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Rationale for describing kurtosis as “peakedness”?
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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?
kurtosis
asked 10 hours ago
Peter WestfallPeter Westfall
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.
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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
asked 10 hours ago
Peter WestfallPeter Westfall
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.
$endgroup$
$begingroup$
what is the "evidence to the contrary" ?
$endgroup$
– Alex C
10 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
6 hours 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
asked 10 hours ago
Peter WestfallPeter Westfall
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.
$endgroup$
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
kurtosis
asked 10 hours ago
Peter WestfallPeter Westfall
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.
asked 10 hours ago
Peter WestfallPeter Westfall
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.
asked 10 hours ago
Peter WestfallPeter Westfall
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.
asked 10 hours ago
Peter WestfallPeter Westfall
1062
asked 10 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
10 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
6 hours ago
add a comment |
$begingroup$
what is the "evidence to the contrary" ?
$endgroup$
– Alex C
10 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
6 hours ago
$begingroup$
what is the "evidence to the contrary" ?
$endgroup$
– Alex C
10 hours ago
$begingroup$
what is the "evidence to the contrary" ?
$endgroup$
– Alex C
10 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
6 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
6 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered 10 hours ago
Alex CAlex C
6,72211123
$endgroup$
$begingroup$
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
8 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
5 hours ago
add a comment |
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1 Answer
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1 Answer
1
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oldest
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active
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votes
$begingroup$
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.
answered 10 hours ago
Alex CAlex C
6,72211123
$endgroup$
$begingroup$
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
8 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
5 hours ago
add a comment |
$begingroup$
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.
answered 10 hours ago
Alex CAlex C
6,72211123
$endgroup$
$begingroup$
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
8 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
5 hours ago
add a comment |
$begingroup$
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.
answered 10 hours ago
Alex CAlex C
6,72211123
$endgroup$
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.
answered 10 hours ago
Alex CAlex C
6,72211123
edited 10 hours ago
answered 10 hours ago
Alex CAlex C
6,72211123
answered 10 hours ago
Alex CAlex C
6,72211123
answered 10 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
8 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
5 hours ago
add a comment |
$begingroup$
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
8 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
5 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
8 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
8 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
5 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
5 hours ago
add a comment |
Peter Westfall is a new contributor. Be nice, and check out our Code of Conduct.
Peter Westfall is a new contributor. Be nice, and check out our Code of Conduct.
Peter Westfall is a new contributor. Be nice, and check out our Code of Conduct.
Peter Westfall is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
what is the "evidence to the contrary" ?
$endgroup$
– Alex C
10 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
6 hours ago