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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 > b) r(Xi) = E(Xi – E(Xi) | Y > 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$9050% Bad$120$10$13049.5% Ugly$120$300$4200.5% 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 Don’t 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” doesn’t matter, once you’re buried, doesn’t 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.”

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