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INTRODUCTION
Value-at-Risk (VaR) is a measure of market risk that analyzes the sensitivity of the portfolio to market changes and solves for the value using probability and statistics. VaR is considered the single best known measure to assess market risk and Basel II uses VaR for the international banking community to set the minimum capital requirements for market risk. Tradionally, the banking industry uses parametric equations to approximate for VaR.
Now with the VaR/T applet comparing parametric VaR with historical VaR is easy and the inaccuracies of parametric VaR become evident. Currently, all methods of VaR have significant limitations when dealing with trending markets and financial crisis. Multiday parametric VaR is defined as, the square root of the number of days multiplied by VaR. Given this equation, Basel II has set the minimum capital requirements for market risk to: 3 times VaR multiplied by the square root of 10.
The newly introduced VaR Trending (VaRT) helps solve VaR's multiday limitations by offering a more accurate method for quantifying trending markets and liquidity risk. Combine this with historical VaR, with more than one year's data, and risk management analysis is further optimized.
The VaR/T applet opens in a new web browser expanded to the size of your screen. The Java applet is optimized for speed and does not communicate with the browser except during loading. The applet opens as a Tab Panel starting with the Histogram panel followed by the Price Chart panel and Data panel. The price chart and data panels are provided for data visualization and the bulk of the analysis is done in the histogram panel. The data employed is the Dow Jones Industrials since 1928 to present. The Dow was chosen for it's universal recognition as an accurate representation of the financial markets.
The first panel has a histogram of the Dow's daily percentage changes for the last year, one year being the industry standard for VaR. The dates for the analysis can be changed to any dates going back to 1928. Notice the current leptokurtic shape of the histogram, higher peak and fatter tails; to make the histogram normally distributed, more than one year's data needs to be used.
The analysis can also be changed to either using Open, High, Low or Close by clicking the radio buttons above the Statistical Analysis. The industry standard is to use daily closes. The statistical analysis, historical analysis and VaR Trending (VaRT) are solved for both as percentages and today's dollars. The analysis includes both parametric VaR and historical VaR, and analyses for major market moves and illiquidity using VaRT. The dates of the Worst Case (WC) for both historical VaR and VaRT are also provided.
The price chart is a standard candlestick open, high, low, close for the Dow Jones Industrials from 1928 to present. By placing your mouse cursor inside the chart, the individual daily open, high, low, close and volume are displayed making it simple to find the data for a particular date. A scroll bar above the price chart displays the left most date and the right most date shown on the chart which makes it easy to scroll to any particular period. The price chart is an invaluable tool for visualizing price movement.
The data spreadsheet is self-explanatory with the daily data starting in October 1, 1928 to present.
The Histogram is the graphic representation of the daily percentage change for either the open, high, low or close. Risk Management uses percentages rather than absolute dollars to make past data relevant to present day. Arbitrage pricing causes markets to move relative to each other in the form of percentage moves, making percentages the common denominator when comparing different securities.
The percentage R is the daily price change , the current price Pc minus the previous day Pc-1, divided by the previous day Pc-1.
Along the X-axis each of the colored bars represents a one-half of one percent band up to ± 4.5%.
The green bars represent positive daily percentage changes and the red bars represent negative
daily percentage changes. Along the Y-axis the height of each bar represents the percentage of total number of days
within the 0.5% band, as all the Y-axis values combined equals 100%.
Above the histogram is the title of the securities analyzed, in this case, the Dow Industrials for either the open, high, low or close, for this example "CLOSES" was used. One of the features making this demonstration so useful is the ability to analyze any date from 1928 to present.
As mentioned before, the current leptokurtic shape, higher peak, fatter tail of the histogram will become more normally distributed when four years are analyzed. More than one year's data should be used when solving for VaR to achieve more normally distributed results. Some institutions maintain excellent credit ratings by valuing their portfolio not with 99% VaR which allows for two to three major losses in a year but rather with 99.9% VaR which allows for one major loss every four years.
Currently the date format is very strict, for example May 4th, 2000 is written "5/4/2000". Error checking for date formatting will be added but until then please enter the date in the proper format or the applet will not operate properly.
Above the Statistical Analysis are radio buttons enabling analysis for either the Open, High, Low or Close. The industry standard is to use daily closes.
The Statistical Analysis uses the industry standard equations for Mean, Standard Deviation and
parametric VaR. The historical VaR and VaRT rely on computing actual price data to solve for the 99% and 99.9%
probability numbers. This is in direct contrast to using statistical analysis or Monte Carlo probability simulations
to approximate VaR.
It is interesting to compare parametric and historical VaR using the applet to see the inaccuracy of
parametric VaR. It is important to note that if financial modeling is to be used to optimize total return
then any inaccuracy will degrade the value of the optimization.
VaRT is used to analyze for major market moves and account for illiquidity. VaRT is offered as a more exact alternative to the Basel II recommendation of using three times the square root of 10 to account for financial shocks and crisis. The dates of the Worst Case (WC) for both historical VaR and VaRT are provided for reference.
The results provided by the analysis are both offered as a percentage and in today's dollars (Current price x %). Arbitrage pricing causes markets to move relative to each other as a percentage, making percentages essential when comparing different securities and their covarience. Today's dollars are provided to account for the exact dollar amount of the portfolio.
The first part of the Statistical Analysis provides the industry standard for Mean, Standard Deviation and parametric VaR. The Mean µ is the summation of a set of results R, divided by the number of results N.
The Standard Deviation measures the degree in which the results vary from the mean. A large standard deviation indicates that there is large variance in the distribution. A small standard deviation indicates that the results are bunched tightly around the distribution's mean. The standard deviation is simple the square root of variance. To solve for variance, divide one by the number of results (1/n) and multiply by the summation of the results minus the mean, squared (R - µ)2.
Parametric VaR assumes the results are Normally distributed and form a Bell Curve. This assumption is based on the fact that Normal distribution is very common in nature. It is described by the Central Limit Theorem, as a large amount of independent, identically distributed, random numbers combined together and whose outcome will tend to be Normally distributed. Parametric VaR uses a covariance matrix which assumes that correlations between risk factors are stable and constant over time. The equation for the Normal Distribution curve, though elegant, assume Skew is zero and the Kurtosis is always three, which makes solving for the probability p[y] overly simplified.
Using this Normal distribution equation, the 99%ile is approximately 2.32 times the standard deviation. Typically parametric VaR is used because it is 100 to 1000 times faster to calculate than historical VaR, however this applet is provided to show it is possible to instantaneously solve for historical VaR.
Historical VaR relies on computational power rather than mathematical approximations to solve for VaR. Faster microprocessors and more efficient algorithms account for the applet's speed in solving for historical VaR. Accuracy is increased with more data and using four years or more is recommended to begin to better account for 99.9% VaR and to achieve a normal distribution curve. Accuracy is the strength of historical VaR and given the newly achieved speeds in computing historical VaR, real-time risk / reward optimization is now possible over either a private network or the Internet.
The Worst Case (WC) date is provided for historical VaR for the dates analyzed, as a reference date to view on the price chart and to be identified as the actual worst case scenario.
VaR Trending (VaRT) helps solve VaR's multiday limitations by offering a more accurate method for quantifying trending markets and liquidity risk. VaRT calculates for illiquid periods by using historical data to compute for negatively trending markets. VaRT searches for negative trends and solves for the worst 1%, 0.1% and worst case. Illiquidity is accounted for during a downtrend and liquidity is returned at the end of the downtrend with a higher daily market. Though some markets are very liquid, allowing disciplined traders to reduce positions, many times when markets trend during financial dis-equilibrium, even very liquid markets lose their liquidity. VaRT is meant to account for these downtrends and summarizes the results using the same percentile benchmarks as daily VaR.
The Worst Case (WC) date is provided for VaRT for the dates analyzed, as a reference date to view on the price chart and to be identified as the actual worst case scenario.
The Price Chart are daily Candle Sticks going back to 1928
3/28/2003
7/7/2004
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- A scroll bar on top of the chart enables searching through out all the dates back to 1928
- Place the cursor in the chart to display that day's open, high, low, close and volume
The Data is a speadsheet of the daily Open, High, Low, Closes and Volume for the
Dow Jones since 1928.
- A scroll bar enables searching through out all the data back to 1928
