Math Problem Statement

The query submitted by the user: Calculate the rate of return for each year and determine the average return, standard deviation of returns, coefficient of variation, and decision regarding inclusion in the portfolio.

Solution

Let's break down the problem step by step.

Part a: Calculate the rate of return for each year

The rate of return for each year is calculated using the formula:

Rate of Return=End PriceBeginning PriceBeginning Price\text{Rate of Return} = \frac{\text{End Price} - \text{Beginning Price}}{\text{Beginning Price}}

2009:

Rate of Return=21.5514.3614.36=7.1914.360.5007 or 50.07%\text{Rate of Return} = \frac{21.55 - 14.36}{14.36} = \frac{7.19}{14.36} \approx 0.5007 \text{ or } 50.07\%

2010:

Rate of Return=64.7821.5521.55=43.2321.552.0056 or 200.56%\text{Rate of Return} = \frac{64.78 - 21.55}{21.55} = \frac{43.23}{21.55} \approx 2.0056 \text{ or } 200.56\%

2011:

Rate of Return=72.3864.7864.78=7.6064.780.1173 or 11.73%\text{Rate of Return} = \frac{72.38 - 64.78}{64.78} = \frac{7.60}{64.78} \approx 0.1173 \text{ or } 11.73\%

2012:

Rate of Return=91.8072.3872.38=19.4272.380.2683 or 26.83%\text{Rate of Return} = \frac{91.80 - 72.38}{72.38} = \frac{19.42}{72.38} \approx 0.2683 \text{ or } 26.83\%

Part b: Calculate the average return

The average return (expected value) over these 4 years is calculated as:

Average Return=(Rate of Return)Number of Years=0.5007+2.0056+0.1173+0.26834=2.891940.722975 or 72.30%\text{Average Return} = \frac{\sum (\text{Rate of Return})}{\text{Number of Years}} = \frac{0.5007 + 2.0056 + 0.1173 + 0.2683}{4} = \frac{2.8919}{4} \approx 0.722975 \text{ or } 72.30\%

Part c: Calculate the standard deviation of returns

The standard deviation measures the variability of returns. First, calculate the squared deviations from the mean for each year, then find the variance, and finally take the square root of the variance to get the standard deviation.

  1. (0.50070.7230)20.04949(0.5007 - 0.7230)^2 \approx 0.04949
  2. (2.00560.7230)21.64119(2.0056 - 0.7230)^2 \approx 1.64119
  3. (0.11730.7230)20.36690(0.1173 - 0.7230)^2 \approx 0.36690
  4. (0.26830.7230)20.20715(0.2683 - 0.7230)^2 \approx 0.20715

Variance:

Variance=0.04949+1.64119+0.36690+0.2071542.2647340.56618\text{Variance} = \frac{0.04949 + 1.64119 + 0.36690 + 0.20715}{4} \approx \frac{2.26473}{4} \approx 0.56618

Standard Deviation:

Standard Deviation=0.566180.7525 or 75.25%\text{Standard Deviation} = \sqrt{0.56618} \approx 0.7525 \text{ or } 75.25\%

Part d: Coefficient of Variation

The coefficient of variation (CV) is calculated as:

Coefficient of Variation=Standard DeviationAverage Return=0.75250.72301.041\text{Coefficient of Variation} = \frac{\text{Standard Deviation}}{\text{Average Return}} = \frac{0.7525}{0.7230} \approx 1.041

Part e: Decision regarding inclusion in the portfolio

Since Mike's rule is to only include securities with a coefficient of variation below 0.90, the Hi-Tech stock with a CV of 1.041 should not be included in his portfolio.


Do you want details or have any questions?

Here are some relative questions to consider:

  1. How would different dividend payments affect the rate of return calculation?
  2. What would happen if one of the years had a negative rate of return?
  3. How can the coefficient of variation be useful in comparing different stocks?
  4. How does the standard deviation impact the risk assessment of an investment?
  5. Could this method be applied to other financial metrics besides stock returns?

Tip: When analyzing investment returns, always consider the risk associated with the standard deviation and coefficient of variation, as they can give you a clearer picture of the investment’s performance stability.

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Math Problem Analysis

Mathematical Concepts

Finance
Statistics
Probability

Formulas

Rate of Return = (End Price - Beginning Price) / Beginning Price
Average Return = Sum of Rates of Return / Number of Years
Variance = Sum of Squared Deviations / Number of Years
Standard Deviation = Square Root of Variance
Coefficient of Variation = Standard Deviation / Average Return

Theorems

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Suitable Grade Level

Advanced High School