ANNEX III
INPUTS, ASSUMPTIONS AND METHODOLOGIES
Part I. Methodology underpinning the presentation of risk and reward
Summary risk indicator
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1. |
PEPP providers shall allocate the Basic PEPP and the individual alternative investment options to four different categories: ‘1’, ‘2’, ‘3’ and ‘4’. The allocation shall be based on:
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2. |
To calculate the risk of not recouping the inflation-adjusted contributions, PEPP providers shall stochastically determine the range of the expected accumulated capital at the end of the accumulation period for generic PEPP savers, generic lengths of accumulation periods and standardised contribution levels. Following a stochastic simulation, the risk shall be expressed as the probability in percentage points, which is translated from the number of observations where the sum of the inflation-adjusted contributions are higher than the expected value of the accumulated capital at the end of the accumulation period, compared to the number of all observations. |
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3. |
The individual investment option’s risk of not recouping the inflation-adjusted contributions shall be allocated to the different categories as follows:
Where the risk category of the investment option diverges between the different accumulation periods, the highest risk category shall be used. |
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4. |
To calculate the expected shortfall, PEPP providers shall stochastically determine the range of the expected accumulated capital at the end of the accumulation period for generic PEPP savers, generic lengths of accumulation periods and standardised contribution levels. Following a stochastic simulation, the risk shall be expressed as the percentage of the expected shortfall in relation to the sum of the inflation-adjusted contributions. The expected shortfall is determined by the observations where the inflation-adjusted contributions are higher than the expected value of the accumulated capital at the end of the accumulation period and the average losses of these observations. |
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5. |
The individual investment option’s risk in terms of expected shortfall shall be allocated to the different categories as follows:
Where the risk category of the investment option diverges between the different accumulation periods, the highest risk category shall be used. |
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6. |
To calculate the expected rewards to reach a certain level of PEPP benefits, PEPP providers shall stochastically determine the range of the expected accumulated capital at the end of the accumulation period for generic PEPP savers, generic lengths of accumulation periods and standardised contribution levels. PEPP providers shall express the rewards in terms of the median accumulated capital at the end of the accumulation period as a multiple of the sum of the inflation-adjusted contributions. |
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7. |
The individual investment option’s rewards to reach a certain level of PEPP benefits shall be allocated to the different categories as follows:
Where the rewards category of the investment option diverges between the different accumulation periods, the lowest rewards category shall be used. |
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8. |
To aggregate the outcomes of the categorisation of the individual investment options to the summary risk indicator, PEPP providers shall:
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Performance scenarios
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9. |
PEPP providers shall stochastically determine the expected PEPP benefits, as appropriate, at the start of, or during the decumulation phase, taking into consideration:
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10. |
The scenario values of the expected PEPP benefits under the different performance scenarios shall be determined in line with the stochastic dispersion of the expected PEPP benefits:
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Part II. Rules to determine the assumptions on pension benefit projections
Annual rate of nominal investment returns
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11. |
PEPP providers shall determine the Basic PEPP’s and alternative investment options’ expected nominal investment returns in an appropriate stochastic approach, mirroring the corresponding investment strategy, the strategic investment allocation and the risk-mitigation technique applied for the individual investment option. |
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12. |
When determining the different elements of the stochastic model, PEPP providers shall use the annual rate of inflation and may consider to take a modular approach for the stochastic calculation of, at least:
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13. |
For the determination of the nominal interest rates, the PEPP provider may use the G2++ short-rate model, as described by Brigo et al. (2006) (1),which is equivalent to the two-factor Hull-White model and allows for negative interest rates. Its behaviour is driven by five parameters, two per factor and one for the correlation. The components of the two-dimensional Wiener process are correlated and a deterministic shift factor allows for a perfect fit of the initial term structure to market rates.
The stochastic differential equations for the two factors x(t) and y(t) are
and
where a, b, σ and η are positive parameters and
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14. |
The risk-neutral valuation using the risk-neutral measure 
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15. |
Using the Girsanov’s theorem, the calculation follows
with λi
being the market price of risk. The dynamics under the
and
The short-rate process r(t) is the sum of the two factors and the deterministic shift, i.e. r(t) = x(t) + y(t) + φ(t), where for the deterministic shift factor φ(t)
holds. In this equation, fM (0, T) denotes the market instantaneous forward rate at initial time 0 with the horizon T. |
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16. |
Following the G2++ model, analytical solutions of the price of a zero coupon bond exist by defining
and
For which the price of a zero coupon bond in the G2++ model is P(t,T) = A(t,T) e –B ( a,t,T ) x ( t ) –B ( b,t,T ) y ( t ). PM (t,T) denotes here the market price of a zero coupon bond at time t for maturity T. |
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17. |
The PEPP provider may use the model prices for determining the returns of risk-free investments in bonds. Further, the short-rate may be used as an input to the modelling of the equity returns and potentially for property returns. |
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18. |
For the determination of credit spreads, the PEPP provider may use the simulation of credit spreads as to combine the risk-free zero coupon bond term structure to yield a credit-risky zero coupon bond term structure. The hazard rates of bonds of different rating classes may be modelled through the use of Cox-Ingersoll-Ross (CIR) processes. The hazard rate πi develops in the risk-neutral measure according to the stochastic differential equation:
together with the condition 2kθ > σ 2 in order to keep π(t) positive for all t. Assuming a market price of risk of the form
the real-world dynamics are given by
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19. |
PEPP providers may model hazard rates for the rating classes AAA (i = 1), AA, A, BBB and BB (i = 5), potentially differentiated for corporate, covered and other bonds. The default probabilities pi
(t,T) are then calculated as the product of the CIR-prices Pi
(t,T) at time t for maturity T, i.e.
where
The spreads si (t,T) are then determined through
with δ being the recovery rate. |
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20. |
For the determination of equity returns, the PEPP provider may use a model for the development of one stock market index through the use of geometric Brownian motion. This model has two parameters: the volatility and the equity risk premium. The nominal interest rate model provides the applicable risk-free rate and the output of the model are yearly annualized returns for investments in the market index.
dSt = (r(t) + λ) Stdt + σStdWt |
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21. |
To determine the yearly volatility, PEPP providers may use the standard deviation of the monthly returns of an appropriate equity index for an appropriate, representative time period to annualise the result. |
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22. |
PEPP providers may apply the equity risk premium λeq
as an implied measure following Damodaran (2020) (2), but calculating it directly on the appropriate equity index without further country risk premia. It is defined as
λeq := E[Rm ] – Rf , where E[Rm ] is the expected market return and the risk-free rate Rf may be chosen as the 10Y spot rate of the ECB’s or National Central Bank’s curve. |
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23. |
For the growth rate g, the PEPP provider may use the long-term growth EPS forecast, whereas γ is the sum of the dividend yield and the buyback yield. Cash flows may be determined using the constant growth rate for five years, after which the final cash flow is a perpetuity with the risk-free rate as the growth rate.
in which PVindex is the present value of the index in this discount dividend model and P 0 is the price of the index at time t = 0. By demanding P 0 = PVIndex, the expected market return can be solved and the equity risk premium can be calculated. |
Annual rate of inflation
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24. |
To calculate the annual rate of inflation, the PEPP provider shall use a one factor Vasicek process. The mean-reverting dynamics of the model are driven by three parameters. The stochastic differential equation of the model is
di(t) = k(θ – i(t))dt + σdW(t), i(0)=i 0, in which i(t) is the inflation rate at time t, k refers to the speed of mean reversion, θ to the level of mean-reversion and σ to the volatility. |
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25. |
The modelling shall target the inflation rate target level of the European Central Bank for the Euro area or, where applicable, of the corresponding central banks for countries outside the Euro area in the medium term, together with the observed standard deviation of the inflation rates. The speed of the mean reversion, together with the current inflation rate, shall be used to fit the model to the current environment and short-term inflation rate forecasts. |
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26. |
The calibration of the inflation rate shall use for the Euro area the European Central Bank’s or, for Member States outside the Euro area, the central bank’s inflation target for the θ-parameter. The monthly Year-on-Year-inflation rate time series of the Member State’s Harmonised Index of Consumer Prices (HICP) shall be used for deriving the standard deviation of the inflation rate in the long-term, which shall be assumed as 100 years. From the same time series, the initial value of the inflation rate at the reference date shall be used. The PEPP provider shall use the inflation projections for the Member State’s HICP, published as the Eurosystem staff macroeconomic bi-annual projections for the Euro area countries, or of the European Commission’s economic forecast for the countries outside the Euro area, unless the corresponding central bank provides for projections. Those inflation projections shall be used for fitting the speed of the mean reversion. |
Trend of future wages
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27. |
To take into account the trends of future wages, where applicable, PEPP providers shall consider the real wage growth in the different Member States, considering Eurostat data and taking into account that real wages increase significantly during the early part of a PEPP saver’s career and experience significantly lower growth or losses in the later parts. The PEPP provider may consider a pattern in the PEPP savers’ real-wage paths partly to reach a plateau closer to the end of the accumulation phase and partly to reach the plateau earlier, which means 20 years from retirement and fall thereafter. |
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28. |
To reflect a large range of possible paths, the PEPP provider may use a real wage index following a quadratic equation with age: wage = a(max – age)2 + b. The coefficient ‘a’ is taken from a uniform distribution between -0,15 and 0,011; max is taken from a uniform distribution between 47 and 64 and corresponds to the age when real wages are at their maximum value; and the coefficient b is solved so that the wage index starts at 100 at age 25. |
Part III. Methodology for the calculation of costs, including the specification of summary indicators
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29. |
In the PEPP KID, the PEPP provider shall present the total annual costs, comprising all costs incurred and chargeable within 12 months in monetary terms and as a percentage of the projected accumulated capital after 12 months. Where necessary, these amounts may be calculated as the average total annual costs over the term of the PEPP contract. The calculation of the compound effect of the costs shall be based on a 40 years’ accumulation period, based on monthly contributions of EUR 100 and on the projected accumulated capital in the best estimate scenario. |
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30. |
In the PEPP Benefit Statement, the PEPP provider shall present the estimated impact of costs on the final PEPP benefits by using the ‘Reduction in Wealth’ approach. The ‘Reduction in Wealth’ shall be calculated as the difference between the projected accumulated savings at the end of the accumulation and the projected accumulated savings at the end of the accumulation period in a cost free scenario. The difference shall be disclosed in monetary and percentage terms relative to the projected accumulated savings. The calculation shall be based on the personalised contribution level of the individual PEPP saver and based on the best estimate scenario of point 10. |
(1) Brigo, D., Mercurio, F.: Interest Rate Models – Theory and Practice, Second Edition, Springer-Verlag Berlin Heidelberg, 2001, 2006.
(2) Damodaran, Aswath, Equity Risk Premiums: Determinants, Estimation and Implications – The 2020 Edition (5 March 2020). NYU Stern School of Business.