Choosing Normal or Lognormal Returns for Parametric VaR

# VaR using normal vol vS lognormal


# VaR using normal vol vS lognormal












We are using a vendor's software to calculate the Parametric VaR (using RiskMetrics approach) that take as input the volatility figure of the risk factors. The volatility used so far was the lognormal. But, due to negative rates (ie EUR swap), we have to switch to normal ones.

Is it OK to just replace the vol figures from the lognormal calc with the ones from the normal?

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/63207

My answer is to a certain extent a question: "Switching from lognormal to normal vol" reminds me pretty much of implied volatility. Could it be that you plug some implied volatility into you parametric VaR?

If this is the case then I would recommend you to change this and to look at the volatility in the physical and not the risk-neutral measure. This means that if your model of the return is: $$ R= -D * \Delta r $$ where $D$ is some duration and $\Delta r$ is the delta of your rate then we can (under certain assumptions) assume that $R \sim N(0,D \sigma)$ where $\sigma$ is the volatility of $\Delta r$. Then you could estimate $\sigma$ from a timeseries $(\Delta r_t)_{t=1}^n$ using basic or more sophisticated methods. For estimating $\sigma$ it does not matter whether $r_t$ is positive or negative.

Furthermore, if you work in the phyiscal (real world) measure, then you can take correlations to other risk factors into account. In the risk-neutral (implied) setting you would need something like basket products that have all you risk factors as underlying and would only getsomething like a global implied correlation [see e.g. https://quant.stackexchange.com/questions/8689/average-correlation-of-index-portfolio].

But for risk measuring and management most probably the implied/risk neutral world is the wrong view.

## Answer by Matt (score 0)

https://quant.stackexchange.com/a/70733

This question is a bit old but I see it was edited a few days ago so I'll talk about this from a risk perspective (I've been in market risk as an analyst and a quant going on 15 years now).

Switching from lognormal to normal returns is done quite often for market risk when returns go negative. This has been done on a regular basis in the power markets for ages. Generally just the time series with negative prices is switched to normal returns, while the rest remain lognormal.

The risk side doesn't care too much about the math being correct (RiskMetrics or GBM MC make their own distribution assumptions - usually untested against actual data). Risk cares about backtesting only - I can say that using normal returns for a few risk factors in my experience does not impact backtesting results. You have more error coming from your volatility and correlation estimates, fat tails, etc. If you have time, the best thing to do is make the change to normal returns, rerun VaR for as many days as you can, and redo your backtesting. In time constrained periods I usually just spot check starting with days that had exceeded VaR originally to see how it performs vs the prior assumptions.

Good luck! This question doesn't realy have an answer anywhere in the literature, so I am just giving you the pragmatic approach a few companies with multi-billion portfolios use in the real world.