Expected Utility Maximization and the Optimal Portfolio of Risky Assets

Classic finance theory advises investors to hold a mix of the risk-free asset and a single portfolio of risky assets: the tangency portfolio that maximizes the Sharpe ratio. Two questions about this ad- vice have remained largely unresolved. Is holding this specific portfolio approximately consistent with expected utility maximization? And if so, what determines how much of it to hold? Existing derivations typically assume either Normally distributed returns or quadratic utility. This paper answers both questions using a second-order expansion of a general concave utility function, without either restriction. The resulting portfolio is exactly proportional to the tangency portfolio, scaled by a term that depends on the maximum Sharpe ratio of the investment opportunity set. The approximation is highly accurate: a fourth-order correction, solved exactly for a two-asset opportunity set and numerically for a ten-asset one, changes portfolio composition only modestly, and the solution captures at least 95% of the achievable utility gain, relative to the numerically optimal CRRA portfolio, over a naive benchmark matched to the same overall risky/risk-free split.