Marin Karaga
AN EXAMPLE OF QUANT’S TASK
IN CROATIAN BANKING
INDUSTRY
Introduction...
 A person has all of hers/his available money invested in
one equity (stock)
 At the same time, she/he needs certain amount of
money (for spending, other investments etc.)
 What can this person do in order to get the money?
First option...
 First option is to close the position in equity (sell the
equity) and use the proceeds from that transaction
 Money is obtained in simple and relatively quick way
 However, there is no longer the position in equity, so the
person is no longer in a position to profit from potential
increase of equity price
Second option...
 Person strongly believes that the equity price will rise in
the near future
 What to do? Person needs the money and yet is reluctant
to sell the position in equity
 Second option could solve this problem:
ask the bank for an equity margin loan!
 What is equity margin loan?
Margin loan
 Equity margin loan is a business transaction between
bank and its client in which client deposits certain
amount of equity in the bank as collateral and receives
the loan.
If the client doesn’t meet hers/his obligations on a loan
(i.e. doesn’t repay the loan) bank has the right to sell the
collateral and use the proceeds from that transaction to
cover its loss from the loan.
Second option...
 Person strongly believes that equity margin loan is the
best solution and approaches the bank with hers/his
equity and asks for a equity margin loan.
 What are the main questions for the bank?
What amount of loan can we issue to the client for a
given amount of equity which is deposited as a collateral
by the client?
What are the risks associated with this loan?
Risks...
 In every moment during the life of loan, bank has to be
able to quickly sell the collateral and receive enough
money from that transaction to cover its loss, should the
client default on a loan (if the loan isn’t fully repaid)
 So, there are two main sources of risks associated with
the equity...
Risks...
1. Uncertainty about movements of equity price
Equity price could fall significantly and bank might not be
able to receive enough money from closure of equity
position...
2. Uncertainty about equity liquidity
The more time it takes you to close the position in equity,
the more time its price has to fall below acceptable
levels...
Risks...
 How to quantify these risks?
A task for bank’s quants!
 What we need to do?
We need to quantify equity price risk and somehow take
liquidity of equity into account.
Equity price risk
St - equity price at the end of day t
St 1
 Let’s look at the ratio ln
St
 Let’s assume that for every t, these ratios are
independent and identically distributed random variables
with following distribution
St 1
ln
~ N(0, 2 )
St
Equity price risk (EWMA)
 Exponentially Weighted Moving Average
 Estimate volatility of random variable by looking at its
observations (realizations) in the past
 How it works?
 Let’s define random variable r and assume the following
r ~ N(0, r )
2
EWMA
 Let’s look at N past observations of this random variable
r1 , r2 , r3 , ..., rN
 Possible estimate of variance (or its square root –
standard deviation, volatility)
 2
r
1 N 2
  rt
N t 1
EWMA
 2
r
1 N 2
  rt
N t 1
 We treat each squared observation equally, they all have
the same contribution toward the estimate of variance
1
N
 Can we improve this reasoning?
EWMA
 Yesterday’s equity price is more indicative for tomorrow’s
equity price that the price from, for example, 9 months
ago is
 So, let’s assign different weights to observations of our
random variable, putting more weight on more recent
observations
EWMA
 Let’s choose the value of factor w, 0 < w < 1, and use it to
transform the series
r1 , r2 , r3 , ..., rN
to
w 0 w1 w2
w N 1
r1 ,
r2 ,
r3 , ...,
rN
F
F
F
F
where we set
N
1

w
F   w i 1 
1w
i 1
N
EWMA
 We have changed the weight assigned to i-th
observation
1
N
w i 1
N
k 1
w


1  w i 1
w
N
1w
k 1
 Let’s see how the series of weights depends on the
choice of factor w
EWMA
N=250
0,018
0,016
0,014
0,012
0,010
0,008
0,006
0,004
0,002
0,000
1
21
41
1/N
61
81
101
w=0,995
121
141
w=0,990
161
181
201
221
w=0,985
 One can understand why factor w is commonly called
decay factor
241
Equity price risk (cont.)
 Using the same formula for variance estimation, now
applied to the EWMA weighted series, we get
 2
r
 2
1 N 2
  rt
N t 1
r
 If we apply this to our ratio ln
1w

1  wN

N
St 1
we get
St
w
t 1
1w

1  wN
t 1
 St 1 
 ln

 St 
2
N
w
t 1
t 1 2
t
r
Equity price risk
 Let X be a random variable, X ~ N( , 2 )
 Let’s define random variable Z,
Z
X 

 Obviously, Z ~ N(0,1)
 Hence, for some α, 0 < α < 1, we have
P Z      
where (.) represents cumulative distribution function
of random variable that has standard normal distribution
Equity price risk
 We have
P Z      
1
 X 

1
P
     
 

P X     1     
P X   1       
Equity price risk
 If we apply the previous formula to our random variable
St 1
ln
~ N(0, 2 )
St
we get
 
 St 1
1
P ln
      
 St

 What this actually tells us?
Equity price risk
 
 St 1
1
P ln
      
 St

 
 St 1

1
P
 exp       


 St
 
 St 1

1
P 
  exp       


 St
 

St 1

1
P 1 
 1  exp       
St



 
 St  St 1

1
P
 1  exp       


 St
Equity price risk

St  St 1
- equity price decrease over one day horizon
St
 
 St  St 1

1
P
 1  exp       


 St
 For α close to zero, we can say that there is only
100   percent chance that the equity price over one day
horizon will fall by more than




1
100   1  exp      percent



 Now we have some measure of equity risk that comes from
the uncertainty about movements of its price
Equity price risk + liquidity
 Let’s assume that it takes us H days to close the position
in equity
 Since it takes us H days to close the position so we are
exposed to movements of equity price for H days
 Using previous notation, we need to examine following
random variable
St H
ln
St
 What is its distribution?
Equity price risk + liquidity
 St H St H 1
St H
St 2 St 1 

ln
 ln

 

St
St 1 St 
 St H 1 St H 2
 St H 
 St H 1 
 St 1 
  ln
    ln

 ln
 St H 1 
 St H 2 
 St 
St 1
 Since for each t we have ln
~ N(0, 2 ) and they are
St
all independent, we have the following
Equity price risk + liquidity
 St H
Var  ln
 St

 St H
St H 1
St 1 
  Var  ln

 ln
   ln
St H 2
St 

 St H 1
H 1
 St 1 i  H 1 2
  
 Var  ln
i 0
 St  i  i 0
that is, we have
St H
ln
~ N(0,H 2 )
St
Equity price risk + liquidity
 Applying the same procedure as before, we get
 
 St H
1
P ln
    H    
 St

and finally
 
 St  St H

1
P
 1  exp    H     


 St
 All that remains is to figure out how to determine
variable H
Equity liquidity
 There are numerous ways to estimate equity liquidity
 We’ll again look at the past observations of equity
liquidity and try to estimate how long it would take us to
close our position in collateral
 The main factor determining how many days it could
take us to close the position is, obviously, the size of
position
 Let’s denote the size of equity position with C (expressed
as market value of equity position; number of equities we
have times its current market price)
Equity liquidity
 Let’s now look at the daily volumes that were traded with
this equity on the equity market during last M days
(daily volume – size of trades with equity during one day,
market value of position that exchanged hands that day)
 Let’s denote the following:
VM – volume that was traded during the first day (the
oldest day) in our M day long history
VM-1 – volume that was traded during the second day
(second oldest day) in our M day long history
etc.
Equity liquidity
 Now, let’s see how many days we would have needed in
order to close the equity position if we had started to
close it on day M
 After first day we have C  VM of our position left,
after second day we have C  VM  VM 1 of our position
left, etc.
 Let’s define TM
J


TM  min J  N : C  VM 1k  0
k 1


Equity liquidity
J


TM  min J  N : C  VM 1k  0
k 1


 TM is the number of days we would have needed in order
to close the equity position if we started to close it on
day M
 In a similar way we can define TM-1
J


TM 1  min J  N : C  V(M 1)1k  0
k 1


as number of days we would have needed in order to
close the equity position if we started to close it on day
M-1
Equity liquidity
 If we continue with these definitions, we will get the
series of numbers
TM , TM 1 , TM 2 , 
all representing number of days we would have needed
in order to close our position if we started to close it on
certain days in the past
 We need to determine our variable H based on the
previous series of numbers, let’s be conservative and set
H  max TM , TM 1 , TM 2 ,  
Equity price risk + liquidity risk
 Now we have everything we need:

 - estimate of equity price volatility
H - estimate of equity liquidity
 Combined measure of risk
 
 St  St H

1
P
 1  exp    H     


 St
Practical use
 Remember what our question was:
What amount of loan can the bank issue to its client for a
given amount of equity which is deposited as a collateral
by the client?
 Let’s assume that the bank wants that in 99% of cases
value of collateral doesn’t fall below the value of the loan
during the selling of collateral
 Expressed in language of our model: α = 0,01
 Next, let’s assume that the bank finds appropriate to set
the decay factor w to be equal to 0,99
Practical use – loan approval
 Let C denote the initial value of position in equity

 Bank calculates H and 
 Then bank looks at the following
 
 St  St H

1
P
 1  exp  0,01 H     0,01


 St
 
 St H

1
P
 exp  0,01 H     0,01


 St




1
P  St H  St  exp  0,01 H     0,01



Practical use – loan approval




1
P  St H  St  exp  0,01 H     0,01



 In 99% cases, St H


1
 St  exp  0,01 H  


 In other words, in 99% of cases, during the selling of
collateral, price of collateral won’t fall below


1
St  exp  0,01 H  


where St is the value of collateral at the start of closure of
equity position
Practical use – loan approval
 So, the bank sets the value of loan


1
L  C  exp  0,01 H  







1
 equivalent ly, L  C  C   1  exp  0,01 H   







“haircut”
 We have solved our problem!
 Important note: once the loan has been issued, L is
constant and C varies, so the client is obliged to maintain
appropriate size of collateral – above relation has to be
true during the entire life of loan
Examples
 C = HRK 10 million
 Using the data from last 250 days (1 year) we get (α =
0,01, w = 0,99):
HT:

 = 0,0127 (1,27%), H = 19


1
exp  0,01 H    87,92%


L  HRK 8.792.000
haircut  100%  87,92%  12,08%

INGRA:  = 0,0339 (3,39%), H = 82


1
exp  0,01 H    48,92%


haircut  100%  48,92%  51,08%
L  HRK 4.892.000
Summary
We have seen:
 “Real life” case from Croatian banking industry
 Identified risks associated with margin loan
 Used EWMA to model equity volatility
 Enhanced EWMA results in order to take equity liquidity
risk into account
 Transformed analytical result into straightforward figure
(haircut) that can be quoted to potential clients
 Two examples of haircut calculation
Final remarks
 Every model is nothing more than just a model
 Check the model assumptions, try to improve it, confirm
its results by comparing them with the results form
different models etc.
 In “historical” model one needs to constantly update the
underlying historical data in order to feed the model with
the most recent information
 Compare the actual losses with the level of losses
predicted by the model – test the soundness of model
Questions
Thank you for your attention!