# Troublesome coefficient of variation question

• th_05
In summary, given the expected profits and standard deviations for investments A, B, and C, the coefficient of variation was calculated to be 0.1, 0.25, and 0.14 respectively. Based on the assumption of a normal distribution, investment A has the least risk as it has the smallest standard deviation of $10, resulting in a narrower range of possible profits. This suggests that investment A may be a more stable and predictable option compared to the other investments. However, further analysis using a diagram may provide more clarity on the relative risk of these investments. th_05 Given the following data for three possibile investments, A, B and C, calculate the coefficient of variation and with the aid of a diagram explain which is the least risky investment. Expected Profit: A - 100 B - 120 C - 140 Standard Devi.: A - 10 B - 30 C - 20 I presume to calculate the COV you divide the standard deviation by the mean, to give you: A: 100/10 = 0.1 B: 30/120 = 0.25 C: 20/140 = 0.14 I am struggling with how/what sort of diagram to use and how to explain which is the least risky investment. Any ideas would be great. In your first calculation for COV, you have 100/10 and you should probably have 10/100. Neither of the statistics books I have has a definition for coefficient of variation, so that's a new one on me. Something that might be helpful is that in a normal distribution, about 69% of the values fall within one standard deviation of the mean. If the profits in these investments are normally distributed, then for investment A, we would expect that 69% of the time the profit would be between$90 and \$110. If you look at each of the other investments in a similar manner you'll get a range of possible profits for each, so maybe you can determine the relative risk in this way.

The coefficient of variation (COV) is a measure of risk that takes into account the variability of a data set relative to its mean. In this case, we are looking at the COV of three different investments, A, B, and C. To calculate the COV, we divide the standard deviation by the mean. This gives us a standardized measure of risk, allowing us to compare the riskiness of each investment.

In this scenario, the COV for investment A is 0.1, for B it is 0.25, and for C it is 0.14. This means that investment B has the highest COV, indicating the highest level of risk among the three options. Investment A has the lowest COV, indicating the lowest level of risk. Investment C falls in between the two, with a moderately low COV.

To better understand the relationship between risk and the COV, we can use a risk-return diagram. This diagram plots the expected return (profit) on the y-axis and the risk (represented by the COV) on the x-axis. The diagram will show the risk-return trade-off, where higher levels of risk are associated with potentially higher returns.

In this case, we can plot the expected profits for each investment (A: 100, B: 120, C: 140) on the y-axis and their corresponding COV values on the x-axis. This will give us three points on the diagram, with investment A having the lowest COV and the lowest expected profit, investment B having the highest COV and the highest expected profit, and investment C falling in between the two.

Based on this diagram, we can see that investment A is the least risky option, as it has the lowest COV and the lowest expected profit. Investment B has the highest potential return, but also the highest risk, making it a riskier choice. Investment C falls in between the two, with a moderate level of risk and a moderate expected profit.

In conclusion, the coefficient of variation is a useful measure of risk that allows us to compare investments with different levels of variability. In this scenario, investment A is the least risky option based on its lower COV, while investment B carries the highest level of risk. The risk-return diagram can further illustrate this relationship and aid in decision making.

## 1. What is the coefficient of variation?

The coefficient of variation is a statistical measure that represents the ratio of the standard deviation to the mean of a dataset. It is used to measure the relative variability of a dataset and is expressed as a percentage.

## 2. How is the coefficient of variation calculated?

The coefficient of variation is calculated by dividing the standard deviation of a dataset by its mean, and then multiplying the result by 100 to get a percentage value. The formula is: CV = (standard deviation / mean) * 100.

## 3. Why is the coefficient of variation important?

The coefficient of variation is important because it allows for the comparison of variability between different datasets, even if they have different scales or units. It also provides a standardized way to measure variability, which can be useful in making decisions and drawing conclusions from data.

## 4. What is considered a high or low coefficient of variation?

A high coefficient of variation indicates that the data has a large amount of variability relative to the mean, while a low coefficient of variation indicates that the data has a small amount of variability. The interpretation of what is considered high or low may vary depending on the context and the type of data being analyzed.

## 5. How can the coefficient of variation be interpreted?

The coefficient of variation can be interpreted as a measure of the relative risk associated with a particular dataset. A higher coefficient of variation indicates a higher risk of variability, while a lower coefficient of variation indicates a lower risk. It can also be used to compare the variability of different datasets and to identify outliers or unusual data points.

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