Finance Homeworkintroduction To Financeweek 7melicher
Finance Homework Introduction to Finance Week 7 Melicher / Norton 14th Edition / 2011 Chapter 12: P1, P2, P3, and P4
P1. From the information below, compute the average annual return, the variance, standard deviation, and coefficient of variation for each asset.
Asset A: 5%, 10%, 15%, 4%
Asset B: -6%, 20%, 2%, -5%, 10%
Asset C: 12%, 15%, 17%
Asset D: 10%, -10%, 20%, -15%, 8%, -7%
P2. Based upon your answers to question 1, which asset appears riskiest based on standard deviation? Based on coefficient of variation?
P3. Recalling the definitions of risk premiums from Chapter 8 and using the Treasury bill return in Table 12.4 as an approximation to the nominal risk-free rate, what is the risk premium from investing in each of the other asset classes listed in Table 12.4?
Table 12.4 - Historical Returns and Standard Deviation of Returns from Different Assets:
Annual Average Return: Treasury Bills (3.8%), Treasury Bonds (5.4%), Stocks (11.1%), Inflation Rate (3.2%)
Standard Deviation: Treasury Bills (3.0%), Treasury Bonds (7.6%), Stocks (20.4%), Inflation Rate (4.0%)
P4. What is the real, or after-inflation, return from each of the asset classes listed in Table 12.4?
Paper For Above instruction
The analysis of investment assets necessitates a comprehensive understanding of various measures of risk and return, which are fundamental to making informed investment decisions. This paper systematically evaluates the average returns, variances, standard deviations, and coefficients of variation for diverse assets, assesses risk levels based on these metrics, and explores the implications of risk premiums and after-inflation returns.
Introduction
In the realm of finance, understanding the behavior and characteristics of different investment assets is crucial for investors aiming to optimize their portfolios while managing risk. Core to this understanding are measures such as average returns, variance, standard deviation, and coefficient of variation, which collectively depict the risk-return profile of assets. This paper addresses these calculations for four assets based on provided historical returns, evaluates their relative risks, discusses risk premiums in relation to a risk-free asset, and estimates after-inflation returns, drawing insights from foundational financial theories and empirical data.
Analysis of Asset Returns and Risk Metrics
Calculating Average Returns
The average annual return provides a central tendency measure of an asset's historical performance. For Asset A, with returns of 5%, 10%, 15%, and 4%, the mean return is calculated as:
Average Return (A)
= (5 + 10 + 15 + 4) / 4 = 8.5%
Similarly, for Asset B, with returns of -6%, 20%, 2%, -5%, and 10%, the average is:
Average Return (B)
= (-6 + 20 + 2 - 5 +10) / 5 = 4.2%
Asset C, with returns of 12%, 15%, and 17%, has an average of:
Average Return (C)
= (12 + 15 + 17) / 3 = 14.67%
For Asset D, with returns of 10%, -10%, 20%, -15%, 8%, and -7%, the average calculates to:
Average Return (D)
= (10 - 10 + 20 - 15 + 8 - 7) / 6 ≈ 1.0%
Variance and Standard Deviation
The variance measures the dispersion of returns around the mean, while the standard deviation is its square root, representing risk in the same units as returns.
Calculations involve determining the squared deviations from the mean and averaging them. For example, Asset A's variance is:
Variance (A) = [(5 - 8.5)^2 + (10 - 8.5)^2 + (15 - 8.5)^2 + (4 - 8.5)^2] / (4 - 1) ≈ 31.67
Standard Deviation (A) = √31.67 ≈ 5.63%
Repeating similar calculations yields:
Asset B: Variance ≈ 108.3, SD ≈ 10.4%
Asset C: Variance ≈ 4.33, SD ≈ 2.08%
Asset D: Variance ≈ 61.16, SD ≈ 7.82%
Coefficient of Variation
The coefficient of variation (CV) is a risk-to-return ratio empowering comparisons across assets with differing means. It is computed as:
CV = Standard Deviation / Average Return
Applying this:
Asset A: CV ≈ 5.63 / 8.5 ≈ 0.66
Asset B: CV ≈ 10.4 / 4.2 ≈ 2.48
Asset C: CV ≈ 2.08 / 14.67 ≈ 0.14
Asset D: CV ≈ 7.82 / 1.0 ≈ 7.82
From this analysis, Asset D exhibits the highest risk relative to its return, followed by Asset B, with Asset C being the least risky per unit of return.
Risk Assessment Based on Standard Deviation and Coefficient of Variation
Standard deviation suggests that Asset D is the riskiest, given its highest SD among the assets evaluated, indicating greater variability in returns. However, the coefficient of variation indicates that Asset D's risk per unit of return is significantly higher, reinforcing its riskiness relative to other assets, particularly Asset C, which has the lowest CV. Such metrics are vital for investors to balance risk and reward effectively.
Risk Premium Calculations
Risk premium signifies the additional return expected from an investment over the risk-free rate, compensating for assumed risk. Using the treasury bill rate of 3.8%, the risk premiums for the assets are computed as:
Asset A: 8.5% - 3.8% = 4.7%
Asset B: 4.2% - 3.8% = 0.4%
Asset C: 14.67% - 3.8% = 10.87%
Asset D: 1.0% - 3.8% = -2.8%
Negative premium for Asset D indicates that its average return falls below the risk-free rate, suggesting it may not provide adequate compensation for risk, or its high volatility may outweigh the return benefit.
After-Inflation (Real) Returns
The real return considers the inflation rate, providing a clearer picture of the purchasing power of returns. It is calculated using the Fisher equation approximation:
Real Return ≈ Nominal Return - Inflation Rate
Applying this to the asset averages in Table 12.4:
Treasury Bills: 3.8% - 3.2% = 0.6%
Treasury Bonds: 5.4% - 3.2% = 2.2%
Stocks: 11.1% - 3.2% = 7.9%
Inflation Rate: 3.2% (by definition)
The real return for stocks is notably higher, reflecting their historical capacity to outpace inflation, thereby preserving and growing investors’ purchasing power. Bonds offer a moderate real return, while treasury bills provide minimal positive after-inflation gains, highlighting the importance of equity investments in combating inflation over the long term.
Conclusion
Understanding the risk and return profile of assets through calculations of average returns, variances, standard deviations, and coefficients of variation informs investment strategies. Assets with higher
variability and risk-to-return ratios, such as Asset D, may require higher expected returns to be appealing. Risk premiums further contextualize these returns relative to a risk-free benchmark, while consideration of after-inflation returns emphasizes the importance of multi-layered analysis in portfolio management. Overall, balancing risk and potential reward remains central to informed financial decision-making, especially considering macroeconomic factors such as inflation.
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