The expected area of the filled planar Brownian loop is π/5

Let B-t, 0 <= t <= 1 be a planar Brownian loop ( 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 C \ 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). As a consequence of this result, using Yor's formula [ 17] for the law of the index of a Brownian loop, we find that the expected area of the region inside the loop having index zero is pi/30; this value could not be obtained directly using Yor's index description.