Stochastic Generational Accounting Applied to Reforms of Dutch Occupational Pensions

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.

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 re-estimated 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, long-term 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

$$\begin{aligned} r_t =\delta _{0r} +\delta _{1r}^\prime X_t \end{aligned}$$

and the instantaneous expected inflation, $\pi$

$$\begin{aligned} \pi _t =\delta _{0\pi } +\delta _{1\pi }^\prime X_t \end{aligned}$$

The dynamics in the state variables govern the autocorrelation in the interest rates and inflation. The state variables follow a mean-reverting process around zero

$$\begin{aligned} \begin{array}{ll} dX_t &{} =\mu dt-KX_t dt+\Sigma _X^{{\prime }} dZ_t \\ &{} K\,\hbox {is}\,2\times 2\,\hbox {and}\,\Sigma _X^{{\prime }} =\left[ {I_{2\times 2} \;0_{2\times 2} } \right] \\ \end{array} \end{aligned}$$

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:

$$\begin{aligned} \frac{d\Pi _t }{\Pi _t }=\pi _t dt+\sigma _\Pi ^{{\prime }} dZ_t\quad \sigma _\Pi \in R^{4}\,\hbox {and}\,\Pi _0 =1 \end{aligned}$$

The stock index $S$ develops according to

$$\begin{aligned} \frac{dS_t }{S_t }=(R_t +\eta _S )dt+\sigma _S^\prime dZ_t\quad \sigma _S \in R^{4}\,\hbox {and}\,S_0 =1 \end{aligned}$$

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 $\phi _t^N$

$$\begin{aligned} \frac{d\phi _t^N }{\phi _t^N }=-R_t dt-\Lambda _t^{{\prime }} dZ_t \end{aligned}$$

with the time-varying price of risk $\Lambda$ affine in the state variables $X$

$$\begin{aligned} \Lambda _t =\Lambda _0 +\Lambda _1 X_t \quad \hbox {and}\,\Lambda _t , \Lambda _0 \in R^{4}\,\hbox {and} \quad \Lambda _1 4\times 2 \end{aligned}$$

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)

$$\begin{aligned} \Lambda _1 =\left[ {{\begin{array}{ll} {\Lambda _{1_{(1, 1)} } }&{} {\Lambda _{1_{(1, 2)} } } \\ {\Lambda _{1_{(2, 1)} } }&{} {\Lambda _{1_{(2, 2)} } } \\ 0&{} 0 \\ {\Lambda _{1_{(4, 1)} } }&{} {\Lambda _{1_{(4, 2)} } } \\ \end{array} }} \right] \end{aligned}$$

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

$$\begin{aligned} Ed\phi ^{N}S=0 \end{aligned}$$

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

$$\begin{aligned} \frac{d\phi ^{N}S}{\phi ^{N}S}&= \frac{d\phi ^{N}}{\phi ^{N}}+\frac{dS}{S}+\frac{d\phi ^{N}}{\phi ^{N}}.\frac{dS}{S} \\&= \left( {\eta _S -\Lambda _t^{{\prime }} \sigma _S^\prime } \right) dt-\left( {\Lambda _t^{{\prime }} -\sigma _S^\prime } \right) dZ_t \end{aligned}$$

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

$$\begin{aligned} \eta _S =\Lambda _t^{{\prime }} \sigma _S \end{aligned}$$

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

$$\begin{aligned} Ed\phi ^{N}P^{N}=0 \end{aligned}$$

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

$$\begin{aligned} Ed\phi ^{N}P^{R}\Pi =0 \end{aligned}$$

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

$$\begin{aligned} \frac{d\phi ^{R}}{\phi ^{R}}&\equiv \frac{d\phi ^{N}}{\phi ^{N}}+\frac{d\Pi }{\Pi }+\frac{d\phi ^{N}}{\phi ^{N}}.\frac{d\Pi }{\Pi }\\&= -\left( {R_t -\pi _t +\sigma _{\Pi }^{\prime } {\Lambda }_t } \right) dt-\left( {{\Lambda }_t^{\prime } -\sigma _{\Pi }^{\prime } } \right) dZ_t\\&= -r_t dt-\left( {{\Lambda }_t^{\prime } -\sigma _{\Pi }^{\prime } } \right) dZ_t \end{aligned}$$

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

$$\begin{aligned} R_t&= r_t +\pi _t -\sigma _{\Pi }^{\prime } {\Lambda }_t\\&= \left( {\delta _{0r} +\delta _{0\pi } -\sigma _{\Pi }^{\prime } {\Lambda }_0 } \right) +(\delta _{1r}^{\prime } +\delta _{1\pi }^{\prime } -\sigma _{\Pi }^{\prime } {\Lambda }_1 )X_t\\&\equiv R_0 +R_1^{\prime } X_t \end{aligned}$$

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: 3-month, 1-, 2-, 3-, 5-, and 10-year maturities, respectively. Three-month money market rates are taken from the Bundesbank (www.bundesbank.de). For the period 1973:I to 1990:II, end-of-quarter money market rates reported by Frankfurt banks are taken, whereas thereafter 3-month Frankfurt Interbank Offered Rates are included. Long nominal yields: From 1987:IV on, zero-coupon rates are constructed from swap rates published by De Nederlandsche Bank (www.dnb.nl). For the period 1973:I to 1987:III, zero-coupon yields with maturities of 1–15 years (from the Bundesbank website) based on government bonds were used as well (15-year 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 2-year 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

$$\begin{aligned} y_t^\tau =\left( {-A(\tau )-B(\tau {)}'X_t } \right) /\tau \end{aligned}$$

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.

$$\begin{aligned} y_t^\tau =\left( {-A(\tau )-B(\tau {)}'X_t } \right) /\tau +\varepsilon _t^\tau \quad \hbox {and}\quad \varepsilon _t^\tau \sim N(0, \sigma ^{\tau }) \end{aligned}$$

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

$$\begin{aligned} Y_{t+h} =\mu ^{(h)}+\Gamma ^{(h)}Y_t +\varepsilon _{t+h}\quad \hbox {and}\quad \varepsilon _{t+h} \sim N(0, \Sigma ^{(h)}) \end{aligned}$$

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 first-order 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.

Table 1 Estimation results for the US and the Netherlands

Table 2 Risk premia and volatilities

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.