Troublesome coefficient of variation question.

[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.

[jemidiah]

I can give guesses, though I haven't done much in economics.

From the Wikipedia page, your COV calculation is correct. The higher the COV, the larger the standard deviation compared to the expected profit. You might call this a risky investment. Say you have a COV of 2. You have a good chance of gaining nothing (even losing some, depending on the underlying probability distribution). Now say COV is 0. You know exactly how much profit you'll get then, so it's much less risky.

So it would seem A is the least risky. You can use a pie chart to explain it. The whole pie would be "expected profit"; one fraction of the pie would be "portion of expected profit covered by 1 standard deviation (in one direction)". You would then see that A has the smallest fraction of the pie.

[th_05]

I can give guesses, though I haven't done much in economics.

From the Wikipedia page, your COV calculation is correct. The higher the COV, the larger the standard deviation compared to the expected profit. You might call this a risky investment. Say you have a COV of 2. You have a good chance of gaining nothing (even losing some, depending on the underlying probability distribution). Now say COV is 0. You know exactly how much profit you'll get then, so it's much less risky.

So it would seem A is the least risky. You can use a pie chart to explain it. The whole pie would be "expected profit"; one fraction of the pie would be "portion of expected profit covered by 1 standard deviation (in one direction)". You would then see that A has the smallest fraction of the pie.

Thank you, that was very helpful. So you don't reckon they were intending me to draw out the standard dev. curve for each investment?

[jemidiah]

So you don't reckon they were intending me to draw out the standard dev. curve for each investment?

I don't think the term "standard deviation curve" has well-defined meaning. I'll guess you meant "normal curve with the given mean [synonymous with expected value] and standard deviation". Of course, that assumes that your class is making the assumption of normally distributed profit, but that seems safe enough--you'd have to assume some distribution for profits to be at all comparable, and the central limit theorem makes the normal distribution emerge from complex interactions remarkably often. (A warning, though: the central limit theorem applies in very specific circumstances and is often overused. I would imagine that if you tried to measure real-life probability distributions for profit in some investments, you'd find that they were only somewhat normal.)

Drawing out normal curves wouldn't help a whole lot. The COV is specifically scaled by expected profit. The COV is unitless (i.e. not in $'s or something like that), but you can think of it as a percent, converted to a fraction. You can even convert it to a % if you want. Anywho, what you want to do is make a diagram comparing these fractions to each other. I suggested a pie chart since it deals only in fractions anyway.


One important thing you might accidentally get out of drawing the normal curves is to see the 68-95-99.7 rule in action. I hadn't heard the name until I searched Wikipedia for it, actually; it may well be non-standard. Regardless, it shows how, given some number of standard deviations away from the mean, you can know the exact probability of picking a point in that range. For one standard deviation, the probability is 68%. Since this is constant across your 3 investments, you can compare them with only standard deviation information.

More empirically, if you invested in A, which you knew was normally distributed with the given mean and standard deviation, you would know absolutely that you had a 68% chance of getting between 90 and 110 units of profit. Investing in B gives the same chance of getting between 90 and 150 units, with C giving between 120 and 160 units. Put in this light, it looks pretty obvious that A is a less risky investment, though the COV in some sense gives you the risk per dollar--which A also happens to win.

[th_05]

Thanks very much for the help but I am struggling in how I would set this pie chart out and how to workout how to split the portions of the pie correctly. Would I just have 3 segments of the pie with the COV of each investment in it? I am not fully understand how you think I should set out the chart. Thanks again and please wb.

[jemidiah]

I'm envisioning three pie charts; you can pick which one you'd like. In either case, you'll need to calculate the fraction of expected profit covered by one standard deviation in one direction. Nicely, this is exactly the COV. The three values, again, are A=0.1, B=0.25, C=0.14. As percents, A=10%, B=25%, C=14%.

1. Make three separate segments of the pie, like this.

2. An overlaid pie chart, like this. It would be much, much less complex, with three levels instead of two but only one segment on each level.

3. An overlaid pie chart, but without separate levels. Instead you would show the excess of each slice. The first slice would be A=10%. Say that the left edge of the slice is aligned with the north radius of the pie. The second slice would be C=14%, with the left edge directly on top of A's left edge. This slice would jut out 4% more than A, though. The B slice would be similar, though it would jut out 11%. From the picture it would be easy to see A as the least risky.

[th_05]

I think i'm with it now, thanks very much for your time, a great help.