Perspectives on Capital Allocation

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Perspectives on Capital Allocation

• Trent Vaughn
• Republic Insurance Group

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Some Brief Notation (borrowed from VMK paper)

• Y ?Xi are (generally) aggregate losses
• P(Y) is the risk measure
• r(Xi) is the allocation of risk to component
• P(Y) ?r(Xi) is an additive allocation rule

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Examples

• XTVaR with cutoff point b
• P(Y) E(Y E(Y) Y gt b)
• r(Xi) E(Xi E(Xi) Y gt b)
• Two common choices for cutoff point
• Variance Method
• P(Y) Var(Y)
• r(Xi) Cov(Xi,Y)
• Often called (incorrectly) the Capm Method

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Simple Thought Experiment
State of World APD Loss Cat Loss Total Loss State Prob.
Good 80 10 90 50
Bad 120 10 130 49.5
Ugly 120 300 420 0.5
Exp. Value 100 11.45 111.45
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Simple Thought Experiment (continued)

• Resulting Prices by Line at 10 Target ROE on
  150 of Supporting Capital (One-Year Horizon w/
  5 Investment Return)
• XTVaR w/ Insolvency Cutoff Point
• APD 95.70 Cat 17.59
• XTVaR w/ Capital Consumption Cutoff Point
• APD 101.90 Cat 11.39
• Variance Method
• APD 98.74 Cat 14.55

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RMK Methods

• Dont require a capital allocation, but still
  require a risk measure (aka riskiness leverage
  ratio, capital call function, etc)
• Example from Mango paper
• Relationship to utility function
• Question Whose risk preferences are we
  measuring shareholders or policyholders?

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Policyholder vs. Shareholder Risk Preferences

• Meyers Only risk that matters to policyholders
  is insurer insolvency
• Probability of insolvency matters
• For insolvency scenarios, degree of insolvency
  (or policyholder deficit) also matters
• Shareholders have different concerns
• Distinguish between various solvency scenarios,
  e.g. does actual return fall short of expected
  return?
• For insolvency scenarios, degree doesnt
  matter, once youre buried, doesnt matter how
  much dirt on top (Kreps)

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Actuarial Allocation Methods

• Premium Discounted (at risk-free) Expected Loss
  Capital Cost x Allocated Capital (e.g. Kreps,
  PCAS 1990)
• For many methods, Allocated Capital is based on
  Shareholder Risk Measure
• Drawbacks
• Both PH and SH risk preferences rolled into a
  single risk measure
• Very difficult to incorporate Shareholder
  portfolio diversification

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Financial Allocation Methods

• Premium Discounted (at Risk-Adjusted Rate)
  Expected Loss Capital Cost x Allocated
  Capital
• Risk-Adjusted Rate reflects some shareholder
  diversification (in practice, risk-free rate is
  often used)
• Zanjani Capital allocation rule is driven by
  consumer attitudes toward risk.