M&A and the Simulation-Based Valuation of Companies with an Uncertain Exit Price and Special Rights

Gleißne , We ne ; Wol um, Ma co; Do lei ne , G ego
A icle
M&A and he Simula ion-Based Valua ion o Companies
wi h an Unce ain Exi P ice and Special Righ s
C edi and Capi al Ma ke s – K edi und Kapi al
P o ided in Coope a ion wi h:
Duncke & Humblo , Be lin
Sugges ed Ci a ion: Gleißne , We ne ; Wol um, Ma co; Do lei ne , G ego (2024) : M&A and he
Simula ion-Based Valua ion o Companies wi h an Unce ain Exi P ice and Special Righ s, C edi
and Capi al Ma ke s – K edi und Kapi al, ISSN 2199-1235, Duncke & Humblo , Be lin, Vol. 57, Iss.
1/4, pp. 185-221,
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Duncke & Humblo · Be lin
M&A and he Simula ion-Based Valua ion o Companies
wi h an Unce ain Exi P ice and Special Righ s
We ne Gleißne *, Ma co Wol um**, G ego Do lei ne ***
Abs ac
This a icle p esen s a new me hodological app oach o alue p i a e equi y in es -
men s based on simula ion. The alua ion elies on ‘impe ec eplica ion’. This me hod
does no p esuppose he pe ec ion o he capi al ma ke and is essen ially buil on meas-
u ing he isk. The app oach u ns ou o be easy o implemen . Fi m speci ic cha ac e -
is ics as well as and exis ing special igh s can be depic ed and modelled. The p oposed
me hodology is o immedia e p ac ical use ulness as i can help o ind decision suppo
o conc e e in es men si ua ions. Also, du ing he in es men pe iod i can be used o
moni o ing. The o iginali y o he esea ch lies in he combina ion o Mon e Ca lo sim-
ula ion, mul iple me hods, ele an isk measu es and isk- alue models.
Keywo ds: Company alua ion, sha e alua ion, exi p ice, isk analysis, Mon e Ca lo
simula ion, incomple e eplica ions
JEL Classi ica ion: G17, G24, G32, G33, G34
Acknowledgemen : We would like o exp ess ou since e g a i ude o an anonymous e-
iewe o hei in aluable eedback and cons uc i e sugges ions. Thei insigh s signi i-
can ly con ibu ed o imp o ing he quali y and cla i y o his pape .
I. In oduc ion and o e iew
A key challenge o p i a e-equi y and en u e-capi al companies is es ima -
ing a ealis ic ange o possible u u e exi p ices o an in es men , pa icula ly
* P o . D . We ne Gleißne is a membe o he boa d o he Fu u eValue G oup AG
and hono a y p o esso o business adminis a ion, esp. isk managemen , a he TU
D esden. He is also a boa d membe o he Eu opean Associa ion o Ce i ied Valua o s
and Analys s (EACVA). E-Mail: w.gleissne @ u u e alue.de.
** Ma co Wol um is a pa ne o Fu u eValue G oup AG and depu y chai man o he
boa d o he RMA Risk Managemen & Ra ing Associa ion e. V. as well as managing di-
ec o o he RMA Ra ing & Risk Academy GmbH. He is a lec u e in isk managemen
a a ious uni e si ies. E-Mail: m.wol um@ u u e alue.de.
*** P o . D . G ego Do lei ne is P o esso o Finance and Di ec o o he Cen e o
Finance a he Uni e si y o Regensbu g.
186 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
in small- and medium-sized en e p ises (SMEs1). The cu en alue o he max-
imum accep able pu chase p ice ( he subjec i e decision alue2) depends on his
ange o possible u u e sale p ices, and speci ically on he expec ed alue o he
sale p ice and he ‘sale p ice isks’, which a e in es men isk, exp essed, o in-
s ance, in he s anda d de ia ion o his p ice. Unce ain u u e sale p ices a e
ele an o he decision, e. g. ega ding he pu chase o a company. The e o e,
adi ional ‘mul iple me hods’, which deal wi h he cu en o m o he alua ion
le el (mul iple) and p o i abili y (EBIT: ea nings be o e in e es and axes) o he
company, a e insu icien on hei own due o hei lack o u u e e e ence.3
In ligh o he empi ically p o en impe ec ions4 o he capi al ma ke , he
p ices ha can be achie ed in an upcoming ansac ion (and also he s ock-ma -
ke p ices) o en de ia e om he company alue as calcula ed based on he
model.5 As many empi ical s udies show, ansac ion p ices o non-lis ed com-
panies, p i a e equi y (PE) and, especially, en u e capi al (VC) a e de e mined
using compa a i e me hods, in pa icula mul iple me hods. The o en- ealised
‘low’ ansac ion mul iples ( ela ed o EBIT o EBITDA) o he high ‘ e u n e-
qui emen s’ show ha he alua ion6 akes in o accoun :
• heabo e-a e agep obabili yo insol ency,whichisusuallypa icula lyhigh
o SMEs and en u e-capi al in es men s, and/o
• company-speci ic(idiosync a ic) isks
Abo e-a e age ea nings and insol ency isks in smalle companies, especially
SMEs, a e one explana ion o he ‘size e ec ’ egula ly ound in empi ical s ud-
ies.7 E e y hing else being equal, smalle companies wi h highe ea nings and
insol ency isk ha e highe cos s o capi al and lowe en e p ise alue.8
The Business Judgmen Rule9 makes i impo an o conside he isks o a
a ge company, including ha o insol ency. When making a business decision
1 On he peculia i ies o alua ing SMEs, see, o ins ance, Coulon (2022) and
Damoda an (2011), who speci ically deal wi h he impo ance o isk analysis and isk
simula ion.
2 See Ma schke (1972).
3 See Bhoj aj/Lee (2002) on c i icisms o mul iple me hods.
4 See, o ins ance, Haugen (2002); G omb/Vayanos (2010); Shlei e /Vishny (1997);
Jegadeesh/ Ti man (2011); Joyce/Maye (2012) and Zhang (2020).
5 See Calhoun (2020).
6 See, o ins ance, Ke ins e al. (2004) and Mülle (2004).
7 See G abowski (2018).
8 See also Bli z (2020) o an o e iew o he cu en empi ical esea ch, as well as
Fama/F ench (2015, 2018) and Elgammal e al. (2020).
9 See e. g. Leach (2014). When buying o selling a sha eholding, a ‘business decision’
wi hin he meaning o he Business Judgmen Rule (Sec ion 93 o he Ge man S ock Co -
po a ion Ac ) mus gene ally be assumed. The applica ion o his p o ision is subjec o
M&A and he Simula ion-Based Valua ion o Companies 187
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
o pu chase a a ge company, he buye ’s boa d o di ec o s should be awa e o
how hei own company’s isk exposu e changes i hey acqui e he a ge com-
pany. This equi es much mo e han jus due diligence o he conside a ion o
s ock- e u n luc ua ions. I necessi a es he analysis and agg ega ion o he isks
o he a ge company, and he conside a ion o addi ional ansac ion-speci ic
isks (e. g., hose a ising due o unce ain syne gies and in eg a ion cos s) and
changes in he isk posi ion as a esul o he inancing o he acquisi ion.10
Me hods o quan i a i e isk analysis, isk agg ega ion and simula ion-based
alua ion can cap u e he implica ions o a decision o pu chase an in es men
and he e ec s on he isk- e u n p o ile o he buye ’s company.11
The simula ion-based me hod we p opose in his a icle, is sui able o ade-
qua ely conside ing he ‘special igh s’ o indi idual in es o s ha a e ypical o
en u e-capi al in es men s when aluing a sha e.12 Based on he unce ain exi
p ice, i is possible o de e mine indi idual in es o s’ sha e o he exi p ice, ak-
ing in o accoun exis ing special con ac ual egula ions such as liquidi y-p e e -
ence egula ions. The sha e o he exi p ice depends on he le el o he exi
p ice. This is he basis o a isk-adequa e assessmen o indi idual ‘equi y
anches,’ including special igh s. We show how ealis ic anges o possible u-
u e sale p ices can be de e mined based on isk analysis and simula ion me h-
ods, and, in u n, how isk-adjus ed discoun a es o he alua ion o a (po en-
ial) in es men can be de e mined.13 Wi h he me hods we p esen , i is possi-
ble, based on he indings o quan i a i e isk analysis, o de i e isk-adjus ed
e u n equi emen s (discoun in e es a es) o cos -o -capi al and undamen al
alues di ec ly om he ‘ea ning isks’, wi hou eso ing o una ailable his o i-
cal capi al-ma ke da a as wi h he capi al-asse p icing model (CAPM).14 The
e ec s o insol ency isks (p obabili y o insol ency, a ing) a e also aken in o
accoun in he alua ion calcula ion.15
When es ima ing p ices, we u ilize he mul iple me hod o ake in o accoun
he in luence o ma ke impe ec ions on he p ices. Ou alua ion a ionally
he condi ion ha he VC company is a co po a ion, bu his should no be a la ge e-
s ic ion in p ac ice.
10 On he me hods, see Gleißne /E ns (2019); Do lei ne /Gleißne (2018); Do lei ne
(2022) and E ns (2022a) on he assump ions and undamen als o he me hods used.
11 See Gleißne (2019) on he isks o M&A ansac ions.
12 See Jenkinson e al. (2019) and Cede bu g/S ough on (2018).
13 The s uc u e o he company in he case s udy is based on Gleißne /Wol um
(2008), bu he alua ion me hods used in his a icle, which we e no known a he ime,
we e no used.
14 See, undamen ally, Gleißne (2011) o he me hod.
15 See Mo is (2009); Gleißne (2010); Knabe (2012); Saha/Malkiel (2012); Lahmann
e al. (2019) and F anken e al. (2020).
188 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
alues p e iously li le-conside ed in o ma ion abou he isks o he alua ion
objec (and does no assume he pe ec ion o he capi al ma ke , unlike, o ex-
ample, a alua ion based on he CAPM). We also ake up he concep s o
semi-in es men - heo e ical alua ion pa adigm, which is based on he me hod
o impe ec eplica ion, as well as simula ion-based alua ion me hods. Sum-
ma izing, he a icle is he i s o
1. demons a e he applica ion o bo h concep s o pu chase decisions in he
con ex o me ge s and acquisi ions,
2. combine he simula ion-based alua ion me hod wi h s anda d ma ke con-
cep s o es ima ing possible ansac ion p ices (mul iple me hod),
3. compa e he alua ion based on he s anda d de ia ion isk measu e wi h
ha based on he semi-s anda d de ia ion, and
4. in eg a e a consis en alua ion o he special igh s o indi idual sha ehol-
de s.
Sec ion II, discusses he heo e ical unde pinnings o ou app oach. Sec-
ionIII p esen s ou model, including he concep o isk-based alua ion using
‘impe ec eplica ion’ and applies his o an unce ain exi p ice wi h s anda d
de ia ion and semi-s anda d de ia ion as isk measu es and a ealis ic model o
he EBIT de elopmen o e ime. The new alua ion app oach is applied in a
case s udy in Sec ion IV. Based on ce ain alua ion assump ions and, we unde -
ake a isk-adequa e assessmen o he company as a whole and a consis en
sha e alua ion ( aking special igh s in o accoun ). A compa ison wi h con en-
ional alua ion me hods illus a es he supe io i y o he new app oach. A gen-
e al discussion in he las sec ion concludes he pape .
II. Valua ion wi h unce ain exi p ices: Theo y
1. Exis ing app oaches
The s anda d heo y o business alua ion is ep esen ed by discoun ed cash
low me hods, which in ol e aking he expec ed cash lows o each pe iod and
discoun ing hem using a isk-adjus ed discoun a e, using he CAPM. This ap-
p oach is well-documen ed in nea ly e e y co po a e inance ex book,16 so we
e ain om p o iding a adi ional li e a u e e iew on he con en ional solu-
ions o his p oblem.
Howe e , o ca ego ise ou app oach, i is impo an o dis inguish be ween
he a ious alua ion unc ions, wi h pa icula emphasis on he a gumen a ion
16 See e. g. Be k/DeMa zo (2023) o a con empo a y p esen a ion and he e e ences
he ein o u he academic li e a u e.

M&A and he Simula ion-Based Valua ion o Companies 189
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
alue and he decision alue, as hey a e especially ele an o he ield o appli-
ca ion unde conside a ion.17 While he a gumen a ion alue is calcula ed o
nego ia ion si ua ions o jus i y one’s own p ice expec a ions, he decision alue
ep esen s he maximum accep able pu chase o sale p ice o he subjec o al-
ua ion, o example, a ma ginal p ice.
Valua ion me hods oo ed in inancing heo y a e ounded on he neoclassical
hypo hesis o pe ec capi al ma ke s. These me hods calcula e discoun a es
based on his o ical luc ua ions in sha e e u ns using he CAPM. The assump-
ions o he CAPM lead o an equi alence be ween p ice and alue.18 In con as ,
ac o modelling me hods conside mul iple ac o s– no jus he be a ac o o
he CAPM– o explain he expec ed sha e e u ns.19 The heo e ical ounda ion
o hese me hods o en d aws upon Ross’s a bi age p icing heo y (1976).
Many me hods in ol e p ice es ima ion.20 Valua ion mul iples a e de i ed
om he p ices o compa able companies, and hese mul iples a e subsequen ly
used o es ima e he alue o he company in ques ion (see II.2.).21
So a , none o he me hods men ioned allow o he calcula ion o decision
alues; his is ins ead acili a ed by in es men - heo e ical alua ion me hods.
These me hods conside he indi idual in o ma ion, ac ionable op ions, and
cons ain s o he asse being alued.22 No ably, hey do no ely on he assump-
ion o pe ec capi al ma ke s. Ins ead, alua ion is based on ‘ alua ion p inci-
ples de i ed om subjec i e alue heo y’23, whe e alue is unde s ood as a ‘sub-
jec -objec -objec ela ionship’.24, 25 Summa izing Olb ich e al. (2015, p. 34), sub-
jec i e business alua ion is g ounded in he concep o alua ion as an en i y
and i s u u e o ien a ion. These aspec s p o ide a obus decision-making basis
o economic subjec s.
The solid heo e ical ounda ion o hese me hods is o se by he signi ican
calcula ion e o equi ed o he ‘ o al model, ’26 which allows o he conside -
a ion o nume ous al e na i e in es men op ions o he alua ion subjec . Con-
17 See Ma schke (1975) on he decision alue and Ma schke (1979) and Olb ich e al.
(2015, pp. 29 – 32) on he ‘ unc ional business alua ion heo y’.
18 Fo c i icism o he app oach, see also Olb ich e al. (2015).
19 See Fama/F ench (1993, 2015, and 2018); Swade e al. (2023).
20 See Schüle (2020); K olle e al. (2005).
21 Fo ‘ heo y-based’ mul iplie me hods, see e. g. Rich e (2005); Kellene s (2004) and
He mann (2002).
22 See He ing (2000).
23 See Olb ich e al. (2015, p. 17).
24 See Olb ich e al. (2015, p. 18), wi h e e ence o Ma schke e al. (2010).
25 See also Olb ich e al. (2015, pp. 25 – 27) on he inadequa e cap u e o unce ain y by
he ß ac o o he CAPM.
26 See He ing/Toll (2013).
190 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
e sely, simple heu is ic a ian s, known as pa ial models27, a e easie o apply.
Howe e , hese pa ial models ely on p ede e mined discoun a es and do no
p o ide a me hodology ha would, o example, inco po a e insigh s om he
company’s isk analysis.
Klingelhö e e al. (2009) p o ide an example o using hese me hods o de e -
mine a ‘maximum a o dable se lemen ’
28, he eby es ablishing a decision
alue.29 The calcula ion equi es linea op imisa ion, which akes in o accoun
exis ing in es men and inancing op ions. Fo he i s ime, He ing e al. (2012)
combined his me hod wi h a Mon e Ca lo simula ion o alue en u e capi al.30
Building on his wo k, He ing e al. (2013a) p esen ed a model o simula i e
company alua ion ha add esses capi al ma ke impe ec ions and ambiguous
cash lows associa ed wi h he asse , ul ima ely de e mining a ange o subjec i e
ma ginal p ices. This alua ion also equi ed linea op imisa ion. In esponse o
c i iques o adi ional alua ion me hods based on DCF and CAPM, Olb ich
e al. (2015) de eloped a simila me hod speci ically designed o M&A alua-
ions.
The semi-in es men - heo e ical alua ion pa adigm31 d aws inspi a ion om
in es men - heo e ical alua ion concep s and in eg a es hem wi h isk- alue
models.32 Howe e , i also accep s ce ain simpli ica ions ha a e common in
inancial alua ion heo y.33
In con as o he s ic ocus on a speci ic business subjec 34, semi-in es -
men - heo e ical alua ion pa adigm emb aces simpli ica ions ega ding he op-
ions conside ed. Fo ins ance, i ypically e alua es wo al e na i e in es men
op ions o he asse : a isk- ee in es men and a b oad s ock ma ke index.
TheMon e Ca lo simula ion agg ega es he company’s isks using quan i a i e
analysis based on co po a e planning ( isk agg ega ion)35, he eby accoun ing
o inancing es ic ions and insol ency isks.36 Based on he simpli ied as-
27 See He ing/Toll (2013).
28 See Klingelhö e e al. (2009, p. 302).
29 This holds especially o he alua ion o changes in o ing igh s.
30 They u ilize a simula ion-based e sion o he s a e ma ginal quo a model.
31 See Gleißne /Folle (2022) and Gleißne /E ns (2024).
32 See Do lei ne /Gleißne (2018), discusses below.
33 See Gleißne /Folle (2022).
34 See Olb ich e al. (2015) and no e II.1.
35 Fo isk agg ega ion using Mon e Ca lo simula ion in isk managemen , see e. g.
Hunzike (2021); Gleißne (2019) and Vanini/Rieg (2021).
36 Fo he basic idea, see Coenenbe g (1970) and Gleißne (2019); Gleißne /E ns (2023)
and E ns (2022a, 2022b). Supplemen a y He ing e al. (2013a) on simula ion in he con-
ex o in es men heo y and Ma schke/B ösel (2013, pp. 277 – 253) on dealing wi h un-
ce ain y.
M&A and he Simula ion-Based Valua ion o Companies 191
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
sump ion o eliable en i onmen al pa ame e s, his app oach yields a single e-
liable company alue, a he han a ange o alues. The in eg a ion o a isk- al-
ue model37, which employs a isk measu e o cap u e he isk con en o cash
lows, acili a es a comp ehensi e ans o ma ion o unce ain y conce ning
hose cash lows. Alongside he limi a ion o al e na i e in es men op ions, his
unce ain y ans o ma ion h ough he isk- alue model is he key ea u e ha
dis inguishes semi-in es men - heo e ical alua ion pa adigm om i s p ede-
cesso s.
E en independen ly o published case s udies, su eys indica e ha he use o
simula ion-based alua ion me hods has inc eased signi ican ly in ecen yea s,
pa icula ly o complex alua ion p oblems. Acco ding o Gleißne e al. (2024),
a su ey o Ge man alua ion p o essionals e ealed ha 80 % o esponden s
al eady u ilise such me hods.
2. Theo e ical backg ound o ou app oach
Ou app oach o he isk-adequa e alua ion o an in es men company o be
sold a an unce ain exi p ice in eg a es alua ion and p ice-es ima ion me h-
ods.38 To ensu e a s ong connec ion be ween heo e ical ounda ions and p ac-
ical ele ance, ou app oach ocuses on h ee heo e ical concep s: mul iple al-
ua ion, isk- alue model alua ion, and Mon e-Ca lo simula ion.
Mul iple alua ion is a s aigh o wa d app oach ha is widely used in p ac-
ice. I elies on de i ing he alua ion om a simila o compa able company
based on he p inciple o p opo ionali y.39 We chose o use his concep no o
i s heo e ical elegance, bu because p o ides an accu a e ep esen a ion o eal-
i y when selling p i a e equi y sha es in a company. Schüle (2020) p o ides an
o e iew o he concep ual equi emen s and implemen a ion o company alu-
a ions using mul iples, highligh ing which mul iples a e sui able o accu a e
alua ions and how he mul iples used in p ac ice a e in e ela ed (e. g., en e -
p ise alue/sales o en e p ise alue/EBITDA). Addi ionally, empi ical s udies
ha e been conduc ed by Cheng/McNama a (2000) and Chullen e  al. (2015).
Le e aged buyou s (LBOs) demons a e ha he o al company alue (en e p ise
alue), based on median igu es, is app oxima ely se en imes he EBITDA. This
inding comes om an analysis o alua ion mul iples o Eu opean LBOs du -
37 See Gleißne /Do lei ne (2018) on he de i a ion o he alua ion equa ions using
incomple e eplica ion.
38 See He ing e al. (2012) on he alua ion o en u e capi al.
39 See Be k/DeMa zo (2023, ch. 9).
192 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
ing he pe iod om 2000 o 2003,40 e ealing signi ican luc ua ions in alua-
ion o e ime.41
Mos olde s udies on mul iple alua ions p ima ily ocus on indus y mul i-
ples, o en inco po a ing addi ional ac o s such as his o ical g ow h.42 P e i-
ous empi ical s udies ha e demons a ed ha he p ice-ea nings a io can be
e ec i ely explained by indus y a ilia ion, as i p o ides a sui able ep esen a-
ion o isk and ea nings g ow h.43 The es ima ion accu acy o indus y- ela ed
alua ion mul iples can be u he enhanced by using e u n on equi y as an
addi ional selec ion c i e ion alongside indus y a ilia ion.44 These in es iga-
ions, which use con olled mul iples based on undamen al ac o s, yield sig-
ni ican ly imp o ed esul s and indica e ha indus y a ilia ion does no con-
ibu e addi ional explana o y powe compa ed o he undamen al explana o-
y ac o s.45
The second, mo e heo e ically o ien ed, concep ual amewo k used in ou
analysis is he isk- alue model o alua ion, as ou lined by Do lei ne /Gleißne
(2018). This app oach in ol es compa ing he in es men o be alued wi h a
e e ence in es men ha has equal isk and expec ed end alue. No ably, his
me hodology a oids un ealis ic assump ions and does no equi e ex ensi e in-
o ma ion. I s g ounding in a ional decision heo y makes i pa icula ly
well-sui ed o add essing he alua ion p oblem a hand.
Fundamen al ideas o isk-adequa e alua ion, which do no ely on he as-
sump ion o a pe ec capi al ma ke , s em om in es men heo y (see II.1.).46
These me hods ha e eme ged in compe i ion wi h alua ion heo ies based on
inancing heo y, which assume he pe ec ion o he capi al ma ke ( o exam-
ple, he CAPM). Due o he high complexi y associa ed wi h classical in es -
men heo y me hods,47 al e na i e alua ion app oaches, known as semi-in-
40 Acco dingly, he a io o he o al en e p ise alue o he di e ence be ween EBITDA
and in es men s in p ope y, plan and equipmen is a ound 10. See Rich e (2005,
p. 181).
41 The inancial in es o s inance he o al company alue wi h an a e age o 63 %
deb , co esponding o 4.6 imes he EBITDA, see also Roosenboom (2012).
42 The bes es ima ion esul s o ma ke p ices can p obably be achie ed ia a combi-
na ion o DCF company- alua ion me hods and ma ke -o ien ed mul iples (He mann,
2002, p. 31; Kaplan/Ruback, 1996, p. 45; DeAngelo, 1989, p. 93; B une e al., 1998, p. 13).
43 See e. g., Al o d (1992).
44 See Al o d (1992).
45 See He mann (2002); Rich e (2005) and Kellene s (2004).
46 See Ma schke e al. (2010); He ing e al. (2013a); He ing e al. (2014); Ma schke/B ösel
(2021).
47 See Ma schke e al. (2010) and Ma schke e al. (2020).
M&A and he Simula ion-Based Valua ion o Companies 199
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
he e o e, ac ual bank up cy).73 Because he exi p ice is paid on he equi y, he
deb capi al T
D

mus be epaid in acco dance wi h he company’s alue. This
esul s in he ollowing equa ion:
(6)

( )
max 0; ,
EXIT TT
T
P m EBIT D= ×-


whe e
m

ep esen s he s ochas ic mul iple. This is a ‘p ice es ima e’ in an im-
pe ec ma ke , and his exi p ice may de ia e om a ‘ easonable’ undamen al
alue (al hough, as is well known, he e is no such hing as a ‘ ue’ alue).74
The exi p ice es ima e is ob ained om capi al-ma ke da a (e. g., s ock-ma -
ke p ices) o ealised ansac ion p ices o ‘compa able’ companies using a
compa ison me hod. The compa ison me hod de e mines a po en ial ma ke
p ice (‘s ock-exchange p ice’) likely o be achie able on he ma ke . Such me h-
ods a e he e o e also e e ed o as ma ke -o ien ed alua ion me hods. In he
mul iplie me hod, he po en ial ma ke p ice
P

is de e mined by mul iplying a
speci ic (size- ela ed) pa ame e X o he company o be alued by a ac o , m,
ha depends on he selec ed e e ence alue and is usually indus y-speci ic.75
=×-
() ,PX m X D
whe e

P
is he es ima ed ma ke p ice, X is he pa ame e ,76 m is he indus-
y-speci ic ac o 77 and D is deb capi al. EBITDA, EBIT, cash low o sales a e
commonly used o ope a ionalize X. In e ms o sui abili y o calcula ing mul-
iples, Liu e al. (2002) ank he possibili ies as ollows: (1) ea nings o ecas s,
(2)his o ical ea nings, (3) cash low and book alue o equi y, and (4) sales.78
73 See Gleißne (2010, 2019) and F anken e al. (2020) on he signi icance o he p ob-
abili y o insol ency, which is only ouched upon he e.
74 Al hough his should no exis in pe ec ma ke s, see Shlei e /Vishny (1997) o
such inco ec alua ions; Haugen (2002) and Campbell/Shille (1998).
75 The deb capi al FK is deduc ed i an en e p ise alue is calcula ed wi h ‘m · X’.
76 Examples a e EBITDA o EBIT.
77 One example is he quasi- ecip ocal discoun ac o . See Coch ane (2011).
78 Bake /Ruback’s (1999) empi ical s udy o indus y mul iples o he S&P 500 index
in 1995 shows ha mul iples based on EBITDA lead o be e es ima es o ma ke p ices
han hose based on EBIT o sales. In addi ion, in his empi ical s udy, he o ma ion o
a ha monic mean u ns ou o be a sui able me hod o calcula ing he alua ion mul i-
ple; see also Kabo h e  al. (2022); He mann/Rich e (2003); Yoo (2006) and Sch eine
(2007).

200 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
3. An al e na i e isk measu e: he semi-s anda d de ia ion
While he s anda d de ia ion measu es posi i e and nega i e luc ua ions
a ound he expec ed alue, he possibili y o nega i e de ia ions can be ega ded
as mo e ele an o he decision o buy o sell a company, as hese de e mine he
company’s equi y equi emen s and, hus, he u ilisa ion o he (sca ce) isk-co -
e age po en ial.79 The e o e, he semi-s anda d de ia ion
(7)
( ) ( )( )
( )
1
22
max ;0 .Z E EZ Z
σ
-
=-
 
Could be mo e app op ia e as a measu e o isk.80 The impo ance o possible
nega i e de ia ions om he plan and, especially, o possible losses in he alua-
ion is well documen ed in esea ch,81 which jus i ies he use o he semi-s and-
a d de ia ion isk measu e.82 This is pa icula ly ue in he con ex o p i-
a e-equi y in es men s, whe e downwa d and upwa d de ia ions a e no sym-
me ically dis ibu ed.
The ma ginal p ice (decision alue) a which a pu chase becomes wo hwhile
om he poin o iew o a po en ial in es o can again be de e mined using
Equa ion (5) wi h
() ()
..
R
σ
-
=. In he case o a log-no mal dis ibu ion,
T
σ
λ
-
can
be de e mined by simula ion (see he case s udy in Sec ion IV). Fo simplici y, i
is assumed he e ha he in es o does no ha e o spend any supplemen al eq-
ui y o he company in addi ion o he pu chase p ice in o de o p o ide i
wi h equi y ha is in line wi h he isk. I his we e he case, he equi ed equi y
capi al would educe he ma ginal p ice. This equi y equi emen can also be
es ima ed di ec ly om he simula ion. Fu he mo e, he p obabili y o insol-
ency is implici ly conside ed in his ma ginal p ice gi en ha he o al loss o
he in es men is also aken in o accoun in he simula ion in he e en o un a-
ou able business de elopmen .
4. Special igh s and sha e alua ion
Wi h he p ocedu e ou lined he e, a isk-adequa e assessmen is possible no
only o he en i e company bu also o indi idual ‘equi y anches’ wi h hei
speci ic igh s. Knowing he comple e equency dis ibu ion o he unce ain
79 See Gleißne (2022, pp. 428 – 488) on he so-called isk-co e age app oach.
80 Fo he impo ance o downside isks and he ‘skewness’ o a dis ibu ion o esul s,
see K aus/Li zenbe ge (1976).
81 See, o ins ance, Kahnemann/T e sky (1979) on he basics o psychology.
82 And i is p ecisely his isk measu e ha can – be e han he s anda d de ia-
ion–cap u e he implica ion o he limi a ions o liabili y and he ‘clipping’ o he pos-
sible losses.
M&A and he Simula ion-Based Valua ion o Companies 201
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
exi p ice makes i easy o depic exis ing con ac ual ag eemen s wi h indi idu-
al in es o s (equi y p o ide s) and he eby independen ly de e mine he ange
o he ‘unce ain e u n low’ o each equi y p o ide . Acco ding o he gi en
ules, he o al exi p ice is spli in o he exi p ice sha es o he n indi idual in-
es o s Exi
i
P (he e, i = 1, …, n):
(8)
1
.
n
Exi Exi
i
i
PP
=
=å
Wi h he me hods o a isk-based assessmen ou lined abo e, conside ing he
expec ed alue and isk o he e u ns o each in es o I = 1, …, n, he in es o s’
subjec i e alue can be calcula ed, aking all special igh s in o accoun . All ha
is necessa y is a b eakdown o he unce ain exi p ice in acco dance wi h he
con ac ual p o isions.
I , o example, an in es o i, who inanced he las equi y inc ease o a com-
pany, has a ‘liquidi y p e e ence’ such ha hey a e ini ially se ed by he unce -
ain exi p ice un il his o he capi al in es men Ii is eached and hen p opo -
iona ely wi h a sha e αi o he exi p ice, he ollowing payo applies:
(9)
{ }{ }
max min , ;
Exi Exi Exi
ii
i
P PI P
α
=×
.
5. A model o he EBIT de elopmen o e ime
In p ac ice, he collec ion o company-speci ic da a o he alua ion is based
on a de ailed due-diligence (including suppo ed) de ec ion o u he isks ha
de e mine he planning secu i y. The mos impo an esul s a e es ima es o he
expec ed alue and isk o he s ochas ic sales g ow h a es
11
, , ,
T
gg
-
¼
and an
es ima e o he expec ed EBIT ma gin and he ypical ex en (s anda d de ia-
ion) o isk- ela ed de ia ions om his o ecas .
Based on sales o he p e ious pe iod 0
R
o €15 million and sales g ow h g0 o
20 % in he p e ious pe iod, Tex i AI, Inc. expec s a linea dec ease in he sales
g ow h a e o 2 % om he e minal alue pe iod (meaning ha planning is
done wi h 5€25
Plan
R= million).83 The coe icien o a ia ion o he sales g ow h
a e (i. e., he a io o he s anda d de ia ion o he expec ed alue) should be
cons an a 25 % annually, wi h de ia ions om he expec ed sales g ow h a e
assumed o be no mally dis ibu ed.
(10)
( ) ( )
( )
11
11
g
R R g R Eg
ε
--
= += + +
  

, whe e
( )
0;
gg
N
εσ
~
.
83 Fo o he models, see, o example, Schwa z/Moon (2000, 2001); Behm (2003) and
Klobucnik/Sie e s (2013).
202 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
A he beginning (i. e., a = 0), he EBIT ma gin (EBITM) is –1 %; in he long
e m, a alue o 7.5 % is expec ed ( 7.5%EBITM =).
The adi ional EBIT planned o = T = 5 wi hou simula ion is he e o e:
· 25 · 7.5%
Plan Plan Plan
TT T
EBIT R EBITM
==
= € 1.875 million.
A mean- e e ing p ocess is assumed o he de elopmen o he EBITM.
Such a isky p ocess ends owa ds he
EBITM
. The u he he alue om he
p e ious pe iod

1
EBITM -
de ia es om his mean, he s onge he mean- e-
e ing endency. Fo mally, his can be desc ibed ecu si ely as ollows:
(11)
 
( )
1.
EBITM
EBITM EBITM a EBITM EBITM
ε
-
=+ -+

De ia ions om he expec ed end alue
( )
ε
EBITM
E a e assumed o be no -
mally dis ibu ed wi h a s anda d de ia ion EBITM
σ
. The s anda d de ia ion o
he EBIT ma gin should dec ease linea ly in he planning pe iod, om 3 % in
he pas o 1 % in he e minal alue pe iod.84
As an al e na i e o he simple s ochas ic p ocess o he EBIT ma gin, mo e
de ailed planning can be ca ied ou . The luc ua ion ange o he EBIT ma gin
can be de e mined using simula ion-based isk agg ega ion me hods. S a ing
om iden i ied and alua ed indi idual isks, hese me hods enable he di ec
calcula ion o he ypical isk- ela ed ange o EBIT de elopmen by calcula ing
a ep esen a i e numbe o isk- ela ed u u e scena ios. The in e es a e on
bo owed capi al i o in e es on bo owed capi al (D) should emain cons an 85
o e he pe iod unde conside a ion and is assumed o be 5.1 %. The simula ion
is ca ied ou ecu si ely, ha is, p og essi ely om –1 o .
The p o i be o e ax in pe iod is de e mined acco ding o he ollowing
equa ion:
(12)


1
EBT R EBITM D i
-
=× - ×

.
Fo he sake o simplici y, axes a e neglec ed he e. G ow h is o be inanced by
e aining he p o i be o e ax, so no payou s a e made. I is assumed ha he cap-
i al employed a he beginning o he planning pe iod, namely
00
o
CE E D=+
,
de elops analogously o sales. In o he wo ds, he capi al u no e emains con-
s an , leading o he ollowing equa ion:
84 Va ious da a sou ces such as compa a i e indus y alues, he esul s o isk analyses
o subjec i e es ima es by expe s can be used o es ima e hese pa ame e s.
85 Assume ha he company has a 5-yea line o c edi wi h a cons an , de e minis ic
in e es a e. S ic ly speaking, he in e es cash lows (I) om he poin o iew o =0
a e a condi ional dis ibu ion, he o m o which depends on he isks ealised up o – 1.
M&A and he Simula ion-Based Valua ion o Companies 203
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
(13)
 
10
0
1
RR
CE CE CE
R
R-
-
==

.
I he EBT (ea nings be o e axes) is no su icien o inancing, addi ional
ou side capi al is aken ou . I he EBT exceeds he necessa y in es men amoun ,
bo owed capi al is epaid. This esul s in he ollowing equa ion:
(14)

 
1
1
D D EBT CE CE -
-
= - +-
 .
The (ne ) bo owed capi al can he e o e also assume nega i e alues. These
can be ega ded as liquid unds. In e es income is hen gene a ed wi h hese,
whe e o simplici y no dis inc ion is made be ween he in e es a e on bo -
owed capi al and he in e es on he c edi balance (each o which = i). The
deb capi al D can hus be in e p e ed as a ne bank liabili y. Gi en ha dis ibu-
ions and capi al inc eases a e excluded, he equi y a he end o pe iod is he
sum o he equi y a he beginning o he pe iod and he EBT:
(15)

1
E E EBT
-
=+
 .
To simula e EXIT
P

acco ding o o mula (6), he EBIT a ime = T mus be
calcula ed. This is done acco ding o he ela ionship al eady used in o mula
(13):
(16)


EBIT R EBITM=×
.
The iangula dis ibu ed mul iple m is s ochas ically simula ed independen -
ly o he o he a iables.
6. Model discussion
Ou app oach can be used o de e mine decision alues o a gumen a ion al-
ues (see II.1.86). To es ablish an a gumen a ion alue in a pu chase nego ia ion,
pa ame e s a e se cau iously, so ha hey esul in a e y low alua ion ha ac s
as he s a ing poin o he p ice nego ia ions. When ealis ic pa ame e s a e
applied om he pe spec i e o he alua ion objec ( he buye ), a decision alue
is ob ained, indica ing he maximum accep able pu chase p ice. In a nego ia ion
con ex , ou app oach has he ad an age ha only a consensus on a p obabili y
dis ibu ion is necessa y conce ning unce ain assump ions. Fo ins ance, his
may pe ain o he u u e sales g ow h a e o he p ice mul iple achie able a
he exi ime.
86 See also Folle e al. (2018).
204 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
When de e mining company alues o p i a e equi y o en u e capi al in-
es men s, i is ad an ageous ha he p ocedu e is based on he amilia me hod
o assessing adequa e p ices using EBIT o EBITDA mul iples. Ex ending his
exis ing app oach o accoun o u u e unce ain ies, such as he sales g ow h
a e and he u u e alua ion le els, ep esen s a signi ican bene i .
I is also ad an ageous ha , unlike alua ions based on inancing heo y using
he CAPM, his me hod does no assume a pe ec capi al ma ke ; ins ead, i
conside s exis ing a ing and inancing es ic ions. The combina ion o a Mon-
e Ca lo simula ion o es ima ing an unce ain exi p ice, combined wi h he
esul s o a quan i a i e analysis o he company’s isks, and he applica ion o
he semi-in es men - heo e ical alua ion pa adigm based on impe ec eplica-
ion, u he enhances he obus ness o his app oach.
In he simple model p esen ed in Chap e III, i may be seen as a disad an age
ha he alua ion does no ully inco po a e in eg a ed co po a e planning.
While adding his elemen would inc ease complexi y, i is ce ainly easible o
implemen .
F om he pe spec i e o inancial alua ion heo y, a no able disad an age is
he absence o he CAPM, which emains widely used in p ac ice, pa icula ly in
legal company alua ions. Howe e , he e is a g owing demand o simula-
ion-based me hods in demanding alua ion cases,87 and he CAPM is inc eas-
ingly subjec o c i ical sc u iny in legal con ex s, such as when calcula ing com-
pensa ion o mino i y sha eholde s in Ge many.88
F om he pe spec i e o in es men alua ion heo y, he model can be c i-
iqued o i s simpli ica ions, pa icula ly ega ding al e na i e in es men op-
ions, which do no accoun o po en ially ele an in o ma ion abou he alu-
a ion subjec ’s oppo uni ies and es ic ions.89 Fo example, he model is no
designed o e alua e a la ge numbe o simul aneously easible in es men op-
po uni ies wi hin a con ex ual amewo k, making i unsui able o de e min-
ing an op imal po olio ha conside s he in e dependencies be ween indi id-
ual VC o PE in es men s.
Fo he sake o simplici y, he alua ion assumes ha all o he alua ion issues
ha e been esol ed and ha only one in es men oppo uni y needs o be e al-
ua ed in isola ion. Howe e , his limi a ion can be elaxed by de e mining he
espec i e ne p esen alue a es o mul iple in es men al e na i es, i. e., cal-
cula ing he en e p ise alue (V) in ela ion o he ini ial in es men (Io), which
acili a es p io i isa ion. F om his pe spec i e, he calcula ed en e p ise alue is
no a decision alue does no se e as a decision alue in he na owe sense, as
87 See he empi ical su ey in Gleißne e al. (2024).
88
Fo example, see Laube (2014); Folle (2019); Quill (2020) and Gleißne /Folle (2022).
89 See again He ing/Toll (2013) and He ing e al. (2013b).

M&A and he Simula ion-Based Valua ion o Companies 205
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
i does no inco po a e all in o ma ion ele an o he alua ion subjec . Ins ead,
he alua ion esul can be iewed as a ‘ ypi ied decision alue’ a he han a
‘subjec i e decision alue’.90
In con as o he app oaches o He ing e al. (2013a) and o he amewo ks in
in es men - heo e ical alua ion heo y, ou model, inspi ed by he ideas o Fa-
ma (1977), de e mines a speci ic alue a he han a ange o alues. This alue
esul s om a comple e ans o ma ion o unce ain y, achie ed h ough he ap-
plica ion o a isk- alue model. The isks associa ed wi h cash lows a e con e -
ed in o a sa e igu e ia he isk measu e, while unce ain ies ela ed o en i on-
men pa ame e s (e. g. he isk- ee in e es a e) a e neglec ed in line wi h he
p inciples ou lined by Fama (1977). The e o e, semi-in es men - heo e ical al-
ua ion pa adigm, simila o inancial- heo e ical alua ion based on CAPM,
aims o de e mine a de ini e alue a he han a alue ange, p o ided he en i-
onmen pa ame e s a e assumed o be s able.
IV. The alua ion o a PE in es men :
A case s udy wi h an unce ain exi p ice
1. Basics and alua ion assump ions
The ollowing simple case s udy o a non-lis ed pa icipa ion in an in es men
und shows how a simula ion-based alua ion can be ca ied ou .91 In he case
s udy, he ma ginal p ice P* is o be de e mined o Tex i AI, Inc., wi h an as-
sumed exi in T = 5 yea s. The ope a ing capi al o € 7.5 million is inanced wi h
equi y a he le el o 20 % (= €1.5 million). The cu en bo owed capi al D0 is
hus € 6 million.
Two pa ne s own he company. The ounde (G) o he company and a en-
u e-capi al (VC) und ha joined a e he company’s ounding each own 50 %
o he sha es. Wi h i s en y in o he company and he acquisi ion o he 50 %
sha eholde sha e, he VC und con ibu ed one million eu os in equi y. I has
ag eed o a special igh (liquidi y p e e ence) o he in ended exi , he sale o a
s a egic in es o , and will ecei e he i s million eu os om he unce ain u-
u e exi p ice
( )
EXIT
p exclusi ely. The excess amoun is di ided in hal .
The owne s ha e wo al e na i e in es men op ions o choose om: quasi-
isk- ee go e nmen bonds wi h an in e es a e o = 3 %, o a b oadly di e -
si ied s ock index wi h an unce ain ma ke e u n

m
. Thei annual dis ibu ion
is i.i.d. (independen and iden ically dis ibu ed) assumed o be log-no mally
90 A dis inc ion ha is also o be made in Ge many wi h he e ision o he IDW S1
alua ion s anda d (IDW ES 1, 2024).
91 See also Gleißne /E ns (2019) and E ns (2022a).
206 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
dis ibu ed, wi h an expec ed alue
e
m
= 8 % and s anda d de ia ion
σ
m
= 20 %.
Equa ion (4) p oduces ma ke p ice o he isk
T
σ
λ
esul s o he isk-measu e
s anda d de ia ion as shown in Table 1.
Table 1
Mul i-pe iod isk p ice depending on he pe iod leng h T using he s anda d
de ia ion as a isk measu e and log-no mally dis ibu ed ma ke e u ns
T 1 2 3 4 5
T
σ
λ
0.25 0.34 0.41 0.45 0.49
These alues (as well as he ones in Table 2) esul om a Mon e Ca lo simu-
la ion wi h he pa ame e s gi en abo e. Wi h he same pa ame e s, he
T
σ
λ
- e-
sul s shown in Table 2 a e ob ained o he isk measu e o he (lowe ) s anda d
de ia ion (σ).
Table 2
Mul i-pe iod isk p ice depending on he pe iod leng h T using he semi-s anda d
de ia ion as a isk measu e and log-no mally dis ibu ed ma ke e u ns
T 1 2 3 4 5
T
σ
λ
-0.33 0.44 0.51 0.57 0.60
In he example case, i is ixed ha
• hep edic edexi  akesplaceinyea T = 5,
• he iskdi e si ica ion ac o isassumed obed = 0.5,92
• a  he imeo exi ,a alua ionmul iple(m) based on EBIT wi h a minimum
o 6, he mos likely alue o 8 and a maximum o 11 is assumed o he a ed
company (modelled wi h a iangula dis ibu ion).
The mos likely alue is conside ed as he adi ional planned alue o he
EBIT mul iple m:
8.m=
In addi ion, a sales mul iple ypical o he indus y is
known, namely mR = 0.7.
Finally, he addi ional unding obliga ions o equi y in es o s a e excluded, so
he exi canno ake nega i e alues. Wi h hese speci ica ions, he uni o m al-
92 This means ha hal o he o e all isk is assumed o be ele an o he assesso a e
di e si ica ion e ec s. In iew o he impe ec di e si ica ion o mos alua ion subjec s,
his use o an a e age isk-di e si ica ion deg ee o d = 0.5 seems accep able.
M&A and he Simula ion-Based Valua ion o Companies 207
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
ua ion equa ion o he ma ginal p ice ( undamen al alue) o he company is
de i ed acco ding o equa ion (5), whe eby only wo a iables ha e o be es i-
ma ed h ough using company-speci ic simula ions: he expec ed alue o he
exi p ice, using a (s ochas ic) mul iplie me hod; and he semi-s anda d de ia-
ion o he exi p ice, as a measu e o he o e all (ope a ional) isk exposu e.
Tha is:
(17)
( ) ( )( )
( )
σ
-
=-
 
1
22
max ; 0
Exi Exi Exi
P E EP P =

( )( )

( )( )
( )
1
22
max max 0; max 0; ;0 .
T T
T T
E E m EBIT D m EBIT D
==
==
= ×--×-


2. Risk simula ion and exi p ice ange
Figu e 1 shows he cou se o he expec ed alues o he capi al i ems. This
g aphic was c ea ed using he alues gi en abo e ia simula ion.93
Figu e 1: Expec ed cou se o he capi al posi ions
Based on he (manageable) in o ma ional inpu om he analysis, we pe o m
a simula ion ollowed by a alua ion. Figu e 2 shows he ange o EBIT de elop-
men . The alues esul om a Mon e Ca lo simula ion ha we ca ied ou in
Mic oso Excel wi h a supplemen al add-in.
93 We use Excel and C ys al Ball.
0
2000
4000
6000
8000
10000
12000
14000
0 1 2 3 4 5
T€
Pe iod (in yea s)
Equi y Deb Capi al Employed
208 We ne Gleißne , Ma co Wol um, G ego Do lei ne
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
Figu e 2: Range o possible EBIT de elopmen s
The ini ial objec i e is o pe o m a isk-based assessmen o he company as
a whole and assess he sha es o he wo sha eholde s, conside ing he special
igh s o he en u e-capi al in es o . The immedia e esul o he simula ions is
he dis ibu ion o Exi
P

a T = 5, calcula ed acco ding o (6), om which he ex-
pec ed alue and (semi-)s anda d de ia ion can be calcula ed. Figu e 3 shows
he esul s o he simula ion.
To de e mine a ma ginal p ice P*, he ange o he unce ain exi p ice mus
be es ima ed. The simula ion o he cash low in he planning pe iod (and he
implici unde lying isk assessmen ) is mainly used o de e mine he possible
s a es o he company (and he en i onmen ) a he ime o exi and o de i e an
exi p ice om hem. The Mon e Ca lo simula ion p o ides he ollowing e-
quency dis ibu ion o EXIT
P

.
-500
0
500
1.000
1.500
2.000
2.500
3.000
0 1 2 3 4 5
T€
Pe iod (in yea s)
EBIT
expec ed EBIT 15% Quan ile 85% Quan ile
M&A and he Simula ion-Based Valua ion o Companies 215
C edi and Capi al Ma ke s, 57 (2024) 1 – 4
y-in es men company ha buys sha es in a company oday wi h he aim o
selling i a a highe p ice in he u u e, he unce ain u u e sale p ice is ele an
o he alua ion. Ou me hodology combines app op ia e alua ion me hods
and u u e-o ien ed p ice-es ima ion me hods in an in eg a i e model ha can
consis en ly and simul aneously alue he company as a whole and he sha es in
he company held by di e en owne s wi h di e en special igh s.
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