Approximating the Numéraire Portfolio by Naive Diversification

Abstract. Estimation theory has shown, due to the limited estimation win-dow available for real asset data, the sample based Markowitz mean-variance approach produces unreliable weights which fluctuate substantially over time. This paper proposes an alternate approach to portfolio optimization, being the use of naive diversification to approximate the numéraire portfolio. The numéraire portfolio is the strictly positive portfolio that, when used as benchmark, makes all benchmarked nonnegative portfolios either mean decreasing or trendless. Furthermore, it maximizes expected logarithmic utility and outperforms any other strictly positive portfolio in the long run. The paper proves for a well-securitized market that the naive equal value weighted portfolio converges to the numéraire portfolio when the number of constituents tends to infinity. This result is model independent and, therefore, very robust. The systematic construction of diversified stock indices by naive diversification from real data is demonstrated. Even when taking transaction costs into account, these indices significantly outperform the corresponding market capitalization weighted indices in the long run, indicating empirically their asymptotic proximity to the numéraire portfolio. Finally, in time of financial crisis, a large equi-weighted fund carrying the investments of major pension funds and insurance companies would provide important liquidity. It would not only dampen the drawdown of a crisis but would also moderate the excesses of an asset price bubble