Economic Foundations of Insurance Pricing UNSW Actuarial Research Symposium 14 November 2003 [email protected].

Economic Foundations of Insurance Pricing

UNSW Actuarial Research Symposium

14 November 2003

[email protected]

PricewaterhouseCoopers

Introduction

• The economic foundations of insurance pricing are being debated actively at present– Regulators and insurance companies are seeking a sound

economic basis to guide pricing decisions

• Myers and Cohn (1981) applied the Capital Asset Pricing Model (CAPM) to an insurance firm

• Many insurance professionals are uncomfortable with the conclusions of the Myers-Cohn approach, as it implies premiums should be set at levels below those considered viable


Outline of this presentation

• We have re-visited the foundations of insurance pricing, using modern economic valuation methods (SDF approach)– Conclusions intuitive to insurance practitioners, within a

rigorous economic framework

• In this presentation, we:– review the Myers-Cohn approach– review economic valuation methods– outline the approach we have taken– summarise our key conceptual findings, and – illustrate how our approach might be applied to guide the

setting of insurance premiums


Summary of our conclusions (1)

• Policyholders will, in aggregate, place a different value on a portfolio of insurance policies than shareholders will

• This value difference – the insurance surplus – gives rise to a range of premiums that policyholders and shareholders will be happy with – the feasible range of premiums

• Our approach, in contrast to CAPM-based approaches, allows the relationship between capital strength and premium to be determined, leading to the determination of the set of feasible combinations of these factors - the feasible region

C0PC0 X0

PremiumP0

X0

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Feasible region, with taxes and expenses

ShareholderNPV 0

PolicyholderNPV 0

C0PC0 X0

PremiumP0

X0

sredloherahS

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Summary of our conclusions (2)

• The feasible region exists due to consumers’ assumed aversion to the insured risks, and the fact that these risks cannot be offset by traded securities – capital market incompleteness

• Consequently, “fair” premiums cannot be determined with reference to capital markets alone – pricing information from consumer insurance markets must be considered also

0.25 0.5 0.75 1 1.25 1.5 1.75PremiumP

0.2

0.4

0.6

0.8

1

sredloherahS

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Implied minimal feasible region

99.9%

99.5%98%


What did Myers and Cohn do ?

1. Discussed premiums with reference to the values of the components of the insurer’s balance sheet

2. Proposed a “fair premium principle”:The premium that makes entering into the insurance contract NPV-zero for shareholders

3. Employed CAPM to calculate the values of the components of the insurer’s balance sheet

Economic Valuation – a brief refresher


Basic terminology and notation

• An asset is defined by the payoff (x) it provides to its owner

• The payoff typically:– occurs in the future– is uncertain (a random variable)

• The price or value (V) of an asset is the amount of cash we would pay today for the right to the asset’s risky future payoff

e.g. The call option, with payoff distribution shown here, has a value of $0.09

1 0 1 2 3Call option payoff

0

0.2

0.4

0.6

0.8

1

evitalumu

Cytilibaborp

Call option payoff distribution


Asset Pricing 101(single time-period)

Old way:

“Risk-adjusted discount rate”

New way:

“Stochastic discount factor”

Cash flow info required

Expected payoff

E(x)

Payoff distribution

x

Discount factor

Deterministic – but different for each asset:1 / (1 + rj)

Stochastic – but prices all assets of interest:

m

Pricing formula V = E(x) / (1 + rj) V = E(m x)


The risk premium is a covariance – with consumption

*)('

*)('

0cu

cum t

),cov()(

),cov()()(

)(

xmR

xE

xmxEmE

xmEV

f

Economic derivation says…

Definition of covariance says…

u’ – marginal utility; c* - optimal consumption; Rf – gross risk-free return


A nested set of equilibrium models

Model Assumptions Discount factor

Stochastic discount factor Expected utility, smooth utility function u

m = A u’(c*)

c* = agent’s optimal consumption

Buhlmann’s model (1980) Exponential utility, closed market

m = A exp(- c),

c = total consumption

Wang’s specialisation of Buhlmann’s model (2003)

Total consumption normal, normal copula with assets

mx = Ax exp(- x h-1(x))

x = h(z), z unit normal,

x = z ,

= (E(Rc) – Rf) / (Rc)

Capital Asset Pricing Model (1965-ish)

Asset payoffs normal m = Ax exp(- x (x-x)/x)

The Insurance Surplus


The shareholders’ perspective

• Insurance companies take risk from policyholders for a fee, and offer the aggregate risk to shareholders– Any risk premium shareholders place on this aggregate risk will

be based on its contribution to the variance of their entire asset portfolio (as a proxy for their consumption)


Consumers are willing to pay a risk premium for insurance

• Some aspects of individuals’ behaviour are not explained by economic valuation models:– We observe individuals holding concentrations of wealth in

particular assets, like the family home– They also have exposure to concentrations of liability, such as

obligations when injury is caused to another person while driving

• This gives rise to risks that can’t be offset in the securities market– There can in no practical sense be traded securities that replicate

the payoff of a particular individual’s house burning down, for example;

• Accordingly, individuals will be willing to pay a risk premium for instruments that mitigate such risks – such as insurance


The insurance surplus

• To understand the impact of the different valuation perspectives of policyholders and shareholders, we introduce the notion of the insurance surplus– This is defined as the difference between the sum of the values

placed by consumers upon a portfolio of insurance policies, and the market value of this portfolio

• It is clear that in practice there must be a positive surplus at the raw liability level – that’s where the expenses and taxes are paid from– Without a surplus, the insurance industry would not exist

• The existence of a positive surplus implies the existence of a range of premiums that consumers and shareholders will be happy with


The feasible range of premiums

• Myers and Cohn defined the “fair premium” to be the premium that makes entering into an insurance contract an NPV-zero proposition for shareholders– They claimed that setting premiums any higher than this would

involve a wealth transfer from policyholders to shareholders– This is not the case – both shareholders and policyholders can

(and must) have their expected utility improved through entering into an insurance contract (a win-win situation)

• We introduce the notion of the feasible range of insurance premiums– This range is defined as the set of premiums that make entering

into the insurance contract NPV-positive for both shareholders and policyholders


Position in the feasible range is determined by competition

• In a “typical” or “realistic” un-regulated insurance market, we would expect premiums to be set in the interior of the feasible range, as there are barriers to entry, such as licensing, capital requirements, systems and skills, and the scale necessary to achieve diversification

• If a regulator has aims other than price minimisation (e.g. stability, accessibility), prices above the lower end will need to be allowed

Low Premium

High Premium

NPVS = 0NPVP > 0

NPVS > 0NPVP = 0

NPVS > 0NPVP > 0

Perfect Competition

MonopolyTypical Market ?


Insurance surplus, mathematically...

• Consumer k faces a set of payoffs Xk, employs a discount factor mk

• All consumers can trade a set of assets X (the securities market)– there exists a market discount factor m– each mk agrees with m on X– mk may be decomposed as mk = m + mk

• Consumer k’s insurance policy has a payoff wk

• Insurance surplus is: = k E(mk wk) – E(m kwk)

• Can show that = k cov(mk, wk) (if Rf traded)

• So insurance surplus will be positive if:– The securities market is incomplete (wk

!= 0), and– Consumers are averse to the non-traded part of their insured risk

(cov(mk, wk) > 0)

Implications for a corporate insurance firm


A simple limited-liability insurance firm

Cash In Cash Out

Shareholders’ Funds Q Expenses X0

Premium P Investments A0

Cash In Cash Out

Investments AT = (P + Q - X0) * Rf Claims paid LT = min(CT, AT)

Taxes GT = …

Equity cash flow AT – LT – GT

Time 0

Time T

• The firm will take on a portfolio of policies to be resolved at time T– The underlying uncertainty is the total amount claimed, denoted CT

– We’ll assume an insurance surplus would exist if claims were guaranteed to be paid


Market value of the claim portfolio (unlimited liability)

Discount factor (m) indicates how much weight the market places on

each loss outcome

Payout distribution ((CT)) describes what we could lose

Value = C0 = E(m CT ) = m(c) c (c) dc

The discount factor shown here places greater weighting on high payouts than low payouts, which makes the liability value bigger

0Aggregate ClaimCT

ytilibaborP

ytisneD

0Aggregate ClaimCT

thgieW

MarketDiscountFactor


0 ATAggregate ClaimCT

AT

tnuom

Adia

PL T

Claims Distribution and Payout Function

Probability Density

We can value the liabilities as derivatives on the claims

The payout is capped by the level of asset

backing:

LT = min(CT, AT)

e.g. NPV from shareholders’ point of view (ignoring tax) is:

NPVS(P, Q)= m(c) max(0, AT(P,Q) –c) (c) dc – Q

...and then calculate the NPV’s for stakeholders, as functions of the initial funds contributed:

PricewaterhouseCoopersC0PC0

PremiumP0

sredloherahS

'sdnufQ Feasible region , with no taxes or expenses

ShareholderNPV 0

PolicyholderNPV 0

C0PC0

PremiumP0

sredloherahS

'sdnufQ Feasible region , with no taxes or expenses

We can determine the impact of variation in asset backing…

Both premiums and shareholders’ funds

provide asset backing:

Insurance surplus is larger when asset backing is

higher

0 0.5 1 1.5 2PremiumP0

0.2

0.4

0.6

0.8

1

1.2

sredloherahS

'sdnufQ Probability of sufficiency contours 85% to 99%


…and expenses and taxes

Expenses eat away initial funding, and require a

certain minimum feasible level of asset backing

Taxes eat away profits, and force a certain

maximum feasible level of asset backing

C0PC0

PremiumP0

sredloherahS

'sdnufQ Feasibleregion, with taxesbut no expenses

ShareholderNPV 0

PolicyholderNPV 0

C0PC0

PremiumP0

sredloherahS

'sdnufQ Feasibleregion, with taxesbut no expenses

C0PC0 X0

PremiumP0

X0

sredloherahS

'sdnufQ Feasibleregion, with expensesbut no taxes

ShareholderNPV 0

PolicyholderNPV 0

C0PC0 X0

PremiumP0

X0

sredloherahS

'sdnufQ Feasibleregion, with expensesbut no taxes


Putting it all together gives the structure of the feasible region

• Taxes and expenses diminish the surplus, and can wipe it out– the surplus is wiped out

by tax if the firm is too strongly capitalised

• Policyholders require a certain strength of asset backing before they will pay for insurance

C0PC0 X0

PremiumP0

X0

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ShareholderNPV 0

PolicyholderNPV 0

C0PC0 X0

PremiumP0

X0

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'sdnufQ



Relationship with the Myers-Cohn approach

• The Myers-Cohn principle yields the lowest premium in the feasible region, for each level of shareholders’ funds

• Myers-Cohn / CAPM calculation method only applies under assumptions of unlimited liability and normally-distributed portfolio payoff– in which case it would

produce the diagonal dashed line

C0PC0 X0

PremiumP0

X0

sredloherahS

'sdnufQ


ShareholderNPV 0

PolicyholderNPV 0

C0PC0 X0

PremiumP0

X0

sredloherahS

'sdnufQ



Next step: attempt to calibrate the model to observed prices

• For example, assume:

• Log-normal claims with Coeff. of var. 24%, expenses 17%, tax 25%

• “zero-beta” liability

• Power-law discount factor

• …and suppose we observe these combinations of sufficiency and premium:

(99.9%, 1.48), (99.5%, 1.43), (98.0%, 1.41)

• Then we can infer the “minimal” aggregate discount factor that the policyholders are using

0.25 0.5 0.75 1 1.25 1.5 1.75PremiumP

0.2

0.4

0.6

0.8

1

sredloherahS

'sdnufQ

Implied minimal feasible region

99.9%

99.5%98%


Summary

• We have argued that policyholders will, in aggregate, place a different value on a portfolio of insurance policies than shareholders will

• This value difference – the insurance surplus – gives rise to a range of premiums that policyholders and shareholders will be happy with

• We have examined how this surplus is affected by the structure of a corporate insurance firm– Our model has produced insights into the workings of such firms,

showing endogenously, for example, that asset backing should be high, but not too high

• “Fair” premiums cannot be determined with reference to capital markets alone – pricing information from consumer insurance markets must be considered also

The author would like to thank Tim Jenkins for many helpful discussions during the development of this paper, Tony Coleman for suggesting the area of research and providing helpful references, and Insurance Australia Group for sponsoring this research.

Economic Foundations of Insurance Pricing UNSW Actuarial Research Symposium 14 November 2003 [email protected].

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