Assume that two investors each hold a portfolio, and that portfolio is their only asset. Investor A's portfolio has a beta of minus 2.0, while Investor B's portfolio has a beta of plus 2.0. Assuming that the unsystematic risks of the stocks in the two portfolios are the same, then the two investors face the same amount of risk. However, the holders of either portfolio could lower their risks, and by exactly the same amount, by adding some "normal" stocks with beta = 1.0.
Question
Assume that two investors each hold a portfolio, and that portfolio is their only asset. Investor A's portfolio has a beta of minus 2.0, while Investor B's portfolio has a beta of plus 2.0. Assuming that the unsystematic risks of the stocks in the two portfolios are the same, then the two investors face the same amount of risk. However, the holders of either portfolio could lower their risks, and by exactly the same amount, by adding some "normal" stocks with beta = 1.0.
Solution
The statement is not entirely correct. Beta is a measure of systematic risk, or market risk, which is the risk that cannot be eliminated through diversification. A beta of -2.0 means that Investor A's portfolio moves in the opposite direction of the market, but with twice the volatility. A beta of 2.0 means that Investor B's portfolio moves in the same direction as the market, but with twice the volatility. Therefore, they do not face the same amount of risk.
Adding "normal" stocks with a beta of 1.0 to either portfolio would indeed lower the portfolio's beta, and therefore its systematic risk. However, the amount by which the risk is lowered would depend on the proportion of "normal" stocks added to the portfolio. The more "normal" stocks added, the closer the portfolio's beta would get to 1.0, and the lower its systematic risk would be.
However, it's important to note that this would not affect the unsystematic risk of the portfolio, which is the risk associated with individual stocks that can be reduced through diversification. Since the unsystematic risks of the stocks in the two portfolios are assumed to be the same, adding "normal" stocks would not change this.
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