Financial insurance, risk capital and option pricing

1
Financial insurance, risk capital and option
pricing

• Lecture 6 - Economics of insurance
• EOCN6053 Selected topic in financial economics
• Raymond Yeung, PhD
• Honorary Assistant Professor
• 29 March 2007

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Background

• Insurance company is a specific form of financial
  intermediaries. Financial firms receive deposit
  or underwrite contingency claim. Similar to
  banks, its profit derives from creating a spread
  between liability and asset holding.
• But such activities are not riskless. The insured
  activities are subject to randomness. Investment
  income from the portfolio is subject to market
  risk. There is also default risk from the company
  management or banks lenders.
• Unlike manufacturing, one can set up a bank or
  insurance company with little initial
  investment upfront. All about whether people
  trust you.

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Background

• Prudential regulator requires financial firms for
  certain amount of regulatory capital to ensure
  they can meet the gross liability safely.
  Insurance companies or banks require working
  capital to meet their operational requirement.
  Customers preference is sensitive to the credit
  standing of the firm.
• The firm profitability is therefore sensitive to
  the cost of capital, which, in turns, depends on
  firms overall risk profile. In a competitive
  financial market, the concept of risk capital
  determines winners and losers.
• Other things being equal, an AAA-rated firm earns
  more than an A-rated firm

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Definition of risk capital

• Risk capital is the smallest amount that can be
  invested to insure the value of the firms net
  assets against a loss in value relative to the
  risk-free investment of those net assets
• Risk capital is equal to the value of providing
  full insurance of the companys net asset
• Such an insurance can take three forms or
  combination of them
• Purchase explicit insurance to ensure asset
  return
• Shareholders guarantees for liability payment
• Sell liability at discounted prices

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Risk capital balance sheet

• A newly established bank can underwrite a
  bridging loan 100 and earns 20. To do so, it
  issues a guarantee note to outside investor
• Suppose a bank can buy insurance from a
  AAAA-rated bond insurer premium 5 and
  guarantee the note a default-free value of 110
• Risky note default-free note note insuance

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Risk capital by seniority

• Risk capital can be provided by shareholder
  (implicit gurantee), debtholder (ordinary
  creditor) and customer liability holders

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Risk capital and profitability

• Without accounting for risk capital,
  profitability would be overstated

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Insurance and option theory

• Financial intermediary can buy an insurance to
  protect its asset it is the right to protect
  from excess claim
• For banks, such insurance can be deposit
  insurance, protecting depositors from risk of
  bankruptcy For insurance companies, they can
  purchase reinsurance or shift the risk to the
  capital market through insurance-linked
  securities
• This risk-bearer of the last resort needs to know
  the cost of provision for guarantee as they also
  need to allocate their own capital or social
  resources

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Put option Default risk guarantee

• An European put option on a common stock is the
  right for the option owner to sell the stock at a
  pre-determined strike price on expiry date
• In expiration date, the option is worthless if
  SgtE. Otherwise, the value of the option is E-S
• The value of option can be written as
• P(T) is the price of option with T, the length of
  time prior to expiratory

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Black-Scholes formula

• Assume that the stock price is distributed
  lognormally, the famous Black-Scholes formula for
  put option is
• where

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Application of option pricing to financial firm
valuation

• Financial firm can be viewed as issuing a promise
  to pay B, provided with the value of asset V. The
  value of equity must be V-B
• The value of debt can be written as MinV,B and
  the value of equity is Max0,V-B
• Now if there is a guarantor to provide full
  insurance up to B, it provides an inflow to the
  firm -min0,V-B or max0,B-V

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Value of guarantee

• The put option can be computed by the formula
• The market value of the risky debt with no
  guarantee plus the value of guarantee is equal to
  riskless debt (note R(T) may change for t/T)

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Value of guarantee

• Current value of riskless deposit can be written
  as
• One can show that

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Risk capital allocation

• Risk capital needs to be allocated to multiple
  business units
• How do we compute risk capital?

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Value of the balance sheet

• At time T, both liability Lt and asset At are
  equal in value with zero return r 0
• The shortfall in net assets at t 0 is
• The default-free financing of the net assets must
  pay, at time T,
• Substitute them into Black-Scholes F(.)

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Risk capital computation

• Assume the standard deviation of individual line
  of business
• The approximation is close to the computed values
  using Black-Scholes formula

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Risk capital computation

• Apply the same calculation to joint business
  units

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Diversified portfolio

• If PC and LH are correlated and they are
  uncorrelated with other, the required risk
  capital for combined business is less.

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Marginal capital of combined business

• The i th business unit has a guarantee option
• which is a convex function
• By Jensens inequalities
• The sum of risk capital of standalone business
  units exceeds the risk capital of combined
  business a portfolio of options always returns
  at least as much as the corresponding option of a
  portfolio of the underlying securities

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Marginal capital of combined business

• The marginal capital for unit i is
• Note that
• By Jensens inequalities

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Marginal capital of combined business

• As f(0)0, we can rewrite
• It follows that

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Implications

• Insurance companies or banks with more business
  units enjoy economy of scale, i.e. externality
  from risk sharing by other business unit
• MA of financial firms increases their scale
  internal risk capital and enjoy lower marginal
  cost of risk capital. They should get a better
  rated
• To evaluate the profitability of new business,
  financial firms should look at marginal risk
  capital requirement instead of allocating total
  risk capital across units

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References

• Merton RC, Perold AF, Theory of risk capital in
  financial firms, in Chew, D, The New Corporate
  Finance, McGraw Hill Press 1993
• Merton RC (1977), An analytic derivation of the
  cost of deposit insurance and loan guarantees,
  Journal of Banking and Finance, vol 1, p.3-11