Assume that the Sharpe-Lintner CAPM holds, so the mean-variance efficient frontier consists of combinations of Treasury bills and the market portfolio. Nonetheless, some households make the mistake of holding undiversified portfolios that contain only one stock or a few stocks. (Empirical evidence on such behavior is discussed in Chapter 10.)

(a) Show that the Sharpe ratio of any portfolio divided by the Sharpe ratio of the market portfolio equals the correlation of that portfolio with the market portfolio.

(b) Suppose the market is made up of identical stocks, each of which has the same market capitalization, the same mean and variance of return, and the same correlation

\(ho>0\), with every other individual stock. Consider the limit as the number of stocks in the market increases. What is the Sharpe ratio of an equally-weighted portfolio that contains \(N\) stocks divided by the Sharpe ratio of the market portfolio? Interpret.