Cornish-Fisher Model (Modified VAR)

Cornish-Fisher model is used in order to calculate modified value-at-risk. It is modified because there is non-normal distribution in data of a portfolio. The main assumption in Cornish-Fisher Model is the portfolio is non-normally distributed. The calculation of value-at-risk is the same as equation (12). On the other hand, the equation 12 differs in Cornish-Fisher model with its way of finding critical z-value. Equation (23) cannot be used. Instead of it, equation (13) is utilized.

zcf = zc + ^

 + ^{ }

 + ^{ }

 (13) Where

zc: critical value in normally-distributed portfolio with given confidence level S: Skewness of the portfolio

K: Kurtosis of the portfolio

zcf: critical value in cornish-fisher expansion model

Then, equation (14) has been applied with found critical z-values in equation (13). The following table 15 indicates the summary of value-at-risks based on Cornish-Fisher model in both Danish and Turkish mutual funds.

Table 15 Summary of VAR Based on Cornish-Fisher Model in Danish Mutual Funds and Turkish Mutual Funds

Average Cornish-Fisher (mVaR)

Danish Mutual Funds

Turkish Mutual Funds

99% Significance Level -5.96% -3.49%

95% Significance Level -1.89% -1.66%

90% Significance Level -0.80% -1.07%

Source: Own Work

As expected, value-at-risks in 1% significance level are the highest and Danish mutual funds are riskier than Turkish mutual funds. However, the VARs in Cornish-Fisher model with 5% and 10%

significance levels, VARs are less than in analytical (parametric) value-at-risk model. In my opinion, the reason is non-normal distribution characteristics of data with wider significance level (5% and 10%). In addition, appendix 5 gives better view of how modified VARs are distributed in described 3 significance levels.

Cornish-Fisher Model is not appropriate for portfolios which indicate normal distribution characteristics. However, as seen in summary of descriptive statistics (Table-4), the kurtosis ratios are quite high and meaning that there is a possibility of non-normal distribution. Therefore, modified value-at-risks have to be calculated for both Danish and Turkish mutual funds with Cornish-Fisher Model.

4.4.3. Risk-Adjusted Performance (Based on mVAR)

Modified VAR based risk-adjustment performance is modified Sharpe ratio. It is modified because instead of standard deviation of the portfolio, modified VAR is used. Modified value-at-risks are found with the help of Cornish-Fisher Model.

The excess return of a mutual fund (return-risk free rate of return) is divided by modified value-at-risk. Gregoriou & Guiete (2003) claim that modified Sharpe ratio gives lower result than classical Sharpe ratio due to consideration of tail risk (skewness and kurtosis) of the portfolio.

In this study, modified value-at-risks at 5% significance level and 5-years data are taken into account due to its clarity and general acceptance as basis in finance. The results for modified Sharpe ratio are given in Table 16 below:

Table 16 Average Modified Sharpe Ratios and Number of Positive and Negative Values in both Danish and Turkish Mutual Funds

Average Values Modified

Sharpe Ratios at 5% Sign. Level Positive Negative

Danish Mutual Funds 2.17 55 5

Turkish Mutual Funds -0.90 10 55

Source: Own Work

The results are extremely different than classical Sharpe ratio calculation. As seen in Table 5, 5-yeats Sharpe ratios in Danish and Turkish mutual funds are -6.52 and -0.55 while they are 2.17 and -0.90, respectively. Unlike Gregoriou & Guieye (2003)’s prediction, although there is not a vast difference, modified Sharpe ratio is lower than classical one in Turkish mutual funds. It is because some Turkish mutual fund samples have really high kurtosis and skewness values. This fact changes modified value-at-risks and so modified Sharpe ratios.

As seen in appendix 6, scatter plot graph indicates that there is a distinction between Danish and Turkish mutual funds in terms of settlement of the results of modified Sharpe ratios. While Danish funds are mainly above zero percent, majority of Turkish funds are in the negative side of the graph.

4.4.4. Equally Weighted Moving Average (EWMA) in Rolling Basis VAR

In EWMA, deviation from the mean of a portfolio is equal to return of the portfolio due to assumption of zero expected return. In addition, constant numbers of observations (rolling past return series) have been chosen in order to see which one gives the best result for Root Mean Squared Error. In EWMA, the estimations of volatility  have been calculated with given equation as the following:

_^ = _^ (16) in which,

rt = Rt – E(Rt) (17)

_^: Estimated volatility (variance) of the portfolio

rt:Deviation from the mean of the portfolio Rt:Return of the portfolio

N: number of observations

E (Rt): expected return of the portfolio (assumed to be zero).

Assumed numbers of observations are 300, 600, 900 and 1200. The observations have descending order meaning that the first selected observation is the return in last day of 2010 and next one is one-day before. Estimation of volatility of a mutual fund is found for each sample sizes with equation (16). The results in averages are given as the following Table 17:

Table 17 EWMA of Danish Mutual Funds and Turkish Mutual Funds in Averages with Given Number of Observations

Danish Mutual Funds Turkish Mutual Funds EWMA (300 Samples) 0.000154324 0.000097481 EWMA (600 Samples) 0.000342662 0.000171998 EWMA (900 Samples) 0.000304028 0.000169441 EWMA (1200 Samples) 0.000262257 0.000163619 *60 Danish Mutual Funds and 65 Turkish Mutual Funds, Source: Own Work

It is seen that equally weighted moving average of 300 samples have the smallest value.

Theoretically, it should give the needed number of observation. However, further study such as root mean squared error is needed to proof.

4.4.4. 1. Root Mean Squared Deviation

Root mean squared deviation or error (RMSD or RMSE) is used to measure goodness of fit of a model to its estimation. The purpose of use of RMSE in the study is to find the best pick-up method in terms of observation with the help of values found in equally weighted moving average model. In other words, finding the most appropriate number of observations can be provided. A rigid amount of observation may cause misdirection and skip of some crucial information. In same manner,

keeping sample size too broad may result in very old and unrelated information in the tests. The formula of RMSE can be written as the following:

RMSD =   _{&$ &} _&^ (19)

Where

n: The number of samples planned to be used in the analysis

_&: Deviation from the mean of a mutual fund in time

_&: Estimated Volatility of a mutual fund in time t

The estimated volatility of a mutual fund (_&) is the value found in equally weighted moving average. The calculation is done for 4 choices of number of selection (300, 600, 900, and 1200).

The results are as the following in Table 18:

Table 18 Root Mean Squared Deviation (Error) Values for both Danish Mutual Funds and Turkish Mutual Funds in Selected Number of Observations

RMSE (Root Mean Squared Error)

Danish Mutual Funds Turkish Mutual Funds

1200 obs. 0.001171562 0.000501499

900 obs. 0.001312238 0.000559868

600 obs. 0.001148587 0.000451884

300 obs. 0.000439174 0.000448397

Source: Own Work

Because the lowest root mean squared error estimation is the best method for picking up, the choice of 300 observations has been selected as rolling past return series.

4.4.5. Log-Likelihood Ratio

The concept of risk model is very important to a portfolio in terms of the risks involved in parameters of it. Therefore, model risk should be deeply analyzed and tested. This aim can be satisfied with “back-testing” and in back-testing, the actual returns of a mutual fund and VAR

amounts are going to be compared (Jing, Zong-Fei & Kai, 2006). VAR amounts that found in analytical (parametric) VAR sections are used in back-testing. Kupiec’s log-likelihood ratio is chosen as back-testing model. The formula of the model is:

LR = -2ln [  '^()'⁾] +2ln [  '^()'⁾] (20) Where

p: p-value (significance level, 1-confidence interval)

p: ⁾₍ (the fraction of number of failure and total number of observations in a portfolio)

T: Total number of observations in a portfolio

f: Number of failures (if an actual return value of a portfolio exceeds parametric VAR amount of a portfolio, it fails) in a portfolio

Log-Likelihood ratio is found for 3 significance levels (1%, 5%, 10%) in purpose of comparison for both Danish and Turkish mutual funds. In each significance level, there are a number of exceptions of VARs which exceeds the significance level of number of total observations. In order to make clearer, the following table has been created:

Table 19 Summary of Number of Observations and Their Acceptance Level in Danish and Turkish Mutual Funds

# of Acceptance (Rounded)

# of Obs. 1% Sign. Level 5% Sign. Level 10% Sign. Level

Danish Mutual Funds 1233 12 62 123

Turkish Mutual Funds 1257 13 63 126

Source: Own Work

Numbers of observations of both countries’ mutual funds have different numbers due to difference in official holidays in those countries but they have same time period of 5-years in total. If an actual return value of a mutual fund is more than VAR amount in a given significance level, there is a failure and a certain level of risk in our financial model we applied. In other words, the fraction of number of failure to total number of observations in a mutual fund should be less than or equal to

H0: p K p

H% p > p

If the null hypothesis cannot be rejected, number of actual returns which exceeds given analytical VAR in each mutual fund is in exception. The following table gives results of the test of the null hypothesis generated.

Table 20 Number of Rejected and Not Rejected Null Hypothesis in Danish and Turkish Mutual Funds

1 % sign. Level 5 % sign. Level 10 % sign. Level

H0 H1 H0 H1 H0 H1

Danish Mutual

Funds 6 54 57 3 59 1

Turkish Mutual

Funds 29 36 62 3 65 0

*60 Danish Mutual Funds and 65 Turkish Mutual Funds, Source: Own Work

In 1% of significance level, number of Turkish mutual funds whose actual returns exceeds calculated VAR amount (failure rate) is more than Danish mutual funds. On the other hand, it is almost same in 5% and 10% significance levels. The explanatory level of failure rate is not enough.

Therefore, the log-likelihood ratios should be calculated according to equation (20) in this paper.

The summary of log-likelihood formula application is below:

Table 21 Average Log-Likelihood Ratios in Danish Mutual Funds and Turkish Mutual Funds Log-Likelihood Ratio (in Average)

Danish Mutual Funds

Turkish Mutual Funds 99% Confidence Level 3.490210008 1.504374957 95% Confidence Level 9.545763683 5.001844134 90% Confidence Level 25.25073191 11.92229706 *60 Danish Mutual Funds and 65 Turkish Mutual Funds, Source: Own Work

Table 21 gives better picture of view about exceeding VAR of actual returns in mutual funds. The larger log-likelihood ratio, the larger amount of actual returns exceed given VAR amount for each mutual fund. In all 3 significance levels, Danish mutual funds have riskier portfolios.

5. COMPARISON

The comparison of Danish and Turkish mutual funds is done in order to see the general picture of their portfolios. In other words, seeing which country has higher amount of risk or better fund performance. The comparison gives us signals about the overall view of money market of Denmark and Turkey in comparison point of view.

First, descriptive statistics of both Danish and Turkish mutual funds should be compared by focusing on skewness and kurtosis values. Higher kurtosis is an indicator of fat tails in a portfolio (Gregoriou, 2003). And skewness shows how the data is distributed in the portfolio, if a portfolio either has left-skewed or right-skewed portfolio, it means there may be a non-normal distribution in the portfolio. Therefore, modified value-at-risk model with the help of Cornish-Fisher model can be needed to investigate the risk of the portfolio. Furthermore, the test of Augmented Dickey-Fuller test is done in order to see whether there is a unit root or not which is a sign of stationary problem (Cheung & Lai, 1998). The augmented version of Dickey-Fuller test is chosen because there is a large amount of data in each mutual fund which is 5 years daily returns of each of them. If there is a stationary problem, this means that the old data in a portfolio may not reflect the change in total.

As indicated in Table 3, in 15 Danish mutual funds out of 60 and 64 Turkish mutual funds out of 65, with 5% significance level, the null hypothesis that says there is a unit root problem (stationary) is rejected. In other words, in 98.4% of Turkish mutual funds and 25% of Danish mutual funds, there is enough significance evidence that there is a stationary problem (See appendix 9). On the other hand, Turkish mutual funds may have some problems with unit root problem. Almost all mutual funds selected in Turkish mutual fund industry shows a feature of stationary. The old data does not reflect the performance of overall performance of the mutual fund. Because of that we can say that selecting a large time span for the analysis may ruin the performance and risk indicators values in Turkish mutual funds. In addition, not all Danish mutual funds have non stationary characteristics. 15 of them have stationary problem but still far better than Turkish mutual funds.

This can be attributed to the economic fluctuations and rapidly changing financial environment in Turkey.

Second, performance indicators are going to be compared in order to see in which country, mutual

Third, risk indicators are going to be compared. In comparison, the distribution of data will be taken into account. To do so, the tests for both normal distributed and non-normal distributed portfolio are formulated. The reason is there are 60 funds in Danish mutual funds and 65 funds in Turkish mutual funds. Each fund has its own characteristic, in order to compare, they should be treated equally.