Fig 1. Investment opportunities including T, the tangency portfolio linking the risk-free return, R_f, and the minimum-variance frontier; D, the aggregate of the portfolios held by misinformed investors; and M, the market portfolio, which is the value-weight combination of T and D.

Fig AI. Expected return on securities underweighted (portfolio H) and overweighted (L) by misinformed investors as a function of θ_H, the fraction of the initial market allocation of H the misinformed choose to sell, for different values of ρ, the correlation between the payoffs on H and L.

Average excess returns and market betas for portfolios formed on market cap, book to market equity, and prior returns, 1927–2004

To form the size portfolios, we sort firms on their market cap (price times shares outstanding) at the beginning of each calender year from 1927 to 2004, and assign 90% of the total beginning-of-year market equity to the Big portfolio (L) and 10% to the Small portfolio (H). We then compute annual value-weight returns from January to December. To form the book-to-market cap (B/M) portfolios, we sort firms at the end of each June from 1926 to 2004 on the ratio of book equity for the previous calendar year divided by market equity for December of that year. We then split the total market cap at the end of June equally between a high B/M value portfolio (H) and a low B/M Growth portfolio (L), and compute monthly value-weight returns from July to the next June. To form the momentum portfolios, we sort firms at the beginning of each month t, from January 1927 to December 2004, on their cumulative returns from month t−12 to month t−2. We then split the total market cap equally between high (H) and low (L) prior return portfolios, and compute value-weight returns for month t. The annual B/M and momentum returns are compounded monthly returns for January to December. The data are from CRSP and Compustat, and each pair of portfolios contains the NYSE, Amex (after 1962), and Nasdaq (after 1972) firms with the data required to construct those portfolios. Thus, the size portfolios for year t include all firms with market cap data at the beginning of year t. The B/M portfolios formed in June of year t include only firms with positive book equity in the previous calendar year and market cap for June of year t and December of year t−1. The excess return for year t is the annual portfolio return minus the compounded one month T-bill return, from Ibbotson Associates. Beta is the slope from a regression of a portfolio's annual excess return on the market's excess return. The Sharpe ratio is the average annual excess return divided by the annual standard deviation. The tangency portfolio (T) is the combination of portfolios (big and small, growth and value, or low and high prior returns) that maximizes the Sharpe ratio.