We use GARCH model to calculate the probability to default. Our innovation is to use two dimensional GARCH model through a formula that combines both the firm's risk and the market risk. The method is calculating the total risk by taking into consideration the different influences of the firm’s and market’s risk, i.e. Beta, using different weights for each one.

1. Finance Project
Default Forcasting
Dr Gal Zahavi
Quantitative Analyst
Global Treasury Department
Bank Hapoalim

2. The main goal
 We want to find a better way to decide
between two stocks.
 Our claim is that the distance to default
measured with Merton’s pricing model isn't
precise enough because it doesn't measure
the direct impact of the market risk which can
be crutial.

3. Our improvement
We use GARCH model to calculate the
probability to default.
Our innovation is to use two dimensional GARCH
model through a formula that combines both
the firm's risk and the market risk.
The method is calculating the total risk by taking
into consideration the different influences of
firm’s and market’s risk, i.e. Beta, using different
weights for each one.

4. Beta
The Beta (β) of a stock or portfolio is a number
describing the correlated volatility of an asset in
relation to the market over a specified time.
β < 0
Asset generally moves in the opposite direction
as compared to the index
β = 0
Movement of the asset is uncorrelated with the
movement of the benchmark
0 < β < 1
Movement of the asset is generally in the same
direction as, but less than the movement of the
benchmark
β = 1
Movement of the asset is generally in the same
direction as, and about the same amount as the
movement of the benchmark
β > 1
Movement of the asset is generally in the same
direction as, but more than the movement of the
benchmark

5. We assume that the market risk, expressed by
the Beta, influences the probability to default
more than the firm's risk.

6. Models review

7. Black & Scholes

8. Merton
 Merton approach is one of the most popular approaches to default
probability estimation using market information.
A simple model of a firm providing a way of relating credit
risk to the firm’s capital structure. By explicitly modeling a
firm’s market value, its volatility and liabilities (both equity
and debt) structure over time using contingent claims issued
against the firm’s underlying assets.
Using this model it is possible to estimate the distance to default (DD).
 The model assumes that a company has a certain amount of zero-
coupon debt and all of it will become due at the same time T in the
future.
Another assumption is that the shareholders receive no
dividends.
σA and r are assumed to be constant.

9. At time T two scenarious might happen:
If AT > D: The lenders are paid the promised amount D and the
shareholders receive the residual asset value (AT - D)
If AT < D: The lenders receive a payment equal to the asset value, and
the shareholders doesn’t get anything.
The firm is defined as defaulted.
The payment to the shareholders: ET = max [AT - D, 0].
Therefore, the equity is a European call option on the firm’s assets
with a strike price equals to the promised debt payment and it can
only be exercised at maturity.

10. The risk-neutral
probability that the
company will default
by time T
The probability that
the shareholders
won’t exercise their
call option

12. KMV
KMV deployed Merton’s framework. Unlike the Merton
model, the KMV model treats the firm as a perpetual
entity that is continuously borrowing and repaying
debt. Moreover, all classes of liabilities, including
equity, may make fixed cash payouts like coupon.
Merton's KMV produces a probability of default for
each firm in the sample at any given point of time .The
methodology of the model subtracts the face value of
the firm’s debt from the market’s estimation for the
firm’s value and then divides this difference by an
estimate of the firm's volatility (both non-observable).

13. After calculating the DD, the KMV departs
from the general framework of the Merton
model and converts the DD into expected
default frequency using empirical mapping
based on 30 years of default history collected
by KMV.

14. Shumny & Baharat
The model includes construction of simple naive
predictor with two objectives, which capture the
same information that the KMV-Merton predictor
uses without simultaneously solving any
equations or estimating any difficult quantities.
For constructing the naive probability, the model
approximates the market value of each firm’s
debt with the face value of its debt:
Naive σD = 0.05 + 0.25 ∗ σE

16. Garch
Generalized Autoregressive Conditional
Heteroskedasticity statistical model.
It is used by financial institutions to estimate the
volatility of stock returns in discrete time. This
information is used to help determine what
stocks will potentially provide higher returns, as
well as to forecast the returns of current
investments to help in the budgeting process.

17. Experiment

18. Financial Stocks
Stocks Choosing Principals: random stocks from SPY 500
Materials: Ball
Financials: American express, Morgan Stanley
Health care: Cardinal health, Pfizer, Medtronic
Energy: Anadarko petroleum, Exxon, Devon energy

19. Consumer Staples: Wal mart stores, P&G, PepsiCo
Industrials: General electric, Union pacific
Information technology: Dell, Intel, Qualcomm
Consumer Discretionary: Mc’donalds, Nike, Starbucks

20. Dell
Date Open ri stock ri spy var stock(i+1) var spy(i+1) f stock f spy f stock * f spy ln (M) ln f stock ln f spy ln(f stock * f spy)
03
-
‫אוג‬
-
12 11.62 -
0.003436429 0.014974 0.000492659 0.000167567 3.3825109 4.17678612
02
-
‫אוג‬
-
12 11.66 -
0.022053481 -
0.01584 0.000107057 0.00200477 29.4446107 65.1561126 1918.496373 7.55929702 3.81415379 2.18774676 7.559297019
01
-
‫אוג‬
-
12 11.92 0.005889794 0.001515 0.000109524 0.00200576 45.3383743 8.91510264 404.1962602 6.00190055 4.67830517 2.18697427 6.001900553
31
-
‫יול‬
-
12 11.85 -
0.015075662 -
0.00043 0.000103255 0.002001499 107.587575 8.90821843 958.4136228 6.86527944 3.83512697 2.22427548 6.865279441
30
-
‫יול‬
-
12 12.03 0.005835781 0.012052 0.000104886 0.002003177 46.299306 9.24678091 428.1195387 6.05940245 6.14790246 2.20098608 6.059402453
27
-
‫יול‬
-
12 11.96 0.022833973 0.007331 0.00010451 0.002001572 467.735263 9.03391725 4225.481662 8.34888854 4.17352971 2.22524306 8.348888536
26
-
‫יול‬
-
12 11.69 0.010318234 0.012215 0.000108005 0.002005886 64.9442822 9.25573227 601.1068886 6.39877277 6.29483149 2.19902989 6.39877277
25
-
‫יול‬
-
12 11.57 -
0.023912326 -
0.00698 0.000105223 0.002002257 541.764544 9.01626246 4884.691317 8.49386137 3.82757109 2.19552908 8.493861373
24
-
‫יול‬
-
12 11.85 0.005924689 0.005563 0.000108415 0.002006343 45.9507924 8.98475349 412.8565421 6.02310018 9.84770722 2.27339374 6.023100177
23
-
‫יול‬
-
12 11.78 -
0.036670776 -
0.01864 0.000103511 0.002001517 18914.9372 9.71230597 183707.6579 12.121101 3.8283933 2.19358492 12.12110096
20
-
‫יול‬
-
12 12.22 0.005744784 -
0.00473 0.000118009 0.002013552 45.9885892 8.96730259 412.3935946 6.02197822 3.78834004 2.2193716 6.02197822
19
-
‫יול‬
-
12 12.15 0.006606135 0.011767 0.000103513 0.002001578 44.1829974 9.20154683 406.5519202 6.00771165 4.61384382 2.18797251 6.007711646
18
-
‫יול‬
-
12 12.07 -
0.013986242 7.35E-05 0.000104519 0.00200164 100.871136 8.91711537 899.4795568 6.80181633 3.69280052 2.19267671 6.801816326
17
-
‫יול‬
-
12 12.24 -
0.002447982 0.004347 0.000104614 0.002002903 40.1571504 8.95916211 359.7744203 5.88547722 4.48709974 2.22029532 5.885477225
16
-
‫יול‬
-
12 12.27 0.013125701 0.011437 0.000103102 0.002001209 88.8633449 9.21005037 818.4358824 6.70739506 3.75328854 2.19142245 6.707395058

21. We Attempt to calculate the variance (i+1) in an
un direct way due to the lack of tools and
knowledge to calculate it directly.
In order to find the next day variance we used
two dimensional GARCH model.
Our two dimensional model is based on
multiplying both the normal function of the
stock's returns and the normal function of
markets returns.

22. If we used one dimensional Garch we would
separately calculate the volatility of the stock
and the volatility of the market. But using the
two dimensional Garch(1,1) helped us calculate
the variance in a combined way which also uses
the influences of the stock and the market.
Calculating them separately wouldn't bring us to
the right numbers because they have mutual
effects on each other.

24. We'll enter the results to Shumway& Bahart
equation of Destance to default (DD) and will
receive the DD of the specific stock today as a
function of both the stock and the market
returns.

25. Thank you