How Much Bitcoin? Spectral Portfolio Allocation Beyond Mean-Variance

We show that mean-variance optimization systematically overestimates the optimal Bitcoin allocation in institutional portfolios. The Markowitz framework assumes returns are Gaussian, but Bitcoin has excess kurtosis \(\kappa_4 \approx 8\)--\(13\) and negative skewness --- making its tails 3--5\(\times\) heavier than the Gaussian approximation. Using the Spectral Fenton Distribution (Nagy, 2026a), we compute the full return distribution of mixed Bitcoin/traditional portfolios as a function of the Bitcoin weight \(w_{\text{BTC}}\), and optimize over all coherent risk measures (Expected Shortfall, spectral risk measures), not just variance. The main result: Markowitz recommends \(w_{\text{BTC}} = 28\%\) while the spectral method recommends \(w_{\text{BTC}} = 10\%\) under the same Expected Shortfall constraint --- an 18 percentage point overallocation. We prove (and formally verify in Lean 4) that this gap is a mathematical consequence of excess kurtosis: for any fat-tailed asset, the spectral optimal weight is strictly less than the Markowitz optimal weight, and the gap increases with kurtosis.