Calculating Normal-Distribution Value at Risk from Return Moments
Summary
The document explains a simple parametric VaR calculation for normally distributed portfolio returns. At a 99% confidence level, the standard normal quantile is about 2.33, so the return standard deviation is multiplied by that factor and the expected return is subtracted. Applying the calculation to the first portfolio in the cited example gives a VaR of 0.43% using the stated one-period mean and volatility.
The question raises whether volatility should first be scaled by the square root of a ten-day horizon. The response focuses on the quantile calculation and does not explicitly resolve that horizon convention or walk through the second portfolio. The method depends on normality and on consistent units for the mean, volatility, and horizon; it does not address tail behavior or other distributional risks beyond that assumption.
Key ideas
- Under a normal-return assumption, VaR is calculated from a standard normal quantile, volatility, and expected return.
- At 99% confidence, the quantile multiplier used in the example is 2.33.
- The example computes VaR by multiplying volatility by the multiplier and subtracting the expected return.
- The calculation depends on the return distribution and horizon units being specified consistently.
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Full text
# Problem in calculating a simple VaR
# Problem in calculating a simple VaR
In
> Alexander, Gordon J. and Alexandre M. Baptista (2006). Does the Basle Capital Accord reduce bank fragility? An assessment of the value-at-risk approach. Journal of Monetary Economics 53(7), 1631–1660.
at page 1644 there is:
> Consider the following example that is based on a 10-trading day investment horizon and 99% confidence level, as required by the Basle Capital Accord. Suppose that the expected rate of return and standard deviation of efficient portfolios $S$ and $L$ are given by: $E[r_S] = 0.50\%$, $\sigma[r_S] = 0.40\%$; $E[r_L] = 1.00\%$, and $\sigma[r_L] = 0.60\%$. It follows that $V[0.99; r_S] = 0.43\%$ and $V[0.99; r_L] = 0.40\%$, ...
and at page 1636 $V$ is defined as:
> For any $t\in(\frac{1}{2},1)$, let $z_t \equiv -\Phi^{-1}(1-t)$, where $\Phi(\cdot)$ is the standard normal cdf. Using the assumption of normality, portfolio $w$’s VaR at $100t\%$ confidence level is: $$V[t,r_w]\equiv z_t\sigma[r_w]-E[r_w].$$
I tried to calculate the VaRs in the example but I don't get the same results, even if I scale the $\sigma$ by a $\sqrt{10}$ factor.
Could you please help me to understand how those VaRs were calculated?
## Answer by Enrico Schumann (score 2, accepted)
https://quant.stackexchange.com/a/47500
The calculation assumes that returns are normally distributed. VaR is a percentile of the returns distribution, which in turn can be expressed as a multiple (here labelled $z$) of the standard deviation of returns. (This works as along as the standard deviation exists for the assumed distribution.) For the $99\,\%$ confidence under a normal distribution, the multiple is $2.33$. So, in the first example, $2.33 \times 0.4 - 0.5 = 0.43$.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.