Myers Read Myers Read Capital Allocation Capital Allocation DiscussionDiscussionCAS Marco Island 2003CAS Marco Island 2003

Gary G VenterGary G Venter

Myers Read Myers Read Capital Allocation Capital Allocation DiscussionDiscussionCAS Marco Island 2003CAS Marco Island 2003

Gary G VenterGary G Venter

Mathematical SetupMathematical SetupMathematical SetupMathematical Setup c iLi = cL,

c iLi is the capital for the ith policy with expected losses Li , and cL is total capital

D/L is value of default put related to losses M-R require that D/Li = D/L D/L is held constant by adjusting capital So ci’s are sought that will change the allocated

capital ciLi by the amount needed to keep D/L constant when there is a small change in Li

The Answer The Answer ((assuming assets and losses are assuming assets and losses are uncorrelated)uncorrelated)

The Answer The Answer ((assuming assets and losses are assuming assets and losses are uncorrelated)uncorrelated)

c i = c + ( i – 1)Z, where Z does not vary with i and i is a correlation measure for Li with L

Several examples in written discussion For instance, a constant risk generally

gets a negative capital charge– It does not add risk but both adds stability

and accepts risk of non-payment

Main ProblemsMain ProblemsMain ProblemsMain Problems1. Some losses pay in the first year, others

take many years to pay– So what is the time period for the option?

2. Don’t know the extreme tail of the aggregate distribution so don’t know what distribution to use for evaluating the option

3. Seems equally valid with other methods that also are completely additive (co-measures)

4. Aimed at allocating frictional costs of holding capital, but tends to be misused as denominator of return on capital calculation by line

1. Time Period1. Time Period1. Time Period1. Time Period Some losses on six month policies pay

within a few months Others take years to pay Current default would affect unpaid

losses from many prior years Could do a sum of three-month

increment default options over 20 years

Formula adapted for volatility smile?

2. Measuring Default 2. Measuring Default ProbabilitiesProbabilities2. Measuring Default 2. Measuring Default ProbabilitiesProbabilities

Usually fit distributions to loss frequency and severity, combine into an aggregate distribution, and extended way out into the tail well beyond the data where the distribution is not known

Actual causes of default are more like court changes in the meaning of contracts, changes in the probabilities of catastrophes, employee fraud, regulatory underpricing, reserve changes showing years of underpricing, asset market collapse, jumps in inflation, etc.

Much of that is difficult to tie to given losses or policies Risk measurements based on default are problematic Partial loss of surplus more relevant due to value of

renewal book as well as easier to calculate

3. Definition of Co-3. Definition of Co-MeasuresMeasures3. Definition of Co-3. Definition of Co-MeasuresMeasures

Suppose a risk measure for risk X with mean m can be defined as: – R(X) = E[(X– am)g(x)|condition] for some

value a and function g– X is the sum of n portfolios Xi each with

mean mi Then co-measure for Xi is:

– CoR(Xi) = E[(Xi– ami)g(x)|condition]– Note that CoR(X1)+CoR(X2) = CoR(X1+X2) and

so the sum of the CoR’s of the n Xi’s is R(X) A risk measure could have equivalent

definitions with different a’s and g’s so alternative co-measures

Example: TVaRExample: TVaRExample: TVaRExample: TVaR

TVARq = E[X|X>q]

Co-TVaRq(Xi) = E[Xi |X>q] Charges each sub-portfolio for its part

of total losses in those cases where total losses exceed threshold value

In simulation, cases where condition is met are selected, and losses of sub-portfolio measured in those cases

Allocates Xi to a constant Xi

Excess TVARExcess TVARExcess TVARExcess TVAR

XTVARq = E[X – m|X>q]

Co- XTVARq = E[Xi – mi|X>q] Allocates average loss excess of

mean when total losses are above the target value

Allocates nothing to a constant Xi

M-R vs. Co-MeasuresM-R vs. Co-MeasuresM-R vs. Co-MeasuresM-R vs. Co-Measures

M-R based on marginal method, which has economic precedent

Co-measures can be adapted to any capital standard

Pretty much of a toss-up

4. Frictional Costs and 4. Frictional Costs and ReturnReturn4. Frictional Costs and 4. Frictional Costs and ReturnReturn

Taxation of investment income Investing conservatively to be more

secure Agency and liquidity costs of letting

someone else control the capital All these accrue even if you don’t write

any business – they are from the intention to write

They are proportional to capital held, so make sense to relate to capital

Risk Bearing CostsRisk Bearing CostsRisk Bearing CostsRisk Bearing Costs Cost for actually bearing risk, not just from

intention to write Not proportional to capital E.g., suppose BIG insurance company wants

its reinsurers to sell it put options on its stock Value of put option is not dependent on

capital needed to support it – except for a reduction in value for credit risk of seller

Reinsurer should charge for value of risk transfer plus allocation of frictional costs of capital

Return on allocated capital should vary across lines, depending on risk bearing costs

Estimating Market Price of Estimating Market Price of Bearing RiskBearing RiskEstimating Market Price of Estimating Market Price of Bearing RiskBearing Risk CAPM might be starting point Company-specific risk needs to be reflected

– Froot-Stein, Mayers-Smith The estimation of beta itself is not an easy matter

– Full information betas Other factors besides beta are needed in actual

pricing– Fama & French Multifactor Explanations of Asset Pricing

Anomalies Heavy tail beyond variance and covariance

– Wang A Universal Framework For Pricing Financial And Insurance Risks

– Kozik and Larson The N-Moment Insurance CAPM PCAS 2001

Impact of jump risk

Alternative Profitability Alternative Profitability Measure: Charge Capital Cost Measure: Charge Capital Cost against Profitsagainst Profits

Alternative Profitability Alternative Profitability Measure: Charge Capital Cost Measure: Charge Capital Cost against Profitsagainst Profits Instead of return rate, subtract cost of

capital from unit profitability Use true marginal capital costs of

business being evaluated, instead of an allocation of entire firm capital– If evaluating growing the business 10%,

charge the cost of the capital needed for that much growth

– If evaluating stopping writing in a line, use the capital that the company would save by eliminating that line

This maintains financial principle of comparing profits to marginal costs

Calculating Marginal Capital Calculating Marginal Capital CostsCostsCalculating Marginal Capital Calculating Marginal Capital CostsCosts

Could use change in overall risk measure of firm that results from the marginal business – but requires selection of the overall risk measure

Or could set capital cost of a business segment as the value of the financial guarantee the firm provides to the clients of the business segment– This could be called capital consumption

Value of Financial GuaranteeValue of Financial GuaranteeValue of Financial GuaranteeValue of Financial Guarantee Cost of capital for subsidiary is a difference

between two put options:– 1. The cost of the guarantee provided by the

corporation to cover any losses of the subsidiary– 2. The cost to the clients of the subsidiary in the

event of the bankruptcy of the corporation Economic value added of the subsidiary is value

of profit less cost of capital– Value of profit is contingent value of profit stream if

positive A pricing method for heavy-tailed contingent

claims would be needed– The individual put-option for the business unit may

be more heavy-tailed than for the company as a whole