PRIIPs Category 3 Methodology for Nonlinear Interest Rate Instruments
Introduction
For interest rate instruments that fall into category 3 of the PRIIPs regulation, extensive simulations have to be performed for determining the SRI (Summary Risk Indicator) and the performance scenarios.
Examples of such category 3 bonds are – the list is by no means exhaustive:
- Floating-rate instruments with caps and floors. The reference rate may be a Libor, a CMS rate, a floating rate in another currency (quanto), an inflation rate or a more complicated rule, such as those that play a role in the case of Snowball bonds, CMS spread bonds or range accrual notes. In the case of these highly structured instruments in particular, these bonds are frequently equipped with early termination rights (callabilities) for the issuer.
- Fixed-rate bonds (without termination rights), on the other hand, are not PRIIPs, even if they are (non-investment grade) associated with a potentially significant risk of default.
In principle, 10,000 paths have to be generated again for the calculation of the PRIIPS key figures (this number is higher for instruments with a longer holding period), and the instrument has to be valuated at the prescribed times (after one year, after half the recommended holding period, after the recommended holding period) along the respective path. The devil is in the details.
The path generation for interest rate instruments
In the case of the Barrier Reverse Convertible (BRC) (Part 4 of the article series), for valuating the down-and-in put option, we needed the price of the underlying at the valuation time, the estimated volatility and information about whether the barrier has been breached in the past.
For the valuation of a Libor floater with cap and floor, we need the yield curve at the time of valuation and an interest model that the manufacturer can (more or less) freely choose, but which of course must not deliver misleading prices. Possible candidates for the interest rate model are short-rate models (e.g. Hull-White, Black Karasinskii, Black-Derman-Toy), the Libor market model (e.g. Brace-Gartarek-Musiela) or an interest rate model with stochastic volatility.
Since the Delegated Regulation does not make any provisions on the volatility structure, our recommendation is to choose a model (1) that is able to valuate instruments with termination rights comparatively fast, (2) that can also handle negative interest rates and (3) which – correctly implemented – has a high numerical stability and robustness. From our point of view, in many cases of structured interest rate instruments for generating PRIIPs key figures, the Hull-White model will be the most suitable model in this sense.
In contrast to equity instruments such as BRCs, the state variable for which the valuation must be carried out is not a scalar, but a vector of interest rates for different tenors. In accordance with the Delegated Ordinance, logarithmic returns are calculated for each of these interest rates.
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Logarithmic returns of interest rates
For example, let us consider the 1-month Euribor between 2014 and 2019
Until September 5, 2014, the Euribor 1m was always positive: on September 4, the (mid-close) rate was 0.05071%, on September 5, 0.00071%. Although this only represents a change of 5 basis points in absolute numbers, in the (questionable) calculation logic of the Delegated Regulation this is a relative drop of 98.6 percent!
On September 8, 2014, the Euribor 1m was negative for the first time. Here, the Delegated Regulation stipulates that the yield curve must be shifted upwards by so many integer percentage points that all values in the time series (normally over the last 5 years) are positive again, i.e. by one percentage point here. After bootstrapping, the generated curves are shifted down again. The almost 99% drop (between September. 4 and 5) then turns into a 5% drop. This means that the paths generated for the Euribor 1m are completely different, depending on whether the 5 years before September 5, 2014 (Friday) or before September 8 (Monday) are used:
The massive drops in the individual paths happen when the bootstrapping procedure draws the change from September 4th to September 5th.
In our view, the only reasonable approach would be not to use quotients of interest rates, but differences. (If the temperature changes from 1 ° Celsius to 0.1 ° Celsius, we also don’t say: “The temperature has fallen by 90%.”)
Simulated yield curves, starting at the end of 2029. Source: Binder-Jadhav-Mehrmann: Model order reduction for the simulation of parametric interest rate models in financial risk analysis. Submitted.
If, based on the data up to the end of 2019, we want to evaluate an interest rate instrument at the end of 2029, then the (10,000) interest rate curve paths of the above picture result.
For the evaluation of a floater with cap and floor under a one-factor Hull-White model, we have to choose the volatility and the reversion speed appropriately. For example, the values of a calibrated model from 2019 could be used.
For a ten-year vanilla floater that pays the Euribor 3m quarterly with a cap at 2.25% (p.a.) and a floor at 0.5% (p.a), we obtain the following values for the performance scenarios (expressed in multiples of the fair value in the beginning:
Performance Scenario |
5 years |
10 years |
Favorable (90th percentile) |
1.014 |
1.101 |
Moderate (50th percentile) |
1.002 |
1.066 |
Unfavorable (10th percentile) |
0.984 |
1.041 |
The values after 10 years (which equals the recommended holding period) are simply the redemption value plus the summed up coupons, the values after 5 years are the coupons up to year 5 plus the present value of the cashflows between year 5 and 10, i.e. the redemption plus the future coupons after year 5. To calculate this present value, the corresponding Hull-White differential equation must be solved.
At a first glance, the histograms of the performance paths are surprising:
Distribution of the redemption values (coupons plus redemption) after 10 years. Due to the construction of the yield curves according to the Delegated Regulation, the Euribor 3m is essentially lognormally distributed, meaning right-skewed. Source: Binder-Jadhav-Mehrmann
Vice versa after 5 years. The main proportion of the value comes from discounting of the redemption with a substantially lognormally distributed 5-year interest rate, which leads to a left-skewed distribution. Source: Binder-Jadhav-Mehrmann