PRIIPs VaR-Equivalent Volatility Below −100%

Article Quant Q&A · Author: glaucon

Summary

The document asks how to convert a Cornish–Fisher VaR in log-return space into PRIIPs VaR-equivalent volatility when the calculated VaR is below −100%. It presents historical return moments for a stock, applies the regulation’s stated formula, and compares the resulting volatility with an alternative that floors VaR at −100% because a simple investment cannot lose more than its initial value.

The answer favors using the unmodified VaR calculation. Its rationale is that PRIIPs defines returns as log returns, which can fall below −100%, so the loss floor for simple returns does not apply directly. The discussion cites the regulation’s annex but provides no detailed derivation or broader regulatory interpretation. It therefore offers a concise answer to this particular conversion question, not a full account of PRIIPs methodology or how to handle other model assumptions.

Key ideas

PRIIPs VaR in log-return space can be below −100%.
The proposed approach applies the VaR-equivalent volatility formula to the calculated log-return VaR without a floor.
A loss floor based on the initial investment applies to simple returns, not necessarily to log returns.
The answer relies on the regulation’s definition of return and does not provide a full derivation.

Tags

Statistics Risk management Volatility

Full text

# PRIIPs KID: if VaR (Return Space) < -1, how to compute VEV (VaR-equivalent volatility)?


# PRIIPs KID: if VaR (Return Space) < -1, how to compute VEV (VaR-equivalent volatility)?












The PRIIPs regulation does not specify how to compute the VaR-equivalent volatility if $VaR_{Return Space} < -1$. What would you do in the following case?

I have the following moments from the historical daily log-returns of a stock:

$M1 = 0.0019$ (Mean)

$\sigma = 0.0378$ (Standard Deviation)

$\mu_1 = 0.9201$ (Skewness)

$\mu_2 = 12.068$ (Excess Kurtosis)

Assume

$T = 1$ (asset’s holding period in years)

$N = 256$ (number of trading periods in days)

Then, the Cornish-Fisher VaR is: \begin{eqnarray} VaR_{Return Space} &=& \sigma \sqrt{N} * (− 1.96 + 0.474 * \mu_1/\sqrt{N} - 0.0687 * \mu_2/N + 0.146* \mu_{1}^2/N) − 0.5 \sigma^2 N \\ &=& -1.3534 \end{eqnarray}

Given that $VaR_{Return Space}$ is below - 1, which of these two VEVs would be the correct one:

(1) Simply apply the formula and obtain: \begin{eqnarray} VEV &=& (\sqrt{3.842-2*VaR_{Return Space}}-1.96)/\sqrt{T} \\ &=& \sqrt{3.842-2*(-1.3534)}-1.96 \\ &=& 0.5990 \end{eqnarray}

(2) Since an investor cannot lose more than the initial investment, put a floor to $VaR_{Return Space} = -1$ and get \begin{eqnarray} VEV &=& \sqrt{3.842-2*(-1)}-1.96 \\ &=& 0.4570 \end{eqnarray}

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/69422

I glimpsed at the regulation: In Annex II, part 1, no 11 and 12, they define the return as log-returns, see screenshot:

Hence, I'd argue that you should use your calculation 1.

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Computing PRIIPs VaR-Equivalent Volatility When Log VaR Is Below −100%

Summary

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