Risk Analysis on European Stock Exchange Indexes

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Introduction[edit]

  • The main question that an investor is permanently faced with is where he should invest his money in order to maximize his profit. However, the most profitable assets on the market are not necessarily the best investment option, due to the risk factor. Conservative ways to invest the money (e.g.: T-bills, bank deposits) offer the lowest possible risk, but a rate of return that is seen as too low. The capital market provides the possibility of combining assets with various levels of return and risk. The goal of this paper is to find out opportunities of obtaining a higher rate of return while having a low level of risk.


  • Stock Indexes generally represent well-diversified portfolios of companies selected according to different criteria and that are listed on a certain market. Due to this diversification, investing in an index may represent a good way for an investor to reach his objective.


  • Our typical investor is risk-averse. Risk-averse behavior can be defined in many ways, such as in the following example.
A fair gamble is one with zero expected return; a risk-averse investor would prefer to avoid fair gambles. If someone had just received his sallary and was given the opportunity to either double his sum or lose it with a flip of a coin, he would probably decline the offer. If the odds would increase from 50-50 to, say 75-25 in his favor, the agent could be tempted to accept the bet. The risk-averse investor can be induced to take fair gambles only if they are sweetened so that they become unfair to the investor's advantage.

Database[edit]

  • The database consists of log returns of daily observations on 6 stock exchange indexes, namely DAX (Germany), CAC-40 (France), FTSE-100 (Great Britain), BET (Romania), RTS (Russia) and SOFIX (Bulgaria). The indexes are representative for the stock exchanges where they are quoted, and the investor has the possibility to choose between 3 mature markets (Germany, France, Great Britain) and 3 emerging markets (Romania, Russia, Bulgaria).


Table 1. Number of companies in each index
DAX CAC-40 FTSE-100 BET RTS SOFIX
No. of companies 31 40 100 10 50 16


  • The period of observation is of 18 months, from 1.11.2005 to 30.04.2007, adding up to a number of 389 daily observations. There are no missing values. However, the different markets have different holidays, and for these days we have considered the values from the last trading day before the holiday. Therefore, we have values of the indexes for all weekdays (Monday to Friday), including the days specified above.


Table 2. Trading days analysis
DAX CAC-40 FTSE-100 BET RTS SOFIX
Trading days 380 378 376 366 368 369
Days with increase 228 207 203 186 213 197
Days with decrease 152 171 173 180 155 172
Holidays 9 11 13 23 21 20


  • Log returns are a good approximation of the daily percentage change in the value of an index. The formula used to calculate them is:
r_{i,t}=ln\frac{P_{i,t}}{P_{i,t-1}} , where
ln\frac{P_{i,t}}{P_{i,t-1}}=ln\frac{P_{i,t}-P_{i,t-1}+P_{i,t-1}}{P_{i,t-1}}
ln\frac{P_{i,t}-P_{i,t-1}+P_{i,t-1}}{P_{i,t-1}}=ln\left(\frac{\Delta P_{i,t}}{P_{i,t-1}}+1\right)
ln\left(\frac{\Delta P_{i,t}}{P_{i,t-1}}+1\right) \simeq \frac{\Delta P_{i,t}}{P_{i,t-1}}

Profitability and Risk[edit]

  • As measures of profitablity, we determined the profitability per trading day, per year and over the whole period (18 months).
  • Risk is estimated by means of standard deviation (volatility).


The used formulas are

  • estimated profitability


\bar{r_i}=\frac{1}{T}\sum_{t=1}^{T}r_{i,t}

  • estimated risk


\hat\sigma_i=\sqrt{\frac{1}{T-1}\sum_{t=1}^{T}(r_{i,t}-\bar{r_i})^2}


Table 3. Profitability
Profitability per trading day in % Annual profitability in % Profitability over the whole period in %
RTS 0.1589 39.73 61.82
SOFIX 0.1073 26.82 41.73
DAX 0.1051 26.28 40.89
BET 0.0895 22.37 34.80
CAC-40 0.0760 19.00 29.56
FTSE-100 0.0483 12.08 18.79


Table 4. Volatility
Volatility per trading day in %
RTS 1.8208
BET 1.2924
DAX 0.9101
CAC-40 0.8736
FTSE-100 0.7519
SOFIX 0.7248



  • The indexes were ordered descendingly with regard to profitability and volatility.
  • Volatility per trading day (in %) was considered as relevant, because if we extend it to longer periods of time, it becomes less accurate.
  • According to economic theory, a high return is associated with a high level of risk. In our case, this is true for the RTS, which is the most profitable but also the riskiest index. Furthermore, FTSE-100 has the lowest profitability, but also a low level of risk. However, SOFIX has a high return but the lowest volatility.

Graphical Tools and Statistical Analysis[edit]

Time Series[edit]

Figure 1. Time Series of Stock Indexes
  • The time series for the indexes has the same scale on both axes, enabling us to easily compare the evolution of the log returns in the respective cases.
  • One may easily see that the RTS is the most volatile index. It was possible to lose almost 10%, but also to gain almost 7% within one day. Comparing this possible gain to the yearly return of a bank deposit, it is clear that investors who like risk will seriously consider the idea of investing on the capital market.
  • The BET is the second most volatile index. It is followed by a group of three indexes (DAX, CAC-40 and FTSE-100) that are comparable from this point of view. We may already assume that there is a positive relationship between the evolution of these three indexes.
  • The least volatile index is the SOFIX, differentiating itself from the other indexes of the emerging markets. At the same time, it can be stated that there is no evident relationship between the evolution of the indexes of the three emerging markets.

Histograms and Jarque Bera Test[edit]

Figure 2. Histograms of the Stock Indexes
  • Histograms are density estimates that give a good idea about the distribution of the data. They are also useful for showing possible multimodality of the data.
  • A very important element when constructing hisograms is the binwidth. The optimal size of the binwidth according to the theory is:


h_{opt}=\left(\frac{24 \sqrt{\pi}}{n}\right)^\frac{1}{3} , where n is the number of observations.

  • The same scale was used for the two axes to insure comparability.
  • What is evident from these pictures is that they have a high kurtosis, the highest one being the SOFIX (as expected, due to the low volatility).
  • The question of normality cannot be answered properly just by looking at this graph; therefore we have performed the Jarque-Bera Test on Normality.
  • The Jarque-Bera Test is a measure of departure from normality, based on the sample kurtosis and skewness, and is defined by:


\mathit{JB}=\frac{n}{6}\left(S^2+\frac{(K-3)^2}{4}\right) ,

where n is the number of observations (or degrees of freedom in general), S is the sample skewness and K is the sample kurtosis.
  • At a significance level of 5%, the critical value is 5.99 .


Table 5. Jarque-Bera Test on Normality
DAX CAC-40 FTSE-100 BET RTS SOFIX
39,74 39,96 68,78 70,42 512,54 458,56


  • At significance level of 5%, none of the indexes is normal distributed.

Boxplots and Extreme Values Analysis[edit]

Figure 3. Boxplots of the Stock Indexes
  • The boxplots are useful in showing the distribution of the log returns, by giving information about the location, skewness, spread, tail length and extreme values.
  • The mean is higher than zero in all of the cases, meaning that an investment in any of the indexes over the whole period generates profit.
  • The F-spread and whiskers are higher in the case of RTS and BET. They are the lowest in the case of SOFIX as expected (the range is directly related to the volatility).
  • The extreme values have similar patterns in the case of DAX, CAC-40 and FTSE-100 respectively, meaning that the extreme values are situated in the proximity of the whiskers. The situation is different in the case of the emerging markets' indexes, where the distance from the whiskers is higher. RTS has the greatest difference between the maximum and minimum values, while the SOFIX has the greatest number of extreme values, with positive values more frequent than negative ones. In all other cases, except for the BET (where the number of positive extreme values is equal to the one of negative values), the negative extreme values predominate.


Figure 4. Extreme Values


Table 6. Extreme Values Analysis
* 0 0 0 1 2 7
o 10 4 6 8 11 15
DAX CAC-40 FTSE-100 BET RTS SOFIX
o 19 9 9 7 14 12
* 3 0 1 2 6 2


  • In figure 4 we have highlighted the extreme values that occured over the analsed period.
  • One can formulate different hypothesis about the developements that took place on particular periods on the stock markets. For example, it would be interesting to analize the factors that affected the RTS index in May 2006.

Scatterplots[edit]

Figure 5. Scatterplot of the Stock Indexes
  • The scatterplots are very useful in showing positive and negative relationships between variables. A downward-sloping scatterplot indicates a negative relationship between the variables, whereas an upward-sloping scatterplot denotes a positive one.
  • From the scatterplot matrix, a strong positive correlation between the DAX, CAC-40 and FTSE-100 is evident. Also, a slightly positive corellation between the above mentioned indexes and the RTS may be observed. In the other cases it is unclear whether there exists a correlation or not and therefore another tool to determine this more precisely is needed- the correlation matrix.

Correlation Matrix[edit]

Table 7. Correlation Matrix
DAX CAC-40 FTSE-100 BET RTS SOFIX
DAX 1 0.95 0.87 0.11 0.46 -0.10
CAC-40 0.95 1 0.89 0.09 0.51 -0.11
FTSE-100 0.87 0.89 1 0.11 0.54 -0.09
BET 0.11 0.09 0.11 1 0.14 0.12
RTS 0.46 0.51 0.54 0.14 1 -0.07
SOFIX -0.10 -0.11 -0.09 0.12 -0.07 1


  • Now, it becomes clear what kind of relationships exist between the indexes. The previous affirmations are now supported by the actual figures of the correlation matrix. The stongest positive correlation is between DAX and CAC-40 (coefficient of 0.95); this reflects the high level of integration between the two economies. They are followed by CAC-40 and FTSE-100 (coefficient of 0.89), then by DAX and FTSE-100 (coefficient of 0.87). So, the coefficients confirm that developed markets have similar behaviour patterns.
  • Also, there is a positive correlation between the RTS and the indexes of the developed markets. This may be a consequence of the fact that there are economic ties between these countries, even though the economies are not sincronized.
  • Athough it was not visible from the scatterplot matrix, the correlation coefficients of BET and SOFIX with regard to the other indexes, show that there is no correlation between them and the other ones. Also, there is no correlation between the BET and the SOFIX, even though the two countries are from the same region. So, movements in the other analysed markets do not have significance influence on these two indexes.

Alternative Portfolios[edit]

Figure 6. Initial portfolios
  • This is the most relevant graph from which one can easily see which portfolio is worth investing in. Risk and profitability are measured on the OX and OY axes, respectively.
  • First, let's consider the group formed by SOFIX, DAX, CAC-40, FTSE-100 and BET. Within this group, the SOFIX has the highest return and at the same time, the lowest degree of risk. Therefore, a rational investor will prefer the SOFIX over the other indexes (SOFIX dominates the other indexes).
  • Now let's focus on the RTS. It has the highest profitability, but also the highest risk. The question is: Which one of these two indexes (RTS or SOFIX) will an investor choose? One can prefer the RTS over the SOFIX (and all the other indexes implicitly) if he/she is an investor with low aversion towards risk, or the SOFIX over the RTS if the agent is more risk-averse.
  • Which is the optimal portfolio? In order to suit the needs of all types of investors, we have built alternative portfolios that are based on different proportions of RTS and SOFIX.
  • In order to determine the portfolio risk, the following formulas are used:


Variance_{portfolio} = X_A^2 \sigma_A^2 + 2 X_A X_B cov(A,B)+ X_B^2 \sigma_B^2,

where
X_{A} is the proportion of the portfolio invested in asset A
X_{B} is the proportion of the portfolio invested in asset B
X_{B} = 1 - X_{A}


Risk = \sigma = \sqrt{Variance_{portfolio}}

  • The portfolio profitability is a weighted average of the profitabilities of the used indexes:


 Profitability_{portfolio} = X_{A} R_{A} + X_{B} R_{B}


Table 8. Profitability and Volatility of the alternative portfolios
SOFIX RTS Profitability Volatility
100% 0% 0.1073 0.7248
90% 10% 0.1124 0.6648
80% 20% 0.1176 0.6599
70% 30% 0.1228 0.7111
60% 40% 0.1279 0.8079
50% 50% 0.1331 0.9363
40% 60% 0.1383 1.0851
30% 70% 0.1434 1.2469
20% 80% 0.1486 1.4174
10% 90% 0.1538 1.5938
0% 100% 0.1589 1.8208


  • In the table alternative portfolios are shown, that contain different proportions of SOFIX and RTS. The profitability and volatility are considered on a daily basis, in percent.
  • We observe that the 100% SOFIX portfolio doesn't have the minimum risk. Altough the RTS is riskier, a diversification effect that reduces the total risk of the new portfolio is visible.
  • A minimum variance portfolio can be computed in the following way:
Figure 7. Alternative portfolios


 min_{X_A} \{X_A^2 \sigma_A^2 + 2 X_A X_B cov(A,B)+ X_B^2 \sigma_B^2\}
which is equivalent to
 min_{X_A} \{X_A^2 \sigma_A^2 + 2 X_A (1-X_A) \rho_{AB} \sigma_{A}\sigma_{B}+ (1-X_{A})^2 \sigma_B^2\}


We solve the first order condition and we get X_{A}=84.65%.
This means that X_{B}=15.35%.
So minimum risk can be achieved if we invest 84.65% in the SOFIX and 15.35% in the RTS.



  • Graphically, the alternative portfolios are represented in Figure 6. The possible combinations are infinite; we have selected steps of 10%. All the possible combinations are situated on the line defined by these points. The Efficient (Markowitz) Frontier is defined by all possible efficient combinations. The Efficient Frontier starts from the Minimum Variance Portfolio (84.65% SOFIX, 15.35% RTS) and goes to the portfolio that contains 100% RTS. The points that are situated on the line but below the Minimum Variance Portfolio are not efficient, because they have the same level of risk as the corresponding points from the Efficient Frontier, but a lower profitability.
Figure 8. Initial and alternative portfolios


  • In Figure 7 are shown the initial indexes and also the alternative portfolios. The difference is visible and suggests clearly that diversification has positive effects. It is now up to the investor to choose his desired portfolio according to his preferences and his utility.
  • However, there are limits to diversification. Total risk is a risk that one bears by holding onto one security only. Portfolio risk is the risk that one still bears after achieving full diversification. Portfolio risk is often called systematic or market risk as well. Diversifiable, unique or unsystematic risk is a risk is a risk that can be diversified away in a large portfolio.
  • To an individual who selects a diversified portfolio, the total risk of an individual security is not important. When considering adding a security to a diversified portfolio, the individual cares about only that portion of the risk of a security that cannot be diversified away. This risk can alternatively be viewed as the contribution of a security to the risk of an entire portfolio.


Other factors[edit]

  • The analysis can be improved by also taking other factors into consideration, e.g. country risk, political factor, transparency, accessibility, market liquidity.
  • A particular one that affects the profitability is the exchange rate. If we consider the case of an investor from the Euro Area, he would be interested in earning as many euros as possible. The profitability of an index from an emerging market is of course relevant, but the investor has to change the euros to the local currency first, then invest on the local market, and finally reconvert the money to euros, after selling the stock. The exchange rate risk that intervenes may limit the investor's gain, or on the contrary, it may be another source of profit.
  • In order to illustrate exactly how much an investor from the Euro Area would have gained (in euros) after investing in the six indexes, we have also taken the exchange rates into account.
  • The exchange rates situation is presented in the following table:


Table 9. Exchange rates
RON/EUR RUB/EUR BGN/EUR
1.11.2005 3.66 34.41 1.96
30.04.2007 3.33 35.08 1.96
Gain/Loss 9.91% -1.91% 0%


  • An investment in the BET has an additional profit, an investment in the RTS has a diminished profit, while an investment in the SOFIX is unaffected by the exchange rate. It is worth mentioning that Bulgaria has a fixed exchange rate with respect to the Euro, so actually there is no fluctuation of this currency.


  • Under these conditions, the total profitability of the indexes is shown in the following table:


Table 10. Profitability with exchange rates
Profitability per trading day in % Annual profitability in % Profitability over the whole period in %
RTS 0.1510 37.74 58.73
BET 0.1238 30.95 48.16
SOFIX 0.1073 26.82 41.73
DAX 0.1051 26.28 40.89
CAC-40 0.0760 19.00 29.56
FTSE-100 0.0483 12.08 18.79


  • The new profitability is calculated from the initial one, compounded by the gain/loss from the exchange rate.
  • The BET jumped from the fourth position to the second position, making the indexes of the emerging markets the most profitable ones.

Conclusions[edit]

  • We performed an analysis of a financial database using XploRe.
  • Our main emphasis was on profitability and risk analysis.
  • We found out that an optimal potfolio can be obtained by investing in RTS and SOFIX, in different proportions, depending on the level of risk aversion of each investor.
  • For an investor from the Euro Area who wants to maximize his return in Euros, it is relevant to also take into consideration the exchange rate. The results that we obtained, including the exchange rate information, show that the most profitable indexes are those of emerging markets.

References[edit]

  • Härdle, W., Klinke, S., Müller, M., XploRe Learning Guide, Springer, 2000
  • Härdle, W., Simar, L.: Applied Multivariate Statistical Analysis, Springer, 2003
  • Elton, E., Gruber, M., Brown, S., Goetzmann, W., Modern Portfolio Theory and Investment Analysis, John Wiley & Sons Inc, 2007
  • Ross, S., Westerfield, R., Jeffey, J., Corporate Finance, McGraw-Hill, 2005

Comments[edit]

  • Profitabilty and Volatility would be nice in one graphic
  • No programs
  • Descriptives of the indices, especially kurtosis and excess?
  • For tests give the empirical significance niveau
  • Nice table for extreme values
  • Typos
  • For Figure 7 and 8 the Profitability should have been on the x-axis
  • What happens to your portfolio when you include the exchange rate?