Allocating the Cost of Capital
CAS Spring Meeting
May 19-22, 2002
Robert P. Butsic
Fireman’s Fund Insurance
Why is Capital Necessary?
• The answer is not obvious:
– We can’t have enough capital to eliminate
insurance insolvency
– So why not have minimal capital and let guaranty
funds protect policyholders?
• Answer is: frictional insolvency costs
– Additional system costs due to insolvency:
– Legal fees, market disruption, extra claims
handling costs
2
Optimal Capital Level
Frictional
Insolvency
Costs
Frictional
Capital
Costs
Capital Amount
3
What is the Cost of Capital?
• Investor supplies capital and expects
return commensurate with risk to which
the capital is exposed
• This return is the cost of capital
– Traditional view of insurance management
– But, look from the modern finance perspective
4
Base Cost of Capital
• A: Investor invests capital in a levered
fund
– borrow cash and invest all assets
– identical to the insurer’s assets
• Investor’s expected return in A is called
Base Cost of Capital
• B: Insurer has same balance sheet
– But insurer has higher COC
5
Frictional Costs of Capital
• The insurance mechanism will introduce
extra costs
– Government, regulation and organization
– Illiquid nature of insurance liability
– Information asymmetry (opaqueness)
• These are frictional costs of capital
– key one is double taxation
– Most easily quantified
6
Double Taxation Example
• Investor can directly invest in security
with 10% return, but invests in ABC
Insurance, who puts money in same
security
• ABC gets 10% return, pays 35% tax
and gives 6.5% net back to investor
• A losing deal unless PH can make up
the difference
7
Other Frictional Costs
• Regulatory costs
– Capital can’t be easily moved, so investment is
illiquid
• Agency costs
– Misalignment of owners’ and managers’ interests
(Enron a classic example)
• Financial distress costs
– Legal fees
– Distressed sale of assets
8
Financial Pricing Model
• Fair premium = total present value of
– loss & LAE (including risk margin)
– UW expenses
– Frictional capital costs
• Note that traditional (base) cost of
capital is embedded in risk margin
9
Risk Margin and COC Example
• Assumptions
– Fair premium is $1000, paid up front, $1040 loss
paid in one year
– Risk-free rate of 6%, $500 of capital required
– No frictional COC, taxes or expenses
• Calculation
– Initial assets of $1500 grow to $1590, leaving
$550 for a 10% return (COC)
– Risk margin is $18.87 = 1000 - 1040/1.06
10
RM and COC Example, Cont.
• Which comes first, the RM or the COC?
– Each implies the other
• In determining a fair premium, it must
be the risk margin:
– Products have different levels of risk
– What COC should a riskless coverage have?
• Thus, the COC is not fixed for an
insurer -- it varies by product
11
Allocating Capital Costs
• For pricing or performance
measurement, must allocate capital
costs to product
• If we know the RM, then we need to
allocate the frictional COC
• If we don’t know the RM, and use a
COC pricing model, then we allocate
both frictional and base COC
12
Capital Allocation
• In order to assess capital costs by
product, we first need to allocate capital
to product
• There are many methods
– Lots of ad hoc models
– Very few economically sound models
• One of them is the general Myers-Read
method
13
Myers-Read Method
• Uses the expected default (PH deficit)
as a solvency measure
– Others, such as default (ruin) probability will also
work (and may be better)
• Major assumptions
– predetermined capital ratios exist: C   ci Li
– A marginal change in the line mix keeps the
default measure at a constant rate:
D Li  d  D / L
14
More on M-R Model
• Other inputs
– Probability distribution of loss and asset values
– Means, correlations and volatilities
• We solve for capital ratio ci
• Result: ci  c  (  i  1) Z
– Beta is covariance/variance
– Z is distribution-dependent
 i   iL /  L2
15
Loss Beta
• Relevant risk measure for capital
allocation is loss beta
– Volatility, correlation with portfolio and weight
determine loss beta
– Strong parallel with asset pricing, CAPM, portfolio
optimization
• Capital allocation is exact; no overlap
– No order dependency
16
Numerical Example
Table 1.1
Loss Beta and Capital Allocation for Numerical Example
Liability
Loss
Loss
Capital/
Value
CV
Beta
Liability
Capital
Line 1
500
0.2000
0.8463
0.3957
197.87
Line 2
400
0.3000
1.3029
0.7055
282.19
Line 3
100
0.5000
0.5568
0.1993
19.93
Total
1000
0.2119
1.0000
0.5000
500.00
17
Application to Coverage Layers
• For policy/treaty, capital allocation to
layer depends on covariance of layer
with that of unlimited loss
• Layer covariance depends only on loss
size distribution
• Layer beta and capital/loss increase
with limits
18
General Layer Beta Properties
• Monotonic increasing with layer, with
zero layer beta at lowest point layer
• Generally unbounded
25.00
Legend: RHS
top to bottom
20.00
15.00
Pareto
Lognormal
Exponential
Gamma
Normal
10.00
5.00
0.00
0
50
100
150
200
250
300
350
400
x
19
Practical Applications
• Best measure of capital is economic
(fair) value
• As an approximation, capital is
proportional to the loss/layer beta
• For allocating a company’s capital, the
relevant time horizon is one year
– Allocation base is reserves plus next year’s AY
incurred losses
20
Summary
• Importance of frictional costs in theory
of solvency and capital allocation
• Myers-Read method is economically
sound, with friendly (to user) results
• We’ve still got a long road ahead before
common agreement on capital
allocation methodology
21
Further Reading
• John Hancock, Paul Huber, Pablo Koch, 2001
The economics of insurance: How insurers create value
for shareholders, Swiss Re Publishing
http://www.swissre.com/
• Myers, Read, 2001, Capital Allocation for
Insurance Companies, Journal of Risk and
Insurance, 68:4, 545-580
• Butsic, 1999, Capital Allocation for
Property Liability Insurers: A
Catastrophe Reinsurance Application.
Casualty Actuarial Society Forum, Fall
1999
22
http://www.casact.org/pubs/forum/99spforu
Further Reading II
• Butsic, Cummins, Derrig, Phillips,
2000,
The Risk Premium Project, Phase I and
II Report,
CAS Website,
http://casact.org/cotor/rppreport.pdf
23