# CFA textbook says positively skewed preferred. Why?

I am studying for a lvl1 of the CFA exam.
One of the chapters deals with normal distributions of returns and goes on to discuss the skewness and I have a few questions about it.

The textbook says in a positively skewed returns:
- mode < median < mean
- investors are more attracted because the mean return falls above the median.

Here is an image of positively skewed distribution from wikipedia.

I don’t understand how the mean is bigger than mode and media.
I don’t understand why the investors are more attracted to positively skewed distribution in general.

Could somebody help me understand these things?

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that graph is ugly, but it does show you the difference

the mean is the average…and a graph is just showing you the distribution of results..if it is positively skewed, then that means there are larger positive results than negative results…if u have larger positive results it pulls the graph to the right…hence, positive skew

..if you were investing in something would you want to invest in something that has a higher probability of giving you high returns, a high probability if giving you low returns, or an equal probability of giving you either?

just remember the mean part..if its positive skew its pulling the average to the right..if its negative skew its dragging the average to the left

then just remember its alphabetical.. so either mode, median, mean or mean, median, mode

Imagine this distribution: {1, 1, 2, 100, 10000000}. Mode = 1. Median = 2. Mean = something very large. Ta da!

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The following is a better picture:

Remember: Positively skewed tails point toward the positive end of the graph heading toward positive infinity.

Negatively skewed tails point toward the negative end of the graph, heading toward negative

infinity, (in most cases 0).

Because neither positively skewed or negatively skewed have a normal distribution (bell curve), their mean, median and modes will not have a central tendancy (all fall in the middle of the bell).

For positively skewed the mode (the number that appears most often) will be less than the median (middle number of the data) because there is more data near 0 than there are heading towards positive infinity; and the mean (average) will be greater than the both because the very few data points that are large, will skew the mean to a larger value positive value.

For negatively skewed the mode (the number that appears most often) will be greater than the median (middle number of the data) because there is more data near the higher positive values than low positive values; and the mean (average) will be lower than the both because the very few data points that are large, will skew the mean to a lower (negative) value.

To exaggerate on ohai’s example:

Find the mode, median, and mean of the following distribution:

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1000000000000}

One huge outlier is all you need.

And to answer your second question (why investors would prefer a positively skewed distr.):

Imagine you’re at a casino. There are two games going on, and you can choose to play either game.

The first game has the following possible payouts:

{\$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1}

The second game has the following possible payouts:

{\$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1000000000000} (positively skewed distr.)

Which game would you play, and why?

Easiest way to remember difference between positvely and negatively skewed distribution:

• Imagine you are a teacher and looking at the test results of your class.
• ‘Skewed’ sounds like ‘Screwed’.
• If class results are postively skewed (heavy on left end), you are positively screwed  :(
• If class results are negatively skewed (heavy on right end), you are negatively screwed  :)

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With the time I studied for CFA, I could have become a doctor.

mode is always the peak, and mean is always in the direction of the skew, while median falls somewhere in between the two.  Mode is easy to remember as it’s the most frequently occuring data point, so after that just remember the order.

I know this question has been posted a long time.  I cannot fully understand the rationale behind the explanation posted by other users.  I personally believe it really depends on the type of investor that we are dealing with.  The data set we are dealing with is unknown so I can argue we can have a data set of -100, -200, 6000, 9000, 10000, 10000.  (a negatively skewed distribution) Or conversely we can have -10000, -10000, -9000, -6000, 100, 200.  This will be positively skewed.  But putting numbers out here just make it more confusing.  I believe it will be useful to just think of this distribution as continuous where all numbers are present with positive or negative infinity on each respective skewed distribution.

I do not know if I am a typical investor, but through the answers and feedbacks by others, my own opinion is that positive skewed distribution (may have?) limited value on the left hand side but infinite upside (skewed to right) and although negative skewed distribution has distribution concentrated on the right hand side, there can be extreme negative value (skewed to the left side, negative infinity). I know, however, a risk adverse person probably will go for positively skewed distribution because they most likely don’t want to see even if there is this slightly chance of losing big(or any) money.

I hope I am right on this.

X is your number of hours of study required to pass the exam. Y is your total time that you put into study. When X-Y=0, your probability of passing the exam will increase. When X-Y=X+Y, it means you did not freaking study at all.

I think that “rational investor” always prefers positive skewed distribution to the negative skewed one. Just think about two games with positive and negative expected outcomes. And in your example you mismatched positive and negative skewed distributions.

-100, -200, 6000, 9000, 10000, 10000 - postively skewed

-10000, -10000, -9000, -6000, 100, 200 - negatively skewed.

No, I think you mismatched positive and negative skewed distribution lol.  Positively skewed means the long tail is on the positive side, in other words, it also means the “raised mountain” is on the other side (negative side).

For negative skewed distributions mean < median < mode.

For positive skewed distribution mean > median > mode.

For simplicity,  -100, -200, 6000, 9000, 10000, 10000

Mean = 5783, Median = 7500 Mode = 10000   this is negatively skewed

-10000, -10000, -9000, -6000, 100, 200

Mean = -5783, Median = -7500 Mode = 10000  this is positively skewed

So do you think you are still “rational”?

X is your number of hours of study required to pass the exam. Y is your total time that you put into study. When X-Y=0, your probability of passing the exam will increase. When X-Y=X+Y, it means you did not freaking study at all.

That’s true with skewness. My bad ))

I found good explanation of the phenomena here:

Why do investors prefer a positive skew over negative skew?

Concave utility function….that’s the answer. Or simply people prefer to avoid large losses even with small probability.

The explanation is clear. We speak in general about the distributions of returns in real world. They are not so skewed as your examples. Maybe your examples are not so good.