Value at Risk Requires a Horizon and a Return Distribution
Summary
The discussion explains that Value at Risk (VaR) is defined for a stated time horizon and confidence level; without a horizon, a VaR figure is incomplete. It outlines a common parametric approach: estimate portfolio return volatility from historical data, an implied variance-covariance matrix, or a combination, then scale volatility to the chosen horizon. A confidence-level quantile is applied to estimate the loss threshold, with some formulations also including expected return.
The answers note that a single asset needs an estimated mean and standard deviation, while a portfolio’s variance depends on the covariance of its holdings. They also mention a theoretical long-run VaR when returns have a well-defined long-run distribution, and a maximum-loss concept for certain mean-reverting returns. These are qualifications rather than practical substitutes for specifying assumptions. The examples describe a normal-distribution method and square-root-of-time scaling; actual return distributions, changing volatility, and model error can limit those approximations. VaR describes a quantile loss threshold, not the size of losses beyond that threshold.
Key ideas
- A VaR estimate needs both a confidence level and a time horizon.
- A parametric VaR approach estimates portfolio volatility and applies a distribution quantile.
- Portfolio variance depends on the covariance among holdings; a single asset has no cross-asset covariance term.
- Volatility can be scaled across horizons using a square-root-of-time assumption.
- Long-run VaR is only meaningful when returns have a suitable stable long-run distribution.
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Full text
# How can I calculate Value at Risk?
# How can I calculate Value at Risk?
Is it possible calculate Value at Risk on an asset without a time horizon?
What kind of variables do you need? Variables that are on the table are value, standard deviation, beta, market return, risk free rate, and its realized return. And, of course, confidence level.
Can someone give me an example of necessary variables for this kind of calculation with a 95% confidence level?
## Answer by user12348 (score 2)
https://quant.stackexchange.com/a/11131
OP is asking if VaR can be calculated without time horizon. @steinbitur has given you the variables to be used in VaR etc. You can google and find links to VaR on wiki. Basically You cannot calculate VaR without time horizon. VaR is defined as worst possible loss in a given time horizon with given level of confidence.
It is possible that your portfolio returns are mean reverting, in which case, you may just have a maximum loss, regardless of time horizon, then it reverts. You could define a Maximum VaR loss theoretically. In reality the markets are random and to think you have such ideal portfolio will fly in the face of risk management. The regulators and Basel all have time horizon associated with it.
## Answer by steinbitur (score 1)
https://quant.stackexchange.com/a/11125
First you decide what timeframe your VaR has to have. Next you decide what level of VaR you are going to have. Many examples use 1 day 99%, but you could use 1 month 95% VaR or whatever as long as it makes sense. When you have chosen those parameters you need to find the standard deviation of your portfolio. The standard deviation is the square root of the portfolio variance which you can see how is calculated here (among other things, e.g.):
http://en.wikipedia.org/wiki/Modern_portfolio_theory
You can either use historical data (which most examples do) or a market implied variance covariance matrix or a hybrid of those things. First I assume the parameters mentioned in your question are not given and you are going to use historical data. Maybe you need modify the data so it's historical returns for a specific time series. Then you can find the variance covariance matrix for your historical data. When you have found the portfolio variance for the data you need to scale for time. So, if your data is monthly and you want 1 day VaR you need to devide with the square root of 21 because that is the average amount of business days in one month, then you have a proxy of the daily standard deviation of your portfolio. You multiply with the square root of time if your data is daily and you want a monthly VaR. The next thing you do is to look up the standard deviate for your choice of VaR level. For example if you want 95% VaR the standard deviate for that is when $\alpha = 5\%$ then the standard deviate is 1.96 and there you have it the VaR: $$VaR = - 1.96 * \sigma_p$$ and some people use the the expected return of the portfolio for the timeframe also in the VaR calculation i.e. $$VaR = \mu_p - 1.96*\sigma_p$$ In your case your portfolio is just one asset an there is there for no covariance matrix, just the expected return $\mu$ and the standard deviation $\sigma$. Usually the default timeframe for the standard deviation is anually. To get a daily VaR from an annual standard deviation you need to devide the annual $\sigma$ with $\sqrt{252}$ to get a daily $\sigma$.
## Answer by user7963 (score 0)
https://quant.stackexchange.com/a/11127
Value at risk means the considering investing in a risky asset.Value at risk is a tool for the risk assessment.Especially in the area of finance and business value of risk is calculated.This is always used in the short form that is VaR or it is abbreviated is a risk measure.There are different approaches for calculating the value at risk.There is a formula for calculating the delta-normal method given below:- VaR=alpha*sigma*Exposure
In the above formula Alpha indicates the normal deviation.Sigma indicates the volatility of the underlying assets. Exposure is the particular risk factor.
## Answer by Drew (score 0)
https://quant.stackexchange.com/a/11139
If the variable of interest shows ergodicity (I.e., exhibits a well - defined long run distribution), one could consider a "long run" VaR in principle. Otherwise, I agree that a time horizon must be specified.
For the ergodic case, the information required to calculate the VaR is a characterization of the long run distribution.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.