Consider The Following Probability Distribution Of Returns

Consider The Following Probability Distribution Of Returns Estimate

Consider the following probability distribution of returns estimated for a proposed project that involves a new ultrasound machine:

- State of Economy: Very Poor; Probability of Occurrence: 0.20; Rate of Return: 0.0%

- State of Economy: Poor; Probability of Occurrence: 0.20; Rate of Return: 0.0%

- State of Economy: Average; Probability of Occurrence: 0.40; Rate of Return: 0.0%

- State of Economy: Good; Probability of Occurrence: 0.20; Rate of Return: 20.0%

- State of Economy: Very Good; Probability of Occurrence: 0.00; Rate of Return: 10.0%

Question 1: What is the expected rate of return on the project?

Question 2: What is the project’s standard deviation of returns?

Question 3: What is the project’s coefficient of variation (CV) of returns?

Question 4: What type of risk do the standard deviation and CV measure?

Question 5: In what situation is this risk relevant?

Sample Paper For Above instruction

The evaluation of a project's risk and return profile is essential in investment decision-making processes. The probabilistic distribution of future project returns provides vital information to quantify expected performance and associated uncertainty, helping investors and managers to make informed choices. This paper aims to analyze a proposed ultrasound machine project by calculating its expected rate of return, standard deviation, coefficient of variation, and discussing the relevance of these risk measures.

Introduction

Investments inherently involve uncertainty, which can be quantified through statistical measures such as expected return and risk metrics like standard deviation and coefficient of variation (CV). These measures assist in balancing the risk-return tradeoff and determining the appropriateness of investment choices based on an individual or institutional risk tolerance. The project under analysis involves deploying a new ultrasound machine, with its estimated returns contingent upon the prevailing economic environment.

Expected Rate of Return

The expected rate of return (ER) is an average return weighted by the probability of various economic states occuring. It is calculated using the formula:

\[ ER = \sum_{i} P_i \times R_i \]

where \( P_i \) is the probability of state \(i\) and \( R_i \) is the rate of return in that state.

Applying this to the given distribution:

- For Very Poor, \( P = 0.20 \), \( R = 0.0\% \)

- For Poor, \( P = 0.20 \), \( R = 0.0\% \)

- For Average, \( P = 0.40 \), \( R = 0.0\% \)

- For Good, \( P = 0.20 \), \( R = 20.0\% \)

- For Very Good, \( P = 0.00 \), \( R = 10.0\% \)

Note: The probabilities sum to 1, but the "Very Good" state has zero probability, indicating it is not expected to occur. Computing ER:

\[ ER = (0.20 \times 0) + (0.20 \times 0) + (0.40 \times 0) + (0.20 \times 20\%) + (0.00 \times 10\%) \]

\[ ER = 0 + 0 + 0 + 0.20 \times 20\% + 0 \]

\[ ER = 0 + 0 + 0 + 4\% + 0 = 4\% \]

Therefore, the expected rate of return for the project is 4%.

Standard Deviation of Returns

The standard deviation (σ) measures the dispersion of returns around the expected value, representing the project’s risk. It is calculated as:

\[ \sigma = \sqrt{\sum_{i} P_i (R_i - ER)^2} \]

Calculating for each state:

- Very Poor: \( (0.0\% - 4\%)^2 = ( -4\%)^2 = 16 \)

- Poor: same as Very Poor, 16

- Average: same as Very Poor, 16

- Good: \( (20\% - 4\%)^2 = (16\%)^2 = 256 \)

- Very Good: \( (10\% - 4\%)^2 = (6\%)^2= 36 \)

Weighted by their probabilities and summing:

\[ Variance = 0.20 \times 16 + 0.20 \times 16 + 0.40 \times 16 + 0.20 \times 256 + 0.00 \times 36 \]

\[ Variance = 3.2 + 3.2 + 6.4 + 51.2 + 0 = 64 \]

Thus, \( σ = \sqrt{64} = 8\% \).

The standard deviation of 8% indicates moderate variability around the expected return, suggesting a certain level of risk inherent in the project.

Coefficient of Variation of Returns

The coefficient of variation (CV) provides a normalized measure of risk per unit of expected return, calculated as:

\[ CV = \frac{\sigma}{ER} \]

\[ CV = \frac{8\%}{4\%} = 2 \]

A CV of 2 signifies that the project’s risk is twice its expected return, which may be considered high or low depending on industry standards and investor risk appetite.

Risk Measures and Their Relevance

Standard deviation and CV measure total risk, encompassing both systematic and unsystematic components; they portray the variability in returns to alert investors to potential deviations from expected outcomes. Such risk measures are especially relevant when comparing projects with different return profiles or in environments where variability significantly impacts decision-making. The risk captured by these measures influences portfolio diversification strategies and risk management policies.

Relevance of Risk Measures in Decision-Making

Understanding the risk associated with a project guides investors in aligning their risk tolerance with potential outcomes. For risk-averse investors, high standard deviation and CV might be deterrent, prompting alternative investments. Conversely, risk-tolerant investors may accept higher variability for potentially higher returns. When considering the ultrasound project, stakeholders must evaluate whether the 8% standard deviation and CV of 2 are acceptable linked to the project's strategic importance and industry standards.

Conclusion

Analyzing the likelihood-weighted returns and their dispersion elucidates the risk-return profile of an ultrasound machine project. While the expected return is modest at 4%, the risk, as indicated by an 8% standard deviation and a CV of 2, demonstrates notable variability. Such insights are pivotal for informed decision-making, especially when balancing risk appetite with expected financial performance. Ultimately, considering risk measures in context ensures a comprehensive understanding of project viability and alignment with investment goals.

References

Allen, F., & Saunders, A. (2004). Introduction to Financial Institutions, Markets, and Managerial Finance. McGraw-Hill.

Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments. McGraw-Hill Education.

Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. Wiley Finance.

Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.

Ross, S. A., Westerfield, R., & Jordan, B. D. (2016). Fundamentals of Corporate Finance. McGraw-Hill Education.