# PRIIPs Category 3 methodology for equity instruments

## Introduction

Many structured investments, which offered to private investors, are non-linear because, e.g., they are equipped with early termination rights (on the side of the issuer or on the side of the investor) or with embedded options. According to the PRIIPs Delegated Regulation, such investments are (see Part 1) typically category 3 instruments, if there is no risk of a total loss (in which case it would be category 1). For such instruments, extensive simulation calculations have to be made.

In this post, we present the PRIIPs the category 3 methodology for equity instruments, in part 5 we discuss the calculation steps for structured interest rate instruments.

## A Barrier Reverse Convertible as a Category 3 PRIIP

### What is a Barrier Reverse Convertible (BRC)?

A BRC is a structured instrument that consists of a bond with relatively high coupon payments and (one or more) short positions in down-and-in knock-in put options. In our specific example here, the underlying share (voestalpine) should have a price of EUR 49,845 at the beginning of the liefteime (December 29, 2017). With a nominal value of EUR 1000, the payout at maturity (here: after one year):

- If the share price never falls below EUR 35 during the lifetime, the investor receives 100% of the bond and a 6% coupon.
- However, If the share price breaks through this lower boundary of EUR 35 at least once during the lifetime (“down-and-in”), the investor will in any case receive the coupon (6%), but the issuer decides on the repayment, either EUR 1000 or 20 shares at maturity. Therefore: If, in the case of the down-and-in, the share price is above EUR 50 again at maturity, the issuer will redeem the bond at the nominal price, but if the share price is below 50, then the investor will be paid 20 shares.

The Delegated Regulation stipulates that 10,000 paths of future developments of the share price must be simulated for both the market risk indicator and the performance scenarios. To do this, first calculate the daily logarithmic returns from the time series of the last 5 years (i.e. from 2013 to 2017), shown in the histogram.

For the path simulation, one draws randomly – with replacement – from these logarithmic returns and applies the change to the current share price. This gives you a path up to the recommended holding period. A total of 10,000 paths must be generated. So 250 trading days times 10,000 paths: dragging 2.5 million times. We obtain

For determining the market risk indicator and the performance scenarios now, we have to consider the share price along each of these paths: If the price falls below 35, the original redemption of the bond is converted into a short positions of 20 put options with a strike of 50. In these cases, at the end of the term, a decision must be made how the bond (cash or share) will be repaid.

We obtain for the redemption value (per share, without 6% coupon):

The 97.5% percentile on the left yields a redemption value of EUR 27.76 + a proportional EUR 3 coupon, i.e. EUR 30.76. Converted to a “VaR equivalent volatility”, this is a volatility value of 23.39% and thus a market risk indicator of 5.

In the above Figure of the simulated equity paths, we have not stated so far, that the risk-free interest rate (for the recommended holding period) should be used as the mean return to calculate the MRI, i.e. it should be calculated in a risk-free manner.

The physical measure is to be used for the performance scenarios, i.e. the actual mean return of the last 5 years for the stocks. As a result, the paths move up a little in our example (because the voestalpine price at the end of 2017 was higher than at the end of 2012), and the payout function for the performance scenarios also changes.

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## Evaluation effort?

In this specific example, the calculation is actually relatively simple: we have to set up 10,000 paths for the market risk indicator, 10,000 paths for the performance scenarios (and 10,000 paths for the stress scenario). The valuation itself is, however, a simple lookup (1) whether the barrier has been breached and (2) if so, whether the stock is above or below 50 at maturity.

It becomes significantly more complicated if the lifetime is longer than a year. Then performance scenarios must also be shown at times in between. This means that at these time points (of intermediate holding periods), depending on the path so far, either a down-and-in put option or a put option must be valuated. For this, 10,000 (performance) + 10,000 (stress) partial differential equations have to be solved.

## Variants

Barrier Reverse Convertibles are also available with several underlyings: For example, it may be sufficient if one in 4 shares falls below 60% of the initial value. In that casethe repayment can be made in the cheapest (compared to the initial price) share at maturity. When building the path, one moves in four-dimensional space. The correlation is automatically taken into account through the process of drawing a day and applying the change of that day to all four stocks. If the maturiry of the PRIIP is longer than a year, a (spatial) four-dimensional differential equation must then be solved or the valuation must be carried out by means of Quasi-Monte Carlo simulation.

If the shares are quoted in different currencies, quanto effects must also be considereed.

**In the next post:** PRIIPs Category 3 Methodology for Nonlinear Interest Rate Instruments.