Introduction

The "VaR" package is a comprehensive Python tool for financial risk assessment, specializing in Value at Risk (VaR) and its extensions. It enables robust financial risk forecasting by incorporating methods like historical, parametric, Monte Carlo, and Parametric GARCH. It also focuses on the Probability Equivalent Level of VaR and Expected Shortfall (PELVE). The package also features backtesting capabilities, distribution fitting, and detailed plotting options for clear visualization. Designed for ease of use, it includes practical examples and is easily installable through pip. With dependencies like numpy and pandas, it's tailored for those seeking advanced risk measurement tools in finance.

Key Features

Calculate, Backtest and Plot the

• Value at Risk,
• Conditional Value at Risk (Expected Shortfall),
• Conditional Drawdown at Risk,

- Probability Equivalent Level of VaR and Expected Shortfall

with different methods, such that:

• Historical
• Parametric
• Monte Carlo
• Parametric GARCH

methods.

Examples

In this example we will show all the key features of the var package. At first we will import all necessary packages.

python from var import VaR, load_data import numpy as np

To quickly test and demonstrate the functions, the package includes a function named load_data, which by default includes the daily returns of stocks TSLA, AAPL and NFLX.

python data = load_data() print(data)

console NFLX AAPL TSLA Date 2016-06-28 0.0056 0.001725 -0.00020 2016-06-29 0.0139 0.001075 0.01012 2016-06-30 0.0057 0.002900 -0.00138 2016-07-01 0.0167 0.001000 0.02072 2016-07-05 0.0271 -0.001000 0.00850 ... ... ... 2021-06-21 -0.0464 0.020000 -0.03650 2021-06-22 0.1028 0.018500 0.05460 2021-06-23 0.0426 -0.000700 0.24570 2021-06-24 0.0010 -0.010400 0.04830 2021-06-25 -0.0177 -0.003500 -0.17710 [1258 rows x 3 columns]

The only important thing in the data preparation is that the columns contain the individual positions of the portfolio, and the rows the date. Another important point is that the column "Date" should be defined as an index, and it must also be formatted as a date.

Now we can define some weights for the individual positions and initialize the VaR class:

```python weights = np.array([0.40, 0.50, 0.10])

var = VaR(data, weights) ```

The standard confidence is at 0.01. Individual confidences can be defined with the parameter alpha:

python var = VaR(data, weights, alpha=0.05) var

console <VaR - μ: 0.05%, σ: 3.5096%, Portfolio σ: 3.511%>

The repr of the class provides the following information:

• μ : The mean return of the portfolio.
• σ : The unweighted volatility of the portfolio.
• Portfolio σ : The weighted volatility of the portfolio.

You can summarize the results of the different methods with the method:

python var.summary().round(2) # or use print(var)

console VaR(0.99) CVaR(0.99) ES(0.99) 2022-04-27 Parametric -0.13 -0.15 -0.64 Historical -0.20 -0.21 -0.64 Monte Carlo -0.13 -0.14 -0.64 Stressed Monte Carlo -0.16 -0.17 -0.64

You can access the different VaR methods by using the methods:

python var.historic()

console VaR(0.99) CVaR(0.99) 2021-06-25 -0.203479 -0.211246

python var.parametric()

console VaR(0.99) CVaR(0.99) 2021-06-25 -0.130036 -0.149269

python var.monte_carlo()

console VaR(0.99) CVaR(0.99) 2021-06-25 -0.132929 -0.13989

python var.monte_carlo(stressed=True)

console VaR(0.99) CVaR(0.99) 2021-06-25 -0.159149 -0.162875

Or access to the Conditional Drawdown at Risk with:

python var.es()

console ES(0.99) 2021-06-25 -0.636892

Fitting Distribution

Thank you @Osman Mahmud Kim for the hint!

By default the VaR class chooses a normal distribution of the data. You can override this with another distribution with:

python var = VaR(data, weights, alpha=0.05, distribution="t") var

console <VaR - μ: 0.06%, σ: 3.4461%, Portfolio σ: 3.4475%>

If the distribution is norm or t, the mean and the standard deviation of the returns are used. The degree of freedom of the t distribution is set to len(returns) - 1. You can define your own parameter with additional keyword arguments. Lets say, you want a f distribution with parameter dfn, dfd

python var = VaR(data, weights, alpha=0.05, distribution="f", dfn=10000, dfd=500) var

console <VaR - μ: 0.06%, σ: 3.4461%, Portfolio σ: 3.4475%>

You can also refine the distribution or fit new distributions with the method

python var.fit_distributions(include_other=True, verbose=True)

```console Fitting 13 distributions: 100%|██████████| 13/13 [00:00<00:00, 15.33it/s]

Best fits:

     sumsquare_error         aic          bic  kl_div  ks_statistic  \

t 156.808460 180.121004 -2598.062910 inf 0.032883 lognorm 955.113977 672.331106 -325.101442 inf 0.126492 f 957.581920 671.103512 -314.717779 inf 0.129057 norm 957.604766 666.417186 -328.962322 inf 0.127633 gamma 985.488639 655.557001 -285.717340 inf 0.138701

        ks_pvalue

t 1.287770e-01 lognorm 5.266286e-18 f 9.986222e-19 norm 2.523508e-18 gamma 1.421048e-21

Best fit:

Distribution t with parameters {'df': 1.6718900963531307, 'loc': 0.0011873373129386235, 'scale': 0.014747352151602807} ```

You can also define only one distribution with

python var.fit_distributions(distribution="gamma")

PELVE

Thank you @Osman Mahmud Kim for the hint!

PELVE is essentially a ratio or a multiplier. It tells us how to adjust the confidence level when switching from VaR to ES so that we get an equivalent measure of risk. The formula for calculating PELVE is ES{1−cɛ}(X)=V aR{1−ɛ}(X), where ε is a small number close to 0, X is a loss random variable, and c is the PELVE.

The idea here is to answer the question: if we replace VaR with ES in our risk models, how will that affect our estimated capital requirements? Will we need more capital to cover potential losses, or less?

The passage provides an example with ε set at 0.01. If the calculated PELVE is more than 2.5, that means using ES at a 97.5% confidence level (ES{0.975}) will estimate a higher risk value than using VaR at a 99% confidence level (VaR{0.99}). In other words, switching to ES would mean we'd need more capital to cover potential losses.

On the other hand, if the PELVE is less than 2.5, that means using ES at a 97.5% confidence level will estimate a lower risk value than using VaR at a 99% confidence level. That means switching to ES would indicate that we need less capital to cover potential losses.

In simple terms, PELVE is a tool to help decide what confidence level to use when switching from VaR to ES, in a way that keeps the estimated level of risk (and hence the amount of capital needed) the same.

In the current version you can calculate the PELVE with the historical or the parametric method:

python var.compute_pelve(method="p")

console (array([8.89189276]), 3.610864030825778e-05)

The first value is the PELVE and the second value is the optimization error.

Backtest

You can backtest the accuracy of each method with the method backtest and the method keys:

• 'h': VaR calculated with the historical method,
• 'p': VaR calculated with the parametric method,
• 'mc': VaR calculated with the monte carlo method,
• 'smc': VaR calculated with the stressed monte carlo method,

python bth = var.backtest(method='h')

console Backtest: Historic Method: 100%|██████████| 1008/1008 [00:03<00:00, 332.53it/s]

Evaluate the backtest results with the method evalutate

python var.evaluate(bth)

```console Amount Percent Mean Deviation STD Deviation Min Deviation \ VaR(0.99) 10 0.009921 -0.023765 0.028671 -0.003515 CVaR(0.99) 10 0.009921 -0.023407 0.028545 -0.003382 CDaR(0.99) 6 0.005952 -0.017702 0.02285 -0.001713

       Max Deviation

VaR(0.99) -0.099554 CVaR(0.99) -0.098803 CDaR(0.99) -0.060682 ```

The table contains the following information:

• Amount : Total amount of exceptions.
• Percent : Total amount of exceptions in relative to all observations (multiply this by 100 to obtain the total amount of exceptions in percent).
• Mean Deviation : The mean value of the exceptions.
• STD Deviation : The standard deviation of the exceptions.
• Min Deviation : The wort overestimation of the value.
• Max Deviation : The worst underestimation of the value.

Plot Backtest

Plot the backtest results via:

python var.var_plot(bth)

python var.cvar_plot(bth)

python var.cdar_plot(bth)

Installation

There are currently different methods to install var.

Using pip

The var package is provided on pip. You can install it with::

pip install var

Standard Python

You can also download the source code package from this repository or from pip. Unpack the file you obtained into some directory ( it can be a temporary directory) and then run::

python setup.py install

Dependencies

• Python: Python 3.7
• Packages: numpy, pandas, scipy, matplotlib, tqdm, seaborn, numba