Determining the size of a total purchasing site to manage the financial risks of rare costly referrals: computer simulation model
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* Max O Bachmann
* Gwyn Bevan

## Abstract

**Objective**: To estimate the financial risks of 15 categories of rare costly referrals for total purchasing sites of different population sizes.

**Design**: Computer simulation of 100 fund years assuming Poisson distribution of referrals.

**Setting**: British general practices that have opted to become total purchasing sites. Referral rates and price estimates were supplied by South and West Devon Health Commission.

**Main outcome measures**: Variation in referral costs to purchasers in relation to size of risk pool (person years at risk).

**Results**: Random variation in referral costs increased as the size of the risk pool decreased. Variation increased greatly below 30 000 person years. The mean simulated cost of the referral categories considered was 2.8% of total NHS hospital and community service costs, and the maximum simulated cost for 7000 person years was 6.8%. Simulated variation was robust to assumptions about prices and referral rates for specific types of referral.

**Conclusion**: Rare costly referrals seem unlikely to bankrupt total purchasing sites. The management of risk is not in itself justification for total purchasing to be based in several general practices in order to generate large populations. There are other ways of managing risk. Sites can easily explore options by simulations using local referral rates and prices.

#### Key messages

*   Insurance arrangements for most risks ought to be organised to extend each practice's risk pool by giving it more years over which to pay ∗ Risk pools of at least 30 000 patient years should be created: a total purchasing practice of 10 000 patients ought to seek at least three years to manage its expenditure

*   Financial risk due to rare costly referrals may easily be modelled if local referral rates and prices are available

*   Special arrangements are needed for cata- strophic insurance for rare and costly hospital care over which general practitioners have little discretion

## Introduction

Arrow's seminal paper on the distinctive characteristics of the economics of health care emphasised the problem of uncertainty and the consequence that institutional arrangements for financing health care are based on insurance.1 The NHS reforms created two kinds of local insurers for hospital and community health services.2 Health authorities are typically responsible for populations of over 250 000, and general practitioners can opt to become fundholders for their populations for a selected range of services, including some hospital services. Now general practices can opt to become total purchasing sites for all hospital and community health services, either as single practices or as groups of practices.

One aim of giving general practitioners budgets is to make them aware of the costs of their referrals.3 4 For the budget to be effective it has to be designed to cope with risk from random variations in expenditure. One determinant of risk is the size of the risk pool expressed in person years: the product of the number of patients and the years over which expenditure has to be kept in line with the budgets. The larger the risk pool the lower the risk because the random variation is comparatively smaller. Managing risks by insurance, in which risks are shared with others, creates the intrinsic problem of moral hazard: insulating individual people from the problems of risk also removes economic incentives for them to be concerned about costs.1

There are several ways for total purchasing sites to manage risk. Several practices could join together to increase the size of the population covered; risks could be spread across several financial years, shared between a site and the health authority, or shared between different sites; and sites could use commercial insurers.

General practice fundholding incorporates three ways of managing risk. Firstly, eligible practices have to be larger than a specified size.

Secondly, practices can defer expenditures on elective inpatient admissions, which account for about 40% of fundholders' total budgets.5 6 7 Additionally, practices are allowed to overspend their budgets by up to 5% in any given year.

Thirdly, there is also a “stop loss” arrangement, in which any referral costing more than £6000 has all of these costs paid for by the patient's health authority rather than the practice budget.

Early comments on the size of population that fundholding general practitioners would require to manage risk were based on the experience of health maintenance organisations in the United States. Scheffler observed that health maintenance organisations with fewer than 50 000 patients had difficulty surviving,8 and Weiner and Ferris argued that a risk pool of only 11 000 patients might be unstable.9 Crump et al examined the risk for fundholders using NHS data on rates and prices for 113 surgical procedures then included in fundholding.10 Their simulations showed that practices of 9000 would exceed their surgical budget by at least 5% in 25 of the 100 years simulated, whereas practices of 24 000 would exceed their budget by this much only in about 5 of the 100 years simulated.

Small risk pools would thus be expected to result in significant overspending by fundholders. In 1992–3, however, only 5% of practices overspent their budgets by more than £100 000 (about 7% of the average practice budget of £1.5m).7 Initially practices required populations of at least 11 000 to become fundholders.2 This threshold has been successively lowered, and from April 1996 the minimum size was 5000 patients.11 The introduction of total purchasing again raises the question of how large sites should be. Because health maintenance organisations cover all services, Scheffler's observation may be a more appropriate guide to the size of total purchasing sites than the experience of general practice fundholding.8

We considered random variation in expenditure caused by rare and costly referrals. We used data on average costs and rates for a selected group of rare costly admissions to examine the influence of the size of the risk pool on random variation in expenditure. We also examined the impact of variations in rates of referral and in hospital prices. Total purchasing is dependent on local arrangements between practices and health authorities, and we used local data to show how those responsible for developing total purchasing in any site could simulate their risks with a standard spreadsheet package on a personal computer for the rare costly admissions that concern them. This method could also be used to examine risks from rare and costly prescriptions of drugs, such as growth hormone.

## Methods

Financial risk due to random variations in 15 categories of rare costly referrals was simulated over 100 fund years for various sizes of risk pool expressed in person years. The categories of referral considered are shown in table 1. These categories were chosen by South and West Devon Health Commission when examining risks for its total purchasing sites. They are similar to, although not as comprehensive as, services purchased by the West Midlands Regional Specialties Agency.12 Data on rates and costs of the referrals were supplied by the health commission.

View this table:
[Table 1](http://www.bmj.com/content/313/7064/1054/T1)

Table 1 
Estimated referral rates and prices used in simulation, with contribution of each type of referral to toal expected cost

Numbers of referrals were simulated on an Excel spreadsheet, using the random number generation tool and assuming Poisson distributions.13 The total cost of the 15 referral categories for each year was calculated by summing the products of the simulated number of each type of referral and the respective hospital price. These costs were expressed in absolute terms, as percentages of the expected cost of rare costly admissions and as percentages of the total purchasing budget. The expected cost was defined as the sum of the products of prices, rates, and risk pool sizes. The total purchasing budget was defined as the product of the population size and the cost per head of hospital and community health services in the NHS in 1994 (£430)—that is, the distinction between standard fundholding and total purchasing was not considered.14 The variation in costs was defined as in the paper by Crump et al as the difference between the 95th and 5th centiles expressed as a percentage of the mean simulated cost10; this measure is more appropriate for Poisson distributions than is the coefficient of variation.

A sensitivity analysis was conducted for a risk pool of 30 000 patients to see how sensitive the simulated variations in total costs were to the assumed rates and prices for each referral category (the base rates and prices in table 1). Hospitals' prices for specific types of admission range between 50% and 150% of the mean price15 and general practitioners' referral rates between about 50% and 200% of the mean rate.16 Simulation was thus repeated four times for each referral category: using a price 50% of the base price, a price 150% of the base price, a rate 50% of the base rate, and a rate 200% of the base rate. For each simulation only one price or rate for one referral category was changed at a time, all other rates and prices being as in table 1.

The annual premiums required to be paid by total purchasing sites for different stop loss arrangements were simulated by examining referrals priced above each amount. We considered stop loss amounts of £6000 (the current amount for standard fundholding), £10 000, and £30 000.

## Results

The distributions of costs simulated over 100 years are shown in table 2. As expected, the degree of random variation decreased as the population increased. A risk pool of 7000 had a comparatively high degree of risk, which decreased appreciably with a risk pool of 30 000, with diminishing marginal returns above this size (summarised as variation in table 2). Mean simulated costs were similar to expected costs (table 2), supporting the validity of the simulation.

View this table:
[Table 2](http://www.bmj.com/content/313/7064/1054/T2)

Table 2 
Distributions of referral costs (£) for different sizes of risk pool (No of person years at risk) simulated over 100 years

The financial risk may also be expressed as another statistic: the discrepancy between simulated and expected costs of rare costly admissions as a percentage of expected costs. Distributions of these statistics are shown in figure 1. The degree of random variation in year to year costs is represented by the intercentile spread. For example, the costs in 90% of simulated years lie between the 5th and 95th centiles. As in table 2, the risk due to random variation increased rapidly with risk pools of under 30 000 patient years. The dotted lines show that for the 95th most costly year to be within 50% of the expected cost, a risk pool of at least 30 000 would be required. For a total purchasing site without an insurance pool to be almost certain of being able to keep within its rare costly admissions budget it would need a contingency reserve. For example, to be 95% certain of being able to cover the cost with a risk pool of 7000 patient years the rare costly admissions budget would have to be 2.1 times the expected value, while with 1 000 000 it would have to be only 1.07 times higher (calculated from table 2).

![Fig 1](http://www.bmj.com/https://www.bmj.com/content/bmj/313/7064/1054/F1.medium.gif)

[Fig 1](http://www.bmj.com/content/313/7064/1054/F1)

Fig 1 
Distributions of differences between simulated and expected referral costs as percentage of expected cost for different sizes of risk pool

The sensitivity analysis showed that the simulated variations in costs were robust to the assumed rates or prices of any particular referral category. In the basic analysis for a risk pool of 30 000 the variation (defined in Methods) was 87% (table 2). In the sensitivity analysis the smallest variation was 75% (obtained when the price of referral to an open ward for forensic psychiatry was halved) and the greatest was 103% (obtained when the price of a referal to a special care baby unit was doubled). Of the 60 sensitivity analyses carried out (four for each referral category, as described in Methods), 30 produced variations between 80% and 90%. The mean simulated costs were most sensitive to assumptions about the rates and prices of special care baby units and psychiatric and forensic psychiatry; this would be expected because these categories make the greatest contribution to the expected cost (table 1). If the rate of referrals to special care baby units was doubled, then in the 95th most costly year these 15 rare costly referrals would cost 22% more than expected; all other sensitivity analyses produced smaller overspends than this.

When expressed as a percentage of the total purchasing budget, the mean costs for all 15 rare costly referral categories were between 2.7% and 2.8%, regardless of fund size. If an insurance fund covering the 15 rare costly admissions categories were created, the premium should thus be about 2.8% of the total purchasing allocation, with modification depending on the stop loss amount, changes over time in referral rates and prices, and administrative costs. The 95th most costly years for funds with risk pools of patient years of 7000, 10000 30 000, 50 000 70 000, 100 000, and 1 000 000 cost 5.8%, 5.5%, 4.1%, 3.8%, 3.6%, 3.3%, and 2.7% of total allocation, respectively. The greatest simulated cost for the smallest risk pool—that is, the 100th centile for 7000 patient years—was 6.8% of the total purchasing budget—that is, 4.0% more than the mean simulated cost. For a risk pool of 10 000 patient years and for stop loss amounts of £6000, £10 000, and £25 000 the annual premiums should be 2.5%, 1.4%, and 1.2% of the total purchasing budget, respectively.

## Discussion

This simulation shows that even small total purchasing sites are unlikely to be bankrupted by these 15 types of rare costly admissions. This conclusion would not be altered if more referral categories had been considered. Increasing the number of referral categories covered would reduce the key problem of managing risk—that is, random variation in cost in relation to expected spend on rare costly referrals. Random variation in costs would decrease because inclusion of more such categories increases the chance that they balance each other. A small total purchasing site, for example, is highly unlikely to require an admission to forensic psychiatry and for renal transplantation and rehabilitation after brain injury in the same year (this combination only occurred in one of 100 simulated years for a risk pool of 7000). This partly explains why the variation obtained for 133 surgical referral categories by Crump et al10 is smaller than the variation obtained for only 15 categories in this exercise (table 2).

### UNCERTAINTY ABOUT RATES AND PRICES

There are intrinsic uncertainties about referral rates and prices for the rarest and most costly admissions. For example, the price of an admission for forensic psychiatry is likely to vary with the length of stay, which is difficult to predict in advance, and the rate estimates are likely to be unstable even when based on district populations. Sensitivity analysis has shown, however, that our main findings are robust to changes in rates of referral and prices charged. The main vulnerability to overspending is due to the comparatively high volume of neonatal special care and to costly forensic and tertiary psychiatry.

In any specific total purchasing site the risk from rare costly admissions will be different because of differences in rates and prices. Anyone with a personal computer and a spreadsheet with a random number generator can estimate risks using local information. When accurate information is lacking, a variety of estimates can be simulated to examine the implications of uncertainty about rates and prices and thus help identify which information is most important. Any putative arrangements can thus be assessed using the methods we have described in this paper. For specific examination of recurrence of referrals in individual patients, a negative binomial distribution may be more appropriate than a Poisson distribution,17 but this was not of central importance to the referral categories considered here.

### EFFECTIVE RISK MANAGEMENT

Our analysis suggests that total purchasing sites ought not to begin by aiming to be large for the purpose of reducing risk. Instead the focus ought to be on how to achieve a common purpose within each site: in agreeing contracts and managing spend against budgets. Having come to a view of the best working arrangements, a site can explore what, if any, extra arrangements might be needed to manage risk.

The appropriate way of increasing a risk pool is to extend it over time to enable a site to have a number of years to pay back any overspends. For other sites to take on this risk creates a problem of moral hazard. It undermines the underlying rationale of giving practices budgets, which is that they are responsible for the economic consequences of their decisions.

Exceptional catastrophic risks—for example, forensic psychiatry—do not naturally form part of practice budgets. These risks ought to be shared with health authorities and financed by a general levy, with payments authorised by a committee representing contributors. In general, therefore, our analysis supports basing total purchasing sites on single practices. We are interested to see whether experience with total purchasing bears this out.

We thank Brian Maynard Potts of South and West Devon Health Commission for supplying the price and referral rate estimates on which this simulation is based, and two anonymous referees for their comments.

## Footnotes

*   Funding No specific funding.

*   Conflict of interest None. We are members of the Total Purchasing National Evaluation Team.

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