## Risk Corridor “Asymmetries”

In this excellent article on risk corridors, Scott Katterman of Milliman writes:

The risk corridor algorithm itself will tend to result in higher insurer receivables, compared to payables, due to an asymmetry in calculating the “target amount” (or expected cost) for each insurer.

The problem here is that the target amount really isn’t the “expected cost”. Actuaries of all stripes have struggled with how to characterize the target. In their seminal “3R” paper in 2013, the American Academy of Actuaries said virtually the same thing in their otherwise informative Risk Corridor Chart:

Below the fold, I show that the “Target Claims” should best be thought of as a function off *actual claims*, with two adjustments. The first adjustment occurs if the plan has profits lower than the amount provided for in the regulation. The second adjustment occurs if administrative expenses exceed the regulation’s cap. If the plan profit falls below 3% (or 5% in 2015), then the target is raised to give a plan a chance to receive money through the risk corridor program. If the plan’s administrative expense, inclusive of profit, exceeds 20% (or 22% in 2015), then the target is lowered, giving the plan a higher probability that they will have to pay money into the program.

Given the nature of competitive markets, it is clearly much less likely that a plan will have administrative expenses (plus profit) greater than 20%. Conversely, it is quite easy to have profits less than 3%; in fact, many plans may have priced for lower profit margins than that. And this is the crux of Katterman’s “asymmetry”; it isn’t so much a direct characteristic of the formula as it is a characteristic for how the formula was calibrated.

It is admittedly mind-blowing to think of a target as being predominantly a function of the actual amount with those two odd adjustments, but the algebra is unforgiving. This is a significant distinction between the Part D risk corridor program, where the target is a function of a plan’s “bid,” which makes a reasonable proxy for “expected”. In the ACA risk corridor, in contrast, if you price for something lower than 3% profit, then you would *expect* claims to be higher than target.

**Target Claims are a Function of Actual Claims: The Algebra**

**Definitions (mostly from the various regulations)**

P=Premium; E=Expenses (excluding taxes or quality expenses); T=Taxes; C=Claims (including Quality)

Target = Premium – Allowable Costs

Profit = P-C-E-T

Regulatory Profit = Max[3%*(P-T),Profit]

Allowable Costs = Min[E + Regulatory Profit, .2*(P-T)] + T

Profit Shortfall = PS = Max[3%*(P-T)-Profit,0)

Excess Admin = EA = Max[E + Profit + PS – .2*(P-T),0]

Assertion: T = C – PS + EA

In English, the Assertion is that the Target = Actual claims minus any profit shortfall relative to the regulation’s target (3% in 2014, 5% in 2015), plus a provision of administrative expenses (plus profit) exceeds the regulation’s 20% (22% in 2015).

**Proof**

Substitute terms:

(1) Regulatory Profit = Profit + PS

Discussion: If Profit is the largest term inside the “MAX” function, then PS=0; if profit is smaller than 3%*(P-T), then PS by definition will be the additional amount you need to increase profit to 3%*(P-T).

(2) Allowable Costs = E + Profit + PS – EA + T

Discussion: If .2*(P-T) is larger than E+Profit+PS, then EA=0 and allowable costs are just E + Profit + PS … in contrast, if E+Profit+PS >.2*(P-T) then subtracting EA is exactly the right amount to reduce the total to .2*(P-T)

Substituting (2) into the Target Equation:

(3) Target = P-E- Profit – PS+EA-T

Inserting the definition of Profit into Equation (3) results in the desired conclusion:

(4) Target = C – PS + EA … QED

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