Signature
SD_N = sigma / sqrt(N)
| Inputs | Definition | Unit |
|---|---|---|
sigma | Standard deviation of the annual claims cost of one covered person | currency per person per year |
N | Number of covered persons whose costs are averaged | persons |
SD_N | Standard deviation of the average annual claims cost per person in a pool of N people | currency per person per year |
|---|
Function
Actuarial claims costing, premium loading and claims reserving function
Maps utilisation rates, unit costs and the size of a covered group to the expected claims cost of a health insurance scheme, turns that expected cost into a gross premium or contribution with loadings, measures how predictable the group's average cost is, and estimates the claims incurred but not yet paid at a reporting date. These are the calculations of the Actuarial Analysis article. The plan generosity measure is covered separately on the Actuarial Value page.
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Implementations
Excel
Pooled standard deviation in one cell
Excel returns the standard deviation of the average cost from named cells holding one person's standard deviation and the number of people in the pool.
=Sigma/SQRT(Members)
Assumptions
Independent costs with a common spread
Members' annual costs are independent and share the same standard deviation. Price rises and new treatments affect everyone at once, so real costs are not fully independent and the formula gives a lower bound on the uncertainty a scheme faces.
Random variation only, with the assumptions taken as correct
The formula measures random variation around the expected cost. It does not include error in the assumptions behind that expected cost, such as the induced use rate.
Worked examples
Pooled spread for 200,000 physiotherapy members
With a standard deviation of £52.31 per member, from the article's example, the average cost across 200,000 members has a standard deviation of about £0.117, about 1% of the expected £12.
sigma = 52.31; N = 200000; SD_N = 0.117
Pooled spread for a scheme of 2,000 members
The same calculation for 2,000 members gives about £1.17, nearly 10% of the expected £12, so a small scheme faces much more random variation in its average cost.
sigma = 52.31; N = 2000; SD_N = 1.17
Common errors
Dividing by the pool size instead of its square root
Dividing £52.31 by 200,000 gives about £0.00026 rather than £0.117, understating the spread of the average by a factor of about 447.
Reading the pooled spread as the full uncertainty of the estimate
For a large scheme the random spread is small, but the assumptions can move the estimate far more: in the article, varying induced use from none to a half moves the net premium from £9.60 to £14.40, against a pooled standard deviation of about £0.12.
Sources
Standard error of a mean as standard deviation over the square root of sample size
Altman DG, Bland JM. Standard deviations and standard errors. BMJ. 2005;331(7521):903. The standard error of the sample mean equals the standard deviation divided by the square root of the sample size.
ILO law of large numbers for the average per capita claim
Cichon M, Newbrander W, Yamabana H, Weber A, Normand C, Dror D, Preker A. Modelling in Health Care Finance: A Compendium of Quantitative Techniques for Health Care Financing. Geneva: International Labour Office; 1999. Issue Brief 3, section 3.2, printed pp. 267-268: the average per capita claim estimates the expected value when the group is large enough, and each risk group must be large enough.
Canonical Identity
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