Principles of financial economics in insurance pricing

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Principles of financial economics in insurance
pricing

• Greg Taylor
• Taylor Fry Consulting Actuaries
• University of Melbourne
• University of New South Wales

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Insurance contracts and premiums

• Insurance contact
• Provides indemnity against an uncertain cost X
• Premium
• The certain consideration to be paid for this
  contract

swap Certain cost P Uncertain cost X
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Evaluating the premium

• Equivalent to evaluating the market price P of
  the swap
• Characteristics of the cash flows to be priced
• Uncertain
• Distributed over time

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Financial economics

• Financial economics tells us how to calculate the
  value of such cash flows
• value market value in a competitive market
  fair value

P St Emt Xt where Xt cash flow at time
t mt stochastic discount factor applicable to
time t - depends on state of the economy
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Premium and profit margin

• Expenses and taxes ignored for the moment
• P St Emt Xt
• Market price is expected value of discounted
  losses
• There is no additional profit margin

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Stochastic discount factors

• Assume
• Individuals are risk-averse expected utility
  maximisers
• No arbitrage possibilities
• Basic properties of factors (Sherris, 2003)

Emt 1/RFt RFt is risk free discount
factor Emt Xt EXt / RFt covmt , Xt
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Pricing with stochastic discount factors

• Emt Xt EXt / RFt covmt , Xt
• P St Emt Xt Expected losses
• discounted risk free
• covariance term (ve or ve)
• Last term is covariance between losses and some
  broader economic variable
• Nature of that variable will depend on the
  particular model of the economy chosen

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CAPM

• One model of economy is CAPM
• Could choose others (but need to say which)
• Pricing formula then becomes
• P St Emt Xt St EXt / RXt
• where
• RXt risk adjusted t-period accumulation factor
  associated with Xt ?s1t (1 rXs)
• rXs rFs ßX ErMs rFs
• rFs risk free ROR
• rMs share market ROR
• ßX covX , rM / EX VrM single period

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Diversifiable and undiversifiable risk

• Risk adjusted ROR
• rX rF ßX ErM rF single period
• For single period
• X / EmX 1 rF ßX rM rF e
• where
• cove , rM 0
• risk free undiversifiable risk
  diversifiable risk
• Only undiversifiable risk taken into account in
  pricing

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Diversifiable and undiversifiable risk

• rXs rFs ßX ErMs rFs
• ßX covX , rM / EX VrM
• Risk adjusted RORs depend on correlation of
  losses with economic wealth (share market
  capitalisation as a proxy)
• Price will be higher (lower) than risk free
  discounted if
• ßX lt (gt)0
• i.e. losses tend to be heavy when economy
  depressed (buoyant)
• How often does this occur?

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Profit margin example

• Hypothetical insurer over single period
• Start of period
• Raises equity
• Underwrites and receives premium
• Invests premium risk free, invests equity in
  share index
• No taxes or expenses
• End of period
• Receives investment return
• Pays claims
• Distributes remaining funds to shareholders

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Profit margin example (contd)

• According to previous theory
• Premium is present value of claims at
  risk-adjusted rate
• No profit margin
• Expected shareholder profit consists of the
  earnings on equities purchased with capital raised

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Why no price for diversifiable volatility?

• Actuaries version of the question
• Why would an insurer assume volatile liabilities
  for no reward over and above that for certain
  liabilities equal in expectation?

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No price for diversifiable volatility (continued)

• This is the wrong question
• Correct question
• Why would a shareholder of an insurer assume an
  exposure to volatile liabilities for no reward
  over and above that for certain liabilities equal
  in expectation?
• Answer
• Because the shareholder can reduce the
  additional exposure to volatility to an
  arbitrarily small quantity if it is diversifiable
  (uncorrelated)

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Expenses and tax

• Expenses
• Just add to claims costs
• Does not change the argument
• Tax
• If complete imputation, shareholder situation is
  exactly as if no tax at company level
• Again no change to argument
• If incomplete imputation, shareholders suffers
  deadweight of double taxation
• Premium must be increased to compensate
  shareholder
• N.B. This is the ONLY source of profit margin

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Financial economic insurance pricing methods

• Myers-Cohn
• Values policy cash flows at risk adjusted RORs
• Premium such that NPV0
• Internal Rate of Return (IRR)
• Values shareholder cash flows at risk adjusted
  RORs
• Premium such that shareholder IRR risk adjusted
  ROR
• Risk adjustment takes account of the debt/equity
  structure of shareholding
• Two approaches can be shown to be different views
  of the same thing (Taylor, 1994)

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Pricing abuses

• Internal Rate of Return (IRR)
• Risk adjusted ROE takes account of the
  debt/equity structure of shareholding
• Lower risk asset allocation yields lower ROE
• Some applications of IRR do not observe this
  nexus
• E.g. assume equity invested risk free but expect
  ROE as if invested in shares

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Main inputs to pricing

• Risk adjusted RORs
• Myers-Cohn
• Losses
• IRR
• Assets
• Equity
• Capital base, since it affects
• Myers-Cohn
• Quantum of tax (on investment earnings of equity)
• IRR
• Net profits payable to shareholders

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Questions arising

• Is CAPM reasonable?
• What are the arguments for assigning a price to
  diversifiable risk?
• What substitute for CAPM might be made?
• Does it affect the results materially?
• What capital base should be assumed for a
  particular LoB
• What is a reasonable total capitalisation of an
  insurer?
• How should this total be allocated by line?
• Should capitalisation mean balance sheet or share
  market capitalisation?