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AN EXAMPLE OF QUANT’S TASK IN CROATIAN BANKING INDUSTRY
Marin Karaga
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 priceEquity price could fall significantly and bank might not be able to receive enough money from closure of equity position...
2. Uncertainty about equity liquidityThe 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
Let’s look at the ratio
Let’s assume that for every t, these ratios are independent and identically distributed random variables with following distribution
t
t
SS 1ln
)N(0, ~ln 21
t
t
SS
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
)N(0, ~ 2rr
EWMA
Let’s look at N past observations of this random variable
Possible estimate of variance (or its square root – standard deviation, volatility)
Nrrrr ...,,,, 321
N
ttr r
N 1
22 1
EWMA
We treat each squared observation equally, they all have the same contribution toward the estimate of variance
Can we improve this reasoning?
N
ttr r
N 1
22 1
N1
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
to
where we set
Nrrrr ...,,,, 321
N
N
rF
wr
Fw
rFw
rFw 1
3
2
2
1
1
0
...,,,,
ww
wFNN
i
i
11
1
1
EWMA
We have changed the weight assigned to i-th observation
Let’s see how the series of weights depends on the choice of factor w
N1
1
1
1
1
11
i
NN
k
k
i
www
w
w
EWMA
One can understand why factor w is commonly called decay factor
0,000
0,002
0,004
0,006
0,008
0,010
0,012
0,014
0,016
0,018
1 21 41 61 81 101 121 141 161 181 201 221 241
N=250
1/N w=0,995 w=0,990 w=0,985
Equity price risk (cont.)
Using the same formula for variance estimation, now applied to the EWMA weighted series, we get
If we apply this to our ratio we get
N
ttr r
N 1
22 1
t
t
SS 1ln
N
tt
tNr rw
ww
1
212
11
N
t t
ttN S
Sw
ww
1
2
11 ln11
Equity price risk
Let X be a random variable,
Let’s define random variable Z,
Obviously,
Hence, for some α, 0 < α < 1, we have
where represents cumulative distribution function of random variable that has standard normal distribution
),N( ~ 2X
X
Z
)1,0N( ~Z
ZP (.)
Equity price risk
We have 1ZP
1XP
1XP
1XP
Equity price risk
If we apply the previous formula to our random variable
we get
What this actually tells us?
)N(0, ~ln 21
t
t
SS
11ln
t
t
SS
P
Equity price risk
11ln
t
t
SS
P
11 exp
t
t
SS
P
11 exp11
t
t
SS
P
11 exp1
t
tt
SSS
P
11 exp
t
t
SS
P
Equity price risk
- equity price decrease over one day horizon
For α close to zero, we can say that there is only percent chance that the equity price over one day horizon will fall by more than
percent
Now we have some measure of equity risk that comes from the uncertainty about movements of its price
t
tt
SSS 1
11 exp1
t
tt
SSS
P
1exp1100
100
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
What is its distribution?
t
Ht
SS ln
Equity price risk + liquidity
Since for each t we have and they are
all independent, we have the following
t
t
Ht
Ht
Ht
Ht
t
t
t
t
Ht
Ht
Ht
Ht
t
Ht
SS
SS
SS
SS
SS
SS
SS
SS
1
2
1
1
1
1
2
2
1
1
lnlnln
lnln
)N(0, ~ln 21
t
t
SS
Equity price risk + liquidity
that is, we have
),H0N( ~ln 2
t
Ht
SS
1
0
21
0
1
1
2
1
1
ln
lnlnlnln
H
i
H
i it
it
t
t
Ht
Ht
Ht
Ht
t
Ht
SS
Var
SS
SS
SS
VarSS
Var
Equity price risk + liquidity
Applying the same procedure as before, we get
and finally
All that remains is to figure out how to determine variable H
H
SS
Pt
Ht 1ln
H
SSS
Pt
Htt 1exp1
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 of our position left, after second day we have of our position left, etc.
Let’s define TM
MVC 1 MM VVC
0:min
11
J
kkMM VCNJT
Equity liquidity
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
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
0:min
11
J
kkMM VCNJT
0:min
11)1(1
J
kkMM VCNJT
Equity liquidity
If we continue with these definitions, we will get the series of numbers
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
,,, 21 MMM TTT
,,,max 21 MMM TTTH
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
H
SSS
Pt
Htt 1exp1
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
01,001,0exp1 1
H
SSS
Pt
Htt
01,001,0exp 1
H
SS
Pt
Ht
01,001,0exp 1
HSSP tHt
Practical use – loan approval
In 99% cases,
In other words, in 99% of cases, during the selling of collateral, price of collateral won’t fall below
where is the value of collateral at the start of closure of equity position
01,001,0exp 1
HSSP tHt
HSS tHt 01,0exp 1
HSt 01,0exp 1
tS
Practical use – loan approval
So, the bank sets the value of loan
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
HCL 01,0exp 1
“haircut”
HCCL 01,0exp1ly,equivalent 1
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
INGRA: = 0,0339 (3,39%), H = 82
%92,8701,0exp 1
H 000.792.8HRKL
%92,4801,0exp 1
H 000.892.4HRKL
%08,51%92,48%100haircut
%08,12%92,87%100haircut
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!
