Allocating the Cost of Capital

CAS Spring Meeting

May 19-22, 2002

Robert P. Butsic

Fireman’s Fund Insurance

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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

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Optimal Capital Level

Frictional Insolvency Costs Frictional

Capital Costs

Capital Amount

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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: – A marginal change in the line mix keeps the

default measure at a constant rate:

ii LcC

LDdLD i /

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More on M-R Model

• Other inputs– Probability distribution of loss and asset values– Means, correlations and volatilities

• We solve for capital ratio

• Result:

– Beta is covariance/variance– Z is distribution-dependent

ic

Zcc ii )1(

2/ LiLi

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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

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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

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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

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General Layer Beta Properties

• Monotonic increasing with layer, with zero layer beta at lowest point layer

• Generally unbounded

0.00

5.00

10.00

15.00

20.00

25.00

0 50 100 150 200 250 300 350 400

x

Legend: RHS top to bottom

ParetoLognormalExponentialGammaNormal

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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

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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

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Further Reading

• John Hancock, Paul Huber, Pablo Koch, 2001 The economics of insurance: How insurers create value for shareholders, Swiss Re Publishinghttp://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 1999http://www.casact.org/pubs/forum/99spforum/99spftoc.htm

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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