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Risk & Volatility Measures - Business Statistics

  1. An investor creates a portfolio with two assets, A and B, which have a perfect negative correlation (ρ=−1\rho = -1). Both assets have a standard deviation of 12%. The investor allocates 50% of the funds to Asset A and 50% to Asset B. What is the standard deviation of this portfolio?
    Answer: 0% The formula for the variance of a two-asset portfolio is σp2=wA2σA2+wB2σB2+2wAwBρABσAσB\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho_{AB}\sigma_A\sigma_B. Given: wA=0.5w_A = 0.5, wB=0.5w_B = 0.5, σA=0.12\sigma_A = 0.12, σB=0.12\sigma_B = 0.12, and ρAB=−1\rho_{AB} = -1. σp2=(0.5)2(0.12)2+(0.5)2(0.12)2+2(0.5)(0.5)(−1)(0.12)(0.12)\sigma_p^2 = (0.5)^2(0.12)^2 + (0.5)^2(0.12)^2 + 2(0.5)(0.5)(-1)(0.12)(0.12) σp2=(0.25)(0.0144)+(0.25)(0.0144)−(0.5)(0.0144)\sigma_p^2 = (0.25)(0.0144) + (0.25)(0.0144) - (0.5)(0.0144) σp2=0.0036+0.0036−0.0072=0\sigma_p^2 = 0.0036 + 0.0036 - 0.0072 = 0. The standard deviation is the square root of the variance, so σp=0=0\sigma_p = \sqrt{0} = 0. A perfectly negatively correlated portfolio can, with the right weights, eliminate all risk.
  2. A market research firm conducts a survey with a sample of n=10 respondents and calculates a sample variance of 18. They later decide to merge this data with a second independent sample of n=10 from the same population, which, by chance, has the exact same data values as the first. What is the sample variance of the combined sample of n=20?
    Answer: 18.95 This requires careful application of the sample variance formula. 1. For the first sample, the sum of squared deviations (SSD) is SSD1=s12×(n1−1)=18×(10−1)=18×9=162SSD_1 = s_1^2 \times (n_1-1) = 18 \times (10-1) = 18 \times 9 = 162. 2. Since the second sample is identical, its mean and SSD are also the same: SSD2=162SSD_2 = 162. 3. When combining the samples, the new mean is the same as the old mean. The total SSD is SSDtotal=SSD1+SSD2=162+162=324SSD_{total} = SSD_1 + SSD_2 = 162 + 162 = 324. 4. The new sample size is nnew=10+10=20n_{new} = 10 + 10 = 20. 5. The new sample variance is snew2=SSDtotal/(nnew−1)=324/(20−1)=324/19≈18.95s_{new}^2 = SSD_{total} / (n_{new}-1) = 324 / (20-1) = 324 / 19 \approx 18.95.
  3. The total variance of a firm's profits is composed of systematic variance (related to the overall economy) and unsystematic variance (specific to the firm). The firm undertakes a major strategic initiative to diversify its operations by acquiring a company in a completely unrelated industry. What is the most likely effect on the components of its profit variance?
    Answer: Unsystematic variance will decrease, while systematic variance will remain largely unchanged. This question applies a core principle of diversification from finance. Systematic risk is market-wide and cannot be eliminated through diversification. It affects all industries. Unsystematic risk is firm-specific or industry-specific. By diversifying into an unrelated industry, the firm reduces its exposure to shocks that affect only its original industry. This action directly targets and reduces the unsystematic component of variance. The systematic component, tied to the broader economy, is not reduced by this type of diversification.
  4. An investment fund manager is comparing two portfolios. Portfolio A has an expected return of 12% with a standard deviation of 8%, while Portfolio B has an expected return of 15% with a standard deviation of 18%. If an investor's primary concern is risk-adjusted return measured by the coefficient of variation, which statement best describes the comparison?
    Answer: Portfolio A is less risky per unit of return because its coefficient of variation is approximately 0.67 compared to Portfolio B's 1.20 The coefficient of variation (CV) = standard deviation ÷ expected return. For Portfolio A: CV = 8% ÷ 12% = 0.67. For Portfolio B: CV = 18% ÷ 15% = 1.20. A lower CV indicates less risk per unit of return, so Portfolio A is preferable. Choice B incorrectly calculates CVs as 0.83 and 1.50. Choice C has the correct CVs but assigns them to wrong portfolios. Choice D reverses the interpretation of which portfolio is better.
  5. A quality control engineer measures the diameter of manufactured bolts and finds that the process has a variance of 0.25 mm². After implementing a new manufacturing technique, the variance decreases to 0.16 mm². What percentage reduction in risk, as measured by standard deviation, has been achieved?
    Answer: 20% reduction because the standard deviation decreased from 0.5 mm to 0.4 mm When you encounter quality control problems involving variance and standard deviation, remember that risk is typically measured by standard deviation (not variance), since it's in the same units as the original measurements. To find the percentage reduction in risk, you need to work with standard deviations. First, convert the variances to standard deviations by taking the square root: 0.25=0.5\sqrt{0.25} = 0.5 mm and 0.16=0.4\sqrt{0.16} = 0.4 mm. The reduction is 0.5−0.4=0.10.5 - 0.4 = 0.1 mm, which represents a 0.10.5=0.20=20%\frac{0.1}{0.5} = 0.20 = 20\% reduction. Choice C correctly identifies this 20% reduction in standard deviation from 0.5 mm to 0.4 mm. Choice A makes a calculation error when computing the new standard deviation, stating it's 0.28 mm instead of 0.4 mm. This leads to an incorrect 44% reduction figure. Choice B confuses the concept by calculating the percentage based on variance reduction rather than standard deviation reduction. While the variance did decrease by 0.09 (from 0.25 to 0.16), risk reduction should be measured using standard deviation since it represents the actual spread in the original units. Choice D incorrectly applies the variance reduction percentage (64%) to standard deviation. Since standard deviation is the square root of variance, the percentage changes are different for these two measures. Remember: when measuring risk reduction in quality control, always work with standard deviation rather than variance, since standard deviation is expressed in the same units as your measurements and provides a more intuitive sense of variability.
  6. A retail chain analyzes monthly sales data for two product categories. Category X has a mean monthly sales of $50,000 with a variance of $900,000,000. Category Y has a mean monthly sales of $75,000 with a standard deviation of $25,000. Which category exhibits greater relative volatility, and what is the difference in their coefficients of variation?
    Answer: Category X has greater relative volatility, with coefficients of variation differing by approximately 0.27 For Category X: Standard deviation = √900,000,000 = $30,000. Coefficient of variation = 30,000/50,000 = 0.60. For Category Y: Coefficient of variation = 25,000/75,000 = 0.333. Category X has greater relative volatility (0.60 > 0.333). The difference in coefficients of variation is 0.60 - 0.333 = 0.267 ≈ 0.27. Choice B incorrectly identifies Category Y as having greater volatility. Choice C gives the CV for Category X rather than the difference. Choice D incorrectly suggests equal volatility.
  7. A dataset of 100 customer satisfaction scores has a standard deviation of 5 points. A data entry error is discovered: one score was entered as 100, but it should have been 50. The mean score for the original dataset was 60. How will correcting this single outlier affect the standard deviation?
    Answer: It will decrease because a value far from the mean has been moved closer to the mean. Standard deviation measures the average distance of data points from the mean. The erroneous score of 100 was 40 points away from the original mean of 60. The corrected score of 50 is only 10 points away from the original mean. By moving an extreme value significantly closer to the center of the distribution, the overall spread of the data is reduced. Since variance and standard deviation are sensitive to outliers, this correction will cause the standard deviation to decrease.
  8. A production line's output is audited. For a sample of 1000 units, the mean weight was 50.0g, the minimum weight was 50.0g, and the maximum weight was 50.0g. What is the variance of this sample?
    Answer: 0.0 Variance measures the spread or dispersion of data points around the mean. If the minimum, maximum, and mean values are all identical, it means that every single data point in the sample is the same value (50.0g). In such a dataset, there is no variation at all. The deviation of each point from the mean is zero, making the sum of squared deviations zero, and therefore the variance is zero.
  9. A fund manager evaluates two stocks. Stock X has a mean monthly return of 2.0% and a variance of 0.0016. Stock Y has a mean monthly return of 1.0% and a variance of 0.0009. Based on the coefficient of variation, which statement is most accurate?
    Answer: Stock Y is 50% riskier than Stock X. When comparing investment risks, you need the coefficient of variation (CV), which measures relative risk by standardizing volatility against returns. The CV equals the standard deviation divided by the mean, allowing you to compare assets with different return levels. For Stock X: Standard deviation = 0.0016=0.04\sqrt{0.0016} = 0.04 or 4%. CV = 0.040.02=2.0\frac{0.04}{0.02} = 2.0 For Stock Y: Standard deviation = 0.0009=0.03\sqrt{0.0009} = 0.03 or 3%. CV = 0.030.01=3.0\frac{0.03}{0.01} = 3.0 Stock Y's CV is 50% higher than Stock X's (3.0 vs 2.0), meaning Stock Y is 50% riskier on a relative basis. This makes answer D correct. Answer A incorrectly suggests Stock X is riskier, when it's actually Stock Y. The calculation would need Stock X's CV to be twice Stock Y's CV, but it's the reverse relationship. Answer B is wrong because the coefficients of variation are clearly different (2.0 vs 3.0), so relative risk levels aren't equal. Answer C falls into the trap of using absolute variance rather than relative risk. While Stock X does have higher variance (0.0016 vs 0.0009), this ignores that Stock X also has twice the expected return. Raw variance doesn't account for the return differential. Remember: When comparing investments with different return levels, always use the coefficient of variation rather than standard deviation or variance alone. CV reveals which investment gives you more "bang for your risk buck."
  10. The variance of monthly sales revenue for a retail store is $16,000,000. To boost sales, the company introduces a new policy that adds a fixed $2,000 bonus to every salesperson's monthly commission, which is a component of sales revenue. Assuming this is the only change, what will be the new variance of the monthly sales revenue?
    Answer: $16,000,000 Variance and standard deviation are measures of dispersion or spread. Adding a constant value to every data point in a set shifts the entire set but does not change its spread. The differences between each data point and the mean remain the same. Therefore, the variance is unaffected. The new variance will remain $16,000,000.
  11. An analyst examines the profits of a small business for the last four quarters: $30k, $50k, $20k, and $40k. The analyst's goal is to measure the volatility of only these specific four quarters to report to the owner. Which of the following is the correct volatility measure and value?
    Answer: Population standard deviation of $11.18k The key phrase is 'only these specific four quarters,' which implies the data set is a population, not a sample of a larger set of possibilities. Therefore, population variance/standard deviation formulas (dividing by N) should be used. 1. Calculate the mean: ($30+$50+$20+$40)/4 = $35k. 2. Calculate sum of squared deviations: (30-35)² + (50-35)² + (20-35)² + (40-35)² = 25 + 225 + 225 + 25 = 500. 3. Population variance: 500/4 = 125k². 4. Population standard deviation: √125 ≈ $11.18k. The sample formulas (dividing by n-1=3) would give variance of 500/3 = 166.67k² and standard deviation of √166.67 = 12.91k, but these are incorrect since we're treating this as the complete population.
  12. A financial analyst is comparing the risk profiles of two mutual funds using their monthly return data. Fund Alpha has monthly returns with a standard deviation of 2.1% and a mean return of 1.4%. Fund Beta has monthly returns with a variance of 0.0009 and a mean return of 0.9%. If an investor prioritizes lower relative risk, which fund should they choose and why?
    Answer: Fund Alpha, because its coefficient of variation of 1.50 is lower than Fund Beta's coefficient of variation of 3.33 When comparing investments with different return levels, you need to measure relative risk rather than absolute risk. The coefficient of variation (CV) standardizes risk by dividing standard deviation by the mean return, allowing fair comparison between investments with different expected returns. First, let's calculate each fund's coefficient of variation. Fund Alpha has a standard deviation of 2.1% and mean return of 1.4%, so its CV = 2.1% ÷ 1.4% = 1.50. Fund Beta has a variance of 0.0009, which means its standard deviation is 0.0009=0.03=3.0%\sqrt{0.0009} = 0.03 = 3.0\%. With a mean return of 0.9%, Fund Beta's CV = 3.0% ÷ 0.9% = 3.33. Fund Alpha's lower coefficient of variation (1.50 vs. 3.33) indicates lower relative risk, making it the better choice for risk-averse investors. Answer D correctly identifies this relationship. Answer A incorrectly states that Fund Beta has lower relative risk when it actually has higher relative risk. Answer B makes the same error, claiming 3.33 is lower than 1.50. Answer C focuses on absolute standard deviation rather than relative risk—while Fund Beta does have higher absolute risk (3.0% vs. 2.1%), this misses the point about relative risk comparison. Remember: when comparing investments with different return levels, always use the coefficient of variation rather than standard deviation alone. The CV tells you how much risk you're taking per unit of expected return, which is the fair way to compare different investment opportunities.
  13. A quality control process has a variance of 0.04 cm² in the diameter of ball bearings. If a new polishing step is implemented that uniformly reduces the diameter of every ball bearing by 0.01 cm, what will be the standard deviation of the diameters after the new step is applied?
    Answer: 0.20 cm This question involves two concepts. First, subtracting a constant from every value in a dataset shifts the mean but does not change the spread. Therefore, the variance and standard deviation remain unchanged. Second, the question provides the variance and asks for the standard deviation. Original Variance = 0.04 cm². The new polishing step does not change the variance. New Variance = 0.04 cm². Standard Deviation = Variance=0.04=0.20\sqrt{\text{Variance}} = \sqrt{0.04} = 0.20 cm.
  14. When presenting volatility metrics to an executive team with a limited statistical background, a financial analyst chooses to report standard deviation rather than variance. What is the primary reason for this choice?
    Answer: The standard deviation is expressed in the same units as the original data, making it more intuitive to interpret. The main advantage of the standard deviation over the variance is its interpretability. Standard deviation is expressed in the same units as the original data (e.g., dollars, units sold, etc.), whereas variance is in squared units (e.g., dollars squared). This makes standard deviation directly comparable to the mean and easier to understand in a practical business context.
  15. A manufacturing company tracks daily production output. Over the past 50 days, the variance in daily output was 144 units². If the company implements a new quality control system that reduces variability by 25%, what will be the new standard deviation of daily output?
    Answer: 10.39 units because reducing variance by 25% results in a new variance of 108 units², and standard deviation equals the square root of variance Original variance = 144 units², so original standard deviation = √144 = 12 units. A 25% reduction in variance means new variance = 144 × (1 - 0.25) = 144 × 0.75 = 108 units². The new standard deviation = √108 ≈ 10.39 units. Choice A incorrectly assumes the standard deviation reduces by 25% rather than the variance. Choice C applies an incorrect square root adjustment. Choice D confuses variance with standard deviation.
  16. The variance of a set of project completion times is 36 days². The project manager needs to report this volatility metric on an international dashboard where the standard unit of time is weeks. What is the standard deviation of the completion times in weeks?
    Answer: 0.86 weeks This is a two-step transformation problem. 1. First, find the standard deviation in the original units (days). The standard deviation is the square root of the variance: σdays=36 days2=6 days\sigma_{days} = \sqrt{36 \text{ days}^2} = 6 \text{ days}. 2. Second, convert the standard deviation from days to weeks. Since there are 7 days in a week, we divide by 7. This is equivalent to multiplying by a constant k = 1/7. The standard deviation scales directly with this constant. σweeks=σdays×(1/7)=6/7≈0.857\sigma_{weeks} = \sigma_{days} \times (1/7) = 6 / 7 \approx 0.857 weeks.
  17. A U.S.-based multinational corporation finds that the standard deviation of its European division's annual profit is €3 million. If the average exchange rate during the period was $1.10 per euro, what is the variance of the annual profit expressed in U.S. dollars?
    Answer: $10.89 million This is a multi-step problem involving unit conversion and the properties of variance. First, convert the standard deviation from euros to dollars. If you multiply a dataset by a constant 'k', the new standard deviation is 'k' times the old standard deviation. Here, k = 1.10. New Standard Deviation (in dollars) = €3 million * 1.10 $/€ = $3.3 million. Second, the question asks for the variance, which is the square of the standard deviation. New Variance (in dollars squared) = ($3.3 million)² = $10.89 million.
  18. Which of the following statements provides the most accurate critique of using standard deviation as the sole measure of investment risk?
    Answer: It fails to distinguish between upside volatility (favorable outcomes) and downside volatility (unfavorable outcomes). When evaluating investment risk measures, you need to understand what standard deviation actually captures versus what investors truly care about. Standard deviation measures total volatility - how much returns spread out from the average in both directions. The key insight is that standard deviation treats all deviations equally, whether they represent gains or losses. If an investment has high volatility because it sometimes delivers exceptionally good returns, standard deviation flags this as "risky" even though upside surprises benefit investors. What investors actually fear is downside risk - the chance of losing money or underperforming expectations. Answer D correctly identifies this fundamental limitation: standard deviation fails to distinguish between favorable and unfavorable volatility. Let's examine why the other options are incorrect. Option A is wrong because standard deviation can be calculated and interpreted for any distribution, not just normal ones - though non-normal distributions may require additional analysis. Option B misunderstands the relationship between standard deviation and variance; standard deviation is simply the square root of variance, making it more interpretable since it's in the same units as the original data. Any rounding errors would be minimal and don't represent a fundamental flaw. Option C is completely false - standard deviation works perfectly well with negative values, including investment losses. Remember this pattern: when you see questions about risk measures, think about whether the measure captures what decision-makers actually need. Standard deviation is useful but incomplete because it doesn't differentiate between "good volatility" and "bad volatility" - a distinction that matters greatly in investment decisions.
  19. A sample of 20 shipping orders has a sample variance of 38 (days²). A 21st order is added to the sample, and its shipping time is exactly equal to the original sample mean. What is the variance of the new sample of 21 orders?
    Answer: 36.10 This problem requires understanding the formula for sample variance. 1. The original sum of squared deviations (SSD) can be found from the sample variance formula: s2=SSD/(n−1)s^2 = SSD / (n-1). So, SSD=s2×(n−1)=38×(20−1)=38×19=722SSD = s^2 \times (n-1) = 38 \times (20-1) = 38 \times 19 = 722. 2. When a new data point equal to the mean is added, the mean of the new, larger sample remains unchanged. The deviation for this new point is (mean - mean) = 0. Therefore, the SSD does not change. 3. The new sample variance is calculated with the same SSD but with the new sample size, n_new = 21. The denominator becomes (n_new - 1) = 20. 4. New Variance = SSD / (n_new - 1) = 722 / 20 = 36.1.
  20. For a sample of 41 customer call durations, an analyst computes the sum of squared deviations from the mean: ∑(xi−xˉ)2=1600\sum (x_i - \bar{x})^2 = 1600 minutes². What is the sample standard deviation?
    Answer: 6.32 minutes This question tests your understanding of sample standard deviation calculation, specifically the relationship between the sum of squared deviations and the final standard deviation value. To find sample standard deviation, you need to use the formula: s=∑(xi−xˉ)2n−1s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} You're given that ∑(xi−xˉ)2=1600\sum (x_i - \bar{x})^2 = 1600 and the sample size is 41. Substituting these values: s=160041−1=160040=40=6.32s = \sqrt{\frac{1600}{41-1}} = \sqrt{\frac{1600}{40}} = \sqrt{40} = 6.32 minutes. Looking at the wrong answers: Choice A (1.00 minute) is far too small and doesn't reflect any reasonable calculation from the given data. Choice B (39.02 minutes) appears to come from using just the degrees of freedom (40) without properly applying the square root - this represents a common calculation error. Choice C (40.00 minutes) is exactly the square root of 1600, which means someone forgot to divide by the degrees of freedom (n-1) before taking the square root. The key insight is that sample standard deviation requires dividing by (n-1), not n, and you must take the square root of the entire variance calculation. Many students either forget the division step or the square root step. Study tip: Remember the acronym "DIVIDE then ROOT" - always divide the sum of squared deviations by (n-1) first, then take the square root. Don't just take the square root of the sum of squared deviations alone.

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