This problem is a numerical illustration of Example 9.4. Consider a single-index model, in which all assets have unit beta. The volatility of each specific risk is \(30 \%\), and you are considering a universe of 20 stocks, half of which have alpha \(+2 \%\) and half have alpha \(-2 \%\). You go long \(\$ 1\) million with an equally weighted portfolio consisting of the stocks with positive alpha, and short \\($1\) million of a similar portfolio of stocks with negative alpha. Note that the resulting portfolio is both dollar-neutral and beta-neutral. What are the expected profit and risk of the long-short portfolio? How does your answer change if you consider 50 or 100 stocks?

Data From Example 9.4

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Example 9.4 A market-neutral long-short portfolio A market-neutral portfolio is a portfolio which is not exposed to systematic risk. The only source of return is specific risk. Such a portfolio is also referred to as beta-neutral, since its betas with re- spect to systematic risk factors are zero. A possible rationale behind such a portfolio is that we may have a view about the relative per- formance of stock shares, but we do not feel safe in making a bet on the direction of the market as a whole. The analysis may suggest that some stocks have positive alpha, and other stocks have negative al- pha. However, investing in the positive alpha stocks may still result in a loss if the market takes a negative turn. We might find only a partial consolation in a portfolio losing less than the index. Thus, we may take a long-short strategy, whereby we short-sell the stocks with negative alpha in order to neutralize the overall beta, i.e., the exposure to portfolio risk. Let us consider a stylized example of the strategy. We have a subset of n assets, characterized by the single-index model = + BRM +i, i = 1,..., n. We assume that a > 0 and are the same for all of these assets. We also have another subset of n assets, characterized by the single-index model = - + BRM + i, in+1,..., 2n, where again we assume that the numerical values of the involved pa- rameters are the same and identical to those of the first subset of as-