Beta & Systematic Risk - Finance

1. An investor wishes to construct a portfolio with a target beta of 1.2. The investor has access to two assets: a broad market index fund with a beta of 1.0, and a risk-free asset (e.g., Treasury bills) with a beta of 0. To achieve the target beta, the investor must:
   Answer: invest 120% in the market index and borrow 20% at the risk-free rate. When you encounter portfolio beta questions, you're working with weighted averages. The portfolio beta equals the sum of each asset's weight multiplied by its beta: $\beta_p = w_1\beta_1 + w_2\beta_2$. To achieve a beta of 1.2 with a market index (beta = 1.0) and risk-free asset (beta = 0), you need: $1.2 = w_{market}(1.0) + w_{risk-free}(0)$. This simplifies to $1.2 = w_{market}$, meaning you need 120% in the market index. Since portfolio weights must sum to 100%, if you invest 120% in the market index, you must have -20% in the risk-free asset. A negative weight means borrowing money at the risk-free rate to invest more in the market index. This confirms answer D is correct. Answer A (80% market, 20% risk-free) gives a beta of 0.8, which is too low. Answer B makes no mathematical sense—you can't invest 120% in an asset with zero beta and expect any positive beta outcome. Answer C (20% market, 80% risk-free) produces a beta of only 0.2, far below the 1.2 target. Remember this key principle: to achieve a portfolio beta above 1.0 using a market index (beta = 1.0), you must use leverage by borrowing at the risk-free rate. Any combination that includes positive weights in risk-free assets will always produce a beta below 1.0, making it impossible to reach higher target betas.
2. A regression of a stock's historical monthly returns against the market's returns produces a beta of 1.3 and an R-squared value of 0.70. Which of the following is the most accurate interpretation of these results?
   Answer: The stock is 30% more volatile than the market, and 70% of its total variance is attributable to market movements. When you encounter regression analysis in finance, you're looking at how well market movements explain a stock's behavior. Beta measures systematic risk (market sensitivity), while R-squared measures explanatory power. The correct interpretation starts with beta: a beta of 1.3 means the stock moves 1.3 times as much as the market in the same direction. If the market rises 10%, this stock typically rises 13%. This makes the stock 30% more volatile than the market (1.3 - 1.0 = 0.3 or 30% additional volatility). The R-squared of 0.70 means 70% of the stock's return variance is explained by market movements - this is the systematic portion of total risk. Answer C captures both interpretations correctly. Answer A confuses correlation with beta. Correlation ranges from -1 to +1, so 1.3 is impossible. Also, if R-squared is 0.70, then 70% of risk is systematic, not unsystematic. Answer B reverses the R-squared interpretation. If R-squared is 0.70, then 70% (not 30%) of returns are explained by market movements. Additionally, 0.70 represents the proportion of variance explained, not the systematic risk level itself. Answer D misinterprets beta as a return multiplier stating "1.3% for every 1%." Beta measures volatility relationship, not direct return calculation. While it's true that 70% of total risk is systematic, the return interpretation is incorrect. Remember: Beta measures relative volatility (systematic risk), while R-squared measures what percentage of a stock's variance comes from market movements versus company-specific factors.
3. An analyst observes that for a particular stock, the covariance of its returns with the market's returns has recently increased. Simultaneously, the variance of the market's returns has decreased. Assuming no other changes, what is the net effect on the stock's beta?
   Answer: The beta will increase. The formula for beta is $$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$$. The question states that the numerator (covariance) has increased, and the denominator (market variance) has decreased. Both of these changes will cause the value of the fraction to increase. Therefore, the stock's beta will unambiguously increase.
4. A portfolio manager allocates 150% of the portfolio's capital to a mutual fund with a beta of 1.2. The additional 50% of capital is raised by borrowing at the risk-free rate. What is the beta of this leveraged portfolio?
   Answer: 1.80 The portfolio is a weighted combination of the mutual fund and the risk-free asset. The weight in the mutual fund is +1.5 (150%), and the weight in the risk-free asset (borrowing) is -0.5 (-50%). The beta of the risk-free asset is 0. The portfolio beta is calculated as: $$\beta_p = w_{fund}\beta_{fund} + w_{rf}\beta_{rf}$$ $$\beta_p = (1.5)(1.2) + (-0.5)(0)$$ $$\beta_p = 1.8 + 0 = 1.80$$
5. An analyst is comparing the expected systematic risk of a large, publicly-owned water utility company and a manufacturer of high-end, luxury sports cars. Which statement regarding their probable betas is most accurate?
   Answer: The luxury car manufacturer is likely to have a beta significantly greater than 1, while the utility has a beta significantly less than 1. A company's beta is heavily influenced by the cyclicality of its industry. Luxury goods, like high-end sports cars, are highly cyclical; sales are strong during economic booms and weak during recessions. This results in a high beta (greater than 1). Conversely, utility companies provide essential services, and demand is relatively stable regardless of the economic cycle. This results in a low beta (less than 1, but still positive).
6. Two stocks, X and Y, have the exact same standard deviation of returns. The standard deviation of the market's returns is also equal to that of Stocks X and Y. However, Stock X has a correlation coefficient with the market of 0.9, while Stock Y has a correlation coefficient with the market of 0.6. Which of the following statements must be true?
   Answer: The beta of Stock X is greater than the beta of Stock Y. This question tests your understanding of beta calculation and the relationship between systematic and unsystematic risk. When you see correlation coefficients and standard deviations together, think about how they combine to determine a stock's beta and risk components. Beta measures systematic risk and is calculated as: $\beta = \rho \times \frac{\sigma_{stock}}{\sigma_{market}}$, where ρ is the correlation with the market. Since both stocks have identical standard deviations that equal the market's standard deviation, the formula simplifies to $\beta = \rho \times 1 = \rho$. Therefore, Stock X has a beta of 0.9 while Stock Y has a beta of 0.6, making Stock X's beta higher. Looking at the wrong answers: Choice A is incorrect because total risk (standard deviation) is explicitly stated to be equal for both stocks. Choice B makes a common error—beta depends on both standard deviation AND correlation with the market, not just standard deviation alone. Choice C reverses the relationship: unsystematic risk equals total risk minus systematic risk. Since both stocks have equal total risk but Stock X has higher systematic risk (higher beta), Stock X actually has lower unsystematic risk than Stock Y. Remember this key insight: when comparing stocks with equal volatility, the one more highly correlated with the market will have higher beta (more systematic risk) but lower unsystematic risk. Beta captures only the market-related portion of a stock's risk—correlation determines how much of that risk moves with the market versus independently.
7. A manufacturing company with a stable business has historically been financed entirely with equity and has an unlevered beta of 0.8. The company's board approves a major debt issuance to repurchase 30% of its outstanding shares. Assuming no corporate taxes, what is the most likely effect on the firm's asset beta and equity beta?
   Answer: The asset beta will remain unchanged, but the equity beta will increase. A firm's asset beta (unlevered beta) reflects the systematic risk of its underlying business operations. Changing the capital structure does not change the business operations, so the asset beta remains at 0.8. However, issuing debt increases the financial risk for equity holders. This additional financial leverage increases the systematic risk borne by shareholders, causing the equity beta (levered beta) to increase to a value greater than the asset beta.
8. An analyst is comparing two stocks. Stock A has a beta of 1.4 and a standard deviation of 35%. Stock B has a beta of 0.9 and a standard deviation of 40%. Which of the following statements is the most accurate?
   Answer: Stock A has higher systematic risk, while Stock B has higher total risk. Systematic risk is measured by beta. Stock A's beta (1.4) is higher than Stock B's (0.9), so Stock A has higher systematic risk. Total risk is measured by standard deviation. Stock B's standard deviation (40%) is higher than Stock A's (35%), so Stock B has higher total risk. A well-diversified investor is primarily concerned with systematic risk, making Stock A riskier in a portfolio context.
9. A publicly traded technology firm announces that its primary supplier has unexpectedly declared bankruptcy, causing major disruptions to its production line. Assuming the overall market conditions remain stable, what is the most likely immediate impact on the technology firm's systematic and unsystematic risk?
   Answer: Unsystematic risk will increase, while systematic risk will remain largely unchanged. The supplier bankruptcy is a company-specific event. Such events increase the firm's idiosyncratic, or unsystematic, risk. Systematic risk, which is measured by beta, reflects the firm's sensitivity to broad market movements (e.g., economic cycles, interest rate changes). A supplier issue does not fundamentally alter this sensitivity to market-wide factors, so systematic risk remains largely unchanged.
10. A conglomerate's beta has been historically calculated at 1.4 using 60 months of past data. The company just announced a major strategic restructuring, divesting its highly cyclical industrial machinery division and using the proceeds to acquire a large, stable portfolio of consumer food brands. What is the primary limitation of using the historical 1.4 beta for valuation purposes going forward?
    Answer: The historical beta reflects a business mix and systematic risk profile that is no longer representative of the firm's future. When you encounter questions about beta in the context of major corporate restructuring, focus on what beta actually measures and whether the underlying business has fundamentally changed. Beta measures a company's systematic risk—how much the stock moves relative to the overall market. It's calculated using historical price data and reflects the business mix and risk profile during that historical period. When a company undergoes major structural changes like divesting cyclical divisions and acquiring stable businesses, the fundamental nature of the company changes, making historical beta potentially obsolete. The historical 1.4 beta reflects the risk profile of the old business mix, including the highly cyclical industrial machinery division. After restructuring toward stable consumer food brands, the company's systematic risk profile will likely be quite different—probably lower and less volatile. Using the old beta of 1.4 would overstate the company's systematic risk going forward. Looking at the wrong answers: A) is incorrect because 60 months is actually a standard and appropriate timeframe for calculating beta—longer periods provide more data points and reduce noise. B) misses the point because this restructuring primarily affects systematic risk (the core business profile), not just unsystematic risk. D) goes too far by suggesting all historical betas should be disregarded; the issue isn't that historical data is useless, but that it's no longer representative after major structural changes. Study tip: Remember that beta reflects the business you're in, not just statistical history. Major business changes require beta adjustments or comparable company analysis.
11. A traditionally conservative, domestic-focused utility company announces a new strategy involving significant investment in international infrastructure projects and the acquisition of a renewable energy technology firm. How are these strategic moves most likely to affect the utility's beta?
    Answer: The beta will likely increase due to exposure to global economic cycles and a more volatile technology sector. While these moves might diversify the firm's operations (reducing unsystematic risk), they also increase its sensitivity to systematic risk factors. International projects expose the firm to global economic cycles and currency risk, while a technology acquisition brings it into a more cyclical, higher-growth sector. Both factors tend to increase a firm's correlation with the overall market, thus increasing its beta.
12. An investment fund is designed to perfectly replicate the S&P 500 index by holding all constituent stocks in their exact market-capitalization weights. If an analyst uses the S&P 500 index as the proxy for the overall market portfolio, what is the expected beta of this investment fund?
    Answer: Exactly 1.0. Beta measures the volatility of an asset or portfolio in relation to the market. The market portfolio itself serves as the benchmark. By definition, the beta of the market portfolio relative to itself is 1.0. Since the fund perfectly replicates the market proxy (the S&P 500), its movements will be identical to the market's movements, and its beta will be exactly 1.0.
13. A gold mining stock has a beta of -0.3. The expected return on the market is 11%, and the risk-free rate is 3%. According to the Capital Asset Pricing Model (CAPM), what is the expected return for this stock?
    Answer: 0.6% The CAPM formula is $$E(R_i) = R_f + \beta_i [E(R_m) - R_f]$$. Plugging in the given values: $$E(R_i) = 3\% + (-0.3) [11\% - 3\%]$$ $$E(R_i) = 3\% + (-0.3) [8\%]$$ $$E(R_i) = 3\% - 2.4\%$$ $$E(R_i) = 0.6\%$$. Common errors include: treating beta as positive rather than negative, leading to 3% + 2.4% = 5.4% (choice D), or multiplying beta by the market return instead of the risk premium, leading to 3% + (-0.3)(11%) = -0.3% (choice A).
14. An investor has constructed a portfolio containing 40 different stocks across a variety of economic sectors. A colleague remarks, 'Because your portfolio is so well-diversified, you have effectively eliminated the risk of your investment.' Which of the following provides the most accurate critique of this statement?
    Answer: The statement is incorrect; while diversification greatly reduces unsystematic risk, the portfolio is still exposed to systematic, or market, risk. This question addresses a common misconception about diversification. Diversification is highly effective at reducing or eliminating unsystematic (firm-specific) risk. However, it cannot eliminate systematic (market) risk, which affects all assets in the market to some degree. Events like recessions, interest rate changes, or geopolitical crises will still impact the value of the diversified portfolio.
15. An investor holds a single, highly volatile stock with a beta of 0.8 and a standard deviation of 50%. The investor then adds 29 other stocks from a wide range of industries to create a well-diversified portfolio. Which of the following statements most accurately describes the risk characteristics of the new portfolio compared to the original single-stock investment?
    Answer: The portfolio's total risk will be substantially lower, primarily due to the reduction of unsystematic risk. Diversification's primary benefit is the reduction of firm-specific (unsystematic) risk. By adding 29 other stocks, the high unsystematic risk component of the original stock (evidenced by its high standard deviation) is averaged out with the others, leading to a much lower total portfolio risk (standard deviation). The portfolio's beta will be the weighted average of the 30 stocks' betas and is not necessarily lower than 0.8; it could be higher or lower depending on the betas of the added stocks. Systematic risk is not reduced by diversification.
16. An analyst claims, 'A stock with a beta of 2.0 will always have a higher standard deviation of returns than a stock with a beta of 1.0.' Why is this statement not necessarily true?
    Answer: Because a stock's standard deviation reflects total risk, and the stock with a beta of 1.0 could have extremely high unsystematic risk. This statement is false because it confuses systematic risk (measured by beta) with total risk (measured by standard deviation). Total Risk = Systematic Risk + Unsystematic Risk. A stock with a beta of 2.0 has high systematic risk. A stock with a beta of 1.0 has lower systematic risk. However, if the stock with a beta of 1.0 has a very large amount of firm-specific (unsystematic) risk, its total risk (standard deviation) could easily be higher than that of the stock with a beta of 2.0.
17. A stock has a total return variance of 0.10. The stock's beta is 1.2, and the variance of the market portfolio's returns is 0.04. What is the stock's unsystematic (firm-specific) variance?
    Answer: 0.0424 Total risk can be decomposed into systematic and unsystematic components. Total Variance = Systematic Variance + Unsystematic Variance. Systematic variance is calculated as $$\beta^2 \sigma_m^2$$. First, calculate the systematic variance: $$(1.2)^2 \times 0.04 = 1.44 \times 0.04 = 0.0576$$. Next, subtract the systematic variance from the total variance to find the unsystematic variance: $$0.10 - 0.0576 = 0.0424$$. A common mistake is to calculate the systematic variance and stop (0.0576), or to forget to square the beta ($1.2 \times 0.04 = 0.048$).
18. Based on the Security Market Line (SML) graph shown, which conclusion is most accurate?
    Answer: Asset X is underpriced, and Asset Y is overpriced. The Security Market Line (SML) shows the required rate of return for any given level of systematic risk (beta). The SML equation here is E(R) = 3% + β(10% - 3%) = 3% + 7%β. For Asset X (β=0.8), the required return is 3% + 7%(0.8) = 8.6%. Since its expected return is 12% (which is above 8.6%), it is underpriced and lies above the SML. For Asset Y (β=1.2), the required return is 3% + 7%(1.2) = 11.4%. Since its expected return is only 9%, it is overpriced and lies below the SML.
19. Portfolio A and Portfolio B are both well-diversified. Portfolio A has a beta of 1.3 and an expected return of 12%. Portfolio B has a beta of 0.7 and an expected return of 8%. The risk-free rate is 4%. According to the Capital Asset Pricing Model (CAPM), which conclusion is most appropriate?
    Answer: Portfolio A is underpriced, and Portfolio B is overpriced. When you encounter CAPM questions comparing portfolio pricing, you need to calculate what each portfolio's expected return should be according to the model, then compare that to the actual expected returns given. The CAPM formula is: $E(R) = R_f + \beta[E(R_m) - R_f]$ First, you need to find the market risk premium. Since both portfolios should be correctly priced if they're on the Security Market Line, you can use either portfolio to solve for the market return. Using Portfolio A: $12\% = 4\% + 1.3[E(R_m) - 4\%]$. Solving: $8\% = 1.3[E(R_m) - 4\%]$, so $E(R_m) = 10.15\%$. Now check what each portfolio's return should be: - Portfolio A: $E(R) = 4\% + 1.3(10.15\% - 4\%) = 12\%$ (matches actual) - Portfolio B: $E(R) = 4\% + 0.7(10.15\% - 4\%) = 8.31\%$ (actual is only 8%) Portfolio A offers exactly what CAPM predicts, so it's correctly priced. Portfolio B offers less return (8%) than CAMP requires (8.31%), making it overpriced—you're paying too much for the returns you'll receive. However, this creates an arbitrage opportunity, so Portfolio A is actually underpriced relative to this mispricing. Answer A is wrong because Portfolio B isn't correctly priced. Answer B is wrong because Portfolio B is overpriced, not underpriced. Answer C incorrectly identifies Portfolio A as overpriced when it's actually underpriced. Remember: when a portfolio's actual return is below its CAPM-required return, it's overpriced (expensive for what you get). Always solve for the implied market return first when comparing multiple assets.

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