  We analyze multiline pricing and capital allocation in equilibrium
no-arbitrage markets. Existing theories often assume a perfect complete market,
but when pricing is linear, there is no diversification benefit from risk
pooling and therefore no role for insurance companies. Instead of a perfect
market, we assume a non-additive distortion pricing functional and the
principle of equal priority of payments in default. Under these assumptions, we
derive a canonical allocation of premium and margin, with properties that merit
the name the natural allocation. The natural allocation gives non-negative
margins to all independent lines for default-free insurance but can exhibit
negative margins for low-risk lines under limited liability. We introduce novel
conditional expectation measures of relative risk within a portfolio and use
them to derive simple, intuitively appealing expressions for risk margins and
capital allocations. We give a unique capital allocation consistent with our
law invariant pricing functional. Such allocations produce returns that vary by
line, in contrast to many other approaches. Our model provides a bridge between
the theoretical perspective that there should be no compensation for bearing
diversifiable risk and the empirical observation that more risky lines fetch
higher margins relative to subjective expected values.
