 Not a goal in itself
 Used to make further calculations, like adequacy of business unit profits, incentive compensation

Can also be used for strategic planning:
– Would growing this business unit by 10% generate enough profits to make up for the cost of the capital it would need?

Best methods to some extent depend on purpose of allocation

1.
Divide return by allocated capital
 Allocate capital by some risk measure and divide
2.
Compare return to price of bearing risk
 Use a theory of market risk pricing to set targets
3.
Charge actual marginal capital costs against profits
 Compare the profits expected from a strategic plan to the cost of the extra total capital the firm needs
 Direct marginal cost – not an allocation
4.
Compare value of float generated by the business to a leveraged investment fund with the same risk
 Do for whole firm, then look at marginal impact of business unit or growth plan

Take one from each column and mix carefully
Risk Measures
– VaR
– EPD
– Tail VaR
– X TVaR
– Standard Deviation
– Variance
– Semi-Variance
– Cost of Default Option
– Mean of Transformed
Loss
Allocation Methods
– Proportional Spread
– Marginal Analysis
• By whole business unit
• Increment of business unit
– Game Theory
– Equalize Relative Risk
– Apply Co-Measure


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 X i each with mean m i
 Then co-measure for X
– CoR(X i
) = E[(X i i is:
– am i
)g(x)|condition]
– Note that CoR(X
1
)+CoR(X
2
) = CoR(X the sum of the CoR’s of the n X
1
+X i
’s is R(X)
2
) and so

A risk measure could have equivalent definitions with different a’s and g’s so alternative co-measures


TVAR q
= E[X|X>q]

Co-TVaR q
(X i
) = E[X i
|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


XTVAR q
= E[X – m|X>q]

Co- XTVAR q
= E[X i
– m i
|X>q]

Allocates average loss excess of mean when total losses are above the target value

Allocates nothing to a constant X i

 Overall capital: target default put cost as % of expected losses
– Limited capital of an insurer gives it an implicit option to put losses that exceed capital to the policyholders
 Allocation method is incremental marginal
– Last dollar of expected loss in a business unit is charged with the capital needed to keep the company put cost ratio constant
– The whole business unit (or policy) gets charged at that ratio to expected losses – a pure marginal method

But all capital is allocated, as the sum of the marginal capital charges equals the whole capital of the firm
– From the additivity of option prices

A constant risk generally gets a negative capital charge
– It does not add risk but both adds stability and accepts risk of non-payment


Myers-Read and Co-XTVaR both additive and reasonable

But pricing to equalize returns on capital so allocated may not tie in to risk pricing standards

Myers-Read do not advocate pricing purely in proportion to allocated capital – a risk charge is also added e.g. for covariance with market

Makes most sense for allocation of frictional costs
– Costs from holding capital even if no risk taken
– Cost is proportional to capital


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 to account for actual risk pricing
– Fama and 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




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


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


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



Insurer can be viewed as a tax-disadvantaged leveraged mutual fund
– Combined ratio less 100% is cost of funds
Measure mean and risk for insurer’s post-tax return

Then find parameters for mutual fund that give same after-tax distribution of return
– Find amount to be borrowed, investment mix, and borrowing rate
 Evaluate financial worth of writing the insurance by the risk-equivalent borrowing rate
– A high rate easy to obtain says writing insurance is not adding value
– A low risk-equivalent rate indicates value is added

Business units can be evaluated based on their marginal impact on the equivalent borrowing rate


Allocating by risk measure straightforward but arbitrary and might be allocating fixed capital costs; works for allocating frictional costs
 Using risk pricing appropriate for profitability comparison but requires a good theory of pricing

Actual marginal surplus most useful for determining economic contributions of business units. This is not the same as allocation in proportion to marginal risk.

Leveraged mutual fund comparison a basically qualitative method for evaluating return on total capital and the marginal contribution of each business unit to that