No. 1/2010
15
COMPARING VARIANCE/COVARIANCE AND HISTORICAL
SIMULATION IN THE CONTEXT OF THE FINANCIAL
CRISIS – DO EXTREME MOVEMENTS HAVE AN
INFLUENCE ONTO PORTFOLIO SELECTION?
Svend Reuse
Masaryk University, Brno / FOM University of Applied Science
Luxemburger Allee 121, 45481 Mülheim, Germany, Svend.Reuse@gmx.de
Abstract: Portfolio theory and the basic ideas of Markowitz have been extended in the recent
past by alternative risk models as historical simulation or even copula functions. The
central question of this paper is if these approaches lead to different results compared to
the classical variance/covariance approach. Therefore, empirical data of the last 10 years
is analysed. Both approaches are compared in the special context of the financial crisis.
The worst case optimization and the Value at Risk (VaR) are defined in order to define the
minimum risk portfolio before and after the financial crisis. The result is that the financial
crisis has nearly no impact onto the portfolio, but the two approaches lead to different
results.
Keywords: Portfolio Theory; Financial Crisis; Historical Simulation; Variance/
Covariance Approach
JEL Classification: G10, G15, G30
1 Introduction and Aim of the Paper
Portfolio theory has been in existence for more than 50 years. But the financial crisis in
2008 led to the discussion of the fact that the extreme movements of diversified portfolios
had not been forecasted. It is often said that this model failure led to high losses in 2008
that might have been prevented if a model that integrates worst cases scenarios would
have been used. Alternative but more complex methods should have been used.
Portfolio theory in extreme market situations is therefore a topic that has to be discussed.
Therefore, the historical simulation and the classical variance/covariance approach are
analyzed. The central question to be answered by this paper is:
Does portfolio optimization by means of historical simulation in the times of financial
crisis lead to other results than the classical variance/covariance approach? Do the
results change if a high confidential level or the worst case is chosen?
The assumption lying behind this is that extreme developments of portfolios cannot be
quantified by means of a normal distribution in the variance/covariance approach. This
has to be proven.
Financial Assets and Investing
16
The structure is as follows. After the introduction, section 2 discusses the theoretical
basics according to portfolio theory. The relevant theories of portfolio management, asset
allocation and risk/return are presented and discussed in the context of portfolio theory in
a worst case scenario. The main aim is to define the used optimizing criteria for section 3.
Section 3 describes the simulation and the parameters. Section 4 deals with the different
results of the variance/covariance approach and the historical simulation. Therefore,
several simulations based on historical data are done. Section 5 sums up the main results,
discusses them critically and answers the critical question of this paper looking at all
aspects.
2 Theoretical Aspects of Portfolio Theory
2.1 Definition of Risk and VaR
Whenever the discussion about portfolio diversification and risk minimization occurs, the
first step is to define risk. In general, risk is the positive or negative unexpected difference
from an expected value (Woll, 1996, p. 605). The positive difference is called chance, so
only the negative difference from the expected value remains (Rolfes, 1999, p. 29; Reuse,
2006, p. 366; Reuse 2008, p. 5). This leads to the result that expected changes from the
actual value of an asset are not defined as risk. They can be calculated ex ante and can be
priced into the asset. An example is the so called spread in the corporate bond market. It is
defined as the difference between the yield of the bond and the maturity equivalent risk
free rate, for example US treasuries or German treasury bonds. The spread becomes
higher, if the credit risks in the corporate bond increases.
One of methods to measure risk is the VaR (Value at Risk). It can be defined as the
unexpected loss that will not be exceeded within a certain time under the assumption of a
defined security level, called confidential level (e.g. Schierenbeck, 2001, p. 17; Reuse,
2006, p. 366). This definition leads to several factors that influence the VaR. In general, it
can be stated that the VaR will increase, if the disposition period becomes longer and the
confidential level gets higher (Reuse, 2006, p. 366). The model used to measure the risk
influences the VaR as well, but this depends on the risk category and not on the basic
assumptions to measure risk (Rolfes, 1999, p. 120).
In this article, three risk measurement variables will be used: the variance (z = 84.13), the
VaR (z = e.g. 99.00) and the worst case (z = 100.00) that is defined as the highest loss
than could have been stated in the historical development of an asset.
2.2 Portfolio Selection
The origin of Portfolio Theory can be found in 1952. Harry M. Markowitz published his
theory in an article in the “Journal of Finance” titled “Portfolio Selection” (Markowitz,
1952, p. 77; Markowitz, 1987, p. 3). Markowitz demonstrates how an investor can reduce
No. 1/2010
17
the risk, or the standard deviation of the portfolio returns by choosing stocks that do not
move exactly together (Discussed in Linnertová & Reuse, 2008, pp. 554–556).
The portfolio theory analyses the situation of investment decisions considering risk using
the principle of : The volatility can be seen as a degree of risk if all shares can be
described by the expected returns ( ) and the volatility ( ) of returns. The theory assumes
risk-averse behaviour of investors, which means that they are willing to renounce with
return in favour of reducing the risk of an investment. A portfolio is consequently risk
efficient if there is no alternative to get less for the same , more for the same or
both more and less .
The expected rate of return resulting from diversification corresponds to the sum of
returns of particular shares weighted by their proportion within the portfolio. Therefore,
the expected rate of return of a portfolio p consisting of n shares i with expected returns
i is:
=
=
n
i
iip x
1
µµ with
=
=
n
i
ix
1
1 (1)
where:
p = expected return of portfolio
i = expected return of share i
n = number of available shares
xi = proportion of share i within the portfolio
The standard deviation of the portfolio diversification is not equivalent to the weighted
sum of every single variance. Additionally, the stochastic dependence of particular shares
measured by the covariance covij has to be taken into account. The basic idea lying behind
this is that a diversified risk is always smaller than the weighted sum of the risks if the
correlations between the assets are smaller than one. Markowitz argues with covariances
and variances of portfolios instead of correlation effects but can be cited as follows: “If
two original Portfolios have equal variance then typically the variance of the resulting
(compound) portfolio will be less than the variance of either original portfolios”
(Markowitz, 1952, pp. 89–90). The formula can be written as follows (Schierenbeck,
2001, pp. 78–79), whereas the standard deviation is multiplied with a certain z-factor in
order to achieve the relevant VaR for the portfolio risk (Hager, n.Y., p. 3).
zxxVaR
n
i
n
j
ijjijiP ⋅⋅⋅⋅⋅=
= =1 1
ρσσ (2)
Financial Assets and Investing
18
[ ] z
x
x
VaR
nnnn
n
n
nP ⋅
⋅
⋅
⋅⋅⋅⋅⋅⋅=
σ
σ
ρρ
ρρ
ρρ
σσσ
:
:
1...
...1......
...1
...1
x;...;x;x
11
2,1,
,21,2
,12,1
n2211 (3)
where:
VaRp = Value at Risk of the portfolio
xi = part of asset i
xj = part of asset j
σi = standard deviation of asset i
σj = standard deviation of asset j
i,j = correlation coefficient between the assets i and j
n = number of assets
z = z factor of the normal distribution (2.326 for 99%, 1.645 for 95%,
1.000 for 84.13%)
ki,j =
ji
ji
σσ ⋅
,cov
This approach will be defined as a variance/covariance or up to now classical Markowitz
approach as it defines the basic idea how to optimize a portfolio – including all advantages
and disadvantages of the assumption of a normal distribution (Reuse 2006, p. 367).
2.3 Historical Simulation – Extending the Variance/Covariance Approach
If historical data is used to define diversification, the historical simulation can be used as
well. While the variance/covariance approach shows the inherent assumption of a standard
normal distribution (Bühler/Korn/Schmidt, 1998, p. 67), the historical simulation goes one
step further. It uses a “real” distribution without converting historical data into a
distribution function. It defines risk as the difference from an expected value. A further
advantage is that extreme changes in the history that exist above the chosen confidential
level can be measured as well as they are not converted into a distribution function. The
historical simulation can be modelled as follows:
[ ])(% isaP rQuartilerVaR −= (4)
=
−1
ln
i
i
i V
V
r (5)
No. 1/2010
19
where:
VaRp = Value at Risk of the portfolio
ri = all historical yields (e.g. 1 / 10 / 250 days) of the used historical data
ra = average of all historical yields (e.g. 1 / 10 / 250 days) of the used historical data
s = security level (e.g. 100%, 99%, 95%, 84.13%) for defining the z value of the
normal distribution
Vi = value of portfolio at ti
The real distribution differs from the normal distribution function. This is shown in figure
1 that represents the lognormal daily changes in a “real” distribution compared with a
transformation into a standard distribution of the German DAX (Deutscher Aktienindex).
Figure 1 shows that a standard normal distribution is only an approximation of the “real”
distribution. The yields show extreme movements in other barriers than expected. So it
would be the best to use the historical yields to measure risk. In addition, the procedure is
less analytical than the variance/covariance approach but the amount of the needed data is
much higher (A critical analysis of several VaR methods can be found in Goebel & Sievi
& Schumacher, 1999, p. 236, discussed in Reuse 2006, p. 367).
Fig. 1 DAX yields – real and transformed distribution
0 3
28
55
155 159
348
175
426
685
255
47
1 0 0
0
100
200
300
400
500
600
700
800
-96.969%
-82.828%
-68.686%
-54.545%
-40.403%
-26.261%
-12.120%
2.022%
16.164%
30.305%
44.447%
58.588%
72.730%
86.872%
101.013%
distributed with 0.5 standard deviations around the mean
frequency(transformedanddiscrete)
Source: VWD, yearly returns, mean = 2.022%, standard deviation = 23.283% p.a. Data:
12/31/1998 up to 11/28/2008
Financial Assets and Investing
20
It works as follows (Bühler/Korn/Schmidt 1998, p. 68): the historical price developments
are transformed into lognormal yields (see equation 5). Yields of a specific history for all
assets build a yield set. All yield sets that are evaluated in this way – typically several
thousands – are added to the existing portfolio. Thousands of virtual portfolio changes can
be defined. They are sorted by height and the needed quartile is defined by counting these
changes. In the case of 100 possible portfolio changes and 99% confidential level, the
second worst value is the VaR that will not be exceeded within the defined period. The
worst case scenario would be defined as the worst simulated yield in order to get the most
extreme historical movements of the past.
Applying this on a portfolio leads to the conclusion that this procedure must be done for
all asset combinations for the given historical yields. The advantage is that extreme
movements can be analyzed specifically and that only changes that appeared in the past
are interpolated into the future.
3 Modelling the Simulation on the Basis of Historical Data
3.1 Asset Classes and Data used
There are many assets, but the most interesting ones are the five assets that are presented
in figure 2.
Fig. 2 Used asset classes
Asset Class Description Reason why the asset class is
used
RexP
(Rentenindex
Performance
index)
This index represents the German
covered bond market and is the
main asset class a risk averse
investor can invest in. For the
asset class “yields”, it is a
representative index as every
maturity is implemented.
Bonds are the basic investments
an investor has. This asset class
has to be diversified.
DAX Represents the biggest 30 shares
in the German market and is a
typical synonym for the asset
class “shares”.
Correlations between shares and
bonds are the oldest histories that
can be analyzed. A diversified
portfolio without shares would
not embrace all aspects.
No. 1/2010
21
Deka Real
Estate
Europa
Estates and the investment in this
asset class are becoming more
popular. The presented
investment fund is one of the
biggest in the German market and
most diversified in Europe. It
may be taken as a synonym for
the asset class “estates”.
The correlation between estates
and all other asset classes is said
to be about 0. This is the reason
why it is used in this analysis.
Money
Market 3M
This index consists of the 3M
Euribor rate and transforms the
yields into a performance. It is a
synonym for “risk free yield”.
As risk is defined as the
difference to the expected or
secure value, a risk free rate in
order to define the zero risk point
has to be used in the analysis as
well.
Dow Jones
AIG
This rollover index consists of 19
physical commodities traded on
the U.S. Exchange with the
exception of aluminium, nickel,
zinc which is traded on the
London Metal Exchange. It may
be taken as a synonym for the
asset class “commodities”.
Commodities are often used to
diversify a portfolio as the
correlations are near 0 or
negative.
Source: author’s own figure
Data is available from 12/31/1998 up to 11/28/2008 on a daily basis (Source: Datastream
2008 and VWD 2008). A 10 year history shall be enough to use reliable data for the
calculation. Further, the beginning of the financial crisis in 2008 and the subprime crisis
can be found in the historical data.
3.2 Parameters of the Simulations
It shall be found out whether the optimization on the basis of the historical simulation
leads to other results than the classical Markowitz approach. Therefore, several
simulations shall be done. As only a free portfolio optimization can prove the central
assumption of this assignment, no volume restrictions are entered. But all other parameters
can be changed. Therefore, the following sets are calculated (Fig. 3).
Financial Assets and Investing
22
Fig. 3 Used parameter sets
Parameter Choice Reason why these parameters are used
Disposition
horizon
1-day-yield
10-day-yield
250-day-yield
These parameters are the typically used
disposition horizons in practice. The
overnight risk and the 10 day (two weeks)
risk period are useful especially for
measuring trading book risk (Schierenbeck,
2001, pp. 358, 447). The 250-day-yield
reflects one year, which is the classical
period in the Markowitz approach. A longer
period does not make sense as the maximal
used period for measuring market price risk
is one year.
History 12/31/1998–12/31/2007
12/31/1998–11/28/2008
The DAX reached its high at the end of
2007. An interesting question is whether a
worst case optimization on the basis of
2007 would have led to other results.
Confidential
Level
99.00%
84.13%
100.00% (Worst Case in
history)
The difference between the classical
standard deviation and the VaR shall be
analyzed critically. Further, the worst case
is analyzed as well.
Source: author’s own figure
The confidential level that has to be defined leads to the definition of risk. In the classical
variance/covariance approach the standard deviation is used to quantify risk. In this
simulation, the confidential level of 99% is chosen. This leads to a different z-factor in the
variance/covariance approach when defining risk – and to another risk level in the
nonnormal distributed historical simulation. This is the first way to define risk in extreme
situations – even though the 99% confidential level might not be enough.
As a consequence, there are 12 scenarios that have to be calculated. As the used histories
and time horizons change, expected yields and correlations change as well for the 12
scenarios. The results are distinguished into those before and into those after the financial
crisis in 2008. Therefore, there are two data periods that lead to 24 scenarios in the end.
3.3 Optimization Problem – Minimizing Risk
This analysis focuses on the risk minimizing portfolios calculated by several confidential
levels in the historical simulation and the variance/covariance approach. In order to solve
the central questions of this work, an Excel based tool is created that is able to simulate
portfolio optimization under different circumstances. The classical variance/covariance
approach as well as the described historical simulation are implemented, including an
No. 1/2010
23
optimization on the basis of the worst case scenario (confidential level 100%). Therefore,
the mathematical framework of section 2.1–2.3 is used.
For both approaches (variance/covariance and historical simulation), an optimization
problem has to be solved. This is done by means of the SOLVER in MS Excel, consisting
of the following framework:
• Iterations: 100
• Exactness: 0.000001
• Tolerance: 5%
• Convergence: 0.0001
• Linear optimization of the Newton approach
For the variance/covariance approach and the historical simulation, the combination with
the lowest risk is chosen in this setup in order to define the minimum risk portfolio
resulting from historical data. Consequently, two minimum risk portfolios for each of the
12 scenarios have to be modelled. These problems can be set up as follows (e.g. Perridon
& Steiner, 2007, p. 244):
Variance /covariance approach
zxxMin
n
i
n
j
ijjiji ⋅⋅⋅⋅⋅
= =1 1
! ρσσ (6)
with:
=
=
n
i
ix
1
1; 10 ≤≤ ix ;
=
=
n
i
iip x
1
µµ
Historical simulation
⋅−
=
5
1
%!
i
iisa rxQuartilerMin (7)
with:
=
=
n
i
ix
1
1; 10 ≤≤ ix ;
=
=
n
i
iip x
1
µµ ; =
−1
ln
i
i
i V
V
r
Financial Assets and Investing
24
4 Results of the Simulations
4.1 Analyzing the Minimum Risk Portfolios
The results are summed up in figure 4. The first result is that the different history – before
and after the financial crisis – does not lead to a different portfolio optimum in both
models. The risk minimized portfolios show nearly the same percentages. Further, the
assumption that the worst case leads to significant different results must be rejected as
well. If the historical simulation is calculated at a high confidential level, the worst case
portfolio combination does not differ too much from e.g. a 99% confidential level result.
Comparing the three historical simulations, it is interesting to see that the combination on
the basis of the standard deviation leads to different results compared to the worst case
and the 99% confidential level setup. The result is that especially in the case of market
price risks a historical simulation has to be done at a high confidential level in order to
measure the risk in an adequate way. Low confidential levels underestimate risk and shall
be omitted.
No. 1/2010
25
Fig. 4 Results of the simulations
Source: author’s own figure
Approach
Assets
Variance/
Covariance
99.00%VaR
Historical
Simulation
84.13%VaR
Historical
Simulation
99.00%VaR
Historical
Simulation
WorstCase
Approach
Assets
Variance/
Covariance
99.00%VaR
Historical
Simulation
84.13%VaR
Historical
Simulation
99.00%VaR
Historical
Simulation
WorstCase
RexP0.28%10.34%2.86%2.98%RexP0.20%5.88%4.97%6.22%
DAX0.01%1.73%1.22%0.97%DAX0.01%2.24%1.52%0.61%
DekaImmoEurope1.55%24.00%14.03%5.80%DekaImmoEurope1.68%25.98%6.83%9.70%
MoneyMarket3M98.13%61.04%81.20%89.24%MoneyMarket3M98.08%64.29%85.23%80.51%
DowJonesAIG0.03%2.90%0.68%1.01%DowJonesAIG0.03%1.60%1.44%2.96%
Risk(VaR/WC)0.30%0.06%0.08%0.09%Risk(VaR/WC)0.32%0.06%0.10%0.14%
Return3.14%1.45%2.14%2.50%Return3.28%1.64%2.40%2.18%
Approach
Assets
Variance/
Covariance
99.00%VaR
Historical
Simulation
84.13%VaR
Historical
Simulation
99.00%VaR
Historical
Simulation
WorstCase
Approach
Assets
Variance/
Covariance
99.00%VaR
Historical
Simulation
84.13%VaR
Historical
Simulation
99.00%VaR
Historical
Simulation
WorstCase
RexP0.17%10.09%6.60%4.66%RexP0.26%8.54%5.59%2.82%
DAX0.10%4.45%2.79%1.97%DAX0.14%1.73%2.87%2.00%
DekaImmoEurope0.00%28.73%14.17%14.81%DekaImmoEurope0.00%31.44%13.45%13.99%
MoneyMarket3M99.67%53.79%73.97%75.80%MoneyMarket3M99.50%53.33%75.24%79.14%
DowJonesAIG0.06%2.94%2.47%2.76%DowJonesAIG0.10%4.95%2.85%2.05%
Risk(VaR/WC)0.42%0.10%0.15%0.16%Risk(VaR/WC)0.44%0.11%0.17%0.18%
Return3.13%1.29%1.81%1.90%Return3.26%1.38%1.94%2.13%
Approach
Assets
Variance/
Covariance
99.00%VaR
Historical
Simulation
84.13%VaR
Historical
Simulation
99.00%VaR
Historical
Simulation
WorstCase
Approach
Assets
Variance/
Covariance
99.00%VaR
Historical
Simulation
84.13%VaR
Historical
Simulation
99.00%VaR
Historical
Simulation
WorstCase
RexP15.01%22.08%18.57%19.38%RexP15.99%25.37%19.59%18.90%
DAX2.17%7.38%6.95%6.94%DAX2.49%8.64%6.94%7.06%
DekaImmoEurope0.00%31.06%30.94%30.20%DekaImmoEurope0.00%18.30%30.17%29.70%
MoneyMarket3M82.07%30.50%35.87%35.84%MoneyMarket3M81.15%39.63%35.38%37.16%
DowJonesAIG0.74%8.98%7.68%7.63%DowJonesAIG0.37%8.05%7.92%7.18%
Risk(VaR/WC)1.49%0.17%0.26%0.28%Risk(VaR/WC)1.55%0.20%0.28%0.30%
Return3.31%0.98%1.00%0.99%Return3.38%1.00%1.02%1.02%
Simulation2008
250day
dispositionhorizon
1day
dispositionhorizon
10day
dispositionhorizon
Simulation2007
Financial Assets and Investing
26
Normally, portfolio optimization is done to create a strategic position. Nevertheless, the
investor often faces the situation that the portfolio loses very much in a short time even
though the long term risk is lower. Therefore, the three different time horizons were
chosen. The results differ both in the variance/covariance approach and in the historical
simulation. While the first approach always achieves more than 98% of money market in
the 1 day and 10 day scenario, the historical simulation only leads to about 80% money
market. At least a small part (5–15%) shall be invested in real estates. When calculating a
250 day scenario, both variance/covariance approach and historical simulation lead to a
smaller part of money market – but the difference in the approaches can be seen most
clearly in the 250 day setup. While the variance/covariance approach offers about 80% of
money market and 15% of RexP, the historical simulation only invests 35% in money
markets but about 30% in real estates, 20% in RexP and about 5–10% in Dow Jones AIG
and DAX.
This difference in the models has to be explained. First, the historical simulation defines
risk as a difference from the expected or calculated yield. The longer the disposition
horizon, the higher is the different definition of risk. But the second argument is more
important: the standard normal distribution cannot be assumed especially in extreme
situations of a market movement. So the diversification effect between real estates and
money market that is quantified with the correlation of 0.8639 in the variance/covariance
approach is much higher when choosing the historical simulation approach.
4.2 Analyzing all Combinations of One Scenario
Besides the optimization presented above, the tool is able to optimize a portfolio
consisting of 5 assets in 10% steps of a portfolio mixture. So the part of every asset class
is between 0 and 100% that makes 11 possible characteristics. This leads to 1,001 possible
portfolio combinations.
The results are visualized in figure 5 that compares the optimization of
variance/covariance approach and historical simulation for 99% confidential level, 250day-yield,
12/31/1998–11/28/2008.
No. 1/2010
27
Fig. 5 Minimum risk portfolio by both approaches
Markowitz-Optimization
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00%
Risk p.a.
Return(calculatedyield)
10%-step portfolio combinations Maximizing Return Minimizing Risk
Historical Simulation - Optimization
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00%
Risk p.a.
Return(calculatedyield)
10%-step portfolio combinations Maximizing Return Minimizing Risk
Source: author’s own figure. 99% confidential level, 250-day-yield, 12/31/1998–11/28/2008
The variance/covariance approach offers several portfolio combinations that did not
appear in the chosen history. This model risk results from the transformation into a
standard normal distribution. The real distribution differs completely from the
standardized one. The portfolio combinations shown in the middle of the
variance/covariance approach have a lower risk in the historical simulation approach.
Further, it can be stated that the extreme values cannot be seen as often as in the
Financial Assets and Investing
28
variance/covariance approach. The result is a different minimum risk portfolio
combination in the historical simulation. The combination suggested by
variance/covariance would have the risk of 1.55%, while the one of the historical
simulation only has 0.28%. A similar result can be stated if the analysis is done with
expected yields. This might be necessary as the DAX yield is too low compared to its
long-term history. The yield should be at least about 8–10%. Even though there are many
portfolio combinations which have nearly the same risk, the historical simulation is better
at optimizing a portfolio.
5. Conclusion
Both variance/covariance approach and historical simulation have been analyzed in this
article. The results can be summed up as follows:
(1) There is nearly no difference in the portfolio mixes including and excluding the
year 2008. The financial crisis has nearly no impacts on the portfolio optimization.
(2) An optimization on the basis of the worst case (100% confidential level) does not
lead to other results, if the confidential level of the chosen simulation is set high
enough, e.g. 99%.
(3) The disposition horizons (1, 10, 250 days) lead to completely different results in
both approaches.
(4) The variance/covariance approach and the historical simulation lead to different
results. These differences increase with the disposition horizon. Two arguments
can be given therefore: first, risk is defined as the difference from the expected
yield in the historical simulation while variance/covariance defines risk as the
difference from the actual value. Second, the transformation into a standard
normal distribution does not reflect the real historical movements of the data.
(5) Therefore, several risk/return combinations offered by variance/covariance do not
exist in the historical simulation. The risk and return that were evaluated by the
historical simulation are lower than in the variance/covariance approach.
(6) The historical simulation leads to a better portfolio mix. The combination
suggested by variance/covariance leads to a higher risk.
(7) The diversification effects of real estates and commodities are quantified better in
the historical simulation as both assets get nearly no percentage in the
variance/covariance approach.
As a result, the answer to the central question of this assignment whether the historical
simulation leads to other results than the variance/covariance approach can be seen as
affirmative. The results differ. On the other hand, an optimization according to the worst
case scenario does not lead to other results if the confidential level is set high enough.
Both approaches are used in practice and lead to reliable results. The final result of this
article is that the historical simulation is better than the variance/covariance approach. But
No. 1/2010
29
both approaches have to be seen critically, as they are based on historical data. The choice
of the history influences especially the yield to a high degree. In the used history, the
DAX has a historical 1 year yield of about 2%. This is too low compared to its long term
history. But the choice of the “right” horizon remains difficult.
Further, it has to be stated that both approaches led to optimized portfolios that were not
able to survive the financial crisis in 2007/2008. Alternative assets as ABS and
commodities were used to realize correlation effects that disappeared in the crisis. The
failure of the models has numerous reasons. First, a simple historical market price analysis
was done in order to define correlations – assuming that an index reflects the development
of a complete asset class. Second, alternative asset classes only have a short term history.
E.g. ABS indices do not exist for a very long time. Third, nearly no investor looked into
the structure of the ABS investments. A simple market price analysis is not enough, if the
assets are very complex in their modulation. The model risk in these asset classes is at
least as high as the market price risk as the development of 2007/2008 has shown
(Frère/Reuse/Svoboda, 2008, p. 16). Fourth, the classical variance/covariance approach
leads to other portfolios than the historical simulation. Without knowing the reasons why
this takes place or under which circumstances which model offers which combination, an
investor uses a tool without understanding how it works.
The main lesson to be learned from the financial crisis is that models should be simple
enough to be understood by the user – as variance/covariance or historical simulation.
Even though more complex approaches as copulas (discussed e.g. Beck & Lesko, 2006;
Beck & Lesko & Schlottmann & Wimmer, 2006; Lesko, 2006; Mashal & Zeevi, 2002)
may lead to results that fulfil statistical requirements better, the collateral damage of using
methods that are not completely understood is higher than the model inexactness.
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