Why Can Financial Firms Charge for Diversifiable Risk?

Andrew Smith, Ian Moran and David Walczak

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Abstract:

It is widely accepted that capital markets do not demand a premium for risk that investors can diversify. On the other hand, insurers’ and banks’ pricing models frequently include an allowance for total risk, diversifiable or not. Why then do competitive product markets not eliminate these pricing margins? We propose an answer to the puzzle, by analysing the frictional costs that financial institutions incur in the course of their risk-bearing function. A series of worked examples demonstrate the implications for pricing, risk management, and financial reporting.

Author Contact Details:

Andrew Smith B&W Deloitte, Horizon House28 Upper High StreetEpsom, SurreyKT18 7LJGreat Britain.

tel +44 1372 824811e: andrewdsmith8@bw-deloitte.com

Ian Moran tel +44 1372 824143 e: imoran@bw-deloitte.com

David Walczak Deloitte & Touche400 One Financial PlazaMinneapolisMN 55402USA

tel +1 612 397 4509 , e: dwalczak@deloitte.com

Introduction

Pricing and accounting have sometimes been uneasy stable-mates in the financial services industry. Firms have been prepared to make an initial accounting loss on a customer, in order to recoup profits later, for example in the sale of life business under UK statutory reporting. In fewer cases, the opposite situation applies and accounting standards permit the recognition of immediate profit when a customer relationship commences, for example when reporting under embedded value techniques.

Fair value accounting will bring financial statements closer than ever before to the assumptions used for product prices. The final form of an insurance accounting standard is some way off, and retail banking is even further from fair value reporting. However, no accounting standard, however well constructed, could comprise a full blueprint for economic pricing. There are good economic reasons for differences between pricing and financial reporting practice. These differences do not imply that one of the components is unsound.

This paper investigates these differences. We focus on the apparent anomaly that diversifiable risk carries a much higher price in retail financial services than in capital markets. We argue that capital market and retail risk pricing are best reconciled using the concept of frictional costs. These are contingent internal costs, which a financial firm faces in managing the risks it retains. Even if capital markets require no premium for diversifiable risk, frictional costs can create the illusion of a non-systematic risk premium for insurance risks.

There are two sets of arguments: demand and supply side. The demand side explains why policyholders may be prepared to pay a premium for diversifiable risk. More subtle arguments are required from the supply side, to explain why competition between insurers does not erode the diversifiable risk margins.

Demand Side Arguments

The first question we must resolve is why policyholders might be prepared to pay for “overpriced” insurance that incorporates margins for risks.

Many financial decision makers do not use a pure expected value approach to the choices they are faced with, such as how much to pay for insurance or whether to purchase one investment vs. another. Instead, individuals seek to balance risk with the possible rewards. For example, utility functions have been proposed for explaining these effects.

The result may very well produce a decision which is not optimal from an expected value or fair value standpoint but allows us to quantify risk averse decision making properly. Investment decisions can be viewed as a utility maximisation problem; investors do not necessarily choose the highest possible mean return but will instead seek to balance risk and return.

The same principle applies to the purchase of insurance. Agents can look at their risk exposures holistically, aggregating traded investment risks and also the specific

insurable risks relative to their circumstances. They may well decide to purchase insurance, if

Doing so materially reduces their total risks The cost is acceptable relative to other means of reducing risk such as

investing more defensively The agent is sufficiently risk averse

Quantifying these effects is helpful because it can indicate an upper bound on how much a policyholder might be prepared to pay for insurance in a monopolistic situation. How much they actually pay, depends on the operation of competition between insurers. As we shall see, this is a much more subtle point to quantify.

Insurance companies themselves also display risk-averse behaviour with regard to using the reinsurance market to mitigate risk. Like insurance policyholders, the company seeking reinsurance is often willing to pay more than the expected value of the reinsurance premium. Views differ as to the best model for explaining this. We could seek to explain corporate behaviour using one of the following:

An aggregate utility function representing the desires of corporate management

A shareholder value measure seeking to value risk via its impact on the business

Taking the utility approach, Borch (1969) described the ‘most efficient’ reinsurance contract as that which minimises the insurance company’s variance of net claim distribution for a given reinsurance premium. Risk based decision making has the following consequences:

A willingness to pay ‘overpriced’ reinsurance premiums Setting a ‘retention limit’ (amount of risk a company is willing to hold on a

single life) Underwriters, compensated by net results, may well purchase reinsurance to protect the risk in their own compensation. Insurance companies may take advantage of hedging programs for the same reasons or because regulatory pronouncements often allow companies to take credit for hedging programs on the balance sheet.

There are problems with treating a corporation as driven by the utility of managers, however. The chief difficulty is allowing for the impact of other competing insurers, and of shareholders. It is hard to see why shareholders would appoint managers who subsequently adopt self-interested strategies in defiance of those they are supposed to serve. Why would shareholders pay a manager to reduce risk if the shareholder can achieve the same effect more easily by diversification?

And this brings us to the real puzzle. If shareholders can diversify, then they should not be worried about diversifiable risk inside the companies they buy. This should flow through to product pricing. If insurers generally charged customers for bearing diversifiable risk, we should expect to see new capital entering the market, until the price of diversifiable risk became consistent with returns required by a diversified investor. The remainder of this paper considers these supply side arguments, and how they may be refined to admit the appearance of a positive price for diversifiable risk.

Supply Side Arguments

Our supply-side argument is based on five steps as follows:Step 1: A frictionless (Modigliani-Miller) insurance marketStep 2: The Myers – Cohn model and frictional costs of investingStep 3: Accounting equity and franchise valueStep 4: Costs of financial impairmentStep 5: Capital optimisation

Part A of the paper describes these steps in further detail, while part B gives a mathematical formulation in the context of a simplified insurance model. Part C offers some tentative conclusions.

Part A: Verbal Reasoning

Step A1: A frictionless (Modigliani-Miller) insurance market

Even in today’s market conditions most economists accept that the mean required return on stocks exceeds that on bonds. Before 1958, practitioners usually concluded that firms could save money by being financed by debt (ie bond issuance) as far as possible, as the return requirements of bondholders are less demanding than those of shareholders.

Modigliani and Miller (1958, henceforth M&M) considered the implications of a frictionless market on a firm’s capital structure. Their frictionless market assumptions were as follows:

No transaction costs in capital markets Individuals can borrow or lend at the risk free rate There are no costs to financial impairment Firms issue two kinds of claims: risk-free debt and (risky) equity No taxes Corporate insiders and outsiders have the same information Managers always maximise shareholders’ wealth

They argued that, as a company increased its degree of debt finance, so the residual shareholders were exposed to greater risk as a result of the leverage. The shareholders’ required return is not some historic average stock return, but instead should be sensitive to the risks in the business reflecting current capital structure. This means the required return on equity increases as the leverage increases. Therefore, there is no automatic reduction in required profits for a firm who shifts financing towards bonds.

A key step in the M&M arguments is the notion of shareholders who can also borrow or invest in bonds, on the same terms as the firm itself. The shareholder who wants a risky investment is indifferent between (i) borrowing money to invest in an unlevered stock or (ii) investing his own cash in a company who is partially financed by debt.

There is a parallel set of arguments in the insurance industry. Here the focus has not been primarily on capital structure, but on investment strategy and capital allocation. The M&M arguments also apply in this context. The implications are:

Investment strategy is irrelevant to value. An insurer who invests his own assets in (other firms’) stocks rather than in bonds, will probably increase risks and returns. The risk premium earned on those stocks in the hands of the insurer is the same as the risk premium in the hands of the insurer’s shareholder. Therefore, there is no overall gain to shareholders when an insurer adopts a risky investment strategy. An insurer can by investing do only what shareholders can do themselves.

Capital allocation is irrelevant to value. An insurer who allocates more capital to a particular line of business makes that business proportionately less risky. Therefore the percentage return required by shareholders is lower, but this now applies to a larger base. The dollar required profit is slightly higher overall, but we also expect a higher profit because of the return on the extra assets. The two effects cancel out and we are back where we started. Therefore, allocating more capital does not imply a need for a higher policyholder premium.

The remainder of this paper seeks to temper these unintuitive findings with increasing doses of business realism.

Step A2: The Myers – Cohn model and frictional costs of investing

To make the M&M propositions more realistic, we allow for corporate taxation. This is covered in Modigliani and Miller (1963). The corresponding insurance reference is Myers and Cohn (1987).

Myers’ and Cohn’s insight came from an exploration of the different terms under which corporations can invest, relative to direct investments by shareholders. In many cases, the corporate investment is subject to double taxation, as income from investments is a source of taxable profit. As a consequence, in order for shareholders to earn an acceptable return, any insurance premiums must be sufficient not only to pay claims and expenses but also to pay tax bills arising from investment returns.

Therefore, contrary to the pure M&M findings, capital does have a cost, and capital efficiency can create value. Recent advancements in North American capital allocation regulation for insurers allows an entity to hold less capital if a reporting unit uses properly matched assets and liabilities from an interest rate risk standpoint. According to the Myers-Cohn model, this is an instance of higher value of an insurance entity resulting from efficient capital allocation

Under the Myers-Cohn model, pricing of insurance by line of business requires an allocation of investments between classes of business. This requires an allocation not only of liability reserves (usually a simple bookkeeping exercise) but also of the assets constituting shareholder equity. And here we have a bigger puzzle, because the net assets are a single legal pool – any part of the equity can be used to meet deficiencies in any line of business. Paper allocation does not imply any sort of ring-fencing of assets to meet claims in specific lines of business.

The need to allocate equity between lines has spawned a whole industry in capital allocation (also called “economic capital”) for performance measurement. Here we encounter an immediate practical difficulty with the Myers-Cohn framework. The

theoretically optimal amount of equity is zero, as this minimises tax. As in practice firms do hold equity, the amount held must be exogenous to the Myers-Cohn model. There is scope for endless argument about how equity should be allocated, with no definitive answer.

Later research has suggested that double taxation is only the tip of the iceberg as regards the cost of holding investments. Usually far greater in magnitude is the risk of poor stewardship of those assets in the hands of corporate managers. For example, managers may be tempted to squander shareholder resources on ambitious acquisitions, which destroy value but enhance the status of the management team concerned. Such effects are called agency costs (see Jensen & Meckling, 1976), arising from conflicts of interest between shareholders and managers. This is a special case of what are more generally called frictional costs. From a shareholder perspective, agency costs act as a form of tax on assets entrusted to third party managers. To justify the existence of an insurance firm, the premiums achievable must not only offset claims, expenses and taxes but also a share of the agency costs associated with the equity in the business.

It may appear circular to expect managers to quantify their own conspiracy against shareholders when setting premiums. We argue that markets allow for such costs when pricing insurance shares; therefore it is natural to use this information also in pricing decisions.

Step A3: Accounting Equity and Franchise Value

Commercial pricing bases usually include an item described as “margin for profit”. This may arise from failure of competition, for example by the operation of cartels or unwarranted entry barriers. The existence of profit margins does not however always point to a market failure. Instead, an insurers’ ability to charge premium margins could be a fair reward on investment in a brand, distribution capability, good customer relationship management or excellent service. The margin for profit is not the same as a loading for risk. The application of economic risk loads to premiums would still not create value for shareholders, as the shareholders also bear the risk taken on. Instead, we are here considering margins for economic profit over and above the cost to shareholders of any risks borne.

In or discussions on the demand side, we noted that the variability of insured loss is one determinant in policyholders’ willingness to pay a premium for insurance. Therefore, loss variability implies an upper bound on the premium we could possibly generate by investment in brand, distribution or service. To this extent, the margin for profit will appear to be risk related. It is tempting, but wrong, to interpret this margin for profit as some sort of market value margin for risk. To see why this is wrong, remember that insurers who have not invested in brand or service capabilities will face a commodity market, which does not permit them to collect the margin for profit.

There is remarkably little in the literature regarding the derivation of margins for profit, save the counterfactual assertion that in a competitive market such margins do not exist. In this paper we make a new proposal relating the margin for profit to the firm’s share price. It requires a step from a single period model of the firm, to a multi-horizon setting.

A cursory glance at the financial pages reveals that the market capitalisation of a firm is not equal to the net equity in its published financial statements. The market capitalisation is typically (but not always) greater than the published equity. We therefore write

market capitalisation = e + f

where e is the published equity and f is the franchise value. The franchise value represents items such as distribution capabilities, business and customer relationships and growth opportunities, which are valuable to shareholders yet not generally recognised as assets by accounting standards.

Many accounting debates concern whether a particular cash flow should or should not be counted as part of equity. Indeed, we shall argue that the question of pricing risky insurance flows falls into this category. For the moment, the detail of this argument is not important; what matters is that standards must draw a line around what is to be recognised as an asset or liability. There is often no unique correct answer. Therefore, we inevitably find a partition of the market value of a firm into e, the portion falling inside the standard, and f falling outside. We should not suppose that a dollar of f is somehow inferior to a dollar of e. Both are equally valuable to shareholders. Management may rationally spend e in order to create more f, for example by investment in a brand. Selling an insurance policy, which at inception fails to recoup its acquisition costs, sometimes called “buying market share”, could be viewed in a similar light.

However, managers have not always been content to report a loss in such circumstances. Various accounting devices have been proposed in order to spread acquisition costs over the term of a policy. These include the net premium method in life assurance, deferred acquisition costs in property / casualty insurance, and the ability to amortise goodwill in acquisition accounting. All these adjustments have the effect of smoothing the income statement by inventing fictitious assets, which circumvent accounting recognition criteria. Modern accounting standards rightly condemn such practices. We will argue that proposals for market value margins in insurance liabilities fall into the same trap, of inventing fictional liabilities in order to smooth income statements.

Let us consider the implications of this equity/franchise split for required returns. An investor who buys a share in an insurer injects not only the equity e, but must also compensate the seller for the franchise value f. Similarly, the shareholder can realise f by selling the shares. While under some accounting standards e might represent a historic sunk cost, this is of little relevance to today’s shareholder. We can reasonably suppose that the shareholder requires a return on the total e+f, not only on e.

Yet insurers and analysts persist in a focus on return on equity. This provides a return on a historic investment e but neglects the return required on the franchise f. Some adjustment to return is required to obtain meaningful performance targets. Grossing-up is required to convert a market return on (e+f) into a target expressed as a fraction of e only. Premiums for insurance risks now need to cover claims, expenses, taxes and frictional costs and also an allowance for maintenance of franchise value. This is

indeed common practice – the maintenance of franchise value being more usually called a loading for profit. It is essential to appreciate that this profit loading is not any sort of market reward for risk, which might be captured by CAPM or coherent risk measures. Instead, it is that part of the return to shareholders arising from their past investments in a franchise structure.

Much effort is expended allocating accounting equity between lines of business, and little on allocating the franchise value. This failure distorts performance measures. It is all very well for a financial service provider to demand a return on risk-based capital, but what about the sunk costs of setting up the service in the first place?

There is an interesting regulatory twist to this discussion. Some competition regulators, both sides of the Atlantic, swear that a loading for profit, of, say $10, is evidence of market failure. The solution, apparently, is regulation of the price of financial services to eliminate the $10 margin. The stated and no doubt well-intentioned aim is to protect consumers from exploitative financial firms.

The predictable effect of such price regulation is to discourage insurers’ investment in brand or service etc that develop the franchise value. This results in reduced customer loyalty and a homogeneous commoditised market with greater switching between insurance providers. And here is the irony – the switching generates additional administrative and acquisition costs of, say, $20, which are recognised in the regulatory formula thereby permitting insurers to pass them on to consumers. The regulators’ impact is to confiscate $10 from consumers, $10 from shareholders and spend the $20 on avoidable administration.

Step A4: Costs of financial impairment

We have now defined equity and franchise value. The next step is to recognise that shareholders are exposed to risks in both components. The currently topical focus on control of balance sheet risk is useful for managing the variability in equity e. Tools for monitoring risks to franchise value, f, are less well developed.

There are several risks to franchise value, including: Loss of confidence in an insurer’s ability to pay claims Damage to a provider’s reputation (or to that of a whole industry) New entrants or distribution channels bringing pressure on margins Adverse effect of regulation of product design or pricing General economic downturn impacting customers’ willingness to spend

In this paper, we focus on the first item, that is, financial impairment. Insurance customers are credit sensitive, and are quick to desert an insurer at the sign of financial peril. Regulators may also intervene to close a marginally solvent insurance business where e falls close to zero. Therefore, risk management of the equity e can be motivated by a desire to protect f. In the broader context, the desire to control costs of financial impairment is often cited as a correction term to the M&M hypotheses. See for example Doherty (1997), Froot & Stein (1998), Hancock et al (2001), Merton & Perold (1999)

These considerations will also impact pricing. If an insurance policy adds to the risk of a firm, it increases the chance of loss to franchise value. This is true whether the insurance risk is diversifiable (by the shareholder) or not. A rationally constructed insurance premium should compensate shareholders for this indirect cost. In our mind, this is the most persuasive rationale for the risk loadings in insurance contracts.

An allowance for risk to franchise value can end up being a proxy for a whole host of other frictional costs associated with risk or variability. These are discussed further by Cumberworth et al (2000) and in the paragraphs below.

Financial distress can be costly due both to direct costs – such as the direct costs of needing to raise fresh capital, and also the indirect costs such as loss of reputation and of future profitable business opportunities. More volatile lines of business are more likely to trigger such financial distress, particularly when these are positively correlated with other lines.

An unexpected loss in one part of the business can result in a shortage of resources, or even termination, for existing projects for which costs are already sunk. A large volume of unexpected claims can overwhelm processing staff, and create the need to recruit for the short term at disproportionate cost. Furthermore, in such circumstances management may be stretched, and may even panic into fighting fires rather than managing the business going forward. On the other hand, an unexpected surplus may lead to a relaxation of financial discipline and control.

Management may underestimate claims costs, and hence under-price, due to an inbuilt optimism about the business they run, and the fact that they do not personally bear the downside risk. It is easier to sustain optimistic plans when the data to challenge those plans is inconclusive, which is more likely to be the case from volatile lines of business. Thus a premium loading, however derived, could be an appropriate counterbalance to optimistic cash flow forecasts.

Having accepted the notion of a load for risk, there are two ways to go about modelling. The implicit approach seeks definitions of risk that satisfy certain axioms, from which to derive premium loads. For example, Borch (1982), Artzner et al (1999), Wang et al (1997) and Wang (2002) follow this axiomatic implicit approach. The explicit approach, which we follow in this paper, constructs a model of the franchise value, which may be impaired by risk, and uses capital market tools to price the impairment. Christofides and Smith (2000) provide a partial reconciliation of the two approaches, showing how a particular form of convex frictional cost function can replicate risk loads using Wang’s 1997 proportional hazard transform.

The accounting treatment of premium loadings is the subject of some debate. A new insurance contract will have a marginal impact on both equity e and franchise f. The pricing of the policy should recognise both of these elements. If an insurance risk threatens the maintenance of franchise value, then this should affect the premium scale. However, the premium loading reflects no additional obligation to external parties. Rather it is the additional profit required by shareholders and should not be recognised as an accounting liability. Therefore, that part of the premium designed to compensate shareholders for risks to franchise value emerges as an accounting profit on inception of the insurance policy to the extent that it is not artificially held back by

devices, such as market value margins on liability measurement discussed for the IASB project on Insurance.

Regretfully, there are precedents for developing artificial devices to obtain the desired accounting result, for example deferred acquisition costs. Standard setters now, rightly in our view, are moving away from the use of deferred acquisition cost assets by acceptance of fair value.

Step A5: Capital optimisation

Until this point, the cost of capital has entered our pricing formulation as an exogenous input. We have now identified the costs of having both too much equity (agency costs) and also of having too little (financial impairment). Somewhere in the middle lies an optimal allocation of equity.

We can identify this by examining the franchise value of the firm, and how it depends on the equity in the business. There are three possible outcomes:

(i) there is a unique optimal level of equity, with both higher and lower levels resulting in lower franchise values. This is the usual state of affairs

(ii) where the underlying business is unprofitable, the optimal equity allocation is driven by the value of the bankruptcy option. This option represents the insurer’s ability to default on a claim that exceeds its total resources. In this case, the optimal amount of capital from a shareholder point of view is zero. It is best to leave minimal shareholder assets at risk, collecting profits when positive and walking away at the first loss. There are several notable cases of this having occurred in the United States, usually due to an asset / liability mismatch or an asset class liquidity problem – Executive Life, Mutual Benefit and Baldwin United come to mind.

(iii) It is also possible to construct examples with two or more local optima. For example, there may be a local optimum at zero capital that maximises the bankruptcy option, and another local optimum, which represents a solvent strategy balancing the costs and benefits of equity. This is the case in our worked example in part B.

This is a major step forward. In contrast to the Myers-Cohn model, we can now establish an optimal amount of capital within a single modelling framework. Thus, we recover a financial approach, which simultaneously delivers a pricing basis and an optimal capital structure.

To conclude: we have succeeded in expressing premiums as the sum of five parts: Expected claims and expenses discounted at the risk free rate Cost of systematic risk (A1) Tax and agency costs (A2) Margin for profit (A3) Impairment costs (A4)

In addition to the pricing arguments, we have considered also which of these items should be recognised as accounting liability at the date of inception. Our tentative conclusions are:

Aspect Recognition as a liabilityExpected claims and expenses discounted at the risk free rate

Recognise as they reflect an obligation to 3rd party

Cost of systematic risk (A1) Recognise as part of the cost the market would pay

Tax and agency costs (A2) There is a case for recognising some of the tax effects as part of doing business;

The agency costs are not an obligation to a 3rd party and should be compensated by the level profits rather than as a liability.

Margin for profit (A3) Like agency costs.Impairment costs (A4) Not a liability, as it is merely a reduction

in an off balance sheet asset, namely franchise value.

Part B: The Mathematical Model

We illustrate the five steps in our argument using the simplest possible model, based on Boulton et al (2002). We take a single investment horizon, and assume the CAPM applies over that period. Our model of the insurance firm may extend beyond the end of the investment period.

Over our single period, we define the following variables, all but the first of which are random.

R = risk free return factor = 1 + risk free rateM = market return factor (gross of taxes)A = asset return factor = market value assets at end less costs and applicable taxes, divided by market value assets at start.L = combined ratio, net of tax = (claims + expenses – tax deduction) / initial premiumG = business growth factor

We use μ to denote mean and σ to denote variance or covariance. Thus μM denotes the mean of the market return factor, and σLM is the covariance between L and M. Note that σMM is the variance of M and not its standard deviation.

For illustrative purposes, we have used the following mean parameters:R 1.05M 1.10A 1.03L 0.98G 1.00

and the following variance-covariance matrix:

M A L GM 0.0400 0.0100 -0.0100 0.0200A 0.0100 0.0050 -0.0013 0.0050L -0.0010 -0.0013 0.0169 -0.0050G 0.0200 0.0050 -0.0050 0.0200

Limitations of our model include the following: We model a single line of business where each policy covers one year All premiums are received at year start and claims settled at year-end. At the

year-end, the firm has only investments and no liabilities. Cash flow and accounting coincide

This avoids the need to determine provisions and accrual items. We therefore overlook the affect of accounting treatment on taxes, dividends and solvency. We will fail to capture the commercial consequences of different accounting treatments.

We have not sought to model the operation of guarantee pools or policyholder compensation schemes that may apply to some classes.

Prospective returns, combined ratios and growth are unchanged from one year to the next. Stochastic effects are independent from one year to the next. There is no allowance for cross-year correlations, for premium cycles or for stochastic interest rates.

Where we have needed to assume a distribution, we have used normal distributions for the random quantities M, A, L and G.

We have used the CAPM for setting financial prices, so that all capital market investments earn an expected return based on their covariance with the market portfolio.

Before we can progress further, we need a statement of the capital asset pricing model. Let S be the return factor on any investment portfolio. Then CAPM states that

The fraction is sometimes called the “beta” but we do not use that terminology here.

Our first corporate model works over a one year horizon, (so G is irrelevant). The shareholder injects initial equity e and the policyholder an initial premium p. At the year end we have

assets (e+p)A liabilities pL dividend = surplus = (e+p)A – pL

In our base case, we assume p = 100 and allocated equity e = 25. We can apply CAPM to determine the competitive equilibrium return for shareholders.

For step B1, we assume the asset returns already solve the CAPM, so that

This would require us to set A = 1.0625. It follows by subtraction that

Therefore, the equilibrium combined ratio will follow the CAPM – in our case giving a combined ratio target of 103.75% so the premium is 96.39% of expected claims and expenses. In effect, the policyholder return in the insurance policy becomes comparable to the return on other assets in the market. As with investments, only systematic risk should be priced in insurance markets. Notice that in this example we need assume nothing about the risk tolerance of policyholders. We assume only the competitive mechanisms in capital markets, which ensure that investors in insurance companies earn a fair return.

Step B2: Now lets allow for taxes and costs on the asset return, in the spirit of Myers and Cohn (1987). Then the condition for equilibrium in the insurer’s shares becomes:

.

Therefore, the combined ratio satisfies the CAPM, less a deduction for any frictional cost on the asset side. In our example, this would result a premium target of 100.31%, up from 96.39% of expected cash flows. This relationship was first pointed out by Taylor (1994) and is discussed further by Sherris (2002).

In our example, for simplicity we have applied the tax and frictional costs to the assets from the ground up. There is an argument for applying these only to those assets representing shareholder equity, and applying a lower or zero loading to assets backing reserves. This could be justified by recognising the regulatory constraints on the investments of technical reserves, which limit the extent to which managers can misuse the assets. This more sophisticated model would result in a lower charge for tax and agency costs (which in our example may appear unrealistically large).

The loading for tax and agency costs depends on the amount of equity allocated to that premium. And here we see the gap in the Myers-Cohn approach – how do we say how much capital is needed?

Step B3: Our next step is to allow for a multi-year model (but still ignoring ruin). We assume there is some profit margin available in the insurance market, and therefore that the firm has some franchise value in addition to the shareholder equity. We look to the market in the insurer’s stock to guide the required profit margin in premium bases.

As before, let e denote the initial equity and f the initial franchise value. As we have a multi-year model, our theory permits f>0. Our objective is to derive an expression for f.

Initially the firm has equity of e plus a premium p. By the year end, it has net assets of (e+p)A – pL. We want net assets of Ge to carry forward for the next year, given a growth

factor G.

Therefore, the firm pays a dividend of e(A-G) + p(A-L). The continuing value of the firm is G(e+f), as the firm at the year end is entirely proportional to the starting position. The total shareholder value (including dividend) at the year end is eA+p(A-L)+fG.

Once again, we assume the capital markets use CAPM to value the firm. This means the total shareholder return must satisfy CAPM:

Eliminating f, we see that:

We can therefore establish the franchise value as follows:

We can rationalise this is the value of the value added on the asset side, minus that on the liability side, projected annually into the future and then discounted.

In practical work, the application more often operates in reverse. We observe a franchise value in the market and calibrate the pricing model to this franchise value in order to derive an appropriate profit margin. We can express a required combined ratio given the franchise value as follows.

We can see that this is the same target that we had before, except for the new term on the second line that relates to the goodwill. The presence of franchise value means insurers must demand lower combined ratios, that is, charge higher premiums than would be the case for a single period model. This is illustrated in the chart below. Each line represents one initial level of equity, the higher lines resulting from holding more equity:

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0 5 10 15 20 25 30 35 40

franchise

prem

ium

Step B4: Allow for ruin and foregoing of future profits

Our next task is to model costs associated with financial distress. This includes very many variables, involving not only the third party costs of running an insurer off, but also the costs of recapitalising in order to prevent a runoff scenario.

In this example we take a very simple approach. We assume that:

The target equity to support the second year is Ge If net assets end up above this target then the excess is distributed as dividend If the net assets end up below this target but still positive then new equity

capital is injected up to the target level If net assets become negative then the firm ceases to exist, all assets are

distributed to policyholders and the shareholders receive nothing

This single pass/fail test is a simple approximation to the curves which would apply in a more realistic example. Therefore, we have the following outcomes for the shareholder:

If (e+p)A ≥ pL then shareholder cash flow= (e+p)A - pL – Ge + G(e+f)

If (e+p)A < pL then shareholder cash flow = 0

Let us denote the shareholder return factor by S, so that the shareholder cash flow is

Our next step is to apply CAPM to S. We need to make some distributional assumptions. To make everything as simple as possible, we assume that the variables M, A, L and G are jointly normally distributed. The necessary integrals are tedious but result in closed form expressions linear in f. A graphical We discover the following

relationship between initial equity and franchise value, based on our illustrative parameters, shown in the chart below:

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0 10 20 30 40 50

equity

fran

chis

e

In practice, regulatory capital requirements may prevent insurers from taking advantage of the default option whose value is apparent for low values of equity.

In this paper we are interested in pricing and the optimisation of capital structure. The same tools are also useful for optimising investment strategy. Optimal strategies are determined by matching and tax considerations (see Cumberworth et al, 2000 or Boulton et al 2002).

Step B5: Optimise capital

We are now in a position to optimise the amount of equity in the firm. Paradoxically, it does not make sense to maximise the total value of the firm, e+f as we could do this just by starting with large initial equity. Instead, we want to maximise the value of the firm minus the amount of capital injected. This is equivalent to maximising the franchise value. The chart above shows a clear optimum at initial equity of around 28.

We can use this to set target loss ratios. Once again we must plot for various levels of starting capital, but now we can see an envelope effect. It is natural to choose the capital level which maximises franchise value for a given premium, or, equivalently, which minimises the necessary premium for a given starting franchise value. These are shown in the chart below:

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104

105

0 5 10 15 20 25 30 35 40

franchise

prem

ium

This is to be compared with the chart of parallel lines at the end of section B3. We are particularly interested in the lower envelope of this bundle of curves. This represents the relationship between franchise and premium given the optimal equity allocation. The zero equity strategy is shown here as a dotted line – we can see that this is optimal if the franchise value is low, although as previously noted regulatory constraints may prevent firms taking advantage of this strategy.

To complete this analysis, we show some premiums decomposed into their constituent loads, for a variety of loss distributions. We consider combined ratios with standard deviations of both 10% and 20%. In each case also, we consider what happens if this variability is entirely systematic or entirely diversifiable. We have developed targets by standardising the franchise value at 30. The results are shown in the bar charts below:

90

95

100

105

110

10%diversifiable

10% systematic 20%diversifiable

20% systematic

prem

ium

Impairment

Profit Margin

Tax + Agency

Systematic

Risk Free

To summarise these charts, we can see that the required premium does increase with risk, but also that the effect of diversifiable risk is much less severe than systematic risk. This is consistent both with accepted market pricing models such as CAPM and also with our observations regarding frictional costs.

Part C: Conclusions

This paper has presented a number of general arguments, backed up by a simplified financial model. Given the simplifications in our model, we need to temper any results with a dose of caution. However, we dare to draw some tentative conclusions as follows.

Frictional costs drive a wedge between the prices we see in capital markets and prices of insurance cover.

Therefore, any search for a universal pricing framework that replicates both markets is a fruitless endeavour. This paper has developed a rationale specifically for insurance pricing.

Accounting practice necessarily decomposes the value of a firm into equity, e, recognised by accounting standards, and an additional franchise value f representing intangible elements outside the scope of accounting standards.

It is common pricing practice to add a “margin for profit”. We rationalise this as the shareholders’ required return on the franchise value f.

Rational product pricing takes account of the impact of a new contract both on the equity and the franchise value of a business. Accounting, by definition, reflects only the change in equity.

A premium is accounted in full, as is the change in equity. Any change in franchise value is not recognised. Any part of the premium corresponding to the change in franchise value will emerge as a profit or loss on inception. This difference between accounting and pricing is an inevitable consequence of the decision to have an accounting framework.

In the past, various devices were used to smooth out the profit on inception. These included the invention of fictional assets (deferred acquisition costs) and fictional liabilities (market value margins).

These devices corrupt the accounting definitions of assets and liabilities, and therefore represent a cure (inconsistent recognition) that is worse than the disease (income volatility).

The accounting imperative is plain. Artificial smoothing devices such as deferred acquisition costs and market value margins deserve no place in fair value insurance accounting. Margins for systematic risk are only a legitimate accounting liability where their calibration is consistent with capital markets.

This is not to deny the importance of risk loadings in setting premiums, which should allow for effects both on accounting equity and non-accounting effects on franchise value.

Acknowledgements

The authors are grateful to many friends and colleagues who have helped us to develop our thoughts in this area or have commented on earlier drafts of this paper. Special thanks are due to David Appel, Roger Boulton, Stavros Christofides, Jon Exley, Paul Huber, Claude Methot, Hui Ming Ng, Dix Roberts, Frances Southall,

Elliot Varnell and Martin White. All opinions expressed and any remaining errors are those of the authors alone.

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