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## Capital Allocation Survey

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Purpose of Allocating Capital

- 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

Approaches Used for Evaluating Profitability of Business Units

- Divide return by allocated capital
- Allocate capital by some risk measure and divide
- Compare return to price of bearing risk
- Use a theory of market risk pricing to set targets
- 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
- 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

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

Allocate by Risk MeasureTake one from each column and mix carefullyDefinition of Co-Measures

- 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: 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

Excess 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

Myers-Read Capital Allocation

- 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

Allocation by Risk Measure

- 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

2. Target to Market Price of Bearing 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 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

3. Charge Capital Cost against 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 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

Value 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

4. Compare to Closed-end Mutual Fund

- 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

Allocation Summary and Evaluation

- 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

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