  Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion
conditioned so that B_0 = B_1). We consider the compact hull obtained by
filling in all the holes, i.e. the complement of the unique unbounded component
of R^2\B[0,1]. We show that the expected area of this hull is Pi/5. The proof
uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). Also,
using the result of Yor about the law of the index of a Brownian loop, we show
that the expected areas of the regions of non-zero index n equal 1/(2 Pi n^2).
As a consequence, we find that the expected area of the region of index zero
inside the loop is Pi/30; this value could not be obtained directly using Yor's
index description.
