After the initial look where the Volatility portfolio appears to be a better choice at first glance, we examine the two portfolios regarding the regulation that the holder of the portfolios face. In the next plots, the VaR at 99,5% confidence has been calculated for both portfolios, as it is the amount of “Own Funds” that the company must keep in order to hold the portfolio. In both plots, the standard formula of 39% (+/- up to 10%
depending on the symmetric adjustment) has been plotted to show the difference. The first plot shows the Volatility portfolio against the standard model, and the second plot shows the Nominal against the standard model:
What is noticeable right away is the large gap up to the standard model from the VaR for both portfolios. This gap is the amount of “Own Funds” the company could freely move to other areas if needed if they applied an internal model estimated the same way as in this paper. It is also noticeable that the volatility, apart from a couple of extreme reads, behaves very controlled for the Volatility model. This is an important feature for an investment manager, as a portfolio with the volatility like the Nominal portfolio is very susceptible to big swings when there is economic turmoil. The look of the two VaR measures comes at no surprise though, as a lot more of the capital weight is placed in bonds with the volatility-weighted portfolio.
To examine the validity of the two VaR series, a backtest on the return data of the total value for the portfolios was conducted. The specific test is Kupiec’s Unconditional
Coverage Test, that is a binomial test that estimates the probability of seeing the amount of Actual Exceedances compared to the Expected number. The expected number for a VaR of 99,5%, is one exceedance every 200 days, which is equivalent to 22 times in the whole data set. With the VaR for the Nominal portfolio reading 34 Actual Exceedances, this is a sign that the model has underestimated the VaR for this portfolio. This also comes to show as the p-value with 95% confidence rejects the hypothesis of the model delivering the right amount of Exceedances. The Volatility portfolio however does not reject the hypothesis, as it even underscores the VaR a little bit as can be seen in the following:
Backtest using Kupiec’s Unconditional Coverage Test
Test Statistics Nom Portfolio VaR Vol Portfolio VaR
Expected 22 22
Actual 34 17
Test Statistic 4.8900 1.5810
Critical Value 7.8794 7.8794
VaR Level 5% 5%
p-value 0.0270* 0.2086
H0-hypothesis "Correct Exceedances" "Correct Exceedances"
Decision "Reject H0" "Fail to Reject H0"
Lastly to determine which model empirically performs best, the Sharpe Ratio was calculated for both models. When the Sharpe Ratio is calculated on daily data, the returns for a daily read is sometimes negative or at least below the risk free rate. This will lead to negative Sharpe Ratios, which does not make much sense regarding the comparison of performance. However with daily data, you do get an idea of how the Sharpe ratio reacts over time, and therefore the Sharpe ratios have both been plotted as time series below, along with a plot of the difference between the two portfolios Sharpe Ratios:
For the Nominal portfolio, the Sharpe Ratio appears to be normally distributed. The same goes for most of the time series for the Volatility portfolio, although it does show a less volatile read of Sharpe Ratio. Even the plot of the difference appears to be normally
distributed, and these plots do not offer a suggestion to either one being better than the other. If one had consistently outperformed the other, the plot of differences would have been mostly on the positive side or mostly on the negative side.
To examine more in detail the two Sharpe Ratio time series, a Welch t-test was made to attain the mean values of both series and further test for difference among them.
Welch Two Sample t-test
Sharpe Ratios Nom PF Sharpe Ratios Vol PF sample mean = 0.04567946 sample mean = 0.0562308
95 percent confidence interval for true difference in mean
0.02274696 -0.04384977
t = -0.62115 p-value = 0.5345
H0-Hypothesis: “True Difference in Means is Equal to 0”
Alternative Hypothesis: “True Difference in Means is not Equal to 0”
The two means revealed to be very close to one another, with the Nominal portfolio having a mean Sharpe Ratio of 4,6 and the Volatility portfolio having a mean of 5,6. This does indicate that the Volatility portfolio indeed does offer more risk-adjusted return per volatility unit and thereby is a better choice. However after conducting the Two Sample t-test, the p-value of .53 leaves us unable to reject the null-hypothesis that the true means are equal. And it can therefore not be concluded with statistical significance that the Volatility portfolio delivers better Sharpe Ratios. With 95% confidence
however, we can say that the true difference in the means lies between 0.02 in favor of
As this was the last of the performance analytical tests, the following chapter will offer perspective on these results along with a discussion and a conclusion of the findings of the paper.