Risk Management Tools
Calculation techniques
There are plenty of approaches to the calculation of VaR and related measures. The methods that are implemented in SmartFolio are described below.
The last four methods utilize block bootstrapping algorithm.
DeltaNormal Method
This method is simplest and the most common in application. Based on the analytical model assumptions, it calculates quantile of the normal distribution with parameters and .
It corresponds to the distribution of portfolio excess growth rate equal to , where denotes the discounted portfolio wealth at time . is then calculated as .
Accordingly,
where denotes the density function of the standard normal distribution.
Unfortunately, DeltaNormal Method (DNM) is far form being precise. The main drawback of DNM is that it doesn’t take into account higher moments of the portfolio returns distribution including fat tails, which are
very common in practice and have critical impact on VaRCVaR values.
Another possible shortcoming of DNM is the assumption that portfolio returns are independent through time.
It leads to the socalled squareroot scaling law for standard deviation, which means that .
In practice it is often the case that instead of square root degree, should be used.
Final DNM weakness is its inability to account for nonlinear relationships between portfolio components, which arise when options are included in portfolio.
As a consequence, quite often DNM seriously underestimates true values for VaR and CVaR, particularly for the extreme values of , exceeding 0.95.
Empirical Distribution
Empirical Distribution approach involves the following steps:
 The array of historical portfolio excess logarithmic returns over the period to is formed. If contains more then one period, then prior implementation of block bootstrapping algorithm must
increase accuracy.
 The obtained array is sorted and the worst values are extracted.
 The best and the average values of the selected worst part are calculated.
 VaR and CVaR are then obtained by transformation of respective values to represent simple returns with the opposite sign using relationship.
If homogeneous historical data of virtually unlimited length was available, then the empirical distribution approach would be ideally suited for the calculation of VaR and CVaR. It accounts for both higher moments
of portfolio returns distribution and nonlinear interdependencies. In reality its use is limited to portfolios, whose components are traded for time long enough (at least, 57 years for daily database).
Implied Normal Distribution
Deltanormal method utilizes the assumption of squareroot growth in portfolio standard deviation as a function of . On the contrary, implied normal distribution approach uses the unique estimate of
for each value of . For this purpose, analogously to empirical distribution approach, the array of historical excess logarithmic returns over the period is formed using block bootstrapping algorithm. Then the sample estimate is obtained and inserted in formulas for deltanormal method.
While the implied normal distribution approach doesn’t assume the squareroot scaling law in standard deviation, it still suffers from two residuary drawbacks, peculiar to deltanormal method: the inability to
account for higher moments of portfolio returns distribution and the nonlinear relationships among portfolio components.
Student’s tDistribution
There is much evidence, coming from recent publications in financial math, that Student’s tdistribution delivers quite satisfactory fit to a wide range of financial assets including stocks, commodities and
currencies. Its attractive feature is a power law of tails behavior, which makes tdistribution an appealing alternative to normal distribution thanks to positive kurtosis excess.
Figure 2. Student’s density vs. Normal density
As before, the VaRCVaR calculation is anticipated with formation of an array of historical excess logarithmic returns over the period by means of block bootstrapping algorithm.
Analytical formulas for VaR and CVaR under the assumption of noncentral Student’s tdistribution with possibly noninteger degrees of freedom are obtained in [Andreev, Kanto; 2004]. Corresponding density
function is defined by the following expression:
where is location parameter, is dispersion parameter and denotes degrees of freedom.
In this case quantity is directly related to kurtosis. Corresponding estimate has the following form:
where is the sample estimate for kurtosis excess (it is assumed that ). Corresponding estimates for and are and respectively.
Then , where and stands for inverse Student’s tdistribution function with parameters , and . Accordingly , where
In the latter expression denotes Student’s tdistribution density function and .
CornishFisher Expansion
CornishFisher Expansion approximates the quantiles of an arbitrary distribution with known moments in terms of quantiles of the standard normal distribution.^{1} The main advantages of applying CornishFisher
expansion in calculation of VaRCVaR are speed and ability to account not only for fat tails as Student’s tdistribution does, but also for asymmetry in returns, measured with skewness.
Algorithm
 An array of historical excess logarithmic returns over the period is created by means of block bootstrapping algorithm.
 Based on the obtained array four moments are estimated: sample mean , sample standard deviation , sample skewness and sample kurtosis excess .
 Let denote quantile of standard normal distribution. quantile , corrected for kurtosis and skewness, is established by means of CornishFisher expansion up to 4th member (for details see [Zangari; 1996]) :

 , where and are the standard normal density and distribution function respectively.
^{1} For more details about CornishFisher expansion visit http://www.riskglossary.com/link/cornish_fisher.htm
 
