Transcript Document

Economic Foundations
of Insurance Pricing
UNSW Actuarial Research Symposium
14 November 2003
[email protected]
PwC
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
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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
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Summary of our conclusions
(1)
• Policyholders will, in aggregate, place a
different value on a portfolio of insurance
policies than shareholders will
• 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
Shareholders' funds Q
• 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
Feasible region, with taxes and expenses
Shareholder
NPV 0
Policy
holder
NPV 0
X0
0
C0 X0
C0P
Premium P
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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
Implied minimal feasible region
99.9 %
Shareholders' funds Q
1
99.5 %
98 %
0.8
0.6
0.4
0.2
0.25
0.5
0.75
1
Premium P
1.25
1.5
1.75
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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
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Economic Valuation –
a brief refresher
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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
Cumulative probability
Call option payoff distribution
1
0.8
0.6
0.4
0.2
0
1
0
1 PricewaterhouseCoopers
2
3
Call option payoff
Asset Pricing 101
(single time-period)
Old way:
New way:
“Risk-adjusted discount rate” “Stochastic discount factor”
Cash
flow info
required
Expected payoff
Payoff distribution
E(x)
x
Discount
factor
Deterministic – but different for
each asset:
1 / (1 + rj)
Stochastic – but prices all
assets of interest:
m
V = E(x) / (1 + rj)
V = E(m x)
Pricing
formula
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The risk premium is a
covariance – with consumption
Economic
derivation says…
Definition of
covariance says…
u' (ct*)
m 
u' (c0*)
V  E (m x )
 E (m) E ( x )  cov(m, x )
E ( x)

 cov(m, x )
Rf
u’ – marginal utility; c* - optimal consumption; Rf – gross risk-free return
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A nested set of equilibrium
models
Model
Stochastic discount factor
Buhlmann’s model (1980)
Wang’s specialisation of
Buhlmann’s model (2003)
Assumptions
Discount factor
Expected utility, smooth
utility function u
m = A u’(c*)
Exponential utility, closed
market
m = A exp(- c),
c* = agent’s optimal
consumption
c = total consumption
Total consumption normal, mx = Ax exp(- x h-1(x))
normal copula with assets
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)
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The Insurance Surplus
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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)
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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
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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
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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
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Position in the feasible range is
determined by competition
NPVS = 0
NPVP > 0
NPVS > 0
NPVP > 0
Low
Premium
Perfect
Competition
Typical
Market ?
NPVS > 0
NPVP = 0
High
Premium
Monopoly
• 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
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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)
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Implications for a
corporate insurance firm
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A simple limited-liability
insurance firm
Cash In
Time
0
Cash Out
Shareholders’ Funds
Q
Expenses
X0
Premium
P
Investments
A0
Cash In
Time
T
Investments
Cash Out
AT = (P + Q - X0) * Rf
Claims paid
Taxes
Equity cash flow
LT = min(CT, AT)
GT = …
AT – LT – GT
• 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
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Market value of the claim
portfolio (unlimited liability)
Discount factor (m) indicates how
much weight the market places on
each loss outcome
Weight
Probability Density
Payout distribution ((CT))
describes what we could lose
Market
Discount
Factor
0
0
Aggregate Claim C T
Aggregate Claim C T
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
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We can value the liabilities as
derivatives on the claims
Claims Distribution and Payout Function
LT = min(CT, AT)
Probability Density
Amount Paid LT
The payout is
capped by the
level of asset
backing:
AT
0
AT
Aggregate Claim C T
...and then calculate the NPV’s for stakeholders, as
functions of the initial funds contributed:
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
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Shareholders' funds Q
We can determine the impact of
variation in asset backing…
Both premiums and
shareholders’ funds
provide asset backing:
Shareholders' funds Q
Feasible region , with no taxes or expenses
Shareholder
NPV
0
Policy
Probability of sufficiency
1.2
contours 85% to 99%
1
0.8
0.6
0.4
0.2
0
0
0.5
1
Premium P
1.5
2
Insurance surplus is larger
when asset backing is
higher
holder
NPV 0
0
C0
Premium P
C0
P
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…and expenses and taxes
Expenses eat away initial
funding, and require a
certain minimum feasible
level of asset backing
Shareholders ' funds Q
Feasible region, with expenses but no taxes
Shareholder
NPV 0
Policy
holder
NPV 0
X0
0
C 0 X0
Premium P
C 0P
Taxes eat away profits,
and force a certain
maximum feasible level of
asset backing
Shareholders ' funds Q
Feasible region, with taxes but no expenses
Shareholder
NPV 0
Policy
holder
NPV 0
0
C 0PricewaterhouseCoopers
C0 P
Premium P
Putting it all together gives the
structure of the feasible region
Shareholders' funds Q
Feasible region, with taxes and expenses
• 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
Shareholder
NPV 0
Policy
holder
NPV 0
• Policyholders require a
certain strength of asset
backing before they will pay
for insurance
X0
0
C0 X0
C0P
Premium P
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Relationship with the MyersCohn approach
Feasible region, with taxes and expenses
Shareholders' funds Q
• The Myers-Cohn principle
yields the lowest premium in
the feasible region, for each
level of shareholders’ funds
Shareholder
NPV 0
Policy
holder
NPV 0
X0
0
C0 X0
C0P
Premium P
• 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
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Next step: attempt to calibrate
the model to observed prices
Implied minimal feasible region
• For example, assume:
• “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)
1
Shareholders' funds Q
• Log-normal claims with
Coeff. of var. 24%,
expenses 17%, tax 25%
99.9 %
99.5 %
98 %
0.8
0.6
0.4
0.2
0.25
0.5
0.75
1
Premium P
1.25
1.5
1.75
• Then we can infer the “minimal”
aggregate discount factor that the
policyholders are using
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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 helpfulPricewaterhouseCoopers
references, and
Insurance Australia Group for sponsoring this research.