Stock indices, such as the S&P 500, are typically "value-weighted." In other words, the weight a firm is assigned in the index depends on its market capitalization. If firm A has twice the market capitalization of firm B, then firm A is assigned twice the weight of firm B in the index. 

Recently, a new product known as a "fundamentals index" has emerged on the financial scene. With fundamentals indices, the weight a firm is assigned in the index depends on some measure of fundamentals, such as sales. So, if firm A has twice the sales of firm B, then firm A is assigned twice the weight of firm B in the index. 

Advocates of fundamentals indices argue that, if markets are inefficient, these indices will earn higher average returns than traditional value-weighted indices. Can you see the argument for why this would be true? 

b)  A closed-end fund is a portfolio of assets that issues a fixed number of shares. Closed-end fund shares are not redeemable from the fund for the underlying portfolio, and are instead listed and traded on a stock exchange. The market price of a closed- end fund is usually lower than the net asset value of the fund's portfolio (i.e., the total value of all assets in the portfolio). One explanation for the closed-end fund discount is noise trader risk. 

What are the two key assumptions in deriving noise trader risk. 

c) A recent study by Frazzini and Pedersen (2013)that an investment strategy that goes long low beta stocks and short high beta stocks generates a positive alpha with respect to the Fama-French three-factor model. However, if momentum factor by Carhart (1997) is added in the analysis, the alpha of the long-short strategy decreases and becomes statistically insignificant. 

What are the implications of this finding for the returns of this betting-against-beta strategy?