Fa ahani, Hamed; Se o a, Ros isla A.
A icle
Asymme y in dis ibu ions o accumula ed gains and
losses in s ock e u ns
Economies
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Fa ahani, Hamed; Se o a, Ros isla A. (2025) : Asymme y in dis ibu ions o
accumula ed gains and losses in s ock e u ns, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13,
Iss. 6, pp. 1-16,
h ps://doi.o g/10.3390/economies13060176
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Ci a ion: Fa ahani, H., & Se o a, R. A.
(2025). Asymme y in Dis ibu ions o
Accumula ed Gains and Losses in
S ock Re u ns. Economies,13(6), 176.
h ps://doi.o g/10.3390/
economies13060176
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A icle
Asymme y in Dis ibu ions o Accumula ed Gains and Losses in
S ock Re u ns
Hamed Fa ahani * and Ros isla A. Se o a *
Depa men o Physics, Uni e si y o Cincinna i, Cincinna i, OH 45221-0011, USA
*Co espondence: [email p o ec ed] (H.F.); [email p o ec ed] (R.A.S.)
Abs ac : We s udied decades-long (1980 o 2024) his o ic dis ibu ions o accumula ed
S&P500 e u ns, om daily e u ns o hose o e se e al weeks. The ime se ies o he
e u ns emphasize majo uphea als in he ma ke s—Black Monday, Tech Bubble, Financial
C isis, and he COVID pandemic—which a e e lec ed in he ail ends o he dis ibu ions.
De- ending he o e all gain, we concen a ed on compa ing dis ibu ions o gains and
losses. Speci ically, we compa ed he ails o he dis ibu ions, which a e belie ed o exhibi
a powe -law beha io and possibly con ain ou lie s. To his end, we de e mined con idence
in e als o he linea i s o he ails o he complemen a y cumula i e dis ibu ion unc-
ions on a log–log scale and conduc ed a s a is ical U- es in o de o de ec ou lie s. We
also s udied p obabili y densi y unc ions o he ull dis ibu ions o he e u ns wi h an em-
phasis on hei asymme y. The key empi ical obse a ions a e ha he mean o de- ended
dis ibu ions inc eases nea -linea ly wi h he numbe o days o accumula ion while he
o e all skew is nega i e—consis en wi h he hea ie ails o losses—and depends li le on
he numbe o days o accumula ion. A he same ime, he a iance o he dis ibu ions
exhibi s nea -pe ec linea dependence on he numbe o days o accumula ion; ha is, i
emains cons an i scaled o he la e . Finally, we discuss he heo e ical amewo k o
unde s anding accumula ed e u ns. Ou main conclusion is ha he cu en s a e o heo y,
which p edic s symme ic o nea -symme ic dis ibu ions o e u ns, canno explain he
agg ega e o empi ical esul s.
Keywo ds: accumula ed e u ns; S&P500; powe -law ails; ou lie s; skewness
1. In oduc ion
Resea ch on asymme y o s ock e u ns has a long and s o ied his o y (Albuque que,
2012;Bekae & Wu,2000;B aun e al.,1995;Campbell & Hen schel,1992;Chak abo i e al.,
2011;Con ,2001;Du ee,1995;F ench e al.,1987;Glos en e al.,1993;Hong & S ein,2003;
Lee & Kang,2023;Neube ge & Payne,2021;Sándo e al.,2016;Si e & Lins,2009;Wu,2001;
Załuska-Ko u e al.,2006). Clea ly, he e a e many aspec s o asymme y and app oaches o
s udy his phenomenon, such as he i s passage ime (Sándo e al.,2016;Si e & Lins,2009;
Załuska-Ko u e al.,2006), di e ences be ween i m-le el and o e all ma ke pe o mance
(Albuque que,2012;Bekae & Wu,2000;B aun e al.,1995;Du ee,1995), and many o he s.
The simples o m o asymme y is ha , o e all, he e is a conside able gain in he s ock
ma ke : inancial ad iso s like o ell hei clien s ha , on a e age, he e is oughly a 10%
annual gain o , mo e p ecisely, 12% gain (see he s aigh line in Figu e 1) minus 2% a e age
in la ion. O cou se, he e a e pe iods o ma ke s agna ion, decline and apid g ow h—such
luc ua ions a ound he o e all g ow h end a e a ibu ed o ma ke ola ili y.
Economies 2025,13, 176 h ps://doi.o g/10.3390/economies13060176
Economies 2025,13, 176 2 o 16
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
log
(
S
/
S
0
) wi h
τ
= 1
linea i wi h slope o
μ
1
= 3.0860 × 10
−4
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
log
(
S
/
S
0
) wi h
τ
= 20
linea i wi h slope o
μ
20
= 3.0890 × 10
−4
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
log
(
S
/
S
0
) wi h
τ
= 50
linea i wi h slope o
μ
50
= 3.0968 × 10
−4
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
log
(
S
/
S
0
) wi h
τ
= 100
linea i wi h slope o
μ
100
= 3.0897 × 10
−4
Figu e 1. Linea i s o =log(S /S0) o =nτ,n=0, 1, . . ., wi h τ=1, 20, 50, 100, espec i ely.
A a mo e in e es ing ques ion is he asymme y be ween gains and losses once he
o e all g ow h ends a e al eady accoun ed o , ha is, when he da a a e de- ended.
In his ega d, o he o he wise nume ous empi ical p ope ies o he aw ma ke da a
(Chak abo i e al.,2011;Con ,2001), ou in e es is cen e ed mainly on asymme y as
ela ed o hea y ails (Taleb,2007) o he dis ibu ions o gains and losses. To his end,
we s udied dis ibu ions o s ock e u ns o he S&P500 index, om daily e u ns o hose
accumula ed o e longe pe iods o ime—a subjec la gely missing om he li e a u e
( o in aday imed ades see (Wa o ek e al.,2021)). Speci ically, we pe o med linea i s
(LFs) o he ails o complemen a y cumula i e dis ibu ion unc ions (CCDFs) o gains and
losses on a log–log scale o es o hei powe -law dependence. We compu ed con idence
in e als (CIs) (Janczu a & We on,2012) o LFs and conduc ed a s a is ical U- es (Pisa enko
& So ne e,2012) in o de o es o possible ou lie s, such as D agon Kings (DKs) (So ne e
& Ouillon,2012) and nega i e D agon Kings (nDKs) (Pisa enko & So ne e,2012).
We also pe o med nume ical measu es o he ull dis ibu ions o e u ns using hei
p obabili y densi y unc ions (PDFs). Speci ically, we e alua ed dependence on he numbe
o days o accumula ion o he mean, a iance, Fishe –Pea son coe icien o skewness, and
i s and second Pea son coe icien s o skewness. The key esul s om hose measu es
a e ha he mean o de- ended dis ibu ions inc eases nea -linea ly wi h he numbe o
days o accumula ion, while he o e all skew is nega i e—consis en wi h he hea ie ails
o losses obse ed om PDF and CCDF—and depends li le on he numbe o days o
accumula ion. A he same ime, he a iance o he dis ibu ions exhibi s nea -pe ec
linea dependence on he numbe o days o accumula ion. While he nea -linea shi o
he mean can be easily accoun ed o phenomenologically, he cu en s a e o heo y based
on con inuous s ochas ic di e en ial equa ions (SDEs) does no p ope ly desc ibe s a is ical
measu es o he dis ibu ions, especially skewness.
This pape is o ganized as ollows: In Sec ion 2.1, we explain he de- ending p oce-
du e o e u ns, p esen he ime se ies o e u ns o he 1980–2024 pe iod, and discuss he
numbe o da a poin s o gains and losses—all in e ms o he numbe o days o accumula-
Economies 2025,13, 176 3 o 16
ion. In Sec ion 2.2, we compa e dis ibu ions o gains and losses using CCDF, including LF
s a is ical es s o ou lie s. In Sec ion 2.3, we s udy ull dis ibu ions o e u ns and hei
s a is ical measu es: mean, a iance, and skewness. In Sec ion 3, we add ess he s a e o
heo y is-a- is ou empi ical obse a ions. Sec ion 4summa izes ou main esul s.
2. Empi ical Resul s
2.1. Ini ial Analysis o Re u ns
Wi h S being he s ock p ice, he linea upwa d end o log e u ns
=logS
S0(1)
is shown in Figu e 1 o
=nτ
,
n=
0, 1, ...wi h
τ=
1, 20, 50, 100 and he plo o slopes
µτ
shown in Figu e 2.
De- ended log e u ns (o simply “ e u ns” below) accumula ed o e ime pe iod
τ
a e hen gi en by
dx =x +τ−x = +τ− −µτ =logS +τ
S −µτ (2)
whe e, om now on, we slide
by one day when ob aining dis ibu ions as a unc ion o
τ
and hus
use
µ=µ1
— he slope o daily log e u ns, al hough, clea ly,
µτ
shows only e y insigni ican
dependence on τ.
0 20 40 60 80 100
τ
3.000
3.025
3.050
3.075
3.100
3.125
3.150
3.175
3.200
μ
τ
×10
−4
μ
τ
Figu e 2. Slopes o linea i s o log e u ns
o
=nτ
,
n=
0, 1,
. . .
, as a unc ion o
τ
. Red do s
co espond o τ=1, 20, 50, 100 as in Figu e 1.
Figu e 3shows he ime se ies o e u ns om 1980 o 2024. No ice he ob ious
simila i y wi h he ime se ies o ealized ola ili y (J. Liu e al.,2024). Clea ly, he la ges
nega i e peaks occu ed du ing Black Monday, he Tech Bubble, Financial C isis, and he
COVID pandemic. No su p isingly, ollowing hose d ops, he la ges posi i e peaks
occu ed ela i ely sho ly a e .
Figu e 4shows he numbe o da a poin s o gains and losses as a unc ions o
τ
, as
well as hei sum— he o al numbe o poin s in he da a se — o he same ime pe iod
(1980–2024) as he ime se ies in Figu e 3. Fo illus a i e pu poses, he numbe s a e
explici ly shown o
τ=
1, 5, 10, 20 in Table 1. Clea ly, he numbe o gains inc eases as a
unc ion o
τ
, while he numbe o losses dec eases. The o al numbe o poin s is gi en by
11,259 −τ+1, whe e 11,259 is he size o he da a se o daily e u ns, τ=1.
Economies 2025,13, 176 4 o 16
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
−0.3
−0.2
−0.1
0.0
0.1
Re u n
Gains,
τ
= 1
Losses,
τ
= 1
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
−0.5
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
Re u n
Gains,
τ
= 20
Losses,
τ
= 20
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
−0.6
−0.4
−0.2
0.0
0.2
Re u n
Gains,
τ
= 50
Losses,
τ
= 50
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
−0.6
−0.4
−0.2
0.0
0.2
0.4
Re u n
Gains,
τ
= 100
Losses,
τ
= 100
Figu e 3. Time se ies o daily e u ns, τ=1, and accumula ed e u ns o τ=20, 50, 100.
1 10 20 30 40 50 60 70 80 90 100
τ
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Numbe o Poin s
Gains (Line)
Losses (Line)
To al (Line)
Gains (Ba s)
Losses (Ba s)
To al (Ba s)
Figu e 4. Numbe o da a poin s o gains and losses and hei sum ( he o al numbe o poin s) as a
unc ion o τ.
Economies 2025,13, 176 5 o 16
Table 1. Summa y o o al poin s, losses, and gains o di e en τ alues.
τTo al Poin s Losses Gains
1 11,259 5455 5804
5 11,255 5167 6088
10 11,250 5063 6187
20 11,240 4871 6369
2.2. Dis ibu ions o Gains and Losses
Figu es 5–8show CCDF, 1
−Fg(x)
and 1
−Fl(x)
, o gains and losses on a log–log scale
o τ=1, 5, 10, 20. He e,
Fg(x) = Zx
−∞ (x)dx Fl(x) = Zx
∞ (x)dx(3)
a e he CDFs o gains and losses, espec i ely, and
(x)
is he PDF o e u ns (see
Figu es 9–12
below). Also shown a e linea i s o he ails, including hei con idence
in e als (CIs) and he esul s o he U-Tes o iden i y ou lie s, such as DK and nDK. CIs
o he i s a e e alua ed ia he in e sion o he binomial dis ibu ion (Janczu a & We on,
2012);
p
- alues a e e alua ed in he amewo k o he U- es , which is based on o de
s a is ics (Pisa enko & So ne e,2012), using he ollowing o mula:
p(xk,n) = 1−B(F(xk,n);k,n−k+1), (4)
whe e xk,nis he k’s membe o numbe s (SR he e) be ween 1 and no de ed by inc easing
magni ude,
F(xk,n)
is he assumed CDF (LF he e), and
B(y
;
a
,
b)
is he incomple e Be a
unc ion (NIST Digi al Lib a y o Ma hema ical Func ions,n.d.). p- alues a e e alua ed in
o de o es he null hypo hesis
H0
: all obse a ions o he sample a e gene a ed by he same
i ing dis ibu ion. The p- alue (4) is de ined as a p obabili y o exceeding he obse ed
alue
xk,n
unde he null hypo hesis. I among he p- alues, he e a e some small alues
(
≤
0.05 he e), hen hose obse ed alues a e iden i ied as DK wi h p obabili y 1
−p
and
ma ked by up iangles. Con e sely, la ge p- alues (
≥
0.95 he e) a e iden i ied as nDK wi h
he p obabili y
p
(Pisa enko & So ne e,2012) and ma ked by down iangles. While daily
e u ns seem o exhibi a he well-de ined linea dependence, o la ge
τ
, he ail beha io
is mo e complex wi h wha migh be called a de eloping shoulde and apid d op-o s
a ail ends. In his ega d, ins ead o hinking o possible DK (pDK) and nDK, pe haps
up and down iangles ob ained om he U- es and c ossing lines o CIs can be simply
an indica o o poo goodness o i . Again, no ice he ob ious simila i ies wi h he ail
beha io o he ealized ola ili y (J. Liu e al.,2024).
2.3. Full Dis ibu ions o Re u ns and Thei S a is ical Measu es
Figu es 9–12 show he PDFs o daily and accumula ed e u ns o
τ=
1, 20, 50, 100.
Clea ly, he PDFs exhibi asymme y and longe ails o losses e sus gains (Paloma ,
2018), which a e becoming mo e p onounced wi h la ge τ.
Economies 2025,13, 176 6 o 16
2.5 × 10
−2
4 × 10
−2
6 × 10
−2
1 × 10
−1
2 × 10
−1
3 × 10
−1
Re u n
10
−5
10
−4
10
−3
10
−2
10
−1
CCDF
Losses,
τ
= 1
pDKs o losses
nDKs o losses
LF o losses
LF CI o losses
Gains,
τ
= 1
pDKs o gains
nDKs o gains
LF o gains
LF CI o gains
Figu e 5. Linea i s o CCDF ails o gains and losses o daily e u ns,
τ=
1, wi h CIs (dashed lines)
and possible DK (pDK), deno ed by up iangles, and nega i e DK (nDK) deno ed by down iangles.
6 × 10
−2
1 × 10
−1
2 × 10
−1
3 × 10
−1
4 × 10
−1
Re u n
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
CCDF
Losses,
τ
= 5
pDKs o losses
nDKs o losses
LF o losses
LF CI o losses
Gains,
τ
= 5
pDKs o gains
nDKs o gains
LF o gains
LF CI o gains
Figu e 6. Linea i s o CCDF ails o gains and losses o
τ=
5 accumula ed e u ns wi h CIs
(dashed lines) and possible DK (pDK), deno ed by up iangles, and nega i e DK (nDK) deno ed by
down iangles.
6 × 10
−2
1 × 10
−1
2 × 10
−1
3 × 10
−1
4 × 10
−1
Re u n
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
CCDF
Losses,
τ
= 10
pDKs o losses
nDKs o losses
LF o losses
LF CI o losses
Gains,
τ
= 10
pDKs o gains
nDKs o gains
LF o gains
LF CI o gains
Figu e 7. Linea i s o CCDF ails o gains and losses o
τ=
10 accumula ed e u ns wi h CIs
(dashed lines) and possible DK (pDK), deno ed by up iangles, and nega i e DK (nDK) deno ed by
down iangles.
Economies 2025,13, 176 7 o 16
7 × 10
−2
1 × 10
−1
2 × 10
−1
3 × 10
−1
4 × 10
−1
Re u n
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
CCDF
Losses,
τ
= 20
pDKs o losses
nDKs o losses
LF o losses
LF CI o losses
Gains,
τ
= 20
pDKs o gains
nDKs o gains
LF o gains
LF CI o gains
Figu e 8. Linea i s o CCDF ails o gains and losses o
τ=
20 accumula ed e u ns wi h CIs
(dashed lines) and possible DK (pDK), deno ed by up iangles, and nega i e DK (nDK) deno ed by
down iangles.
−0.2 −0.1 0.0 0.1 0.2
Re u n
0
10
20
30
40
50
60
PDF
τ
= 1
−0.25 −0.20 −0.15 −0.10 −0.05
0.0
0.1
0.2
0.3
0.4
0.5
0.05 0.10 0.15 0.20 0.25
0.0
0.1
0.2
0.3
0.4
0.5
Figu e 9. PDF o daily e u ns, wi h inse s showing ails o he dis ibu ion.
−0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4
Re u n
0
2
4
6
8
10
12
14
PDF
τ
= 20
−0.40 −0.35 −0.30 −0.25 −0.20 −0.15
0.0
0.1
0.2
0.3
0.4
0.5
0.15 0.20 0.25 0.30 0.35 0.40
0.0
0.1
0.2
0.3
0.4
0.5
Figu e 10. PDF o τ=20 accumula ed e u ns, wi h inse s showing ails o he dis ibu ion.
Economies 2025,13, 176 8 o 16
−0.4 −0.2 0.0 0.2 0.4
Re u n
0
1
2
3
4
5
6
7
8
PDF
τ
= 50
−0.50 −0.45 −0.40 −0.35 −0.30 −0.25 −0.20
0.0
0.1
0.2
0.3
0.4
0.5
0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.0
0.1
0.2
0.3
0.4
0.5
Figu e 11. τ=50 accumula ed e u ns, wi h inse s showing ails o he dis ibu ion.
−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6
Re u n
0
1
2
3
4
5
6
7
PDF
τ
= 100
−0.6 −0.5 −0.4 −0.3 −0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.2 0.3 0.4 0.5 0.6
0.0
0.1
0.2
0.3
0.4
0.5
Figu e 12. τ=100 accumula ed e u ns, wi h inse s showing ails o he dis ibu ion.
Nex , we add ess he mean,
m1
, a iance,
m2
, and skewness o dis ibu ions in
Figu es 9–12.
Fo he la e , we employed he Fishe –Pea son coe icien o skewness and
he i s and second Pea son coe icien s o skewness, de ined, espec i ely, as ollows:
ζ=m3
m3/2
2
(5)
ζ1=(m1−m)
m1/2
2
(6)
ζ2=3(m1−e
m)
m1/2
2
(7)
whe e
m3
is he hi d cen al momen o he dis ibu ion,
m2
is he a iance,
m1/2
2
is he
s anda d de ia ion,
m
is he mode, and
e
m
is he median. The eason o using
ζ1
and
ζ2
is
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