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. Introduction.

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Allocation of Surplus Based Upon Right Tail Deviation

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Allocation of surplus based upon right tail deviation

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.

Surplus allocation issues

Surplus Allocation Issues

(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”.

Right tail deviation of x with parameter r 0 r 1

Right Tail Deviation of X with Parameter r, 0<r<=1

  • 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.

Intuitive explanation

Intuitive Explanation

  • 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.

Basic properties of right tail deviation rtd

Basic Properties of Right Tail Deviation (RTD)

  • 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.

Important properties of rtd

Important Properties of RTD

  • 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.

Interesting properties of rtd

Interesting Properties of RTD

  • 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.

Layers of fixed width

Layers of Fixed Width

  • 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.

Parameterization of rtd

Parameterization of RTD

  • 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.

Surplus allocation proposal

Surplus Allocation Proposal

  • 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.

Benefit of diversification

Benefit of Diversification

  • 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.

Return on equity

Return on Equity

  • 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.

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