Why Can Financial Firms Charge for Diversifiable Risk?
Andrew Smith, Ian Moran and David Walczak
DRAFT Discussion Paper
Comments welcome
Please do not quote without our permission
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
Ian Moran
David Walczak
B&W Deloitte,
Horizon House
28 Upper High Street
Epsom, Surrey
KT18 7LJ
Great Britain.
tel +44 1372 824811
e: andrewdsmith8@bw-deloitte.com
Deloitte & Touche
400 One Financial Plaza
Minneapolis
MN 55402
USA
tel +1 612 397 4509 ,
e: dwalczak@deloitte.com
tel +44 1372 824143
e: imoran@bw-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 market
Step 2: The Myers – Cohn model and frictional costs of investing
Step 3: Accounting equity and franchise value
Step 4: Costs of financial impairment
Step 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 ringfencing 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 multihorizon 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. Grossingup 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 wellintentioned 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
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)
Recognition as a liability
Recognise as they reflect an obligation to
3rd party
Recognise as part of the cost the market
would pay
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.
Like agency costs.
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 rate
M = 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 premium
G = 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
1.05
M
1.10
A
1.03
L
0.98
G
1.00
and the following variance-covariance matrix:
M
0.0400
0.0100
-0.0010
0.0200
M
A
L
G
A
0.0100
0.0050
-0.0013
0.0050
L
-0.0100
-0.0013
0.0169
-0.0050
G
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
Cov ( S , M )
E(M )  R
E( S )  R 
Var ( M )
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.
(e  p)  A  p L
(e  p) AM  p LM
 M  R
R
e
e MM
For step B1, we assume the asset returns already solve the CAPM, so that
 AM
 M  R
 MM
This would require us to set A = 1.0625. It follows by subtraction that

 L  R  LM  M  R
 MM
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.
A  R 
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:




e 
 L  R  LM  M  R 1   R  AM  M  R  A  .
 MM
p 
 MM


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(AL)+fG.
Once again, we assume the capital markets use CAPM to value the firm. This means
the total shareholder return must satisfy CAPM:
e A  p(  A   L )  f G
e  p( AM   LM )  f GM
 M  R
 R  AM
e f
(e  f ) MM
Eliminating f, we see that:
e  p( AM   LM )  f GM
 M  R
e A  p(  A   L )  f G  (e  f ) R  AM
 MM
We can therefore establish the franchise value as follows:






(e  p )   A  R  AM  M  R  p   L  R  LM  M  R
 MM
 MM




f 
 GM
 M  R   G
R
 MM
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.




e 
 L  R  LM  M  R  1    R  AM  M  R   A 
 MM
p 
 MM



 GM
f 
 M  R   G 
R 
p
 MM

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:

105
premium
104
103
102
101
100
0
5
10
15
20
25
30
35
40
franchise
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
eA  p( A  L)  fG (e  p) A  pL
(e  f ) S  
0
(e  p) A  pL

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:
20
franchise
18
16
14
12
10
0
10
20
30
40
50
equity
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:
105
premium
104
103
102
101
100
0
5
10
15
20
25
30
35
40
franchise
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:
110
premium
105
Impairment
Profit Margin
Tax + Agency
Systematic
Risk Free
100
95
90
10%
diversifiable
10% systematic
20%
diversifiable
20% systematic
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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