1. Introduction
The question of what constitutes a fair profit margin for insurance products is receiving increased attention from regulators worldwide. Insurers have faced scrutiny for failing to meet consumer expectations, including insufficient justification for profit margins and instances of excessive margins for some products. In response, regulators are implementing consumer fairness frameworks that involve fair value assessments. Certain pricing practices, including “price walking” of loyal customers, are being criticised. Enhancing premium transparency and providing justification for profit margins is key for property and casualty insurers to rebuild social licence.
Frameworks for determining a fair premium, and hence profit margin, typically involve an extension of the capital assets pricing model (CAPM) to accommodate insurance (see Turner, Reference Turner, Cummins and Harrington1981). This results in a profit margin that is reflective of the cost of the insurer’s capital, which has been embedded within actuarial premium rating practices (see GRIP Report IFoA, 2007 and Robb et al., Reference Robb2012).
The CAPM framework produces profit margins considered by some stakeholders to be too low to fulfil shareholder expectations and often lower than what has been consistently observed in competitive markets. Hence, the framework does not provide an adequate justification for the profit margins charged by insurers. This has led to debate over whether the framework may have missing elements, such as not reflecting all risks and frictions involved in manufacturing and delivering insurance products.
2. Fair Premiums in Insurance
2.1 Accepted Components of a Fair Premium
In 2010, the Committee on Theory of Risk of the Casualty Actuarial Society issued their Risk Premium Project (RPP) Update Phase II Report. Together with the Phase I Report, the RPP represents a comprehensive literature review of nearly 1,000 actuarial and financial sources. The report concluded that the construct of a fair premium included the components, slightly modernised from the original, shown in Table 1. This framework remains generally accepted among actuaries for constructing a fair premium.
Components of a fair premium
2.2 Risk Adjustment, Components A+B
The consumer utility maximisation economy model extended for insurance (see Turner, Reference Turner, Cummins and Harrington1981) treats insurance losses the same as other risky asset cashflows. The expected loss cashflows are therefore risk-adjusted in the same manner as for other assets within the economy, that is, with a risk-adjusted discount rate based on the covariance to market systemic risk. Assume assets of type
$k$
experience an identical loss distribution of
${X_k}$
. The model shows that the risk-adjusted discount rate to apply to the expected value
${\bar X_k}$
is
$${{{\bar r}_{{X_k}}} = {r_f} + {\beta _{{X_k}}}\left( {{{\bar r}_M} - {r_f}} \right)},$$
where
${\bar r_{{X_k}}}$
is the risk-adjusted discount rate,
${r_f}$
is the risk-free rate,
${\beta _{{X_k}}}$
is the underwriting Beta and
${\bar r_M}$
is the expected market return.
The underwriting Beta is defined as,
$${{\beta _{{X_k}}} = {{{\rm{Cov}}\left[ {{r_{{X_k}}},{r_M}} \right]} \over {{\rm{Var}}\left[ {{r_M}} \right]}},}$$
where
${\rm{Cov}}\left[ {{r_{{X_k}}},{r_M}} \right]$
is the covariance of the consumer’s return on their investment in a policy insuring losses
${X_k}$
, that is, the loss ratio minus one, and the market return.
This approach to assessing the risk adjustment is applied in Myers & Cohn (Reference Myers, Cohn, Cummins and Harrington1981) and was accepted as fair under regulation in the US state of Massachusetts. Given the complexities in directly justifying an underwriting Beta, this approach gave way to an internal rate of return approach that utilised an estimate of the insurer’s weighted average cost of capital (see Cummins, Reference Cummins1990). These two approaches can be shown to be equivalent with consistent economic assumptions for the market and the insurer (see Taylor, Reference Taylor1994).
2.3. Insurance Frictional Costs, Components A+B+C
Insurers play a vital role in the economy, typically leading to regulation and supervisory oversight. Insurers are required to maintain strong capital levels, have in place a comprehensive reinsurance programme for major losses and have high standards for risk management across capital, underwriting, pricing and claims. These requirements result in higher capital levels and expenses than a free market would dictate. It is generally accepted that these “frictional costs” form part of the fair premium.
It has been argued that since frictional costs arise from enhanced capital and operational management of an insurer’s total risk, then both systemic and non-systemic risk should be rewarded (see Exley & Smith, Reference Exley and Smith2006). With the risk adjustment only addressing systemic risk, new insurance pricing models have been developed to consider profit margins arising from total risk of the insurer and how these profit margins are assigned to individual policyholders (see Wang, Reference Wang2002). Some pricing models use a CAPM-like framework, substituting the market risk premium with a measure of total insurer risk and a Beta based on the covariance of a policyholder’s loss with the aggregate insurer’s loss (see Furman & Zitikis, Reference Furman and Zitikis2017).
The suggestion that frictions contribute to profit beyond the risk adjustment remains a subject of debate. The additional expense arising from frictions represents production costs of the insurer and is included in policyholder premiums. Since insurer frictions are widely known and understood within the market, it is reasonable to assume that the weighted average cost of capital reflects the impact of these frictions. Therefore, this paper does not include an additional allowance for frictions.
2.4 Default Option, Component D
Limited liability means that if aggregate losses for an insurer exceed assets then the policyholders are not paid in full, but receive a share of assets proportional to their losses. The market value of this deficiency is termed the default option.
In practice, the default option is often ignored owing to an insurer’s potential to raise capital in the event of insolvency, as well as the presence of industry guarantee funds and government support that activate upon insurer insolvency.
The default option is also ignored for the purpose of this paper. Appendix A sets out an approach for calculating the default option should this not be immaterial to a fair premium.
2.5 Tax Compensation, Component E
Shareholders should be compensated for additional tax when investing in insurers compared to other companies. For example, earnings on capital invested in the market by insurers are taxed before reaching shareholders, ultimately resulting in double taxation compared to a direct investment in the market.
An internal rate of return pricing model will incorporate tax cashflows and utilise or solve for a net of tax cost of capital, thus implicitly including tax compensation as part of the profit margin. Where a risk-adjusted discount rate is used for pricing, not only does the underwriting Beta need to be adjusted for the impact of tax, but an explicit allowance for double taxation of capital earnings is required. An appropriate allowance for this double taxation, using Myers’ Tax Theorem derived in Appendix B, may be calculated as:
$${{\delta _k}{{{r_f}} \over {1 + {r_f}}}\left( {{\tau \over {1 - \tau }}} \right),}$$
where
${\delta _k}$
is the capital allocated to a policy insuring an asset of type
$k$
, expressed as a proportion of risk-adjusted losses, and
$\tau $
is the company tax rate.
2.6 Expenses, Component F
The insurer’s total anticipated expenses are fully allocated to policyholders on a cost basis. Significant system investments, such as those associated with new policy or claim management systems, are typically amortised over an appropriate period. A substantial portion of expenses relate to underwriting and distribution, which are generally incurred at policy inception or renewal. Loss adjustment expenses are recognised throughout the claim management process until all liabilities are settled.
Expenses allocated to a policy, assuming that all consumers receive the same service for a policy insuring asset type
$k$
, may be expressed as:
$$\mathop \sum \limits_m {{\mathbb E}_{km},}$$
where
${{\mathbb E}_{km}}$
is the amount of expense allocated to a policy insuring an asset of type
$k$
for a service element
$m$
. Services elements might include distribution, underwriting and pricing, claims and corporate overheads for example. Some expense items may be allocated on a per policy basis and others proportional to the expected loss.
2.7 The Fair Insurance Pricing Formula
Based on the preceding component review, an insurance pricing formula for determining a fair premium
${P_k}$
for a policy insuring an asset of type
$k$
is,
$${P_k} = {{{{\bar X}_k}} \over {\left( {1 + {{\bar r}_{{X_k}}}} \right)}}\left( {1 + {\delta _f}{{{r_f}} \over {1 + {r_f}}}\left( {{\tau \over {1 - \tau }}} \right)} \right) + \mathop \sum \limits_m {{\mathbb E}_{km}.}$$
3. Consumer Pricing Using Marginal Utility
3.1. Insurance is Both a Risky Investment and Bundle of Consumable Services
The economic model expanded for insurance assumes a consumer maximises utility by distributing their finite wealth between investment in risky assets and consumption. The fair premium framework and insurance pricing formula in Equation (5) only considers insurance as a risky cashflow investment.
However, an insurance policy encompasses more than just investment features. It may incorporate various services, including product design, underwriting and pricing, distribution, claim management and corporate services. These services provide utility to consumers that is separate from risk transfer utility. As a result, it is argued that the insurance services included in a policy are part of the consumption aspect within the consumer utility maximisation model and should be priced accordingly.
3.2 Marginal Utility Pricing Theory
Partial differentiation of a consumer’s utility function, constrained by finite wealth, leads to the familiar CAPM equation for determining prices of risky assets and insurance risks. These differential equations also give rise to the marginal utility pricing equation for consumable goods and services. The marginal utility pricing equation is derived in Appendix C.
Marginal utility pricing theory infers that, in equilibrium, a consumer’s utility-to-price ratio is the same for all goods and services they purchase. If not, consumers will adjust their choices to maximise their overall utility, restoring equilibrium. A consumer
$h$
who receives utility of
${{\mathbb U}_{hg}}$
from good or service
$g$
and has a utility-to-price ratio of
${\lambda _h}$
will price the good or service as
${{\mathbb P}_{hg}}$
as,
$${{\mathbb P}_{hg}} = {{{{\mathbb U}_{hg}}} \over {{\lambda _h}}}.$$
3.3 Marginal Utility Pricing and Profit Margins
Assume that expenses to produce the good or service
$g$
equal
${{\mathbb E}_g}$
, then the profit margin
${\eta _{hg}}$
proportional to the expenses will be,
$${\eta _{hg}} = {{{{\mathbb P}_{hg}}} \over {{{\mathbb E}_g}}} - 1,$$
$${\eta _{hg}} = {{{{\mathbb U}_{hg}}} \over {{{\mathbb E}_g}{\lambda _h}}} - 1.$$
Companies are rewarded with profit only when the utility their goods or services provide to the consumer exceeds production costs adjusted by the consumer’s utility constant. Companies are motivated to deliver greater consumer utility using fewer resources, as this results in increased profitability and contributes to enhanced enterprise value. Although the utility assessment of goods and services by consumers and their utility constants are difficult to measure, the profits on expense margins
${\eta _{hg}}$
are observable both in the market for similar goods and services as well as historically within the company.
4. Utility from Insurance and Claim Fulfilment Services
4.1 Modular Services
Goods and services are increasingly offered as modules or components in many industries, including financial services (see Alchin et al., Reference Alchin2016). This approach boosts efficiency and specialisation, while giving consumers more options to customise their purchases for maximum utility.
Insurance has long exhibited elements of modularisation, with services delivered directly to consumers independently of insurance risk. This includes brokers who provide distribution and underwriting services for companies and high net worth individuals; underwriting and management agencies that typically underwrite and distribute insurance on behalf of one or more insurers or syndicates; and companies such as banks or supermarkets that offer white-labelled insurance products under their own brands to their customer base. As noted above, an insurer’s services could be considered as comprising four modules: product design, underwriting and pricing; distribution; claim management; and corporate services.
Focussing on the insurer as providing distinct service modules, each delivering utility to consumers and profit to the insurer, brings multiple advantages. This approach improves transparency on the specific modules and grades of services that are used by each line of business or segment, thereby improving the allocation of expenses. It also deepens the examination of how much utility each module creates for consumers relative to its costs, encouraging innovation and enterprise value creation. In addition, a clearer focus on expenses and utility strengthens the rationale for the profit margins applied to the service modules. Ultimately, this modular perspective equips the insurer to engage more effectively in a modularised market, ensuring that the profit margins are fair for all stakeholders.
It is hence proposed that a fair premium for an insurance policy that provides service modules includes the market utility value of those services estimated as:
$$\mathop \sum \limits_m {{\mathbb E}_{km}}\left( {1 + {\rho _{km}}} \right),$$
where
${\rho _{km}}$
represent fair profit on service margins on expenses that have been benchmarked historically within the insurer and against profit on service margins for comparable services provided by other companies. For simplicity, it has been assumed that the profit on service margins are identical for all policies insuring asset type
$k$
. This means consumers will choose among offerings from insurers and select the best match to their utility-to-price ratio.
4.2 Claim Fulfilment Services
Beyond offering cash settlements for losses, insurers frequently engage suppliers to provide claim fulfilment services such as smash repairs after motor accidents, construction and materials to restore damaged homes and commercial properties, and medical and rehabilitation assistance for insured personal injuries.
Insurers may enter into supply chain agreements with companies that provide claim fulfilment services. These arrangements can result in different ratios of utility to expense for claim fulfilment than if the policyholder selects and engages their own claim fulfilment service after receiving a cash settlement. As a result, there may be profit available for both the insurer and the service supplier through this partnership. This profit will be relative to the portion of the expected loss that is addressed by the claim fulfilment service.
It is proposed that a fair premium for an insurance policy may include the following additional profit that emerges from partnership arrangements with providers of claim fulfilment services,
$${{{{\bar X}_k}} \over {\left( {1 + {{\bar r}_{{X_k}}}} \right)}}\mathop \sum \limits_l {\upsilon _{kl},}$$
where
${\upsilon _{kl}}$
is the insurer portion of the fair, utility profit margin for claim fulfilment service
$l$
, expressed as a percentage of total risk-adjusted losses, for an insured asset of type
$k$
.
5. Proposed Components of a Fair Premium and Insurance Pricing Model
Table 2 summarises the proposed amendments (in bold) to the currently accepted components of a fair premium.
Proposed amendments to components of a fair premium
These proposed amendments produce the following revised insurance pricing model,
$${P_k} = {{{{\bar X}_k}} \over {\left( {1 + {{\bar r}_{{X_k}}}} \right)}}\left( {1 + {\delta _f}{{{r_f}} \over {1 + {r_f}}}\left( {{\tau \over {1 - \tau }}} \right) + \mathop \sum \limits_l {\upsilon _{kl}}} \right) + \mathop \sum \limits_m {{\mathbb E}_{km}}\left( {1 + {\rho _{km}}} \right).$$
6. Worked Examples
6.1 Introduction
These worked examples provide a simplified application of the proposed fair insurance premium pricing model to two consumer insurance policy types, motor comprehensive (MC) and motor bodily injury (BI). The assumptions are rounded and approximate and should be taken as illustrative.
6.2 Pricing Assumptions
The assumptions required for the pricing model for MC and motor BI are shown in Table 3.
Illustrative assumptions for proposed fair pricing model
6.3 Utility Services and Claim Fulfilment Profit Margins
For the worked examples, historical benchmarking information is used to support the profit on service margins and claim fulfilment margins rather than a direct assessment of utility. The two largest listed insurers in Australia and New Zealand (IAG and Suncorp) provide an underlying net insurance margin in their market results. This is intended to be a result normalised for variability from natural perils losses and reserve releases. From this underlying net insurance margin, the cost of capital and an allowance for claim fulfilment profit is deducted. The remaining margin is assumed to represent the total profit margin on services. This analysis is summarised in Table 4.
Observed utility services profit margins for the two largest Australian insurers 2010–2022
Notes: [1] The underlying margin as a percentage of net earned premium was taken from market investor reports for IAG and Suncorp. Due to material quota share arrangements, an adjustment has been made to the margins to reflect exchange commission arrangements, where appropriate.
[2] Insurance risk capital has been assumed to be 50% of the net premium for IAG and 45% for Suncorp. The cost of capital for these insurers is assumed to be 9.5% over the observation period and earnings on capital is assumed to be 5.75% before tax. The tax rate is a uniform 30%. An estimate of the cost of capital as a percentage of the net premium for IAG is therefore ((9.5%)/(1−30%)−5.75%)×50%=3.9%.
[3] Claim fulfilment profit has been assumed to be 2% for these insurers based on research into individual lines of business and considering the mix of these large composite insurers. Many portfolios will have negligible to low claim fulfilment profit on service margins.
[4] Expense ratios have been estimated based on the latest APRA industry data adjusted to allow for the assumed impact of quota share exchange commission arrangements. This includes the underwriting expenses plus an assumed allowance for claim handling expenses.
It is also appropriate to benchmark profit margins against companies offering services similar to those provided by insurers, but without assuming insurance risk. This comparison can be challenging, with insurance brokers generally regarded as the closest comparable entities. Brokers provide a variety of services to both customers and insurers, including product design, distribution, underwriting and claim management. Table 5 presents operating margins for the five largest publicly listed international brokers and two Australian-listed brokers. In addition, several prominent management consulting firms have been included to offer a perspective on margins associated with broader professional advisory services.
Observed operating margins for international and Australian-listed brokers and international consultants
As indicated, insurance brokerage services deliver between 10 and 35% profit on service margins proportional to expenses. Management consulting services appear to command lower profit on service margins in the 10 to 18% range.
Based on the above analysis, the assumed profit on service margins are shown in Table 6.
Assumed profit margins on services
In addition, a 7% claim fulfilment margin on risk-adjusted losses has been assumed for MC after considering the value generated from the supply chain agreements and the appropriate share for the insurer to retain. This is applied to the entire risk premium with supply chain arrangements covering both collision repairs and total losses. A 0% profit on service margin has been assumed for motor BI with limited scope for utility uplift given most claim payments relate to statutory weekly benefits or common law payouts.
6.4 Fair Premium for a Motor Comprehensive Policy
Table 7 and Table 8 bring together the derived assumptions to demonstrate the calculation of a fair premium for a MC policy.
Motor comprehensive discounted cashflows, capital requirements and tax compensation
Notes: [a] A constant risk-free rate of 3.5% has been used, hence the discount factor is
${(1 + 3.5{\rm{\% }})^{ - 0.25{\rm{t}}}}$
.
[b] The risk-adjusted rate is
$3.5{\rm{\% }} - 0.375 \times 6{\rm{\% }} = 1.25{\rm{\% }}$
. Hence, the risk-adjusted discount factors are
${(1 + 1.25{\rm{\% }})^{ - 0.25t}}$
.
[c] Loss payments assumed to be paid at the end of each quarter indicated.
[d] Sum product of loss payments and the risk-free discount factors (a)×(c).
[e] Sum product of remaining loss payments and the risk-adjusted discount factors (b) × (c).
[f] Capital for each quarter is then the capital ratio applied to risk-adjusted liabilities 0.5 × (e).
[g] Tax compensation in each period equals the opening capital (f) mutliplied by
$(\left( {1 + 3.5{\rm{\% }}{)^{0.25}} - 1} \right) \times 30{\rm{\% }}{(1 - 30{\rm{\% }})^{ - 1}}$
using Myers’ Tax Theorem.
[h] Sum product of tax compensation and the risk-free discount factors (a)x(g).
Calculation of fair premium for a motor comprehensive policy
Notes: [a] Present value of losses at the risk-free discount rate.
[b] Expenses are from Table 6 with the claim handling expense assumption applied to the losses discounted at the risk-free rate.
[c] The CAPM risk adjustment is the difference in the loss cashflows discounted at the risk-adjusted rate and the risk-free rate, from Table 7(e) less Table 7(d).
[d] Present value of tax compensation from Table 7(i).
[e] Total of the profit margins from utility for services and claim fulfilment; aligns with the definition of Contract Service Margin under IFRS17. Claim fulfilment is proportional to the present value of losses plus the CAPM risk adjustment.
6.5 Fair Premium for a Motor BI Policy
Similarly, Table 9 and Table 10 bring together the derived assumptions to demonstrate the calculation of a fair premium for a motor BI policy.
Motor bodily injury discounted cashflows, capital requirements and tax compensation
Notes: [a] A constant risk-free rate of 3.5% has been used, hence the discount factor is
${(1 + 3.5{\rm{\% }})^{ - t}}$
.
[b] The risk-adjusted rate is
$3.5{\rm{\% }} - 0.375 \times 6{\rm{\% }} = 1.25{\rm{\% }}$
. Hence, the risk-adjusted discount factors are
${(1 + 1.25{\rm{\% }})^{ - t}}$
.
[c] Loss payments assumed to be paid at the end of each year indicated.
[d] Sum product of loss payments and the risk-free discount factors (a)×(c).
[e] Sum product of remaining loss payments and the risk-adjusted discount factors (b) × (c).
[f] Capital for each year is then the capital ratio applied to risk-adjusted liabilities 0.5 × (e).
[g] Tax compensation in each period equals the opening capital (f) mutliplied by
$3.5{\rm{\% }} \times 30{\rm{\% }}{(1 - 30{\rm{\% }})^{ - 1}}$
using Myers’ Tax Theorem.
[h] Sum product of tax compensation and the risk-free discount factors (a)x(g).
Calculation of fair premium for a motor bodily injury policy
Notes: [a] Present value of losses at the risk-free discount rate.
[b] Expenses are from Table 6 with the claim handling expense assumption applied to the losses discounted at the risk-free rate.
[c] The CAPM risk adjustment is the difference in the loss cashflows discounted at the risk-adjusted rate and the risk-free rate, from Table 9(e) less Table 9(d).
[d] Present value of tax compensation from Table 9(i).
[e] Total of the profit margins from utility for services and claim fulfilment; aligns with the definition of Contract Service Margin under IFRS17.
7. Further Thinking on Consumer Utility
The utility profit margins derived for the worked examples presented in this paper use observed profit outcomes from comparable service providers rather than directly measuring consumer utility. This section presents a possible framework for the direct measurement of consumer utility. Using Equation (8), this value for consumer utility is divided by expenses adjusted for the consumer’s assumed utility constant (subtracting one) to arrive at the profit margin as a proportion of expenses.
7.1 A proposed Model for Direct Measurement of Consumer Utility
A model for the direct measurement of consumer utility is proposed that brings together two key theories, intrinsic utility theory and confirmation theory.
Intrinsic utility theory emphasises measurement of the extent of the tangible and intangible holistic value that a good or service provides to the consumer. Almquist et al. (Reference Almquist, Senior and Bloch2016) of Bain & Co. present a comprehensive and contemporary model of the intrinsic utility provided by a good or service. Their model identifies 30 elements of intrinsic value grouped into a four-layered pyramid adapted from Maslow’s hierarchy of needs. These four layers of intrinsic utility are described in Table 11.
Intrinsic utility in relation to four layers of need
Confirmation theory examines the discrepancy between a consumer’s expectations regarding how their needs will be met by a good or service, and their perceptions of the actual performance and quality of the good or service. This discrepancy ultimately influences whether satisfaction is perceived as positive or negative (see Oliver, Reference Oliver1980). The satisfaction assessment has been summarised into four considerations shown in Table 12. In the proposed model, it is assumed that these satisfaction assessments have a multiplicative impact on the intrinsic value and are termed relativities.
Satisfaction assessment considerations
A proposed model for the direct assessment of consumer utility that combines the two theories is depicted in Figure 1.
Model for the direct measurement of consumer utility.
7.2 Linking Consumer Utility to Profit
Heskett et al. (Reference Heskett1994) is an early example of the proposition of proportionality of consumer utility, consumer loyalty and profitability, which led to extensive research on the linkage. Also at this time, the American Customer Satisfaction Index (ACSI) was developed that showed a link between an increased ACSI and performance (see Fornell et al., Reference Fornell and Johnson1996). Fornell et al. (Reference Fornell and Mithas2006) also examines the link of increased ACSI to enhanced market value and lower volatility.
Bain & Co. have also surveyed consumers and conducted interviews to confirm their various hypotheses associated with their 30 elements of value model. Some key findings, with few surprises, are as follows:
-
The more of the defined elements of utility that are fulfilled from the consumer’s perspective, the greater the net promoter score and revenue growth for the company.
-
The extent of value and utility to be gained from the individual elements varies, for example, “Quality” was found to be the element that most impacted consumer advocacy.
-
Digital aspects of goods and services are more highly valued by consumers than traditional propositions in the fulfilment of many of their functional elements.
-
Traditional, higher interaction distribution platforms appear to better connect consumers to goods and services fulfilling higher order needs and enhance utility.
-
The higher the order need being fulfilled by the good or service the higher the consumer advocacy.
7.3 Potential Application of Utility Model
Assume a MC policy issued by an insurer with a distribution cost of $100. The insurer has a strong, highly considered brand that is viewed as selling premium products with a price to match. After extensive customer research into the specific distribution service of the insurer using the above utility model, it is concluded that customers obtain material utility from the following need layers and elements of value from the proposed direct measurement of utility model:
-
Function: avoids hassle; reduces effort; saves time; quality; and reduces risk.
-
Emotion: reduces anxiety; badge value.
Estimates of the utility value for each of the seven elements are also derived from the research. The research also focusses on the impact of relative utility, and this identifies that the distribution service is assessed by customers as follows for the four utility relativities:
-
Expectation: Typically exceeds expectations of product options and suitability as well as a high grade of service received through digital and traditional distribution channels.
-
Comparison: The service compares very well to competitor experiences, being easier to navigate with a higher degree of transparency of the product and its features.
-
Fairness: There is a strong sense that the insurer treats customers equally in terms of the distribution services, all customers receiving the same attention to detail in matching their needs to product features.
-
Value for money: This is one area where customers feel that the insurer has a slight sense of arrogance around knowing they are among the best and looking to extract healthy profits for their shareholders.
Estimates of how these relativities impact intrinsic value have also been derived from the research. The insurer then proceeds to assess the utility and the price that is fairly charged for the distribution service as shown in Table 13.
Assessed utility and price for the distribution service for a motor comprehensive policy
Notes: • For simplicity, the relative utility percentages have been added and total utility assessed as
$190 \times 1.05 = 200$
.
• The price equals the utility value divided by the assumed consumer utility constant
$200 \div 1.5 = 133$
.
• The profit on service margin, expressed as a proportion of expenses, is
$\left( {133 - 100} \right) \div 100 = 33{\rm{\% }}$
Considerable additional research and investigation is required to be able to craft a justifiable set of assumptions using this model. However, it seems very likely that a focus in accordance with the proposed direct measurement utility model would lead to more justifiable profit margins and more optimal outcomes for consumers and shareholders.
8. Conclusion
The commonly accepted elements of a fair insurance premium do not fully justify the profit margins observed in competitive market prices charged to consumers. Furthermore, the assertion that insurance frictions allow for the pricing of non-systemic risk lacks support within the CAPM economic theory.
This paper argues that economic theory supports the use of utility market pricing for consumable goods and services, which are presently incorporated into insurance pricing models only at cost. A new insurance pricing model is presented that integrates utility market pricing of services associated with insurance policies, as well as the value derived from the supply chain agreements involved in the fulfilment of claims. The updated pricing model is intended to offer a more robust rationale for profit margins. This justification has become increasingly necessary as regulatory authorities intensify their scrutiny of the adoption, execution and evaluation of customer fairness frameworks.
Data availability
The data (within the worked examples) and code (formulae) are as presented in the body of the paper.
Acknowledgements
The author thanks Antonie Jagga FFA and Timothy Lee FIAA for their support and assistance in the publication of this paper. Review and input was also received from Ben Wang FIAA and Nic Baker FIAA.
Competing interests
The author discloses their employment with Insurance Australia Group Ltd. The author confirms that there are no relevant financial or non-financial competing interests to report. This paper has been prepared in his personal capacity as a Fellow of the Institute of Actuaries of Australia. This paper expresses the views of the author and not necessarily those of his employer or affiliations. The information and expressions of opinion contained in this publication are not intended to be a comprehensive study, nor to provide actuarial advice or advice of any nature and should not be treated as a substitute for specific advice concerning individual situations. The author’s employer and affiliations do not endorse any of the views stated, nor any claims or representations made in this publication and accept no responsibility or liability to any person for loss or damage suffered as a consequence of their placing reliance upon any view, claim or representation made in this publication.
Funding statement
There was no external funding.
Appendix A. Default option
Limited liability gives shareholders of insurers an exchange option. In a one-period model, a shareholder’s payoff at the end of the period for their investment in an insurer is the value of the asset minus losses. However, should losses exceed assets (insolvency), shareholders exchange with policyholders the value of the assets for those losses for a payoff of zero. The default option is the exchange option minus the shareholders’ initial capital, representing the expected losses policyholders would not recover due to insolvency.
In a one-period model, Fischer (Reference Fischer1978) gives the market value of the default option
${{\mathbb D}_f}$
for insurer
$f$
at the beginning of the period as,
$${{\mathbb D}_f} = {{\mathbb A}_f}{\rm{\Phi }}\left[ {{{\rm{d}}_ + }} \right] - {{{{\bar X}_f}} \over {\left( {1 + {r_f}} \right)}}{\rm{\Phi }}\left[ {{d_ - }} \right] - {K_f},$$
where
${{\mathbb A}_f}$
is the total investment assets equal to premium plus capital,
${\bar X_f}$
is the total expected losses of the insurer,
${K_f}$
is the initial amount of capital injected,
${\rm{\Phi }}\left[ \bullet \right]$
is the cumulative normal distribution function, and,
$${{\rm{d}}_ \pm } = {{{\rm{log}}\left[ {{{{{\mathbb A}_f}\left( {1 + {r_f}} \right)} \over {{{\bar X}_f}}}} \right]} \over {{\sigma _f}}} \pm {{{\sigma _f}} \over 2},$$
where
${\sigma _f}$
is the instantaneous proportional standard deviation of the log of the ratio of assets to liabilities of the insurer. The end period values of assets and liabilities are assumed to follow a lognormal distribution having evolved through the period as a Weiner process. Taylor (Reference Taylor1995) explores the default option further, leading to a market equilibrium insurer capitalisation equation.
Appendix B. Myers’ tax theorem
Appendix B.1. Myers’ tax theorem for a one-period model
Assume that for an insurer
$f$
we have capital per unit premium of
${\delta _f}$
at the beginning of the period. This capital is invested and experiences a stochastic return over one period of
${\tilde j_1}$
with the return subject to tax of
$\tau $
payable at the end of the period. Tax for the period is therefore,
$${{\delta _f}\tau \left( {{{\tilde j}_1} - 1} \right),}$$
$${ = {\delta _f}\tau {{\tilde j}_1} - {\delta _f}\tau .}$$
The present value of these terms at the beginning of the period, noting that the second term is non-stochastic and discounted at the risk-free rate of
${r_f}$
, is therefore,
$${\left( {{\delta _f}\tau - {{{\delta _f}\tau } \over {1 + {r_f}}}} \right)},$$
$${ = {\delta _f}\tau \left( {{{{r_f}} \over {1 + {r_f}}}} \right).}$$
The present value of the tax therefore does not involve the risk and expected return of the underlying asset, rather it involves the tax rate and risk-free rate of return.
An interpretation of Equation (A6) is that it represents the market value of the taxation that is transferred from the shareholder to the taxing authority before compensation activity by the policyholder. The policyholder is required to compensate shareholders by returning them to the same risk and return profile for their investment in the insurer compared to the direct investment of their capital into the market – or at least fund costs associated with the mechanism to do so.
Appendix B.2. Alternative hedging derivation
As in Appendix B.1, assume shareholders inject capital of
${\delta _f}$
per unit of premium at the beginning of the period into an insurer. The insurer invests this capital and incurs tax of
$\tau $
on stochastic earnings at the end of the period before passing those earnings onto shareholders.
Faced with this tax, one mechanism for the insurer to replicate the risk and return profile of the shareholders as though they invested directly in the market is to increase the amount invested to
${\delta _f}\tau {(1 - \tau )^{ - 1}}$
. The net of tax outcome on this increased investment is then identical to the shareholder investing
${\delta _f}$
directly in the market.
The insurer could borrow (hedge), at the risk-free rate of
${r_f}$
, the amount of
${\delta _f}\tau {(1 - \tau )^{ - 1}}$
to add to the capital
${\delta _f}$
for investment. The policyholder need only compensate for the risk-free cost of borrowing this amount as the shareholder takes the risk and return associated with the total investment of
${\delta _f}{(1 - \tau )^{ - 1}}$
and repays the initial loan amount at the end of the period.
This compensation paid by the policyholder is part of the premium that is taxable within the insurer. However, there is a deduction available for the risk-free borrowing cost and hence no further gross up for tax is required. It follows that the compensation required by the policyholder to pay with their premium at the beginning of the period is,
$${{{{r_f}} \over {1 + {r_f}}}{\delta _f}\left( {{1 \over {1 - \tau }} - 1} \right),}$$
$${ = {\delta _f}{{{r_f}} \over {1 + {r_f}}}\left( {{\tau \over {1 - \tau }}} \right).}$$
This is simply, but importantly, Myers’ tax theorem Equation (A6) grossed up for tax with a factor of
${(1 - \tau )^{ - 1}}$
.
Suppose that the insurer decides not to borrow and therefore does not hedge the tax on investment earnings. Instead, reduced risk and a fair replacement for the tax on stochastic earnings are accepted. Policyholders are reducing the risk associated with total investment earnings on capital by replacing the tax on stochastic earnings with a risk-free fixed amount. Hence, the compensation can be argued to be
${\delta _f}{r_f}\tau $
that is, the tax assuming risk-free earnings on capital. As there is no tax deduction associated with this amount when paid by the policyholder as premium, this compensation needs to be grossed up for tax, that is, by a factor of
${(1 - \tau )^{ - 1}}$
. Discounted risk-free to the beginning of the period this also results in Equation (A8) for the tax compensation.
Appendix C. Marginal utility pricing equation
Assume that consumers are utility maximisers and use their allocated consumption amount to maximise their aggregate consumer utility over the period. If consumer
$h$
obtains utility
${{\mathbb U}_{hng}}$
from the nth good or service
$g$
acquired, then the aggregate utility of the consumer will be,
$$\mathop \sum \limits_g {n_{hg}}\mathop \sum \limits_{{n_{hg}}} {{\mathbb U}_{hng}.}$$
This is simply the product of the number of goods or services acquired by the utility obtained from each good or service. The double summation is required as utility may vary with the second, third or more additional acquisition of a particular type of good or service. Economists generally propose that there is diminishing utility associated with consuming more of the same type of good or service. Since the household has the constraint of a consumption budget
${C_h}$
, then,
$${C_h} = \mathop \sum \limits_g {n_{hg}}{{\mathbb P}_g}.$$
The Lagrangian function to maximise is,
$${\cal L}\left( {{n_{hg}},{\lambda _h}} \right) = \mathop \sum \limits_g {n_{hg}}\mathop \sum \limits_{{n_{hg}}} {{\mathbb U}_{hng}} - {\lambda _h}\left( {\mathop \sum \limits_g {n_{hg}}{{\mathbb P}_g} - {C_h}} \right),$$
where
${\lambda _h}$
is a Lagrangian multiplier. The first order equation of the partial derivative with respect to the final good or service of each type acquired, that is,
${{{\partial L\left( {{n_{hg}},{\lambda _h}} \right)} \over {\partial {n_{hg}}}}}$
is,
$$0 = {{\mathbb U}_{hng}} - {\lambda _h}{{\mathbb P}_g,}$$
$${{{{\mathbb U}_{hng}}} \over {{{\mathbb P}_g}}} = {\lambda _h.}$$
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