A Probabilistic Approach to Worst Case Scenarios
A Probabilistic Approach to Worst Case Scenarios
By Giovanni Barone-Adesi
University of Alberta, Canada
and
City University Business School, London
Frederick Bourgoin
Millennium Global Investment, London
&
Kostas Giannopoulos
University of Westminster, London
March 1997
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A Probabilistic Approach to Worst Case Scenarios
Historical simulation is a natural setting for scenario analysis, but it must pay attention to current market conditions
Value at Risk (VaR) is increasingly popular as a management and regulatory tool.
To further its acceptance it is necessary to assess its reliability under conditions
likely to be encountered in financial markets. A logical venue to investigate this
issue is through the use of historical simulation.
Historical simulation relies on a uniform distribution to select innovations from
the past. These innovations are applied to current asset prices to simulate their
future evolution. Once a sufficient number of different paths has been explored it
is possible to determine a portfolio VaR without making arbitrary assumptions
on the distribution of portfolio returns. This is especially useful in the presence
of abnormally large portfolio returns.
From the early days of modern finance large returns are known to cluster in
time. The resulting fluctuations in daily volatility make the confidence levels of
VaR computations that ignore clustering unreliable. This is the case with VaR
measurements based on the variance-covariance matrix and Monte-Carlo methods, that typically ignore current market conditions to produce flat volatility forecasts for future days. Moreover the use of the covariance matrix of security returns or the choice of an arbitrary distribution in the Monte-Carlo method usually destroys valuable information about the distribution of portfolio returns.
To make our historical simulation consistent with the clustering of large returns
we model the volatility of our portfolio as a GARCH process. Past daily portfolio
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Barone-Adesi, Bourgoin & Giannopoulos
returns are divided by the GARCH volatility estimated for the same date to obtain standardised residuals. A simulated portfolio return for tomorrow is obtained multiplying a randomly selected standardised residual by the GARCH
forecast of tomorrow volatility. This simulated return is used to update the
GARCH forecast for the following day, that is then multiplied by a newly selected standardised residual to simulate the return for the second day. Our recursive procedure is repeated until the VaR horizon (i.e.,10 days) is reached,
generating a sample path of portfolio volatilities and returns. We repeat our procedure to obtain a batch of sample paths of portfolio returns. A confidence band
for the corresponding portfolio values is built by taking the Kernel (empirical)
frequency distribution of values at each time. The lower 1% area identifies the
worst case over the next ten days.
To illustrate our procedure we constructed a hypothetical portfolio, diversified
across all thirteen national equity markets of our data sample. To form our portfolio each equity market is weighted proportionally to its capitalisation in the
world index as on December 1995. The portfolio weights are reported in the table
below:
country
Denmark
France
Germany
Hong Kong
Italy
Japan
Netherlands
Singapore
Spain
Sweden
Switzerland
UK
US
Table 1 : Portfolio
our portfolio
0.004854
0.038444
0.041905
0.018918
0.013626
0.250371
0.024552
0.007147
0.010993
0.012406
0.036343
0.103207
0.437233
2
weights
world index (dec 95)
0.004528
0.035857
0.039086
0.0176450
0.012709
0.233527
0.022900
0.006667
0.010254
0.011571
0.033898
0.096264
0.407818
A Probabilistic Approach to Worst Case Scenarios
Hence these weights are held constant for the entire 10 year period and multiplied by the thirteen local index returns. Since market risk needs to quantify
eventual portfolio losses in one currency all local portfolio returns are measured
in US dollars. The descriptive statistics together with the Jarque-Bera normality
test are reported on table 2. Figure 1 displays the empirical distribution of portfolio's returns.
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Table 2 : Descriptive statistics of the equally weighted portfolio
mean (p.a.)
std. dev (p.a.)
skewness kurtosis normality
p value
10.92%
12.34%
-2.828
62.362
3474.39
0.000
The last column is the probability that our portfolio returns are generated from a
normal distribution.
The rejection of normality in table 1 and the pattern of clustering visible in figure
1 led us to model our portfolio returns, r , as a GARCH process with asymmetries, with daily volatility h given by eq.1:
t
t
 P 
rt = 100 ⋅ Ln t  + ε t
 Pt −1 
(
h t = ω + α ε t −1 + γ
1
)
2
( )
ε t ~ N 0 , ht
+ βh t −1
(1.b)
The test for normality is the Jarque-Bera test,
((
)
)
JB = skewness / 6 + (Kurtosis − 3 ) / 24 law
→ χ 2 df
2
(1.a)
2
2
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Barone-Adesi, Bourgoin & Giannopoulos
Fig 1 : World Capitalisation weigthed Portfolio Returns
0.1
0.05
0
-0.05
14/02/96
27/09/95
10/05/95
21/12/94
03/08/94
16/03/94
27/10/93
09/06/93
20/01/93
02/09/92
15/04/92
27/11/91
10/07/91
20/02/91
03/10/90
16/05/90
27/12/89
09/08/89
22/03/89
02/11/88
15/06/88
27/01/88
09/09/87
22/04/87
03/12/86
16/07/86
26/02/86
09/10/85
22/05/85
-0.15
02/01/85
-0.1
Therefore our portfolio volatility is modelled to depend on the most recently observed portfolio return. The combination of GARCH volatility and portfolio historical returns offers us a fast and accurate measure of the past, current (and future) volatility of the current portfolio. No estimation of the correlation matrix of
security returns is required. Furthermore our
VaR
method contains fewer "bad
surprises", since GARCH models allow for fat tails on the unconditional distribution of the data. The effects of our choice become apparent if the returns in figure 1 are compared with figure 2, where the returns have been scaled by their
daily volatility, as in equation 2:
zt =
rt
(2)
ht
Clustering of returns in time is reduced by volatility scaling and the distribution
of returns now appears to be more uniform. However the large number of returns still exceeding 3 standard deviations suggests that our scaling does not
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5
28/12/95
10/08/95
23/03/95
03/11/94
16/06/94
27/01/94
09/09/93
22/04/93
03/12/92
16/07/92
27/02/92
10/10/91
23/05/91
03/01/91
16/08/90
29/03/90
09/11/89
22/06/89
02/02/89
15/09/88
28/04/88
10/12/87
23/07/87
05/03/87
16/10/86
29/05/86
20/12/95
02/08/95
15/03/95
26/10/94
08/06/94
19/01/94
01/09/93
14/04/93
25/11/92
08/07/92
19/02/92
02/10/91
15/05/91
26/12/90
08/08/90
21/03/90
01/11/89
14/06/89
25/01/89
07/09/88
20/04/88
02/12/87
15/07/87
25/02/87
08/10/86
21/05/86
01/01/86
-15
09/01/86
A Probabilistic Approach to Worst Case Scenarios
make returns normal. Our annualised portfolio volatility, in figure 3, varied from
7 to 21 % over 10 years.
Fig 2 : Porfolio Stress Analysis
(Standardised Residuals)
10
5
0
-5
-10
Fig 3 : Annualized volatility of the portfolio
25
20
15
10
5
0
Barone-Adesi, Bourgoin & Giannopoulos
The scaled returns are the foundation of our simulation. To simulate portfolio returns over next 10 days we select randomly 10 returns from figure 2. We then
construct iteratively the daily portfolio volatility that these returns imply according to equation 1. We use this volatility to rescale our returns. The resulting returns reflect therefore current market conditions rather than market conditions
associated with returns in figure 1.
To obtain the distribution of our portfolio returns we replicated the above procedure 10,000 times. The resulting -normalised- distribution is shown in figure 4.
The normal distribution is shown in dots in the same figure for ease of comparison.
Fig 4 : Normalized Estimated Distribution of Returns in 10 days
versus the normal density (10,000 Simulations)
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
5%
0.05
0.00
-7.5
-5.0
-2.5
0.0
Smoothed Density
Normal Density
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2.5
5.0
7.5
A Probabilistic Approach to Worst Case Scenarios
Not surprisingly, simulated returns on our well-diversified portfolio are almost
normal, except for their steeper peaking around 0 and some clustering in the
tails. The general shape of the distribution supports the validity of the usual
measure of
VaR
for our portfolio. However a closer examination of our simula-
tion results shows how even our well-diversified portfolio may depart from normality under worst case scenarios. There are in fact several occurrences of very
large negative returns, reaching a maximum loss of 9.52%. Our empirical distribution implies losses of 3.38% and 2.24% at confidence levels of 1% and 5% respectively.
Fig 5 : Estimated Distribution of Portfolio VaR in 10 days
(10,000 Simulations)
180
160
140
120
mean = 3.25%
100
80
60
40
20
0
0.025
0.050
0.075
0.10
The reason for this departure is the changing portfolio volatility and thus portfolio
VaR,
shown in figure 5. Portfolio
VaR
over next 10 days depends on the
random returns selected in each simulation run. Its pattern is skewed to the
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Barone-Adesi, Bourgoin & Giannopoulos
right, showing how large returns tend to cluster in time. These clusters provide
realistic worst case scenarios consistent with historical experience. Of course our
methodology may produce more extreme departures from normality for less diversified portfolios.
In conclusion, our simulation methodology allows for a fast evaluation of
VaR
and worst case scenarios for large portfolios. It takes into account current market
conditions and does not rely on the knowledge of the correlation matrix of security returns.
Our computations were performed using the RiskClock software.
A full description of it is available from the authors.
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