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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
• 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”
Simple Thought Experiment
State of
World
Good
APD
Loss
$80
Cat
Loss
$10
Total
Loss
$90
State
Prob.
50%
Bad
$120
$10
$130
49.5%
Ugly
$120
$300
$420
0.5%
Exp.
Value
$100
$11.45
$111.45
Simple Thought Experiment
(continued)
• Resulting Prices by Line at 10% Target
ROE on $150 of Supporting Capital (OneYear 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
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?
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)
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
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.”