Which Investment is Better?

nasi112

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If I have three investments. The first one has a mean of \(\displaystyle $35\) and a standard deviation of \(\displaystyle $11.18\). The second has a mean of \(\displaystyle $35\) and a standard deviation of \(\displaystyle $10.25\). The last one has a mean of \(\displaystyle $26\) and a standard deviation of \(\displaystyle $26.81\). Which investment is better?

In this kind of problems, I always look for the highest mean, so 1st and 2nd investments will be chosen? But, is it better to take larger standard deviation? In this case, choosing investment 1. Is my thinking correct?
 

JeffM

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A mean of what variable? A standard deviation of what variable?
 

nasi112

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Jomo

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I agree with JeffM? A mean of what variable?
 

nasi112

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nasi112

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The test is after 10 hours. Can anyone answer this question?
 

JeffM

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Well, i still don’t understand the question. If your goal is to have the best chance of maximizing your gain, you should at least consider the investment at 26. There is a reasonable probability that you will make a profit in excess of 52.

If your goal is to have the best chance of avoiding major loss, you should choose, the 35 with standard deviation 10.25.

The point is that there is no mathematical reason to prefer one over the other. It depends on your appetite for gain versus your aversion for or appetite for risk. Economists often make assumptions about the “normal“ views of people on such matters and assume that those views are universal. If that is the context of your question, you have not told us what criteria you are to base your decision on.
 

nasi112

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Thank you JeffM for passing by. They have given us three investments to find their mean and standard deviation. Then, they asked us. Do you prefer investment 1 or investment 2? Why? Then, they asked, do you prefer investment 1, investment 2, or investment 3? Why?

This is how the question was!
 

JeffM

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Thank you JeffM for passing by. They have given us three investments to find their mean and standard deviation. Then, they asked us. Do you prefer investment 1 or investment 2? Why? Then, they asked, do you prefer investment 1, investment 2, or investment 3? Why?

This is how the question was!
Ahh. Well that is a sensible question because they are asking your preferences. They are not saying that this can be solved purely by mathematics.

So notice that the 26 dollar payoff is very risky. Yes, you can make a bundle if it works out well, but lose your shirt if it does not.

The other two investments are far less risky. There really is not a lot of difference between them. One has a slightly greater potential for gain and a slightly greater potential for loss than the other. How do you FEEL about that.

Now we can do a lot of math with this sort of data about constructing efficient portfolios. But we must first assume something about how the owner of the portfolio feels. See Markowitz Model. It expresses those feelings in mathematical terms.
 

nasi112

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How did you decide that the $26 investment is risky? Is it because 1 standard deviation is far from the mean? One standard deviation is $26 + 26.81 = $52.81.

Does this mean we should not look how big is the mean, but we look how big is the standard deviation? Does this mean the near the standard deviation to the mean the less the risk is?
 

JeffM

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First, the problem as you have presented it is odd. In reality, investments come with a price. On the assumption that you know the distribution of each payoff’s distribution, you would take differences in price into account. That is, if one of the 35 investments has a current price of 3 and the other has a current price of 30, most people would prefer to buy the chance to get somewhere between 35 - 33 = 2 and 35 + 33 = 58 for 3, where you are almost guaranteed to make a profit, than at 30, where there is a decent probability that you will incur a loss. A problem on selection of investments where the price of the different investments is unknown but the distribution of possible future returns is known is a problem so far removed from reality that I have trouble treating it seriously.

in practice, price is known with certainty and the distribution of future returns is unknown. I once was offered an investment in exploiting a used oil well using a new technique to recover residual oil. I was given a table of oil recovered by that technique at different abandoned wells. That provided a solid basis for creating a distribution of physical returns. But monetary returns also depended on future oil prices averaged over many months. The best that could be done was to create a probability distribution that was plausible.

Second, you can attack this problem on the assumption that the prices are equal. Now if you can only buy one, you would almost certainly buy one of the 35 return investments because the expected return is higher. If you can buy more than one, you would want to look at covariances among the investments. It will almost certainly be the case that a combination of the investments will generate an expected value with less variance.

Ceteris paribus, higher expected return is better, and, for most people, lower variance is better. The practical problem is that in most cases everything is not the same and neither the expected value nor the variance is better than an educated guess.
 
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nasi112

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That's beautiful JeffM. Thanks a lot for the explanation. This means each situation has a lot factors and we cannot just confirm one is better than the other just by looking at the mean and standard deviation. I think that the world of distributions is a big sea and we are just learning a dot of that. And I also think that even expert people in this field will not give an accurate answers, but their educated guess would be better than others.

Thanks a lot again.
 
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