Down. ASTIN Bulletin, 15(2), 89–101. https://doi.org/10.2143/ast.15.2.2015021

Casualty Actuarial Society. (1988). Statement of Principles Regarding Property and Casualty Insurance Ratemaking.

McClenahan, C. L. (1999). Insurance Profitability. Actuarial Considerations Regarding Risk and Return, 113–124.

Mildenhall, S. J., & Major, J. A. (2022). Pricing insurance risk: Theory and practice. John Wiley & Sons, Inc.