What is Value-at-Risk (VaR)?
Value-at-Risk, or VaR, is a measure of the potential losses and riskiness of a given portfolio, or a single investment. It’s a statistical measure that quantifies the possible loss that could occur in a portfolio over a certain timeframe, essentially the probability of loss. It’s widely used in the world of investment management, with many fund managers, risk managers, and investors all using it to improve and optimize their portfolios.
How to compute/measure VaR?
There are 3 components in VaR of a portfolio or an asset: the timeframe, the potential loss, and the confidence interval. For example, if a portfolio has a VaR of $100,000 at a two-day, 90% confidence interval, it indicates that there is a 10% chance that the portfolio will lose more than 100,000 dollars over the two-day interval. Similarly, if the VaR is negative, it indicates that the portfolio has a chance of gaining more than the specified amount over a specific timeframe.
There are 3 main ways to compute the VaR measure: the historical method, the parametric method, and the Monte Carlo method.
Historical method:
Source: https://www.investopedia.com/articles/04/092904.asp
The simplest method for computing this measure, the historical method uses past returns to calculate the VaR for each day.
To do the calculation, gather historical prices of the asset you’re analyzing, such as prices over the past 100 days.
Then, calculate the daily returns based on the historical prices. The daily return is calculated as:
Then, sort the returns from best to worst. Choose the percentile of the sorted data that corresponds to your confidence level, and select the return on that day. For a 90% confidence interval, the 10th percentile contains the corresponding return. In the case that you have collected data over 100 days, the 10th percentile would simply be 10% * 100 = 10. This means that the return on the 10th value of the sorted list is the 10th percentile of returns. Multiply this return by your current portfolio value to get the VaR. If the 10th percentile in the sorted list is -5%, and your portfolio value is 10,000 dollars, the VaR for the next day for your portfolio at a 90% confidence interval is 500 dollars.
Parametric Method:
Source: https://quantpedia.com/an-introduction-to-value-at-risk-methodologies/
Another method of calculating the VaR measure is the parametric method. This method assumes that the returns of the asset obey a normal distribution. Generally, this method is suited for situations where statistical measures(e.g., mean, standard deviation) of the data are known, and the sample size is large. Here’s how we calculate VaR:
First, similar to the historical method, calculate the daily returns for every day. Then, find the mean and standard deviation of those returns.
The mean is given by:
The standard deviation is given by:
Choose a confidence level (90%, 95%, 99%, etc), and find the corresponding z-score of that confidence level from the normal distribution.
% Z-score formula:
% Value at Risk (VaR) formula:
For example, suppose our portfolio has a value of 100,000 dollars. Suppose the mean for the returns is 5%, and the standard deviation is 0.75%. If we choose the 95% confidence interval, that corresponds to a z-score of approximately 1.96.
So our VaR is given by:
$100,000 * (0.05 + 1.96*0.0075) = $6470.
This means that there is a 5% chance that we can expect to lose more than 6470 dollars during the next trading day.
Monte Carlo Method:
Source: https://www.simtrade.fr/blog_simtrade/monte-carlo-simulation-method-var-calculation/
The last method used to calculate the VaR measure is the Monte Carlo method. It’s extremely versatile and powerful, and is able to account for a wide range of scenarios.
To calculate the VaR using the Monte Carlo method, gather historical data on returns for the assets in the portfolio. Fit the data into a normal distribution, and record the mean and standard deviation.
Next, choose the confidence interval for the VaR (95%, 99%, 90%, etc.). Generate random returns for the assets in the portfolio, and using the randomly simulated returns, predict the future portfolio value by applying the statistical properties of the randomly-simulated returns to the current portfolio.
Repeatedly simulate returns at a larger number of times (e.g. 10,000 times). Once the outcomes have been simulated, sort them from worst to best, and select the return that corresponds to the confidence interval chosen.
The formula for the random return of an asset is given by:
, where Z is a random number chosen from the normal distribution N(0,1).
For example, if we select a 90% confidence interval for our daily VaR, run 100,000 simulated outcomes, fit it into a normal distribution, we now choose the 10th percentile (corresponding to the 90% confidence interval) to obtain the VaR. If the 10th percentile corresponds to a 15% loss, and our portfolio is valued at 100,000 dollars, there is a 10% chance we lose more than 15,000 dollars the next day.
Extension: Expected Shortfall:
The Expected Shortfall is an extension of the VaR measure. It’s calculated by averaging the returns that are worse than the confidence interval of the VaR. For example, if we calculate the VaR at a 90% confidence interval, the Expected Shortfall is calculated by averaging the worst 10% of returns. It’s a useful tool, and when combined with the VaR, offers a comprehensive breakdown of the risk of a portfolio or an asset.
VaR is one of the most powerful risk measures in finance, and can be extended to offer even more insight into the riskiness of a portfolio. If used correctly, one can optimize their portfolio by controlling losses.
More Resources:
- Blog on Data Science in Risk Management
- On-Demand Video Course – Quantitative Stock Trading Course Part 1: Quartz Trader™
- Live Masterclass – Quantitative Stock Trading Masterclass
About The Author
Sushant Pisupati competed in numerous finance, mathematics, and physics competitions, and has successfully translated his skills into the field quantitative finance. At Aries Profits, his goal is to help others learn about the fascinating field of quantitative trading/investing, and make this subject easy and accessible for everybody.
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