Sorry -- you guessed wrong!  (But who would know about this
stuff without looking it up?)  
Minnesota's largest HMO uses 19%
of it's revenue to support overhead, compared to Medicare's 2%
.  
Although private enterprise is supposed to function more
efficiently than anything connected with government,
private-sector health insurance companies seem to be an
exception.  The normal range for administration and overhead
costs for these firms is 15-35%.  Government-run health
insurance is much more efficient.

Medicare's mission is to help older people receive medical care
as a right), and has established fixed reimbursement schedules
for medical services.  In contrast, for-profit private health
insurance companies have a motivation to dispute as many
charges as possible, to improve the bottom line.  In order to
control payments to service providers, private health insurance
companies maintain much larger administrative staffs than
Medicare, to challenge, question, and (when possible) refuse
reimbursement for services.  This forces hospitals and clinics to
maintain larger staffs, as well, to respond to the reluctance of
private health insurance companies to pay.  Costs are increased
on both sides.

The level of waste detailed above accounts for a large
portion of the gap between excessive American health
care spending and the uneven performance of the
American health care system
.  For this reason, it may be
impossible to achieve "Universal Health Care" in the U.S. without
changing the basic way that money is raised and spent. That's
why the "Single Payer" idea has been gaining support.  
A "Single
Payer" system would basically expand Medicare coverage to
everyone, add some budgetary restraints on hospital spending,
and act to limit drug prices
.

Sources for overhead percentages:  Medicare annual reports; for Medicaid,
Minnesota Office of the Legislative Auditor,
The Financial Impact of
Alternative Health Care Models on Administrative and Benefits Costs in
Minnesota
, 1995; for Medica, Chapter 6, "Administrative Expenses", in an
untitled report by Minnesota Attorney General Mike Hatch.  All as cited in
notes of Kip Sullivan of the Minnesota Universal Health Care Coalition.