premija za osiguranje depozita na primeru bankarskog sistema

originalni
naučni
rad
dr Dragan Jović
Centralna banka Bosne i
Hercegovine,
Glavna banka Republike Srpske
[email protected];
UDK 368.025.1:336.717.22(497.6)
PREMIJA ZA
OSIGURANJE DEPOZITA
NA PRIMERU
BANKARSKOG SISTEMA
BOSNE I HERCEGOVINE
Rezime
Ključne riječi: osiguranje depozita, stopa premije za osiguranje depozita,
varijabilnost, put opcija, Merton 1977, teorija vjerovatnoće, prekidni rasporedi,
Bosna i Hercegovina
JEL: E44, G21, G33
Rad primljen: 11.09.2012.
Odobren za štampu: 27.09.2012.
bankarstvo 1 2013
Razmotrili smo dilemu u pogledu uvođenja promjenljive stope premije
za osiguranje depozita u bankarskom sektoru Bosne i Hercegovine/BiH.
Promjenljiva stopa/premija je alternativa fiksnoj, koja je u BiH već 6 godina
0,3%. Rezultat istraživanja nije jednoznačan. Od četiri upotrebljena modela dva
sugerišu smanjenje premije, a dva povećanje. Prema modelu “Merton 1977”
premija čak nije ni potrebna, jer je tržišna vrijednost aktive banaka u prosjeku
veća od bankarskih dugova. Rezultate dobijene primjenom hipergeometrijskog
rasporeda takođe interpretiramo kao prjedlog za smanjenje premije. Međutim,
drugi prekidni raspored, binomni raspored, sugeriše povećanje premije
osiguranja. Motiv povećanja premije je rast loših kredita koji utiče na povećanje
vjerovatnoće difolta banke, što vodi ka pomjeranju binomnog rasporeda u
desnu stranu. I linearni pristup u istraživanju opravdanosti fiksne premije
traži povećanje premije tj. varijabilnu premiju. Istraživanje nije dalo konačan
i nedvosmislen stav u odnosu na dilemu da li stopa premije za osiguranje
depozita u bankarskom sektoru BiH treba biti promjenljiva ili fiksna. Modeli
se značajno razlikuju po smjeru, ali i po veličini, predloženog varijabiliteta.
Zato na bazi modela dobijeni varijabilitet premije ne tumačimo na način da
premija treba biti varijabilna, već na način da nismo u potpunosti uspjeli
dokazati da premija ne treba biti fiksna. Bez novih istraživanja nije moguće
dokazati hipotezu o potrebi uvođenja varijabilne stope premije osiguranja.
Pošto hipoteza nije potvrđena, tj. samo je djelimično potvrđena, i dalje ostaje
stav da stopa premije za osiguranje depozita u BiH treba biti fiksna.
76
UDC 368.025.1:336.717.22(497.6)
DEPOSIT INSURANCE
PREMIUM IN THE
BANKING SECTOR
OF BOSNIA AND
HERZEGOVINA
original
scientific
paper
Dragan Jović, PhD
Central bank of Bosnia and
Herzegovina,
Glavna banka of the
Republic of Srpska
[email protected];
bankarstvo 1 2013
Summary
77
The paper examines the dilemma concerning the implementation of variable
deposit insurance premium in the banking sector of Bosnia and Herzegovina/
B&H. The variable rate/premium is an alternative to the fixed rate, which has
in B&H for six years already amounted to 0.3%. The results of our research
are not uniform. Out of the four used models, two suggest a reduction of the
premium, whereas the other two suggest an increase. According to the “Merton
1977” model, the premium is not even required, given that the market value
of the banks’ assets is on average higher than the banks’ debts. The results
received through the implementation of hypergeometric distribution have also
been interpreted as a suggestion of the premium reduction. However, another
discrete distribution - binomial distribution, suggests the insurance premium
to be increased. The motive for the premium increase is the growth of nonperforming loans, which has caused higher probability of banks’ default, and,
in turn, led to the binomial distribution being skewed to the right. The linear
approach to examining the justifiability of fixed premium has also asked for
an increased premium, i.e. a variable premium. The research, however, has
not provided a final and unambiguous conclusion concerning the dilemma
of whether the deposit insurance premium rate in the banking sector of B&H
should be variable or fixed. The models considerably differ as to the direction,
but also the size of the proposed variability. Therefore, based on the applied
models, our interpretation is not that the variability of the premium from our
research suggests that the premium should be variable, but rather that we
have not fully proved that the premium should not be fixed. Without any
new surveys, it is impossible to prove the hypothesis about the necessity of
introducing a variable insurance premium rate. Given that the hypothesis has
not been proven, i.e. that it has only been partially confirmed, there remains
the position that the deposit insurance premium rate in B&H should be fixed.
Key words: deposit insurance, deposit insurance premium rate, variability,
put option, Merton 1977, probability theory, discrete distributions, Bosnia and
Herzegovina
JEL: E44, G21, G33
Paper received: 11.09.2012
Approved for publishing: 27.09.2012
Uvod
U okviru tržišno-orijentisane reforme
bankarskog sistema u bankarskom sektoru BiH
(u daljem tekstu BSBiH) je uvedeno obavezno
osiguranje depozita. Be-ha banka koja nije
osigurala depozite ne može dobiti dozvolu za
rad.
Predmet istraživanja je stopa premije za
osiguranje depozita (u daljem tekstu: POD)
u BSBiH. Od početka instaliranja šeme za
osiguranje depozita, do danas, ona iznosi 3
promila, ili 0,3%. Određena kao fiksna, takva
je i ostala. Naš cilj je da odredimo da li POD
treba biti fiksna ili varijabilna/promjenljiva.
Zalažemo se za drugu opciju, promjenljivu
POD, to je hipoteza istraživanja. POD treba biti
promjenljiva, a ne konstantna. U širem smislu
istraživanje se može tretirati i kao doprinos
staroj dilemi: fiksna ili varijabilna POD pro i
contra. Obuhvaćen je period od 12/2007. do
12/2011. sa ishodištem u 12/2000. g.
Najduže smo se zadržali na modelu
“Merton 1977”, jer je on najsloženiji. Pored
njega upotrebljen je binomni raspored i
hipergeometrijski raspored. Model linearizacije
je najprostiji primjenjeni metod. Poslije
prezentovanja metodologije slijedi prikaz
rezultata i rasprava o njima. Završna riječ je
predstavljena u obliku zaključnih razmatranja.
je bazična aktiva (underlying asset), a tržišna
vrijednost duga/osiguranih depozita (D) je
cijena izvršenja opcije (strike price). Naplata POD
proizvodi obavezu institucije za osiguranje
depozita da održi njihovu likvidnost, tj. da u
krajnjoj liniji isplati deponente, čiji su depoziti
osigurani. U osiguranju depozita banka je
kupac, a institucija za osiguranje depozita
prodavac put opcije. Banka ima pravo da proda
aktivu banke po bilo kojoj cijeni, tj. po tržišnoj
vrijednosti depozita, a AOD ima obavezu da
isplati osigurane depozite. Za kupovinu put
opcije banka plaća premiju - POD.
Glavne varijable “Merton 1977” su: A tržišna vrjednost aktive, D - tržišna vrjednost
duga (u primjeni ovoga modela korporativni
dugovi korespondiraju depozitima, Laeven,
2002. p. 7), T - vrijeme do dospjeća bankarskog
duga, t - frakcija vremena, σ - standardna
devijacija stope prinosa na aktivu/ROA, g POD, δ- prinos od dividende (dividend yield), N
- funkcija kumulativne normalne distribucije.
Sama specifikacija modela je bitna u mjeri
uticaja varijabli na POD.
Formula 1. Premija osiguranja *
(
)
 (1 − δ ) A 
g = N σ * T − t − ht − 
 * N (− ht )
 D 
gdje je
 (1 − δ )A  σ
(T − t )
ln 
+
D  2

ht =
σ T −t
2
Model “Merton 1977”
Model određivanja POD “Merton 1977” je
šire objašnjen i obrazložen u literaturi (Laeven,
2002), a na ovome mjestu ukratko, u skladu sa
ciljem rada.
Kao i kod svakog drugog osiguranja, i za
osiguranje depozita se plaća premija. Banka
uplaćuje POD instituciji za osiguranje depozita.
U BSBiH to je Agencija za osiguranje depozita
(u daljem tekstu AOD). Bilansna protustavka
osiguranim depozitima je aktiva banaka. Iz
ovoga odnosa Merton je izvukao analogiju
između osiguranja depozita i put opcije:
osiguranje depozita je jednako pravu, ali ne i
obavezi banke da proda svoju aktivu instituciji
za osiguranje depozita po cijeni koja je jednaka
vrijednosti obaveza u osiguranim depozitima.
Sa aspekta put opcije imovina/aktiva banke (A)
Izvor: Laeven, Luc. Pricing of Deposit Insurance.
World Bank Policy Research Working Paper
2871, July 2002. p.7.
* U orginalnoj formuli Vt (tržišna vrijednost
aktive) je zamijenjeno, radi lakše notacije sa A.
Uopšteno, prema “Merton 1977” POD je
obrnuto proporcionalna odnosu između tržišne
vrijednosti aktive i tržišne vrijednosti duga tj.
osiguranih depozita. Logika je jasna, sve dok
je tržišna vrijednost aktive veća od tržišne
vrijednosti duga ne postoji interes banke da
aktivira put opciju i da ustupi aktivu za manji
iznos nego što ona zaista vrijedi. Dakle, pod
pretpostavkom da je sve ostalo jednako, viši A
/ D daje nižu POD.
Između standardne devijacije stope ROA
bankarstvo 1 2013
Materijal i metode
78
Introduction
As part of the market-oriented reform of the
banking system, obligatory deposit insurance
was introduced to the banking sector of B&H
(hereafter to the referred to as: BSB&H). Any
bank in B&H which failed to insure the deposits
cannot obtain an operating license.
The subject of our research is deposit
insurance premium rate (hereafter to be
referred to as: DIP) in the BSB&H. Since the
implementation of the deposit insurance
scheme, until today, this rate has amounted to
3 per mills, or 0.3%. Originally defined as such,
the rate has remained fixed. Our objective in
this paper is to determine whether DIP should
be fixed or variable. We are in favour of the
latter option - the variable DIP, which is the
main hypothesis of our research. DIP should
be variable, not constant. In a broader sense,
this research may be treated as a contribution
to addressing the old dilemma: fixed or variable
DIP - pros and cons. The paper has covered the
period from 12/2007 to 12/2011, the starting
point being 12/2000.
We have devoted most of our time to the
“Merton 1977” Model, given that it is the most
complex one. In addition, we have resorted to
binomial and hypergeometric distributions.
Linearization model is the simplest of the applied
models. The presentation of methodology is
followed by the review of achieved results and
a relevant discussion. This is accompanied by
the concluding remarks.
Subject Matter and Methods
between deposit insurance and put option:
deposit insurance equals the right, but not the
obligation of a bank to sell its assets to a deposit
insurance institution, at a price equal to the
value of liabilities in the insured deposits. From
the perspective of a put option, the bank’s assets
(A) are the underlying asset, and the market
value of debt/insured deposits (D) is the strike
price. The collection of DIP generates obligation
for the deposit insurance institution to maintain
their liquidity, i.e. to ultimately pay out the
deponents, whose deposits were insured. In
the process of deposit insurance, the bank is the
buyer, whereas the deposit insurance institution
is the seller of a put option. The bank has the
right to sell the bank’s assets at any price, i.e. at
the deposit’s market value, whereas the DIA has
the obligation to disburse the insured deposits.
In order to purchase the put option, the bank
has to pay the premium - DIP.
The main variables of “Merton 1977” are the
following: A - market value of assets, D - market
value of debt (in the application of this model
corporate debts correspond to deposits, Laeven,
2002, p. 7), T - time until maturity of the bank
debt, t - fraction of time, σ - standard deviation
of the rate of return on assets/ROA, g - DIP, δ dividend yield, N - function of the cumulative
normal distribution. The specification of the
model itself is relevant in terms of the extent of
variables’ impact on DIP.
Equation 1. Insurance premium*
(
)
 (1 − δ ) A 
g = N σ * T − t − ht − 
 * N (− ht )
 D 
bankarstvo 1 2013
where
79
“Merton 1977” Model
As a model for defining DIP, “Merton 1977”
has been extensively presented and explained
in reference literature (Laeven, 2002), and here
we will only present it briefly, in line with the
objective of this paper.
As with any other form of insurance, there
is a premium to be paid for deposit insurance.
A bank pays DIP to a deposit insurance
institution. In the BSB&H that is the Deposit
Insurance Agency (hereafter to be referred to
as: DIA). The balance sheet counter-item to the
insured deposits are the banks’ assets. Based
on this relationship, Merton drew the analogy
 (1 − δ )A  σ
(T − t )
ln 
+
D  2

ht =
σ T −t
2
Source: Laeven, Luc. Pricing of Deposit Insurance.
World Bank Policy Research Working Paper
2871, July 2002. p. 7.
Note: * Vt (market value of assets) from the
original equation has been replaced with A, for
the purpose of easier notation.
In general, according to “Merton 1977”, DIP
is indirectly proportionate to the ratio between
Za volatilnost od 1% POD je nula za A / D ≥
1,05, isto tako i za volatilnost od 3% pri A / D
≥1,10, i tako sve do para A / D, σ (1,15, 5%). Za
održavanje POD na niskom nivou, pri visokoj
volatilnosti ROA, potrebna je naknada u obliku
rasta A / D. Za A / D ≤ 1, bez obzira na nivo
volatilnosti, POD je veća od nule. Par A / D σ (1,1,
7%) približava POD vrijednosti POD u BSBiH
0,29% tj. 0,3%. POD raste sa rastom volatilnosti
i padom A / D, a u eksperimentalnom uzorku
(Grafikon 1) za najekstremnije vrijednosti A /
D = 0,75 i σ = 21% iznosi 25,71%, što bi značilo
da banka, čija je tržišna vrijednost aktive svega
75% tržišne vrijednosti duga, a volatilnost ROA
21%, trebala uplatiti na ime POD 0,25 novčanih
jedinica za svaku jedinicu osiguranog depozita.
Banke sa A / D ≥ 1 i stabilnim i uravnoteženim
ROA bi plaćale nižu POD od banaka sa A / D <
1 i visokom volatilnošću ROA. Model određuje
POD u zavisnosti od rizika difolta banke, a on
je funkcija ROA i A / D.
Ako je A / D konstantan, npr. 1 (grafikon
U idealnim uslovima nulte volatilnosti POD
je uvijek 0% - ukoliko je A / D ≥ 1 (Grafikon
1). Logičko obrazloženje bi moglo biti da, kada
je tržišna vrijednost aktive jednaka, ili veća
od duga banke, uz stalno isti ROA, AOD nije
izložena riziku isplate osiguranih depozita,
jer banka nema apsolutno nikakav interes da
aktivira put opciju pošto je tržišna vrijednost
aktive banke veća od tržišne vrijednosti duga,
a ROA iz godine u godinu ima iste, pozitivne,
vrijednosti. Do nulte POD se dolazi i na višim
nivoima volatilnosti, ali uz A / D veće od jedan.
2), POD se povećava sa povećanjem volatilnosti
ROA. Volatilnost ROA od nula daje POD
od nula, dok volatilnost ROA od 21% daje
POD od 8,36%. Isti A / D uz uslov da je sve
ostalo jednako, vodi ka linearnoj vezi između
volatilnosti ROA i POD.
Rasporedi vjerovatnoće
Koristili smo dva rasporeda vjerovatnoće:
binomni (u daljem tekstu BR) i hipergeometrijski
raspored (u daljem tekstu HGR). Oba rasporeda
su prekidna. BR ima samo dva ishoda, npr:
bankarstvo 1 2013
(σ) i prinosa od dividende (δ), sa jedne strane i
POD ,sa druge strane, postoji pozitivna veza. I
ovdje je logika više nego jasna. Veći varijabilitet
stope ROA izlaže banku većem riziku od
gubitka, posebno u uslovima visokog leveridža.
Npr. ako je leveridž (leveridž/multiplikator
dioničkog kapitala = bilansna aktiva/akcijski
kapital) 30, a prinos na aktivu (ROA) 1%, ROE je
30%, dok je ROE je - 30% kada je ROA - 1%. Zato
je POD pozitivno korelisana sa varijabilitetom
ROA.
Veći prinos od dividende znači manju
neraspoređenu dobit i manji pripis dobiti
akcijskom kapitalu. Odsustvo rasta, sposobnosti
banke da eventualni gubitak pokrije iz kapitala
prenosi rizik od gubitka na dug, deponente, tj.
instituciju za osiguranje depozita. Veći rizik
traži veću naknadu za osiguranje, tj. veću POD
i otuda pozitivna korelacija između δ i ROA.
Pod pretpostavkom T = 1, t = 0 i δ= 0 odnos
između A / D, volatilnosti (σ), ROA i POD je
kao na grafikonu 1.
80
bankarstvo 1 2013
market value of assets and market value of
debt, i.e. insured deposits. The logic behind it
is clear - as long as the market value of assets is
higher than the market value of debt, it is not
in the bank’s interest to activate a put option
and sell the assets for a lower amount than
they are actually worth. In other words, under
the assumption that all other variables are the
same, higher A/D implies lower DIP.
There is a positive correlation between the
standard deviation of the ROA rate (σ) and
dividend yield (δ) on one hand, and DIP, on
the other. The logic is also more than clear.
Higher variability of the ROA rate exposes
the bank to the higher risk of loss, especially
when the leverage is high. For example, if the
leverage (leverage/equity multiplier = balance
sheet assets/equity) is 30, and return on assets
(ROA) amounts to 1%, ROE is 30%, and if ROA
amounts to -1%, then ROE is -30%. This is why
DIP is positively correlated to the variability of
ROA.
Higher dividend yield implies lower
retained earnings and lower allocation of profit
to shareholders’ equity. Lack of growth and
bank’s ability to cover potential losses from its
capital transfers the risk of losses to depositors,
i.e. to the deposit insurance institution. Higher
risk calls for higher insurance fee, i.e. higher
DIP, which leads to the positive correlation of
δ and ROA. Under the assumption that T = 1,
t = 0 and δ = 0, the relationship between A/D,
volatility (σ), ROA and DIP is as shown in Chart
1 below.
81
In the ideal circumstances of zero volatility,
DIP is always 0% - provided that A/D ≥ 1 (Chart
1). A logical explanation could be that, when
the market value of assets is equal to or higher
than the bank’s debt, with the same values
of ROA, DIA is not exposed to the risk of
insured deposits disbursement, given that the
bank has absolutely no interest in activating
the put option when the market value of the
bank’s assets is higher than the market value
of debt, and ROA has recorded the same,
positive amounts for years. Zero DIP can also
be achieved at the higher level of volatility, but
when A/D > 1. When volatility is 1%, DIP is
zero for A/D ≥ 1.05, and also when volatility
is 3% and A/D ≥ 1.10, until the pair A/D, σ
(1.15, 5%). In order to maintain DIP at the low
level, when ROA is highly volatile, we need
compensation in the form of increased A/D.
For A/D ≤ 1, regardless of the volatility level,
DIP is higher than zero. The pair A/D, σ (1.1,
7%) approximates DIP to the value of DIP in
the BSB&H of 0.29%, i.e. 0.3%. DIP increases as
the volatility grows and A/D falls, reaching the
amount of 25.71% for the most extreme values
in the experimental sample (Chart 1) of A/D =
0.75 and σ = 21%. This would imply that a bank
whose market value of assets is only 75% of the
market value of debt, and whose volatility of
ROA is 21%, should pay in the name of DIP
0.25 monetary units for each unit of the insured
deposit. The banks whose A/D ≥ 1 and whose
ROA is stable and balanced would pay lower
DIP than the banks whose A/D < 1 and whose
Formula 2. Binomni raspored
n!
n x
n− x
=
p x (1 − p ) n − x
  p (1 − p )
x!(n − x)!
 x
Izvor: Žižić et al. 1992. Metodi statističke
analize. Savremena administracija, Beograd. p.
112.
VD banke određujemo na osnovu relativne
vrijednosti loših kredita (u daljem tekstu LK,
eng: nonperforming loan), a prema dva scenarija:
pesimističkom i optimističkom. LK je udjel loših
kredita u ukupnim kreditima.
2000. g. je godina bankarske krize u BiH, sa
nastavkom u 2001. g. i 2002. g. Mnoge banke
su bile nelikvidne, a neke i nesolventne. Difolt
se nije pominjao, iako je bio rasprostranjen.
Kriza je “liječena” tolerisanjem nelikvidnosti
banaka. Kada to nije pomoglo, usljedila je
državna (tj.entitetska) intervencija: neke banke
su otišle u stečaj, jedan dio je transformisan, a
najviše ih je privatizovano. Prema podacima
be-ha emisione banke, tokom 2000. g. LK
dostižu svoj maksimum od 21%. Taj broj smo
uzeli kao reper za difolt tj. stvaranje obaveze
AOD da isplati osigurane depozite - i dali
mu vrijednost VD od 1. U trenutku kada LK
dostigne 21%, vjerovatnoća da će AOD morati
preuzeti obavezu isplate depozita pojedinačne
banke je 100% - ovo je osnovna pretpostavka
pesimističkog scenarija. Na osnovu reperne
vrjednosti LK, odredili smo VD pojedinačne
banke u narednim godinama - djeljenjem LK sa
LK u baznoj godini (VD20..=LK20../LK2000).
Optimistički scenario se razlikuje “samo” po
vrijednosti LK u baznoj godini. Vrijednost LK je
hipotetička - iznosi 40%. Po istom mehanizmu
kao kod pesimističkog scenarija (djeljenjem LK
sa LK u baznoj godini) bazni LK od 40% daje
nižu VD pojedinačne banke. Zato smo ovaj
scenario nazvali optimistički ili zbog upotrebe
značajno višeg LK, hipotetički.
HGR je sekundarni alat u istraživanju. Ne
može biti primaran zato što osim broja banaka,
ne uzima u obzir niti jednu drugu empirijsku
varijablu iz bankarskog sektora. Elementi HGR
(formula 2) su: N - ukupan broj banaka, N1 broj velikih banaka, N2 - broj ostalih banaka,
n - veličina uzorkA, x - broj velikih banaka
u uzorku, n-x - broj malih banaka u uzorku.
Nalazi BR su pouzdaniji, od nalaza HGR, jer
se pesimistički scenario po BR u potpunosti
zasniva na empirijskim podacima, a djelom i
optimistički po BR, u djelu podataka o LK - osim
za 2000. g. Podaci iz ovih scenariJa se koriste u
konstrukciji BR, ali ne i u izradi HGR.
Formula 3. Hipergeometrijski raspored (HGR)
N 1!
N2
 N 1  N 2 

 
 x  n − x  = x!( N 1 − x)! (n − x)! ( N 2 − (n − x))!
N!
N
 
n!( N − n)!
n
Izvor: Ibid. p. 117.
Model linearizacije
Zasniva se na pretpostavljenoj linearnoj vezi
između VD i POD.
Formula 4.
POD2008, 2009... =V
D
2008, 2009...
/V
D
2007
*0,3.
Rezultati i diskusija
“Merton 1977”
Na osnovu modela “Merton 1977” ne
možemo odrediti jednu, tačkastu, vrijednost
POD. Ali ono što možemo odrediti dovoljno
je da zaključimo da postoji potreba prelaska sa
režima fiksne na režim varijabilne POD. Podaci
o standardnoj devijaciji ROA su empirijski
(drugi red tabele 1), ali nemamo podatke o
tržišnoj vrijednosti aktive i duga pojedinih
banaka pa ne možemo odrediti individualne
POD.
bankarstvo 1 2013
pozitivno/negativno,
para/grb,
uspjeh/
neuspjeh. Prilagođeno cilju istraživanja ishodi
su difolt/odsustvo difolta. Difolt znači da AOD
mora isplatiti osiguranu sumu.
BR (formula 2) ima tri elementa: n - broj
banaka, p - vjerovatnoća difolta banke (u daljem
tekstu VD), x - broj banaka koje su proglasile
difolt. Dobijeni raspored pokazuju vjerovatnoću
da jedna, ili više banaka, proglase difolt.
82
ROA is highly volatile. The model determined
DIP depending on the bank’s risk of default,
and it is the function of ROA and A/D.
If A/D is constant, for instance it equals 1
(Chart 2), DIP is increasing as the ROA volatility
increases. Zero ROA volatility results in zero
DIP, whereas ROA volatility of 21% results
in DIP of 8.36%. The same A/D, under the
condition that all other variables remain the
same, leads towards the linear correlation
between volatilities of ROA and DIP.
Probability Distributions
We applied two probability distributions binomial (hereafter to be referred to as: BD) and
hypergeometric (hereafter to be referred to as:
HGD). Both of these distributions are discrete.
BD has only two outcomes, for instance:
positive/negative, heads/tails, success/failure.
Adjusted to the objective of our research, the
possible outcomes are default/no default.
Default implies that the DIA must disburse the
insured amount.
BD (Equation 2) has three elements: n number of banks, p - the bank’s probability
of default (hereafter to be referred to as PD),
x - number of banks in default. The received
distribution indicates the probability of one or
more banks declaring default.
Equation 2. Binomial distribution
n!
n x
n− x
=
p x (1 − p ) n − x
  p (1 − p )
x!(n − x)!
 x
bankarstvo 1 2013
Source: Žižić et al. 1992. Metodi statističke analize.
Savremena administracija, Beograd, p. 112.
83
The bank’s PD is determined based on
the relative value of non-performing loans
(hereafter to be referred to as: NPLs), according
to the two scenarios - pessimistic and optimistic.
NPLs are a share of non-performing loans in
total loans.
2000 was the year of a banking crisis in
B&H, and it continued in 2001 and 2002. Many
banks were illiquid, some even insolvent.
Default was not being mentioned, although
it was widespread. The crisis was “cured”
by tolerating illiquidity of banks. When this
did not help, there ensued a state (i.e. entity)
intervention: some banks went bankrupt,
some of them were transformed, most of them
privatized. According to the data of the B&H
issuing bank, in 2000 the NPLs reached their
peek at 21%. We took this percentage as the
benchmark for default, i.e. for the obligation of
DIA to disburse insured deposits - and assigned
it the value of PD = 1. At the moment when NPLs
reach 21%, the probability that DIA will have to
assume its obligation of disbursing deposits of
an individual bank is 100% - this is the basic
assumption of the pessimistic scenario. Based
on the benchmark value of NPLs, we determined
the PD of an individual bank in the forthcoming
years - by dividing NPLs with NPLs in the basic
year (PD20.. = NPLs20.. / NPLs2000).
The optimistic scenario differs “only” in
the value of NPLs in the basic year. The value
of NPLs is hypothetical - it amounts to 40%.
According to the same mechanism as in the case
of the pessimistic scenario (dividing of NPLs
with NPLs in the basic year), the basic NPLs of
40% result in lower PD of an individual bank.
This is why we called this scenario optimistic,
or, due to the application of considerably higher
level of NPLs, hypothetical.
HGD is a secondary tool in this research.
It cannot be primary because, except for
the number of banks, it does not take into
consideration any other empirical variable
from the banking sector. The elements of
HGD (Equation 3) are the following: N - total
number of banks, N1 - number of large banks,
N2 - number of other banks, n - sample size,
x - number of large banks in the sample, n-x
- number of small banks in the sample. The
findings of BD are more reliable than the
findings of HGD given that the pessimistic BD
scenario is fully based on empirical data, which
is the case of the optimistic BD scenario, too, in
the part concerning the data on NPLs - except
for the year 2000. The data from these scenarios
are used in the concept of BD, but not in the
preparation of HGD.
Tabela 1. Primjena modela “Merton 1977” u BSBiH - odnos
tržišne vrijednosti aktive i duga (A/D) i POD
Razdoblje
A/D,
Volatilnost
2000 - 2011.
2000 - 2002.
2003 - 2007.
2008 - 2011.
0,83%
0,75%
0,20%
0,56%
0,75
25,00%
25,00%
25,00%
25,00%
0,8
20,00%
20,00%
20,00%
20,00%
0,85
15,00%
15,00%
15,00%
15,00%
0,9
10,00%
10,00%
10,00%
10,00%
0,95
5,00%
5,00%
5,00%
5,00%
1
0,33%
0,30%
0,08%
0,22%
1,03
0,00%
0,00%
0,00%
0,00%
1,07
0,00%
0,00%
0,00%
0,00%
1,14
0,00%
0,00%
0,00%
0,00%
1,2
0,00%
0,00%
0,00%
1,25
0,00%
0,00%
0,00%
Izvor: www.cbbh.ba (Obradio autor)
Na nivou BSBiH do POD se dolazi u
nekoliko koraka (Tabela 2). Neto dobit iz 2011.
g. diskontujemo sa očekivanom/zahtjevanom
stopom prinosa, dobijamo
tržišnu vrijednost dioničkog
kapitala, za iznos razlike
umanjujemo knjigovodstvenu
vrijednost aktive i tako
dobijenu tržišnu vrijednost
aktive stavljamo u odnos
sa
knjigovodstvenom
vrijednošću duga. A/D za
diskontne stope od 0,2,
0,1 i 0,05 je 1,03, 1,07 i 1,14
respektivno. Za te vrijednosti
A/D POD je nula.
Modeli binomnog
rasporeda(BR)
0,00%
Prema
pesimističkom
scenariju, a koristeći podatak
o LK u baznoj godini, VD
u 2007. g. je 0,143 (3/21).
Vrijednost LK u 2011. g. daje VD 4 puta veću od
one u 2007. g.; 0,562 (11,8/21) u odnosu na 0,143.
0,00%
Tabela 2. A/D, BSBiH 31.12.2011. godine*
Neto
dobit
Očekivani
prinos
(u%)
Dionički
kapital
T.V.**
Dionički
i drugi
kapital
K.V.***
“Gubitak”
Aktiva
K.V.
Aktiva
T.V
(A)
Pasiva
K.V.
(D)
(A/D)
1
2
3
4
5=4-3
6
7=6-5
8
9=7/8
141.780
20
708.900
3.058.746
2.349.846
23.726.000
21.376.184
20.667.284
1,03
141.780
10
1.417.800
3.058.746
1.640.946
23.726.000
22.085.084
20.667.284
1,07
141.780
5
2.835.600
3.058.746
223.146
23.726.000
23.502.884
20.667.284
1,14
bankarstvo 1 2013
Izvor: Ibid.
* U svim kolonama, osim u koloni 2 i 9 podaci su u 000 KM.
**TV - tržišna vrijednost.
*** KV - knjigovodstvena vrijednost.
84
Equation 3. Hypergeometric distribution (HGD)
Results and Discussion
N 1!
N2
 N 1  N 2 
“Merton 1977”

 
 x  n − x  = x!( N 1 − x)! (n − x)! ( N 2 − (n − x))!
Based on “Merton 1977” model we cannot
N!
N
determine
a single point value of DIP. However,
 
n
!
(
N
n
)
!
−
n
 
what we can determine is sufficient for us to
Source: Ibid. p. 117.
Linearization Model
This model is based on the assumed linear
correlation between PD and DIP.
Equation 4.
DIP2008, 2009… = PD2008, 2009… / PD2007 * 0.3
conclude that there is a need for transition
from the fixed DIP regime to the variable DIP
regime. The data on standard deviation of ROA
are empirical (second row in Table 1 below), but
we do not possess the data on market value of
assets and debt of certain banks, which is why
we cannot determine individual DIPs.
At the level of the BSB&H, DIP is calculated
in several steps (Table 2 below). Net profit for
2011 is discounted by the expected/targeted rate
of return; we get the market value of equity,
deduce the accounting value of assets by the
difference of these two amounts, after which
we put thus received market value of assets
into relation with the accounting value of debt.
A/D for discounting rates of 0.2, 0.1 and 0.05
amounts to 1.03, 1.07 and 1.14, respectively. For
these values of A/D, DIP equals zero.
Table 1. Implementation of “Merton 1977” model in the BSB&H
- relation of market values of assets and debt (A/D) and DIP
Period
A/D,
Volatility
2000 - 2011.
0,83%
0,75%
0,20%
0,56%
0,75
25,00%
25,00%
25,00%
25,00%
0,8
20,00%
20,00%
20,00%
20,00%
0,85
15,00%
15,00%
15,00%
15,00%
2000 - 2002.
bankarstvo 1 2013
2008 - 2011.
0,9
10,00%
10,00%
10,00%
10,00%
0,95
5,00%
5,00%
5,00%
5,00%
1
0,33%
0,30%
0,08%
0,22%
1,03
0,00%
0,00%
0,00%
0,00%
1,07
0,00%
0,00%
0,00%
0,00%
1,14
0,00%
0,00%
0,00%
0,00%
1,2
0,00%
0,00%
0,00%
0,00%
1,25
0,00%
0,00%
0,00%
0,00%
Source: www.cbbh.ba (Prepared by the author)
85
2003 - 2007.
Tabela 3. Pesimistički scenario
bankarstvo 1 2013
je u suprotnosti sa osnovnim principom
osiguranja kao djelatnosti.
2000.
2007.
2008.
2009.
2010.
2011.
Scenariji, pesimistički i optimistički, isto
LK
21
3
3,1
5,9
11,4
11,8
kao i pojedinačne vrijednosti VD, otkrivaju
VD
1
0,143
0,148
0,281
0,543
0,562
rast vjerovatnoće da će AOD morati isplatiti
Izvor: Ibid.
jedan dio osiguranih depozita. U 2007. g.
u pesimističkom scenariju vjerovatnoća
da neće bankrotirati niti jedna banka
U optimističkom scenariju, takođe raste VD,
je 0,028 (ili 2,8%) (Grafikon 3). VD za jednu
ali je on manji zbog većeg LK u baznoj godini.
i više banaka je 0,972. VD dvije, tri i četiri
Pretpostavili smo da banka proglašava difolt
kada LK dođe tek do 40. VD u 2007. g. od 0,075
banke je 0,20, 0,23 i 0,19 respektivno. Srednja
(3/40) je duplo manji od VD po pesimističkom
vrijednost BR je 3,3, što prema teoriji BR znači
scenariju, isto kao i u 2011. g. 0,295 (11,8/40).
da je najveća vjerovatnoća da će AOD morati
isplatiti osigurane depozite u ukupno
3 banke. VD za 7 i više banaka je svega
Tabela 4. Optimistički scenario
0,045. U 2007.g. nije postojala opasnost od
2000.
2007.
2008.
2009.
2010.
2011.
masovnog difolta banaka, pa prema tome
LK
40
3
3,1
5,9
11,4
11,8
ni od masovnog angažmana AOD.
VD
1
0,075
0,078
0,148
0,285
0,295
U 2011. g. stvari se mjenjaju (Grafikon
Izvor: Ibid.
4). Sve do brojke od 8 banaka VD je nula,
ili blizu nuli. Vjerovatnoća da će difolt
proglasiti 8 i manje banaka je 0,007. VD
VD raste u oba scenarija. Ako se visina
od 12 do 16 banaka tj. vjerovatnoća masivnog
POD veže za rizik/vjerovatnoću dešavanja
difolta je 0,65. Grafički, do rasta vjerovatnoće
osiguranog slučaja, onda zbog rasta rizika mora
masovnog difolta banaka dolazi zbog
rasti i POD. Svaki drugačiji način razmišljanja
pomjeranja BR u desnu stranu.
86
Table 2. A/D, BSB&H as of 31.12.2011*
Net
profit
Expected
return
(u%)
Equity
M.V.**
Equity
and other
capital
A.V.***
“Loss”
Assets
M.V.
Assets
M.V.
(A)
Liabilities
A.V.
(D)
(A/D)
1
2
3
4
5=4-3
6
7=6-5
8
9=7/8
141.780
20
708.900
3.058.746
2.349.846
23.726.000
21.376.184
20.667.284
1,03
141.780
10
1.417.800
3.058.746
1.640.946
23.726.000
22.085.084
20.667.284
1,07
141.780
5
2.835.600
3.058.746
223.146
23.726.000
23.502.884
20.667.284
1,14
Source: Ibid.
* In all columns, except in columns 2 and 9, the data are in KM (convertible marks) thousand.
** M.V. - market value
*** A.V. - accounting value
Binomial Distribution Models
According to the pessimistic scenario, based
on the data on NPLs in the basic year, PD in 2007
amounted to 0.143 (3/21). The value of NPLs in
2011 results in the PD four times higher than the
one in 2007: 0.562 (11.8/21) compared to 0.143.
Table 3. Pessimistic scenario
2000.
2007.
2008.
2009.
2010.
21
3
3,1
5,9
11,4
1
0,143
0,148
0,281
0,543
NPLs
PD
Source: Ibid.
In the optimistic scenario PD also increases,
but it is somewhat lower due to the higher
level of NPLs in the basic year. We assumed
that the bank does not declare default until
the NPLs reach 40. PD in 2007 in the amount
of 0.075 (3/40) is two times lower than the PD
according to the pessimistic scenario, and the
same goes for 2011, when the PD amounted to
0.295 (11.8/40).
Table 4. Optimistic scenario
2000.
NPLs
PD
2007.
2008.
2009.
2010.
40
3
3,1
5,9
11,4
1
0,075
0,078
0,148
0,285
bankarstvo 1 2013
Source: Ibid.
87
PD increases in both scenarios. If the amount
of DIP gets pegged to the risk/probability of
occurrence of the insured case, then, due to the
risk growth, DIP also has to increase. Every
other way of thinking contradicts the main
principle of insurance as a business activity.
These scenarios, the pessimistic and the
optimistic one, just like the individual values
of PD, reveal the growth of probability that
the DIA will have to disburse one portion
2011.
of the insured deposits. In 2007, according
11,8
to the pessimistic scenario, the probability
0,562
that none of the banks would go bankrupt
amounted to 0.028 (or 2.8%) (Chart 3). PD
for one or more banks was 0.972. PD for
two, three and four banks was 0.20, 0.23, and
0.19, respectively. The mean value of BD was
3.3, which, according to the BD theory, implies
that the highest probability that the DIA would
have to disburse the insured deposits was in 3
banks. PD for 7 and more banks was only 0.045.
In 2007 there was no danger of mass default of
banks; hence there was no mass engagement
of the DIA.
In 2011 things changed (Chart 4). Up
to the number of 8 banks, PD was zero,
or close to zero. Probability that 8 or less
2011.
banks would declare default was 0.007.
11,8
PD for 12 to 16 banks, i.e. the probability
0,295
of mass default, amounted to 0.65. In
graphic terms, the growth of mass default
probability occurred because BD skewed
to the right.
Najveći VD ima brojka od 8 banaka (7,7). Uprkos
povećanju LK na 40, tj. procjeni da banke mogu
izdržati LK od 40, duplo veći od onoga iz 2007.
g., i optimistički model BR poručuje da rizik od
isplate osiguranih depozita raste.
Optimistički scenario je mnogo tolerantniji u
pogledu VD i LK. Prag tolerancije je postavljen
na mnogo višem nivou, a VD pojedinačne banke
je 1 tek kada LK dostigne 40. Zato uprkos rastu
LK, VD u optimističkom scenariju, u 2011. g. ne
prelazi jednu trećinu. U 2007. g. VD nula banaka
je izuzetno visoka 0,16. VD za 3 i manje banaka je
0,91. U ovome scenariju opasnost od masovnog
difolta/pomora banaka je ispod 0,007 (više od
6 banaka). Aritmetička sredina rasporeda je
1,7. U 2011. g. raspored je gušći, pomjeren u
desnu stranu, simetričan i teži normalnom.
Vjerovatnoća da će bankrotirati između 6 i
10 banaka je 0,7, a između 4 i 11 banaka 0,92.
Model hipergeometrijskog rasporeda (HGR)
U 2007. g. VD samo jedne banke, i to velike
banke, je 0,174, a VD dvije velike banke 0,024
(Tabela 6). Par koji se sastoji od jedne velike i
jedne male banke ima šansu od 30% da se obrati
AOD (VD = 0,3). Difolt četverca “dvije velike
i dvije male banke” je moguć u malo više od
desetine slučajeva. (VD = 0,116).
Tabela 5. Parametri BR*
Pesimistički
scenario
Optimistički
scenario
2007.
2011.
2007.
2011.
Aritmetička sredina
3,3
14,6
1,7
7,7
Standardna devijacija
1,7
2,5
1,3
2,3
bankarstvo 1 2013
Izvor: Ibid.
* O načinu određivanja sredine i devijacije BR vidjeti Žižić et al., 1992. p.
115.
88
bankarstvo 1 2013
The optimistic scenario is much more
tolerant in terms of PD and NPLs. The tolerance
threshold was set at a much higher level, and
PD of an individual bank equals 1 only when
NPLs reach 40. Thus, despite the growth of
NPLs, PD in the optimistic scenario in 2011 does
not exceed one third. In 2007 PD for zero banks
is extremely high, and reaches 0.16. PD for 3 and
less banks is 0.91. In this scenario, the danger of
mass default of banks is below 0.007 (more than
6 banks). Arithmetic mean of the distribution is
89
1.7. In 2011 the distribution is denser, skewed
to the right, symmetric, and with a tendency
towards normal. Probability that between 6
and 10 banks would go bankrupt is 0.7, and
between 4 and 11 banks 0.92. The highest PD is
for 8 banks (7.7). Despite the increase of NPLs
to 40, i.e. despite the assessment that banks may
survive the NPLs rate of 40, the twice as much
as the one from 2007, the optimistic BD model
also suggests that the risk of insured deposits
disbursement increases.
Tabela 6. VD jedne ili više banaka prema HGR
2007.
2008.
2009.
2010.
Razlika
2011-2007.
2011.
Jedna velika banka
0,174
0,167
0,167
0,160
0,154
-0,020
Dvije velike banke
0,024
0,022
0,022
0,020
0,018
-0,005
Jedna velika i jedna mala banka
0,300
0,290
0,290
0,280
0,271
-0,030
Dvije velike i dvije male banke
0,116
0,107
0,107
0,100
0,093
-0,023
23
24
24
25
26
4
4
4
4
4
Broj banaka
Broj velikih banaka
Izvor: Ibid.
U 2011. g. VD u sve četiri kombinacije je
smanjena: “jedna velika banka” 0,154; “dvije
velike banke” 0,018; “jedna velika i jedna
mala banka” 0,271; “dvije velike i dvije male
banke”0,093. Smanjenje VD je posljedica
povećanja broja banaka, uz isti broj velikih
banaka - tj. promjene elemenata u formuli za
HGR. Prema rezultatima HGR u 2011. g. VD za
četiri odabrane kombinacije banaka je smanjen
u odnosu na 2007. g. Razloga za povećanje
POD nema. VD je smanjen za 0,02, 0,005, 0,030,
0,023 - u prosjeku za 0,02. Isključiva i dosljedna
primjena principa matematičke vjerovatnoće,
HGR, zahtjeva smanjenje a ne povećanje POD.
Ali i prema HGR premija osiguranja mora biti
promjenljiva.
2009. g. ona je 0,59%((0,28/0,14)*0,3), a u 2011.
g. iznad 1; 1,18%. Ovaj model ima logiku, ali je
prilično naivan. Na veličinu POD ne utiče visina
VD već relativni odnos između VD. Pošto su
relativni odnosi koji se formiraju između VD
i u pesimističkom i optimističkom scenariju
jednaki, POD ostaje ista bez obzira na skoro
duplo viši bazni VD u optimističkom scenariju
(tabela 8). Za ekstremne vrijednosti scenarija,
LK = 21,1, odnosno LK = 40, stopa premije je 2,1%
odnosno 4% (zadnja kolona u tabelama 7 i 8).
Tumačenje ekstremnog scenarija je da ukoliko
je VD pojedinačne banke 1, što je u teorijskom
okviru BR nemoguć događaj (vjerovatnoća 0),
POD treba, prema pretpostavkama modela
linearizacije, povećati na 2,1% , odnosno 4%.
Na ove brojke ne utiče odnos A/D jer model
linearizacije uopšte ne obuhvata te varijable.
Iako model linearizacije u jednom dijelu sadrži
empirijske podatke (podaci o LK), on predstavlja
izuzetno uproštenu sliku stvarnosti.
Model linearizacije
Pod pretpostavkom potpune pozitivne
korelacije (koeficjent korelacije 1) između
vrijednosti VD i POD, POD raste (Tabela 7). U
Tabela 7. POD u modelu linearizacije, pesimistički scenario
2000.
2001.
2002.
2003.
2004.
2005.
2006.
2007.
2008.
2009.
2010.
2011.
Pretpostavka
LK u%
21,0
17,9
11,0
9,3
6,1
5,3
4,0
3,0
3,1
5,9
11,4
11,8
21,0
VD
1,00
0,85
0,52
0,44
0,29
0,25
0,19
0,14
0,15
0,28
0,54
0,56
1
0,3
0,31
0,59
1,14
1,18
2,1
POD (u%)
Izvor: Ibid.
2000.
NPL u%
VD
POD (u%)
Izvor: Ibid.
2001.
2002.
2003.
2004.
2005.
2006.
2007.
2008.
2009.
2010.
2011. Pretpostavka
40
17,9
11
9,3
6,1
5,3
4,0
3,0
3,1
5,9
11,4
11,8
40
1,00
0,45
0,28
0,23
0,15
0,13
0,10
0,08
0,08
0,15
0,29
0,30
1
0,3
0,31
0,59
1,14
1,18
4
bankarstvo 1 2013
Tabela 8. POD u modelu linearizacije, optimistički scenario
90
Table 5. Parameters of BD*
Linearization Model
Under the assumption of full positive
Pessimistic
Optimistic
correlation
(correlation coefficient of 1)
scenario
scenario
between the values of PD and DIP, DIP
2007.
2011.
2007.
2011.
increases (Table 7). In 2009 it amounted
Arithmetic mean
3,3
14,6
1,7
7,7
to 0.59% ((0.28/0.14)*0.3), whereas in
Standard deviation
1,7
2,5
1,3
2,3
2011 it exceeded 1, amounting to 1.18%.
Source: Ibid.
This model rests on a sound logic, but is
* For more details about determining arithmetic mean and standard
deviation, see Žižić et al., 1992. p. 115.
rather naïve. The amount of DIP is not
impacted by the level of PD, but by the
Hypergeometric Distribution Model
relative correlation among PDs. Given that the
In 2007 PD of only one bank, a large one, was
relative correlations formed among PDs both
0.174, and PD of two large banks 0.024 (Table 6).
in the pessimistic and the optimistic scenarios
The pair consisting of one large and one small
are equal, DIP remains the same, regardless
of the almost two times higher basic PD in
bank had 30% chance of addressing the DIA
(PD = 0.3). Default of “two large and two small
the optimistic scenario (Table 8). For extreme
banks” was probable in a bit more than a dozen
scenario values, NPLs = 21.1, or NPLs = 40,
cases (PD = 0.116).
the premium rate is 2.1%, or 4%, respectively
Table 6. PD of one or more banks according to HGD
2007.
One large bank
2008.
0,174
2009.
0,167
2010.
0,167
Difference
2011-2007.
2011.
0,160
0,154
-0,020
Two large banks
0,024
0,022
0,022
0,020
0,018
-0,005
One large and one small bank
0,300
0,290
0,290
0,280
0,271
-0,030
Two large and two small banks
0,116
0,107
0,107
0,100
0,093
-0,023
23
24
24
25
26
4
4
4
4
4
Number of banks
Number of large banks
bankarstvo 1 2013
Source: Ibid.
91
In 2011 PD in all four combinations was
reduced: “one large bank” 0.154; “two large
banks” 0.018; “one large and one small bank”
0.271; “two large and two small banks” 0.093.
Lower PD was the result of the increased
number of banks, with the same number of
large banks - i.e. the change of elements in the
HGD equation. According to the HGD results,
in 2011 PD for four selected combinations of
banks was reduced compared to 2007. There
was no reason to increase DIP. PD was reduced
by 0.02, 0.005, 0.030, 0.023 - on average by 0.02.
The exclusive and consistent implementation
of mathematic probability principle, HGD,
requires a reduction, not an increase of DIP.
But, even according to the HGD, the insurance
premium has to be variable.
(as shown in the last column in Tables 7 and
8). According to the interpretation of the
extreme scenario, if the PD of an individual
bank equals 1, which is an impossible event
within the theoretical framework of BD (zero
probability), pursuant to the linearization
model assumptions, DIP should be increased to
2.1%, or 4%. These figures are not influenced by
the A/D ratio, because the linearization model
does not include these variables at all. Although
the linearization model contains empirical data
in one part (data on NPLs), it is an extremely
simplified representation of reality.
Dilemu fiksna ili varijabilna POD smo
testirali preko 4 modela. Jedan od njih (“Merton
1977”) je posuđen, a ostala tri smo konstruisali
prema prjedlozima teorije vjerovatnoće i
linearne algebre.
Namjera modela “Merton 1977” - da
smanjenje tržišne vrijednosti aktive banke
nadoknadi povećanjem premije - je očigledna.
Tako za A/D od 0,75 premija ide na 25%, da bi
se tržišna vrijednost aktive i tržišna vrijednost
duga ponovo doveli u ravnotežu (1:1). Prema
toj logici za volatilnost ROA od 0 i A/D = 1
nema potrebe za POD, jer su A i D u savršenoj
ravnoteži. Pri A/D >=1,15 uz nisku volatilnost
ROA (do 5%) takođe ne postoji potreba za
POD. Poruka modela u slučaju BSBiH, pod
pretpostavkom realnosti izabranih diskontnih
stopa i načina određivanja tržišne cijene aktive
i duga je da BSBiH uopšte ne treba POD, tj.
osiguranje depozita, jer je A/D BSBiH iznad 1
(za diskontne stope od 0,2, 0,1 i 0,05, A/D je 1,05,
1,09 i 1,16 respektivno). BSBiH je još vitalniji
nego što se to iz podataka o A/D vidi, jer smo
prema specifikaciji modela “Merton 1977” u
D (debt, dug) unijeli sve dugove, a ne samo
osigurane depozite koji su u BSBiH značajno
manji od ukupnog duga.
Rasporedi vjerovatnoće, BR i HGR, takođe
predlažu varijabilnu POD. Prvi raspored
(BR) se ne slaže sa “Merton 1977” u pogledu
smjera varijabiliteta, a drugi (HGR) se slaže.
BR kao raspored VD određenog broja banaka
razvijamo u dvije krajnje varijante: pesimističku
i optimističku. VD je u prvoj računata na bazi
LK od 20,1 (2000. g.) za koju smo vezali VD,
odnosno vjerovatnoću isplate osiguranih
depozita od 1. U drugoj, optimističkoj varijanti,
LK u 2000. g. smo postavili na visokih 40. U oba
scenarija rasporedi vjerovatnoće su zbog rasta
LK pomjereni značajno udesno u odnosu na
2007. g. U desnom djelu rasporeda se nalaze
obilježja veće vrijednosti, ili prevedeno na
jezik statistike: raste VD većeg broja banaka. U
2007. g. VD za više od 6 banaka je zanemarljiva,
ispod 1% (optimistički scenario), ali je zato u
optimističkom scenariju 10%. U 2011. g. stvari
su bitno izmjenjene. BR se pomjerio u desno u
odnosu na onaj u 2007. g. Vjerovatnoća da AOD
mora isplatiti depozite u više od 12, a manje
od 16 banaka, je 0,658. Rast VD zahtjeva rast
POD - ako se pridržavamo osnovnog načela
osiguranja, da veći rizik povlači veću premiju
osiguranja.
Rezultati primjene HGR na četiri kombinacije
između, sa jedne strane četiri velike banke, i sa
druge strane banke koje po metodologiji AOD
nisu velike, zahtjevaju smanjenje POD. U sva
četri slučaja VD je smanjena - 2007. g. u odnosu
na 2011. g. Vjerovatnoća da će bankrotirati jedna
velika banka i niti jedna banka iz grupe “ostale
banke” je sa 17,4% (2007), smanjena na 15,4%
(2011), isto kao i vjerovatnoća da će se dvije
velike banke obratiti AOD za isplatu depozita:
0,024% (2007) i 0,018% (2011). Do ovih promjena
je došlo zbog rasta broja banaka članica sistema
za osiguranje depozita sa 23 (2007. g.) na 26
(2011. g.), uz isti broj velikih banaka.
Model linearizacije, kao i model “Merton
1977” daje numeričke vrijednosti POD, ali
ne uključuje niti jedan parametar iz modela
“Merton 1977”. Pretpostavljena linearna veza
(savršena linearna korelacija između LK i VD) u
2008, 2009, 2010. i 2011, daje POD od 0,31, 0,59,
1,14 i 1,18 respektivno. Prema ovom modelu
kada LK dođu do 21,2 odnosno 40 zahtjevana
POD je 2,1 odnosno 4%.
U radu smo koristili istorijske vjerovatnoće
- od 12/2011 pa unazad. Rizici su porasli, BP je
pomjeren udesno, ali AOD nije intervenisala.
Kako to objasniti? To što neki događaj, prema
modelu, ima veliku vjerovatnoću javljanja ne
mora značiti da će se on zaista i desiti, isto
kao što se može realizovati neki događaj sa
malom vjerovatnoćom javljanja. Rasporedi
vjerovatnoće kazuju vjerovatnoću i u vezi sa
tim rizike sa kojima se suočava AOD. Promjena
rasporeda vjerovatnoće i percepcije rizika
sugeriše korekciju POD makar se ti veći
rizici i nematerijalizovali. U tome je smisao
primjene teorije vjerovatnoće u analizi POD i
u cijelom bankarstvu. Odnos u trouglu modelvjerovatnoća-stvarnost se najbolje vidi iz
primjera hedž fonda LTCM (Long Term Capital
Market). Gubitak koji je hedž fond realizovao
tokom 1998. g. je bio 14 standardnih devijacija
udaljen od očekivanog/prosječnog prinosa na
portfolio (Dowd, 2002). Desio se događaj čija je
vjerovatnoća dešavanja ravna nuli!
Primjenjeni modeli se ne slažu u pogledu
veličine POD, smjera njenih promjena.
bankarstvo 1 2013
Zaključna razmatranja
92
Table 7. DIP according to the linearization model, pessimistic scenario
2000.
2001.
2002.
2003.
2004.
2005.
2006.
2007.
2008.
2009.
2010.
2011.
Estimate
NPLs in %
21,0
17,9
11,0
9,3
6,1
5,3
4,0
3,0
3,1
5,9
11,4
11,8
21,0
PD
1,00
0,85
0,52
0,44
0,29
0,25
0,19
0,14
0,15
0,28
0,54
0,56
1
0,3
0,31
0,59
1,14
1,18
2,1
DIP in %
Source: Ibid.
Table 8. DIP according to the linearization model, optimistic scenario
2000.
NPLs in %
PD
2001.
2002.
2003.
2004.
2005.
2006.
2007.
2008.
5,9
2010.
11,4
2011.
11,8
Estimate
17,9
11
9,3
6,1
5,3
4,0
3,0
1,00
0,45
0,28
0,23
0,15
0,13
0,10
0,08
0,08
0,15
0,29
0,30
1
0,3
0,31
0,59
1,14
1,18
4
DIP in %
3,1
2009.
40
40
Source: Ibid.
bankarstvo 1 2013
Concluding Remarks
93
In this paper we tested the dilemma “fixed or
variable DIP” through 4 models. We borrowed
one of them (“Merton 1977”), whereas the
remaining three we constructed in line with
the principles of probability theory and linear
algebra.
The intention of the “Merton 1977” model
- to offset the reduction in market value of a
bank’s assets by increasing the premium - is
obvious. Thus, for A/D of 0.75 the premium
goes up to 25%, in order to regain the balance
(1:1) between market value of assets and
market value of debt. According to this logic,
for ROA volatility of 0 and A/D = 1, there is no
need for DIP, given that A and D are perfectly
balanced. When A/D ≥ 1.15, with low ROA
volatility (up to 5%), there is also no need for
DIP. The message conveyed by the model in
case of BSB&H, under the assumption that
the selected discounting rates and manner of
determining market values of assets and debt
are realistic, is that BSB&H does not need DIP,
i.e. deposit insurance, at all, given that A/D in
the BSB&H is above 1 (for discounting rates of
0.2, 0.1, and 0.05, A/D equals 1.05, 1.09, and 1.16,
respectively). BSB&H is even more buoyant than
it might be indicated by the A/D data, bearing
in mind that, according to the specification of
the “Merton 1977” model, within the variable D
we recorded all debts, and not only the insured
deposits, which are in the BSB&H considerably
lower than the total debt.
Probability distributions, BD and HGD, also
suggest a variable DIP. The former distribution
(BD) does not comply with the “Merton
1977” concerning the direction of variability,
whereas the latter one (HGD) does comply. As
a distribution of PDs of a certain number of
banks, BD is developed in two ultimate variants:
pessimistic and optimistic. In the former, PD is
calculated on the basis of NPLs rate amounting
to 20.1 (2000), for which we pegged the PD,
i.e. the probability of disbursement of insured
deposits equalling 1. In the latter, optimistic
variant, we set the NPLs rate in 2000 to the
high level of 40. In both scenarios probability
distributions, due to the growth of NPLs,
considerably skewed to the right, compared
to 2007. The right section of the distribution is
marked by higher values, or, in the language of
statistics - the PD increases in a larger number
of banks. In 2007 for more than 6 banks PD was
negligible, i.e. below 1% (optimistic scenario),
but within the pessimistic scenario it reached
10%. In 2011 things considerably changed. BD
skewed to the right compared to the situation
from 2007. Probability that DIA must disburse
deposits in more than 12, and less than 16
banks, was 0.658. The growth of PD requires an
increase of DIP - if we follow the main principle
of insurance, that higher risk entails higher
insurance premium.
The results of implementing HGD on four
combinations, including “four large banks”,
on one hand, and banks that are not large
according to the DIA methodology, on the
other hand, require a reduction of DIP. In all
four cases the PD was reduced - comparing
1. Dowd, Kevin. Measuring Market Risk, John
Wiley & Sons Ltd, 2002. p.11.
2. Laeven, Luc. Pricing of Deposit Insurance.
World Bank Policy Research Working Paper
2871, July 2002.
3. Žižić et al. Metodi statističke analize.
Beograd. Savremena administracija, 1992.
4. Zakon o osiguranju depozita, Službeni
glasnik BiH 20/02, 18/05, 100/08 i 75/09.
5. Odluka o visini stope premije za 2007. g.,
Službeni glasnik BiH br. 89/06.
6. Odluka o visini stope premije za 2012. g.,
Službeni glasnik BiH br. 89/11.
7. Centralna banka Bosne i Hercegovine,
www.cbbh.ba.
8. Agencija za osiguranje depozita Bosne i
Hercegovine, www.aod.ba.
Iako rezultati svih modela govore u korist
promjenljive POD, velike razlike u smjeru
i veličini predloženog varijabiliteta POD
tumačimo na način da nemamo dovoljno
dokaza da predložimo varijabilnu umjesto
fiksne POD. Smatramo da hipotezu o potrebi
uvođenja varijabilne POD nismo dokazali,
odnosno da smo je samo djelimično dokazali,
pa zato nemamo još uvijek dovoljno razloga i
dokaza da osporimo, i predložimo zamjenu,
fiksne POD promjenljivom. Istraživanje smo
započeli sa navođenjem činjenice da je POD
u BSBiH fiksna. Nismo u potpunosti uspjeli
dokazati da bi trebalo da bude varijabilna.
Jedan model čak smatra da POD treba biti nula,
pa zbog svega toga fiksna POD i dalje ostaje
kao važeći model “varijabiliteta” POD u šemi
za osiguranje depozita u BiH.
Ne treba se zanositi mišljenjem da je
moguće potpuno precizno odrediti cijenu
osiguranja depozita/POD. Ali isto tako ne treba
prestati razvijati modele za odredjivanje POD,
analizirati već postojeće modele, a nove i stare
modele upoređivati i prilagođavati stvarnosti
i poželjnim karakteristikama bankarskog
sistema. Nije uvijek najkomplikovaniji model
i najbolji model. Najbolji model je onaj koji
najbolje i najpribližnije podražava stvarnost,
ma koliko prost, ili složen, on bio. Neka ovo
istraživanje bude jedan od malih koraka u
smjeru razvoja što boljeg modela za određivanje
POD.
bankarstvo 1 2013
Literatura / References
94
bankarstvo 1 2013
2007 with 2011. Probability that one large bank
and no banks from the group of “other banks”
would go bankrupt came down from 17.4%
(2007) to 15.4% (2011), just like the probability
that two large banks would address DIA for
disbursement of deposits - from 0.024% (2007)
to 0.018 (2011). These changes were caused by
the increased number of banks members of
the deposit insurance system from 23 (2007)
to 26 (2011), while the number of large banks
remained the same.
Linearization model, just like “Merton
1977”, provides numerical valued of DIP, but
it does not include a single parameter used in
the “Merton 1977” model. The assumed linear
relation (perfect linear correlation between
NPLs and PD) in 2008, 2009, 2010 and 2011
results in DIP amounting to 0.31, 0.59, 1.14,
and 1.18, respectively. According to this model,
when NPLs reach 21.2, or 40, the required DIP
amounts to 2.1%, or 4%.
In the paper we used historical probabilities starting from 12/2011 backwards. The risks have
grown, PD was skewed to the right, but DIA
failed to intervene. How can we explain that?
The fact that a certain event, according to some
model, has a high probability of occurrence
does not necessarily mean that it will actually
occur, just like an event with low probability
of occurrence might actually occur. Probability
distributions indicate the probability, and, in
this respect, the risks DIA is facing. Changes
in probability distribution and risk perception
suggest a correction of DIP, even though these
high risks may not actually materialize. This is
the point of implementing probability theory in
the analysis of DIP and in banking overall. The
relationships in the triangle model-probabilityreality are best illustrated by the example of a
hedge fund in the long-term capital market
(LTCM). The loss this hedge fund recorded in
95
1998 was by 14 standard deviations removed
from the expected/average return on portfolio
(Dowd, 2002). An event with zero probability
of occurrence actually occurred!
The applied models differ in terms of the
size of DIP or the direction of its changes.
Although the results of all models are in favour
of the variable DIP, there are huge differences
in the direction and extent of the proposed
DIP variability, which we interpret as having
insufficient evidence to actually propose a
variable DIP instead of the fixed one. We believe
that the hypothesis about the necessity of
introducing a variable DIP has not been proven,
i.e. that it has been only partially proven, and
hence we still do not have sufficient reasons and
evidence to dispute the fixed DIP and propose
its replacement with a variable DIP. We started
this research by stating the fact that DIP in the
BSB&H is fixed. We failed to fully prove that it
should be variable. One model even suggested
that DIP should be zero. Taking all this into
account, the fixed DIP still remains the valid
model of DIP “variability” within the deposit
insurance scheme in B&H.
One should not get carried away thinking
that it is possible to precisely determine the
price of deposit insurance/DIP. However,
one should not stop developing models for
determining DIP, analyzing the already existent
models, comparing the old models with the
new ones, and adjusting them to the reality
and desirable characteristics of the banking
sector. The most complicated model is not
always the best one. The best model is the one
which simulates reality in the best and closest
way, no matter how simple or complex it may
be. Let this research be a small step in the path
of development of the best possible model for
determining DIP.