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Calculating Multi-Day Value at Risk from Mean and Volatility

Article Quant Q&A · Author: Ankita Datta
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Summary

The document explains a parametric calculation of value at risk over a multi-day horizon using a portfolio’s average daily return and standard deviation. Its formula scales the mean linearly with time and volatility with the square root of time, then combines both with a confidence-level quantile to express the loss threshold as a positive VaR amount.

The example asks for five-day VaR at 99% confidence, given a portfolio value and daily return statistics, but the answer does not work through the arithmetic or provide a numeric result. It emphasizes that the drift and volatility terms must use matching time horizons and that VaR is conventionally stated as a positive loss despite describing a lower-tail outcome. This is a simplified parametric method; the excerpt does not discuss distributional assumptions, estimation uncertainty, or alternatives for non-normal returns.

Key ideas

  • Scale the mean return linearly with the number of days in the horizon.
  • Scale standard deviation by the square root of the horizon length.
  • Combine drift, volatility, and the confidence quantile to estimate VaR.
  • Report VaR as a positive loss amount, while interpreting it as a lower-tail outcome.

Tags

StatisticsRisk managementVolatility
Full text
# How to calculate VaR given mean and sd?


# How to calculate VaR given mean and sd?












Sarah manages a hedge fund with a portfolio valued at \$2,000,000. The portfolio's daily returns have a standard deviation of \$3,000 and an average daily return of \$1,200. Calculate the five-day VAR at a 99% confidence level for Sarah's portfolio.

## Answer by KaiSqDist (score 1, accepted)

https://quant.stackexchange.com/a/77427

It is really simple. The formula is just:

$VaR_{\alpha,T} = -\mu T + Z_{\alpha} \sigma \sqrt{T}$

Take note the time horizons should match between the drift and the vol terms. Also VaR is usually represented positively despite being a loss. Therefore, the PnL distribution is "reversed".

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.