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Allocation of Surplus Based Upon Right Tail Deviation

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Allocation of Surplus Based Upon Right Tail Deviation

Bob Bear

PXRE Corporation

CARE 6/99

Expected Policyholder Deficit has become a widely accepted method of assessing adequacy of surplus to support a book of business. It is an understandable way to quantify management’s risk tolerance level.

(1) EPD is not additive for layers (Shaun Wang, “An Actuarial Index of the Right-Rail Risk”, NAAJ, April 1998). Refer to 1998 PCAS paper by Wang.

(2) EPD produces counterintuitive results (e.g., EPD is bigger for a Gamma distribution than a Pareto); see 8/98 CARE presentation on “Getting to E in ROE” by Todd Bault.

- A good risk load procedure is a good candidate for a surplus allocation approach, with different parameterization.
- Right Tail Deviation has the desirable properties of a good risk load procedure. Refer to Wang’s papers, and the 10/98 ASTIN paper by Christofides on “Pricing for Risk in Financial Transactions”.

- Definition: D(X;r)=Int{S(t)^r -S(t)}, where S(t) is the probability that the variable of interest X exceeds t (1-CDF). The integration is performed over all values of t for which X is defined.
- Note that this definition is based on Wang’s original definition of a ph-transform, rather that his Right Tail Deviation definition with r=.5.

- The integral of S(t) yields the expected value of X.
- The integral of S(t)^r yields a risk-adjusted expected value.
- D(X;r) is the area between these curves, providing the needed risk load or surplus allocation.

- The variable X of interest is underwriting loss attributable to a portfolio at end of planning horizon or contract expiration, reflecting only funds supplied by underwriting (with interest and taxes).
- This is the negative of the underwriting profit definition. Thus, losses are positive and profits are negative. The integration is performed over all real values.

- Shift and Scale Invariant: D(aX+b;r)=a*D(x;r), as for standard dev.
- Subadditivity: D(X+Y;r)<=D(X;r)+D(Y;r) as for standard deviation.
- If X and Y are comonotonic (including perfect correlation), then D(X+Y;r) =D(X;r)+D(Y;r); additive for layers, unlike standard deviation and EPD.

- Preserves stop-loss ordering of risks (second-order stochastic dominance). EPD also has this property, but standard deviation does not.
- When r declines from one to zero, the risk loaded loss (or loss plus allocated surplus) increases from the expected loss to the maximum possible loss.

- For a small layer at the right tail, the standard deviation and the Right Tail Deviation with r=.5 converge to each other. D(X;.5)<=SD for a small layer.
- Widely used distributions are ordered consistent with relative tail thickness. For a fixed mean and CV, the RTD would rank following from most to least risky: Pareto, Lognormal, Weibull, Gamma.

- The risk adjusted loss (or loss plus allocated surplus) declines (doesn’t increase) as the attachment increases.
- The risk load (or allocated surplus) increases as a percentage of expected loss as the attachment increases.
- More underwriting profit is needed to meet ROE target, as attachment goes up.

- Use EPD or other management criteria to quantify surplus required to support the portfolio, assuming risk-free interest.
- Calculate r for which the RTD equals the required surplus. The variable X is defined to be the underwriting loss attributable to a portfolio at the end of the planning horizon or at contract expiration, with interest and taxes.

- For each business segment, calculate the reduction in RTD for the portfolio if the particular segment were to be excluded.
- Allocate surplus AS(s) to the various segments in proportion to these reductions in RTD.
- Calculate RTD(s) for each business segment as a stand-alone portfolio.

- Calculate Diversification Factor d from AS(s) = d*RTD(s). The difference between AS(s) and RTD(s) reflects the benefit of diversification.
- This process can be repeated to allocate surplus to contract within each business segment. In the end, surplus is allocated to a contract by applying a Diversification Factor d to the RTD for the contract.

- Given allocated surplus at beginning of planning horizon or at contract inception, the surplus at end of horizon or at contract expiration may be calculated.
- For a contract, future profits are discounted to contract expiration date.
- ROE may be calculated as the average annual percentage change in surplus.

- RTD has all the desired properties of a risk load procedure.
- RTD may also be used for surplus allocation and ROE estimation purposes.
- RTD may be parameterized to reflect management’s risk tolerance level.