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Perspectives on Capital Allocation

Perspectives on Capital Allocation. Trent Vaughn Republic Insurance Group. 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. Examples.

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Perspectives on Capital Allocation

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  1. Perspectives on Capital Allocation Trent Vaughn Republic Insurance Group

  2. 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

  3. 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”

  4. Simple Thought Experiment

  5. 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

  6. 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?

  7. 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)

  8. 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

  9. 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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