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Rationale for describing kurtosis as “peakedness”?

0

$begingroup$

Despite plenty of evidence to the contrary, many quantitative finance sources of information, including teaching resources such as CFA prep, persist in defining kurtosis as a measure of "peakedness." Can anyone give a logical rationale for this characterization in terms of distributions of asset returns?

kurtosis

share|improve this question

asked Apr 20 at 16:25

Peter WestfallPeter Westfall

1072

New contributor

Peter Westfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  what is the "evidence to the contrary" ?
  $endgroup$
  – Alex C
  Apr 20 at 17:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Not trying to be a jerk here, but "Peakedness," "Tailedness" who cares? Is the characterization even important? For any metric, understanding use and interpretation are what matters.
  $endgroup$
  – amdopt
  Apr 20 at 21:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Downvoted for not citing this evidence.
  $endgroup$
  – Bob Jansen♦
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Wikipedia and sources therein.
  $endgroup$
  – Peter Westfall
  2 days ago
  

  
  
  
  

add a comment | 

0

$begingroup$

Despite plenty of evidence to the contrary, many quantitative finance sources of information, including teaching resources such as CFA prep, persist in defining kurtosis as a measure of "peakedness." Can anyone give a logical rationale for this characterization in terms of distributions of asset returns?

kurtosis

share|improve this question

asked Apr 20 at 16:25

Peter WestfallPeter Westfall

1072

New contributor

Peter Westfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  what is the "evidence to the contrary" ?
  $endgroup$
  – Alex C
  Apr 20 at 17:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Not trying to be a jerk here, but "Peakedness," "Tailedness" who cares? Is the characterization even important? For any metric, understanding use and interpretation are what matters.
  $endgroup$
  – amdopt
  Apr 20 at 21:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Downvoted for not citing this evidence.
  $endgroup$
  – Bob Jansen♦
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Wikipedia and sources therein.
  $endgroup$
  – Peter Westfall
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

Despite plenty of evidence to the contrary, many quantitative finance sources of information, including teaching resources such as CFA prep, persist in defining kurtosis as a measure of "peakedness." Can anyone give a logical rationale for this characterization in terms of distributions of asset returns?

kurtosis

share|improve this question

asked Apr 20 at 16:25

Peter WestfallPeter Westfall

1072

New contributor

Peter Westfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|improve this question

asked Apr 20 at 16:25

Peter WestfallPeter Westfall

1072

New contributor

Peter Westfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|improve this question

asked Apr 20 at 16:25

Peter WestfallPeter Westfall

1072

New contributor

Peter Westfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Apr 20 at 16:25

Peter WestfallPeter Westfall

1072

New contributor

Peter Westfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Apr 20 at 16:25

Peter WestfallPeter Westfall

1072

1 Answer
1

active

oldest

votes

4

$begingroup$

Quailtatively a (zero skewness) Leptokurtic distribution, after being standardized to have zero mean and unit variance shows three features when you plot the density and compare it to a standard normal N(0,1) distribution: higher peak, higher (fatter) tails, and lower mid-range(*). All three properties go together, even though people sometimes mention only one of them ("a fat tailed distribution"). After all there has to be an area of 1 under the curve, and a variance of 1, so a deficit in the mid-range has to be made up by an excess elsewhere, i.e. in the tails and near the centre. Otherwise it would not be a probability distribution or would not have unit variance. The peakedness in the centre "balances" the thickness in the tails while staying with a unit variance.

So "peakedness in the centre" and "fat in the tails" describe exactly the same thing. You can't have one without the other.

(*) By mid-range is meant the two areas, one on each side, located approximately one standard deviation from the centre.

share|improve this answer

edited Apr 20 at 17:13

answered Apr 20 at 16:56

Alex CAlex C

6,74211123

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I understand where you are coming from but I think this is fruitless. Distributions are functions, i.e. they have infinite degrees of freedom. Were your statement still true if the distribution is multimodal or had point masses?
  $endgroup$
  – g g
  Apr 20 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Fat tails are a problem because so many of the models we use assume that the data are Gaussian IID. Fat tails means an increased probablility of events that we considered rare. For example the Black Scholes option pricing model assume your data are Gaussian. If they have a high kurtosis, this problematic.
  $endgroup$
  – Jacques Joubert
  Apr 20 at 22:14
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  These comments are simply not true. It has been known since Kaplansky (1945) that higher peak does not correspond with higher kurtosis. Also, distributions such as beta(.5,1) have infinite peaks but negative excess kurtosis. (More support for my "evidence to the contrary" statement which for some reason got me downvoted.) Also, if you take a uniform(0,1) and mix it with a N(0,1000000), with .0001 probability on the latter, you get a distribution that is perfectly flat-topped over 99.99% of the observable data but has very high kurtosis. So the argument connecting tail to peak is not logical.
  $endgroup$
  – Peter Westfall
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But my question is not about the math, since that is already settled. My question is whether there is a finance rationale for the "peakedness" characterization, say in terms of return distributions of assets or portfolios.
  $endgroup$
  – Peter Westfall
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  are you simply looking to argue semantics? equity returns are pretty widely held to display excess kurtosis, with a peak around the mean and fatter than normal tails. peaked-ness is often used as a stand-in description for this, though references to fat tails are also made.
  $endgroup$
  – Chris
  yesterday
  

  
  
  
  

 | 
show 3 more comments

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4

$begingroup$

Quailtatively a (zero skewness) Leptokurtic distribution, after being standardized to have zero mean and unit variance shows three features when you plot the density and compare it to a standard normal N(0,1) distribution: higher peak, higher (fatter) tails, and lower mid-range(*). All three properties go together, even though people sometimes mention only one of them ("a fat tailed distribution"). After all there has to be an area of 1 under the curve, and a variance of 1, so a deficit in the mid-range has to be made up by an excess elsewhere, i.e. in the tails and near the centre. Otherwise it would not be a probability distribution or would not have unit variance. The peakedness in the centre "balances" the thickness in the tails while staying with a unit variance.

So "peakedness in the centre" and "fat in the tails" describe exactly the same thing. You can't have one without the other.

(*) By mid-range is meant the two areas, one on each side, located approximately one standard deviation from the centre.

share|improve this answer

edited Apr 20 at 17:13

answered Apr 20 at 16:56

Alex CAlex C

6,74211123

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I understand where you are coming from but I think this is fruitless. Distributions are functions, i.e. they have infinite degrees of freedom. Were your statement still true if the distribution is multimodal or had point masses?
  $endgroup$
  – g g
  Apr 20 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Fat tails are a problem because so many of the models we use assume that the data are Gaussian IID. Fat tails means an increased probablility of events that we considered rare. For example the Black Scholes option pricing model assume your data are Gaussian. If they have a high kurtosis, this problematic.
  $endgroup$
  – Jacques Joubert
  Apr 20 at 22:14
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  These comments are simply not true. It has been known since Kaplansky (1945) that higher peak does not correspond with higher kurtosis. Also, distributions such as beta(.5,1) have infinite peaks but negative excess kurtosis. (More support for my "evidence to the contrary" statement which for some reason got me downvoted.) Also, if you take a uniform(0,1) and mix it with a N(0,1000000), with .0001 probability on the latter, you get a distribution that is perfectly flat-topped over 99.99% of the observable data but has very high kurtosis. So the argument connecting tail to peak is not logical.
  $endgroup$
  – Peter Westfall
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But my question is not about the math, since that is already settled. My question is whether there is a finance rationale for the "peakedness" characterization, say in terms of return distributions of assets or portfolios.
  $endgroup$
  – Peter Westfall
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  are you simply looking to argue semantics? equity returns are pretty widely held to display excess kurtosis, with a peak around the mean and fatter than normal tails. peaked-ness is often used as a stand-in description for this, though references to fat tails are also made.
  $endgroup$
  – Chris
  yesterday
  

  
  
  
  

 | 
show 3 more comments

4

$begingroup$

Quailtatively a (zero skewness) Leptokurtic distribution, after being standardized to have zero mean and unit variance shows three features when you plot the density and compare it to a standard normal N(0,1) distribution: higher peak, higher (fatter) tails, and lower mid-range(*). All three properties go together, even though people sometimes mention only one of them ("a fat tailed distribution"). After all there has to be an area of 1 under the curve, and a variance of 1, so a deficit in the mid-range has to be made up by an excess elsewhere, i.e. in the tails and near the centre. Otherwise it would not be a probability distribution or would not have unit variance. The peakedness in the centre "balances" the thickness in the tails while staying with a unit variance.

So "peakedness in the centre" and "fat in the tails" describe exactly the same thing. You can't have one without the other.

(*) By mid-range is meant the two areas, one on each side, located approximately one standard deviation from the centre.

share|improve this answer

edited Apr 20 at 17:13

answered Apr 20 at 16:56

Alex CAlex C

6,74211123

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I understand where you are coming from but I think this is fruitless. Distributions are functions, i.e. they have infinite degrees of freedom. Were your statement still true if the distribution is multimodal or had point masses?
  $endgroup$
  – g g
  Apr 20 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Fat tails are a problem because so many of the models we use assume that the data are Gaussian IID. Fat tails means an increased probablility of events that we considered rare. For example the Black Scholes option pricing model assume your data are Gaussian. If they have a high kurtosis, this problematic.
  $endgroup$
  – Jacques Joubert
  Apr 20 at 22:14
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  These comments are simply not true. It has been known since Kaplansky (1945) that higher peak does not correspond with higher kurtosis. Also, distributions such as beta(.5,1) have infinite peaks but negative excess kurtosis. (More support for my "evidence to the contrary" statement which for some reason got me downvoted.) Also, if you take a uniform(0,1) and mix it with a N(0,1000000), with .0001 probability on the latter, you get a distribution that is perfectly flat-topped over 99.99% of the observable data but has very high kurtosis. So the argument connecting tail to peak is not logical.
  $endgroup$
  – Peter Westfall
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But my question is not about the math, since that is already settled. My question is whether there is a finance rationale for the "peakedness" characterization, say in terms of return distributions of assets or portfolios.
  $endgroup$
  – Peter Westfall
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  are you simply looking to argue semantics? equity returns are pretty widely held to display excess kurtosis, with a peak around the mean and fatter than normal tails. peaked-ness is often used as a stand-in description for this, though references to fat tails are also made.
  $endgroup$
  – Chris
  yesterday
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

Quailtatively a (zero skewness) Leptokurtic distribution, after being standardized to have zero mean and unit variance shows three features when you plot the density and compare it to a standard normal N(0,1) distribution: higher peak, higher (fatter) tails, and lower mid-range(*). All three properties go together, even though people sometimes mention only one of them ("a fat tailed distribution"). After all there has to be an area of 1 under the curve, and a variance of 1, so a deficit in the mid-range has to be made up by an excess elsewhere, i.e. in the tails and near the centre. Otherwise it would not be a probability distribution or would not have unit variance. The peakedness in the centre "balances" the thickness in the tails while staying with a unit variance.

So "peakedness in the centre" and "fat in the tails" describe exactly the same thing. You can't have one without the other.

(*) By mid-range is meant the two areas, one on each side, located approximately one standard deviation from the centre.

share|improve this answer

edited Apr 20 at 17:13

answered Apr 20 at 16:56

Alex CAlex C

6,74211123

$endgroup$

share|improve this answer

edited Apr 20 at 17:13

answered Apr 20 at 16:56

Alex CAlex C

6,74211123

answered Apr 20 at 16:56

Alex CAlex C

6,74211123

answered Apr 20 at 16:56

Alex CAlex C

6,74211123