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Rationale for describing kurtosis as "peakedness"?

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Rationale for describing kurtosis as “peakedness”?



Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
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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?










share|improve this question







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
    9 hours ago










  • $begingroup$
    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
    4 hours ago
















1












$begingroup$


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?










share|improve this question







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
    9 hours ago










  • $begingroup$
    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
    4 hours ago














1












1








1





$begingroup$


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?










share|improve this question







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.







$endgroup$




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







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







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




share|improve this question






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 9 hours ago









Peter WestfallPeter Westfall

1062




1062




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.





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.






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












  • $begingroup$
    what is the "evidence to the contrary" ?
    $endgroup$
    – Alex C
    9 hours ago










  • $begingroup$
    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
    4 hours ago


















  • $begingroup$
    what is the "evidence to the contrary" ?
    $endgroup$
    – Alex C
    9 hours ago










  • $begingroup$
    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
    4 hours ago
















$begingroup$
what is the "evidence to the contrary" ?
$endgroup$
– Alex C
9 hours ago




$begingroup$
what is the "evidence to the contrary" ?
$endgroup$
– Alex C
9 hours ago












$begingroup$
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
4 hours ago




$begingroup$
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
4 hours ago










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











$endgroup$













  • $begingroup$
    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
    7 hours ago










  • $begingroup$
    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
    3 hours ago














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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






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











$endgroup$













  • $begingroup$
    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
    7 hours ago










  • $begingroup$
    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
    3 hours ago


















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











$endgroup$













  • $begingroup$
    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
    7 hours ago










  • $begingroup$
    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
    3 hours ago
















4












4








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











$endgroup$



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














share|improve this answer



share|improve this answer








edited 8 hours ago

























answered 9 hours ago









Alex CAlex C

6,72211123




6,72211123












  • $begingroup$
    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
    7 hours ago










  • $begingroup$
    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
    3 hours ago




















  • $begingroup$
    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
    7 hours ago










  • $begingroup$
    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
    3 hours ago


















$begingroup$
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
7 hours ago




$begingroup$
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
7 hours ago












$begingroup$
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
3 hours ago






$begingroup$
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
3 hours ago












Peter Westfall is a new contributor. Be nice, and check out our Code of Conduct.










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