Abstract
This paper examines stochastic or ‘value based’ generational accounting as a method to assess the intergenerational redistributive impact of pension reform. The analysis is applied to three policy changes to the regulation of Dutch occupational pensions during the years 2012 and 2013 that mark the transition from defined benefit pensions to ‘defined ambition’ pension schemes.
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Notes
See e.g. Bovenberg and Mehlkopf (2014).
This follows from the assumption of complete markets. This and other limitations will be discussed in the conclusion of this paper.
We apply a partial equilibrium framework in this paper in which there is an exogenous price of risk that is not affected by the intergenerational contract. In a general equilibrium framework, in which current generations can trade with future generations, market prices would adjust. Such ‘fictional’ trading between nonoverlapping generations would change the market prices of risk, thereby redistributing resources between agents, see Ball and Mankiw (2007).
This gives rise to the discussions about the socalled ultimate forward interest rate.
A number of studies have explored valuation techniques for nontraded risk factors in pension schemes. De Jong (2008) examines the valuation of wagelinked cash flows in an incomplete market setting in which the wage index cannot be hedged perfectly with financial market instruments. He discusses several methods to find a value in such incomplete markets and advocates utilitybased valuation. Geanakoplos and Zeldes (2010) derive the value of wagelinked cash flows on the basis of an assumed theoretical longrun relationship between wages and stocks.
See Bovenberg et al. (2014) for a more elaborate description of the Dutch pension system and analysis of recent developments.
For a detailed description of the pension contract, see Lever et al. (2012a).
A thorough presentation can be found in Draper (2012).
See Lever et al. (2012a) for the results for these alternative scenario sets.
For details see Lever et al. (2012a).
See ‘UFR method for calculating the term structure of interest rates’, http://www.toezicht.dnb.nl/en/binaries/51226788.pdf.
Participants are assumed to save any reductions in contributions and to consume them after retirement. This simplifying assumption allows us to limit the analysis to income (consumption) after retirement.
For details, see Lever and Bonenkamp (2013).
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The authors acknowledge helpful assistance by Jan Bonenkamp, Mark Brussen, Pascal Janssen, and André Nibbelink.
Appendix: The Capital Market Model
Appendix: The Capital Market Model
This appendix contains a summary of the capital market model. The model builds on Koijen et al. (2010) and is reestimated on data for the Netherlands. The appendix subsequently describes the model, the data, the estimation procedure and the estimation results.
1.1 Model Specification
The model distinguishes four assets: a stock index, longterm nominal and real bonds and a nominal money account. The real interest rate and the instantaneous expected inflation are modeled using two state variables, which are collected in vector \(X\). More precisely, the instantaneous real interest rate, \(r\), follows
and the instantaneous expected inflation, \(\pi \)
The dynamics in the state variables govern the autocorrelation in the interest rates and inflation. The state variables follow a meanreverting process around zero
where \(Z\) denotes a four dimensional vector of independent Brownian motions. Four sources of uncertainty can be identified: uncertainty about the real interest rate, uncertainty about the instantaneous expected inflation, uncertainty about unexpected inflation and uncertainty about the stock return. Correlation between the real interest rate and inflation is modeled using \(\delta _{1r}^\prime \) and \(\delta _{1\pi }^\prime \). The actual inflation, \(\Pi \) is equal to expected inflation, \(\pi \), except for unexpected shocks:
The stock index \(S\) develops according to
where \(R\) is the nominal instantaneous interest rate, which is determined by the real interest rate and the inflation process, and \(\eta _S \) the equity risk premium. The model is completed with the specification of the nominal stochastic discount factor \(\phi _t^N \)
with the timevarying price of risk \(\Lambda \) affine in the state variables \(X\)
A theoretical justification of this stochastic discount factor can be found in Merton (1992) and Cochrane (2001). The price of risk depends on the risk aversion of investors. Assume no risk premium for unexpected inflation, that is the third row \(\Lambda _1 \) contains zeros only. This restriction is imposed because unexpected inflation risk cannot be identified on the basis of market data (see Koijen et al. 2010)
1.2 Parameter Restrictions
The stochastic discount factor is used to determine the value of cash flows in a complete market setting. For instance, the fundamental valuation equation (Cochrane 2001) of the equity index
implies that the expected value of the discounted stock price does not change over time. This equation implies a restriction. Using the Itô Doeblin theorem gives
because in the limit \(dt\) tends to \(0\), the \(dt^{2}\) and \(dtdZ\) terms disappear and the \(dZ^{2}\) term tends to \(dt\). Taking expectations gives the restriction
which implies \(\sigma _S^{{\prime }} \Lambda _0 =\eta _S \) and \(\sigma _S^{{\prime }} \Lambda _1 =0\). This restriction is imposed on the model.
1.3 Nominal and Inflation Linked Bonds
The fundamental pricing equation for a nominal zero coupon bond is
i.e. the expected discounted value of the price of a nominal bond does not change over time. The condition implies for inflation linked bonds
i.e. the discounted value of the inflation corrected price of real bonds doesn’t change over time. Define the real stochastic discount factor as \(\phi ^{R}\equiv \phi ^{N}{\Pi }\). Using the Itô Doeblin theorem we derive for the real stochastic discount factor
because in the limit \(dt\) tends to \(0\). The \(dt^{2}\) and \(dtdZ\) terms disappear and the \(dZ^{2}\) term tends to \(dt\). The nominal rate can thus be written as
1.4 Data
The data for the Netherlands are taken from Van den Goorbergh et al. (2011).

Inflation: From 1999 on, the Harmonized Index of Consumer Prices for the euro area from the European Central Bank data website (http://sdw.ecb.europa.eu) is used. Before then, German (Western German until 1990) consumer price index figures published by the International Financial Statistics of the International Monetary Fund are included; note that the exchange rate of the Dutch guilders was pegged to the Deutschmark since 1983.

Yields: Six yields are used in estimation: 3month, 1, 2, 3, 5, and 10year maturities, respectively. Threemonth money market rates are taken from the Bundesbank (www.bundesbank.de). For the period 1973:I to 1990:II, endofquarter money market rates reported by Frankfurt banks are taken, whereas thereafter 3month Frankfurt Interbank Offered Rates are included. Long nominal yields: From 1987:IV on, zerocoupon rates are constructed from swap rates published by De Nederlandsche Bank (www.dnb.nl). For the period 1973:I to 1987:III, zerocoupon yields with maturities of 1–15 years (from the Bundesbank website) based on government bonds were used as well (15year rates start in June 1986). No adjustments were made to correct for possible differences in the credit risk of swaps, on the one hand, and German bonds, on the other. The biggest difference in yield between the two term structures (for the 2year yield) in 1987:IV was only 12 basis points.

Stock returns: MSCI index from Fact Set. Returns are in euros (Deutschmark before 1999) and hedged for US dollar exposure.
1.5 Estimation Procedure
The model implies relations for the yield curve linear in the state variables. Assume, two yields are observed without measurement error. For those yields it holds
These observations can be used to determine the state vector \(X,\) given a set parameters which determine \(A\) and \(B\). The other four yields are observed with a measurement error by assumption.
Assume no correlation between the measurement errors. This system of measurement equations is extended with the equations for the state, inflation and equity, which can be written as a vector autoregressive system
The likelihood is maximized with respect to the parameters using the method of simulated annealing of Goffe et al. (1994) to find the global optimum.
1.6 Estimation Results
The estimation results for the Netherlands are presented in Table 1. For reference, the table also includes the estimates using the same model for the US (using the data in Koijen, Nijman and Werker). The significance of the estimates for the Netherlands is lower in general. The unconditional expected inflation is in the US (4.2 %) larger than in the Netherlands (2.2 %). This explains the 2 % larger unconditional nominal interest rate in the US. The instantaneous short rate and expected inflation are increasing in both \(X1\) and \(X2. \quad X2\) is more persistent than \(X1\). Moreover, the persistence is larger in the Netherlands than in the US for both variables, which explains partly the less significant parameter estimates. The firstorder autocorrelation on an annual frequency equals 0.503 and 0.861 for the US and 0.725 and 0.906 for the Netherlands, respectively. The equity risk premium \((\eta _S )\) is 5.4 % for the US and 3.5 % for the Netherlands. Table 2 reports the risk premium on nominal bonds along with their volatilities. The risk premium for bonds with a maturity of 10 years is 65 basis points higher in the Netherlands than in the US.
Together these estimates determine the parameters of the model underlying the simulations of the pension cash flows and their valuation in the analysis of policy reforms in this paper.
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Draper, N., van Ewijk, C., Lever, M. et al. Stochastic Generational Accounting Applied to Reforms of Dutch Occupational Pensions. De Economist 162, 287–307 (2014). https://doi.org/10.1007/s106450149232x
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DOI: https://doi.org/10.1007/s106450149232x