# Which calculations have to be carried out for category 2 products?

## Introduction

In Part 1 of the series we showed an example of a PRIIPs KID gPRIIP KID and introduced the categorization of PRIIPs. In this post, we use a specific example to show the calculations steps for the key figures of a category 2 PRIIP. Again, the relevant legal framework is the Commission Delegated Regulation (EU) 2017/653 of March 8, 2017.

Category 2 PRIIPS are packaged linear investments in an underlying. We demonstrate the calculation steps on a fictitious tracker certificate on the Swiss Market Index SMI. At the time of the KID generation, this certificate should start at a rate of CHF 10,000 and accurately reflect the (relative) change in value of the SMI. For the sake of simplicity, we (here) disregard entry and exit costs as well as ongoing costs. The calculation logic of category 2 PRIIPs prescribed in the Delegated Regulation assumes that information about future value development can be obtained from the past performance in a probabilistic sense.

## Calculation of the market risk measure (MRM)

### Step 1: Determine historical data

In the case of the SMI, this is simple: we need daily closing prices for the last 5 years. The following figure shows the performance of the SMI between April 2015 and the end of March 2020. We can clearly see the COVID-19-related crash at the end of the time series.

A remark: If the underlying is an instrument for which prices are available less than daily or for which there is no price time series available for the last 5 years (yet), e.g. for new issues for which proxies can be found, the Delegated Regulation also regulates the procedure in the event of poor data. We refer to the relevant EU texts.

### Step 2: Determine historical returns and their statistical moments

The Delegated Regulation assumes as a model for future price developments that the stochastic movement of the underlying behaves in the future in an analogous way as in the past, more precisely: In the case of a category 2 instrument, the model assumes that

- daily returns are statistically independent,
- these future daily returns have the same probability distribution as the historical returns for the past 5 years.

A note on the second point: For the performance scenarios, the historical return is also extrapolated into the future; for the market risk indicator of category 2 instruemtns, in order to avoid an overly optimistic risk estimate, a future return of 0 on average is assumed.

We calculate the following distribution of logarithmic historical daily returns for the SMI

where *X _{i} *is the closing price on day

*i*and

*log*is the natural logarithm. This historical distribution results in the following histogram for the SMI.

A calculation of the higher statistical moments (we will need them for the Cornish-Fisher expansion) results in the mean value of 0.0000115656, the standard deviation of 0.010167 for the period under consideration, a skewness of -0.94 (the distribution is skewed to the left: the peak is flatter on the left and steeper to the right) and a kurtosis of 14.83 (the distribution is slender in the middle and has more frequent extreme events than a normal distribution whose kurtosis is 3). We define: excess kurtosis = kurtosis – 3, so that the excess kurtosis of the normal distribution is 0.

If we now generate simulated paths into the future on the basis of these historical returns (here: for 5 years), then these paths can look like this, for example. We are showing 200 paths here, with a starting value of 100 percent.

#### Interested in more information?

Please contact us.

## The Cornish-Fisher-Expansion

Under the assumptions from above (identically distributed, independent returns) it can be deduced from the central limit theorem that the sum of the logarithmic returns after **N** trading days comes closer and closer to the normal distribution and the skewness of this sum equals the skewness of the individual returns **/** _{}, the Excess kurtosis of the sum equals the excess kurtosis of the individual returns **/ N**.

The regulation takes a Cornish-Fisher ansatz for Category 2 instruments. (Cornish, E.A, and Fisher, R.A. “Moments and Cumulants in the Specification of Distributions.” Revue De L’Institut International De Statistique/ Review of the International Statistical Institute, vol. 5, no. 4, 1938, pp. 307–320.), that uses a suitable transformation to convert the cumulative distribution of the normal distribution into a distribution with different skewness and different kurtosis. In this way, quantiles can be determined very easily, because only quantiles of the normal distribution are required.

So if we want to know the (simulated) distribution of the SMI after 5 years, we only need to insert into thee formulas of Number 12 and 13 of the Delegated Regulation Annex II, Part 1 (more on this and examples of a few bad traps in Part 3 of this blog series) and can determine the 97.5% quantile for the market risk indicator. With a recommended holding period of 5 years, as assumed in the simulated paths, this results in a VaR-equivalent volatility of 16.2 % and therefore a **market risk indicator of 4**.

## Calculation of the credit risk measure (CRM)

If the manufacturer of the PRIIP is a company that has a rating from a rating agency, the credit risk value is determined from the table that is set out in the Commission’s Implementing Regulation (EU) 2016/1800. For example, an “A” rating from Moody’s means a credit risk class of CR2, a “BBB” from Standard & Poor’s a CR5. Manufacturers without a rating receive – roughly speaking – a default value of 3 if they are supervised by a financial market authority (e.g. banks without a rating, insurance companies without a rating), otherwise a value of 5.

## Information on liquidity risk

If the PRIIP is difficult to liquidate (because there is no secondary market, because prices are very seldom determined or because the denominations are very large), the manufacturer must emphasise this in a text module. Tracker certificates on major indices should normally be very liquid.

## Calculation of the performance scenarios

The performance scenarios (cf. Delegated Regulation Annex IV) for category 2 instruments are determined in a similar way. In the example considered here, with an investment of CHF 10,000 and a time horizon of 5 years (without taking costs into account) the results are:

- In the pessimistic scenario (10% are worse): 5,990 CHF
- In the moderate scenario (median): 9,524 CHF
- In the optimistic scenario (90% are worse): 15,062 CHF

The example shows that the price development of the SMI in the first quarter of 2020 significantly influences the performance scenarios, i.e. has led to a massive reduction of yield expectations.

If the key information sheet had been created at the beginning of 2020 (based on the prices from January 2015 to December 2019), the performance scenarios would have been:

- In the pessimistic scenario (10% are worse): 7,339 CHF
- In the moderate scenario (median): CHF 11,258
- In the optimistic scenario (90% are worse): CHF 17,176

## The stress scenario

For the stress scenario of category 2 instruments, the Delegated Regulation prescribes the following steps:

- Instead of the average volatility of the 5 years of the observation period, a stress volatility should be used. In our example, the average volatilities are to be determined over all 3-month periods (i.e.: January 1st to April 1st, January 2nd to April 2nd, and so on). Of the 1193 volatility values that result in this way (corresponds to 4 years and 9 months as the start time of the window), the 119th largest (90% quantile) is to be used.
- With this greater volatility, the 5% quantile (5% is worse) can be determined in the Cornish-Fisher expansion. Use a return of 0 (and not the actual historical return).

Based on the data up to March 2020, this results in a value for the

- Stress scenario of 4,916 CHF,

And, based on the pre-COVID data a value for the

- Stress scenario of 5,159 CHF.

The greater difference between the pessimistic and stress scenario value in the pre-COVID case is explained by the prescribed calculation method, which pushes the historical (positive) return value in the stress scenario down. In both cases (before and after COVID) the stress volatility is significantly greater (+ 33% before COVID, + 30% after COVID).

The stress scenario becomes even more extreme if the recommended holding period is less than one year: Then 21 instead of 63 days are to be used as the observation window (in step 1 above) to determine the stress volatility, and for the stress volatility (also in step 1) the 99% quantile is used. The 1% quantile is also used in step 2.

** In the next post: **The Cornish Fisher Expansion and its Limitations.