Interpret Risk Statistics - Series 65

1. Calculate mean return for $r_1=5\%$, $r_2=-1\%$, $r_3=2\%$.

   $\bar{r}=2\%$. The mean is computed by summing the returns and dividing by the number of observations to find the average performance.

2. Compute Jensen's $\alpha$ if $R_i=12\%$, $R_f=2\%$, $R_m=8\%$, and $\beta=1.5$.

   $\alpha=1\%$. Alpha subtracts the CAPM-expected return from actual return to measure excess performance adjusted for systematic risk.

3. What is the formula for arithmetic mean return for returns $r_1...r_n$?

   $\bar{r}=\frac{\sum_{i=1}^{n} r_i}{n}$. The arithmetic mean return averages the sum of periodic returns over the number of periods to measure central tendency.

4. State the formula for beta of asset $i$ relative to market $m$ using covariance and variance.

   $\beta_i=\frac{\operatorname{Cov}(r_i,r_m)}{\operatorname{Var}(r_m)}$. Beta is derived as the ratio of covariance to market variance, normalizing the asset's sensitivity to market volatility.

5. Which risk does beta measure relative to a market index?

   Systematic (market) risk; sensitivity to market movements. Beta quantifies an asset's exposure to market-wide risks, indicating how its returns correlate with broader market fluctuations.

6. What is the relationship between standard deviation and variance?

   $\sigma=\sqrt{\sigma^2}$ and $s=\sqrt{s^2}$. Standard deviation is the square root of variance, providing a measure of dispersion in the same units as the original data.

7. State the formula for variance using $n$ observations and mean $\bar{x}$ (population form).

   $\sigma^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n}$. Population variance quantifies average squared deviation from the mean, dividing by the total number of observations for the entire population.

8. What does a negative beta (for example, $\beta=-0.5$) indicate?

   Tends to move opposite the market (inverse relationship). Negative beta implies an inverse correlation, where the asset's returns typically move counter to market trends for hedging potential.

9. What does $\beta=1.0$ imply about an asset's expected movement versus the market?

   Moves in line with the market (same sensitivity as the index). A beta of 1.0 indicates the asset's returns are expected to fluctuate proportionally with the market index's movements.

10. If variance is $0.04$, what is the standard deviation?

    $\sigma=0.2$. Standard deviation is the positive square root of variance, converting it back to the scale of the original data units.

11. What does $\beta>1.0$ imply about an asset relative to the market?

    More volatile than the market; amplified response to market moves. Betas exceeding 1.0 signify higher systematic risk, where asset returns exaggerate market changes due to greater sensitivity.

12. Identify which measure is total risk: standard deviation, beta, or alpha.

    Standard deviation. Standard deviation encompasses all sources of return variability, unlike beta which isolates market-related risk or alpha which assesses performance.

13. State the formula for covariance between asset $i$ and market $m$ using expectations.

    $\operatorname{Cov}(i,m)=E[(r_i-E[r_i])(r_m-E[r_m])]$. Covariance measures the expected joint variability between an asset's returns and the market's, capturing directional co-movement.

14. If $\operatorname{Cov}(i,m)=0.03$ and $\operatorname{Var}(m)=0.02$, what is $\beta$?

    $\beta=1.5$. Beta is calculated by dividing the covariance by market variance to quantify relative sensitivity to market movements.

15. If market return changes by $-10\%$, estimate asset return change when $\beta=0.8$.

    $-8\%$. The asset's return change is approximated by multiplying beta by the market's return change, reflecting systematic risk exposure.

16. State Jensen's alpha formula using actual return $R_i$ and CAPM expected return.

    $\alpha_i=R_i-[R_f+\beta_i(R_m-R_f)]$. Jensen's alpha computes the difference between actual return and CAPM-predicted return to evaluate manager skill in generating excess returns.

17. State the CAPM expected return formula using $\beta$, $R_f$, and $E[R_m]$.

    $E[R_i]=R_f+\beta_i(E[R_m]-R_f)$. The CAPM formula calculates expected return as risk-free rate plus beta-adjusted market risk premium to reflect systematic risk compensation.

18. Using CAPM, find $E[R_i]$ if $R_f=3\%$, $E[R_m]=9\%$, and $\beta=1.2$.

    $E[R_i]=10.2\%$. CAPM adds the risk-free rate to the product of beta and market risk premium to estimate required return.

19. How do you interpret a positive alpha (for example, $\alpha=+2\%$)?

    Outperformed the risk-adjusted benchmark by $+2\%$. Positive alpha indicates superior performance, exceeding the return expected given the asset's beta and market conditions.

20. How do you interpret a negative alpha (for example, $\alpha=-1\%$)?

    Underperformed the risk-adjusted benchmark by $-1\%$. Negative alpha signals underperformance, falling short of the risk-adjusted return predicted by the CAPM model.

21. Calculate population standard deviation for returns $[1\%,1\%,1\%]$.

    $\sigma=0\%$. When all returns are identical, deviations from the mean are zero, resulting in no variability in the return series.

22. What does $\beta<1.0$ imply about an asset relative to the market?

    Less sensitive than the market; dampened response to market moves. Betas below 1.0 reflect lower systematic risk, with asset returns responding less intensely to market fluctuations.

23. Which risk does standard deviation measure for an investment return series?

    Total volatility (dispersion of returns around the mean). Standard deviation captures the overall variability in returns, reflecting both systematic and unsystematic risks inherent in the investment.

24. State the formula for sample variance using $n$ observations and mean $\bar{x}$.

    $s^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}$. Sample variance estimates population variance by dividing the sum of squared deviations by $n-1$ to account for sampling bias.

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