Shares of company A are sold at 10 per share. Shares of...

Shares of company A are sold at $$\$ 10$$ per share. Shares of company B are sold at $$\$ 50$$ per share. According to a market analyst, 1 share of each company can either gain $$\$ 1$$, with probability 0.5 , or lose $$\$ 1$$, with probability 0.5 , independently of the other company. Which of the following portfolios has the lowest risk: (a) 100 shares of $\mathrm{A}$ (b) 50 shares of $\mathrm{A}+10$ shares of $\mathrm{B}$ (c) 40 shares of $\mathrm{A}+12$ shares of $\mathrm{B}$

Shares of company A are sold at $$\$ 10$$ per share. Shares of company B are sold at $$\$ 50$$ per share. According to a market analyst, 1 share of each company can either gain $$\$ 1$$, with probability 0.5 , or lose $$\$ 1$$, with probability 0.5 , independently of the other company. Which of the following portfolios has the lowest risk:
(a) 100 shares of $\mathrm{A}$
(b) 50 shares of $\mathrm{A}+10$ shares of $\mathrm{B}$
(c) 40 shares of $\mathrm{A}+12$ shares of $\mathrm{B}$

Key Concepts

Variance

Variance is a statistical measure that quantifies the spread of a set of data points around their mean. In the context of finance, it represents the dispersion of returns of an asset or portfolio, which directly relates to the level of risk. A higher variance indicates that returns can deviate more widely from the expected return, implying higher uncertainty or risk.

Portfolio Diversification

Portfolio diversification is the strategy of holding a variety of assets to reduce the overall risk exposure. By combining assets whose returns do not move perfectly in sync, investors can achieve a reduction in unsystematic risk. This concept is key in portfolio management because a diversified portfolio may experience a lower overall variance compared to the variance of its individual components.

Independence of Random Variables

The independence of random variables means that the outcome of one variable does not affect the outcome of another. In the context of asset returns, if the gains or losses of different assets occur independently, the covariance between them is zero. This simplifies the calculation of the overall portfolio variance and aids in understanding how combining independent risky assets can lead to risk reduction.

Risk Measurement

Risk measurement in finance involves quantifying the uncertainty associated with the returns on an investment. Common statistical measures include variance and standard deviation, which help investors understand the potential for losses or gains. Assessing risk is essential for making informed investment decisions and optimizing portfolio allocation based on the investor's risk tolerance.

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