2.1 Life-cycle investing

The goal of life-cycle portfolio allocation problems is to determine the optimal consumption and investment choices of an investor with total wealth consisting of human capital, financial wealth, and other real assets, such as housing property. Therefore, the selection of the optimal asset allocation in these settings incorporates also the illiquid and the real components of investors' wealth.

Formally, human wealth is the present value of all future (uncertain) labor income earned over the remaining lifetime. Alternatively, one can consider labor income as the liquid dividend of an illiquid stock of capital, which endows the newborn and eventually grows by investing in education. Given the inalienable, intangible, and at least partially idiosyncratic nature of human capital, the quantitative description of its risk and return profile is both a challenge and essential to shape the overall allocation strategy over the life-cycle.

One of the reasons to hold liquid financial wealth over the life-cycle is to partially hedge the systematic and predictable variations of human wealth. The benchmark advice that the amount of equity held in the portfolio should decline with age has a simple foundation in Samuelson (Reference Samuelson1969) and Merton (Reference Merton1971) model settings.

Bodie et al. (Reference Bodie, Merton and Samuelson1992) study the hedging potential of financial wealth for risky human wealth under the assumption that exogenous labor income shocks are perfectly spanned by traded asset return shocks. In their model, households are able to fully hedge the effects of exogenous labor income shocks on their total wealth, if they are not constrained by short-selling or borrowing constraints. However, in reality, an important fraction of labor income risk is idiosyncratic and thus unhedgeable. Moreover, labor income and human capital investment are endogenous choice variables for households. Therefore, households may also actively react to weak financial wealth performance by adjusting their labor supply and retirement date.

The risk–return properties of endogenous labor income have been addressed by a second direction in the literature. Cocco et al. (Reference Cocco, Gomes and Maenhout2005) argue that labor income is similar to an implicit holding of safe assets, whereas Benzoni et al. (Reference Benzoni, Collin-Dufresne and Goldstein2007) document that labor and capital income are positively correlated in the long run. Under various assumptions on the correlation between liquid and illiquid wealth shocks, Schwartz and Tebaldi (Reference Schwartz and Tebaldi2006) study optimal portfolio allocations in a model where human capital risk is not fully hedgeable. Importantly, housing property is both a consumption and an investment asset to the household. Moreover, its share of total wealth is extremely costly to adjust due to its intrinsic illiquidity. Kaplan and Violante (Reference Kaplan and Violante2014) introduce the notion of ‘wealthy hand-to-mouth’ investors to characterize households with substantial assets in the form of housing and retirement accounts, but with little liquid wealth or credit facilities to offset short-term income falls. Pension investment is the main component of financial investment for most households. Hence, it is an essential instrument to diversify risk over the life-cycle, for example, by hedging intertemporal variations in human wealth or other illiquid components of households' wealth. From this perspective, it seems a very reasonable approach the one pioneered by Swanson (Reference Swanson2012), which implies that households are allowed to adjust flexibly their pension allocations to risky assets during the last part of their life-cycle in accordance with their risk tolerance and their total wealth composition.

Various authors have tried to characterize quantitatively the optimal portfolio strategies and the welfare implications of life-cycle allocations. Under different assumptions, Cocco et al. (Reference Cocco, Gomes and Maenhout2005), Gomes and Michaelides (Reference Gomes and Michaelides2005), and Gomes et al. (Reference Gomes, Kotlikoff and Viceira2008) show that the empirically observed stock market participation rates and asset allocations can be reproduced by a calibrated life-cycle model with plausible specifications of uninsurable labor income risk and risk aversion. Fagereng et al. (Reference Fagereng, Gottlieb and Guiso2017) provide related empirical support on life-cycle portfolio allocations, by documenting a double (optimal) adjustment as households’ age: a rebalancing of the portfolio composition away from stocks as they approach retirement and a stock market exit after retirement. In a life-cycle model similar to Cocco et al. (Reference Cocco, Gomes and Maenhout2005), but with flexible labor supply, Gomes et al. (Reference Gomes, Kotlikoff and Viceira2008) investigate optimal consumption, asset accumulation, and portfolio decisions. Importantly, they quantify the welfare costs of suboptimal life-cycle allocations that mimic popular default investment choices in defined-contribution pension plans. They document that life-cycle funds designed to match investor risk tolerance and investment horizon have small welfare costs, that is, a small reduction in the utility of the household. In contrast, all other policies induce substantial welfare costs.

In summary, this literature documents potentially large welfare costs of popular default investment choices in defined-contribution pension plans that deviate from optimal life-cycle allocations and instead rely on investments in fixed income or in other ‘safe’ instruments with low yields.

2.2 Guaranteed investment strategies

Various financial intermediaries, often insurers, offer pension products with upside performance linked to an equity benchmark and downside performance limited by a contractual long-term capital guarantee, in the form of a minimum nominal guaranteed rate of return.

Therefore, the payout of these products has a typical option-like form with respect to the benchmark investment: if the benchmark investment performance falls below a minimum floor level, the contract payoff equals the floor; if the benchmark investment produces a performance above the floor, then a fraction of the payout of the policy will raise with it. There are different categories of products explicitly designed to achieve this goal. The simplest one is an index-linked or unit-linked insurance contract, which guarantees to the policyholder a return that is the maximum between the index performance and a minimum guaranteed rate (see, e.g., Hipp, Reference Hipp1996).

From the point of view of the insurance, a policy with a minimum guaranteed rate induces a long-term liability whose present value depends on the level of interest rates. When interest rates are low, the price of the liability rises and the buffer of capital that can be invested in riskier assets is reduced. This problem, known with the name of ‘lock-in’, limits the performance of an index-linked strategy in periods of low rates, inducing a payout similar to the one offered by a long-term bond. Such an allocation is known to be optimal for an investor with arbitrary low-risk propensity (see Wachter, Reference Wachter2003).

A more advanced approach to structured products with minimum guarantees can be achieved by implementing an asset-liability management strategy (see, e.g., Consiglio et al., Reference Consiglio, Saunders and Zenios2006; Committee Global Financial System, 2011). Following the traditional actuarial approach, insurance companies actively use their balance sheet to reconcile the targets of policyholders, who play the role of insurance liability holders, and shareholders, who provide funding to keep adequate levels of risk capital required by regulators to minimize insolvency risk.Footnote ³ The provision of capital by shareholders is compensated through dividend distributions. The contributions paid by the policyholders of traditional guaranteed products, which may be sold in different years and for different guarantee levels, are pooled together and invested into a corresponding portfolio of assets. The cash flows generated by the insurer's asset portfolio in any given year can then be used to fulfil the obligations deriving from the portfolio of outstanding policies. As long as these cash flows exceed the sum of all existing minimum guarantees, all insurance contracts participate to the asset portfolio excess performance.

From a broader perspective, the organization and regulation of the insurance sector are explicitly designed to optimize the diversification of individual risks through pooling and collective management. In the specific case of market risk, which is undiversifiable, whether such a collective management can be beneficial is still a subject of debate (see Hombert and Lyonnet (Reference Hombert and Lyonnet2022) and the references therein). When comparing index-linked and traditional participating life-insurance contracts, the latter allow for additional flexibility, as temporary provisions of (costly) capital may reduce the lock-in effect determined by low interest rates. Similarly, when the asset portfolio performance is high, a bonus reserve can be created, which may be used to re-inject capital when needed in the future. Maurer et al. (Reference Maurer, Mitchell, Rogalla and Siegelin2016) discuss in depth how life insurers use accounting and actuarial techniques to smooth reporting of firm assets and liabilities, seeking to transfer surpluses in good years to cover benefit payouts in bad years. The final effect of these dynamic adjustments is to reduce the policyholder return volatility. This reduction is typically offered in a uniform way throughout the full policy life-cycle, not only in some proximity of the policy liquidation date. Therefore, these investment vehicles stand apart at least in part from some of the insights of traditional financial advices on life-cycle and retirement investing.

In our analysis, for simplicity, we will largely abstract from the economic reasons that may explain the policy provider's selection of a particular form of minimum guarantee investment and the corresponding asset portfolio.Footnote ⁴ Instead, we take the perspective of a potential policyholder who wants to quantify the welfare tradeoffs of a choice between a given minimum guarantee investment policy and other benchmark life-cycle investment strategies, such as a Poterba life-cycle strategy (see Poterba et al., Reference Poterba, Rauh, Venti and Wise2006). To achieve this purpose, we basically need to model plausible asset-liability management plans that are implementable by insurance companies in order to produce the contractual cash flows of minimum guarantee investment products.

3.1 Basic specification of the VAR model

Consider the continuously compounded security market returns from time t to time t + 1, ${\bf r}_{t + 1}$, and denote with ${\bf \mu }$ the conditional expected log return given information up to time t:

(1)$${\bf r}_{t + 1} = {\bf \mu } + {\bf u}_{t + 1}, \;$$

where ${\bf u}_{t + 1}$ is the unexpected log return. Then, the τ-period cumulative return from period t + 1 through period t + τ is given by

(2)$${\bf r}_{t, t + \tau } = \sum\limits_{i = 1}^\tau {{\bf r}_{t + i}} .$$

The term structure of risk is defined as the conditional variance of cumulative returns, given the investor's information set, scaled by the investment horizon

(3)$$\Sigma _r( \tau ) \equiv \displaystyle{1 \over \tau }{\rm Var}( {\bf r}_{t, t + \tau }{\rm \mid }D_t^{{\rm Mkt}} ) , \;$$

where $D_t^{{\rm Mkt}} \equiv \sigma \{ {z_\tau^{{\rm Mkt}} \colon \tau \le t} \}$ consists of the full histories of returns as well as predictors that investors use in forecasting returns. Following Barberis (Reference Barberis2000) and Campbell and Viceira (Reference Campbell and Viceira2002), we describe asset return dynamics by means of a first-order vector autoregressive or VAR(1) model. We choose a VAR(1) as the inclusion of additional lags, even if easily implemented, would reduce the precision of the estimates:

(4)$${\bf z}_t^{{\rm Mkt}} = \Phi _0^{{\rm Mkt}} + \Phi _1^{{\rm Mkt}} {\bf z}_{t-1}^{{\rm Mkt}} + \nu _t^{{\rm Mkt}} , \;$$

where

$${\bf z}_t^{{\rm Mkt}} = \left[{\matrix{ {r_{0t}} \cr {{\bf x}_t^{{\rm Mkt}} } \cr {{\bf s}_t^{{\rm Mkt}} } \cr } } \right]$$

is a (m × 1) vector, with r _0t being the log real return on the asset used as a benchmark to compute excess returns on all other asset classes, ${\bf x}_t^{{\rm Mkt}}$ being the (n × 1) vector of log excess returns on all other asset classes with respect to the benchmark, and ${\bf s}_t^{{\rm Mkt}}$ is the ((m − n − 1) × 1) vector of returns predictors. In the VAR(1) specification, $\Phi _0^{{\rm Mkt}}$ is a (m × 1) vector of intercepts and $\Phi _1^{{\rm Mkt}}$ is a (m × m) matrix of slopes. Finally, $\nu _t^{{\rm Mkt}}$ is a (m × 1) vector of innovations in asset returns and returns' predictors for which standard assumptions apply:

(5)$$\nu _t^{{\rm Mkt}} \sim {\cal N}( {0, \;\Sigma_\nu^{{\rm Mkt}} } ) , \;$$

where $\Sigma _\nu ^{{\rm Mkt}}$ is the (m × m) variance–covariance matrix:

$$\Sigma _\nu ^{{\rm Mkt}} = \left[{\matrix{ {\sigma_0^2 } & {\sigma_{0x}^{\prime} } & {\sigma_{0s}^{\prime} } \cr {\sigma_{0x}} & {\Sigma_{xx}} & {\Sigma_{xs}^{\prime} } \cr {\sigma_{0s}} & {\Sigma_{xs}} & {\Sigma_{ss}} \cr } } \right].$$

Assuming that the VAR is stationary and thus that the moments are well-defined, the unconditional mean and variance–covariance matrix of ${\bf z}_t$ can be represented as follows:

(6)$$\mu _z^{{\rm Mkt}} = ( {I_m-\Phi_1^{{\rm Mkt}} } ) ^{{-}1}\Phi _0^{{\rm Mkt}} $$

(7)$$vec( {\Sigma_{zz}^{{\rm Mkt}} } ) = ( {I_{m^2}-\Phi_1^{{\rm Mkt}} \otimes \Phi_1^{{\rm Mkt}} } ) ^{{-}1}vec( {\Sigma_\nu^{{\rm Mkt}} } ) .$$

The conditional mean of the cumulative asset returns at different horizons are instead

(8)$$\eqalign{E_t( z_{t + 1}^{{\rm Mkt}} + \ldots + z_{t + \tau }^{{\rm Mkt}} ) = & \left({\mathop {\mathop \sum \limits_{i = 0} }\limits^{\tau -1} ( {\tau -i} ) {( {\Phi_1^{{\rm Mkt}} } ) }^i} \right)\Phi _0^{{\rm Mkt}} \cr & \quad + \left({\mathop {\mathop \sum \limits_{\,j = 0} }\limits^\tau {( {\Phi_1^{{\rm Mkt}} } ) }^j} \right){\bf z}_t^{{\rm Mkt}} , \;} $$

and their variance is:

(9)$$\eqalign{{\rm Va}{\rm r}_t( z_{t + 1}^{{\rm Mkt}} + \ldots + z_{t + \tau }^{{\rm Mkt}} ) & = \Sigma _\nu ^{{\rm Mkt}} + ( I + \Phi _1^{{\rm Mkt}} ) \Sigma _\nu ^{{\rm Mkt}} ( I + \Phi _1^{{\rm Mkt}} ) ^{\prime} \cr & \quad + ( I + \Phi _1^{{\rm Mkt}} + ( {\Phi_1^{{\rm Mkt}} } ) ^2) \Sigma _\nu ^{{\rm Mkt}} ( I + \Phi _1^{{\rm Mkt}} + ( {\Phi_1^{{\rm Mkt}} } ) ^2) ^{\prime} + \ldots \cr & \quad + ( I + \Phi _1^{{\rm Mkt}} + \ldots + ( {\Phi_1^{{\rm Mkt}} } ) ^{\tau -1}) \Sigma _\nu ( I + \Phi _1^{{\rm Mkt}} + \ldots + ( {\Phi_1^{{\rm Mkt}} } ) ^{\tau -1}) ^{\prime}.} $$

Once the conditional moments of excess returns are available, the following selector matrix extracts, for each period, τ-period conditional moments of log real returns

(10)$$M_r = \left[{\matrix{ 1 & {0_{1 \times n}} & {0_{1 \times ( {m-n-1} ) }} \cr {\iota_{n \times 1}} & {I_{n \times n}} & {0_{n \times ( {m-n-1} ) }} \cr } } \right]$$

which implies

(11)$$\displaystyle{1 \over \tau }\left[{\matrix{ {E_t( {r_{0, t + 1}^\tau } ) } \cr {E_t( {{\bf r}_{t + 1}^\tau } ) } \cr } } \right] = \displaystyle{1 \over \tau }M_rE_t( z_{t + 1}^{{\rm Mkt}} + \ldots + z_{t + \tau }^{{\rm Mkt}} ) .$$

(12)$$\displaystyle{1 \over \tau }\left[{\matrix{ {{\rm Va}{\rm r}_t( {r_{0, t + 1}^\tau } ) } \cr {{\rm Va}{\rm r}_t( {{\bf r}_{t + 1}^\tau } ) } \cr } } \right] = \displaystyle{1 \over \tau }M_r{\rm Va}{\rm r}_t( z_{t + 1}^{{\rm Mkt}} + \ldots + z_{t + \tau }^{{\rm Mkt}} ) M_r^{\prime} .$$

Using the estimated VAR coefficients, it is possible to derive unconditional and conditional moments for returns and excess returns at all different investment horizons. These moments deliver the dynamics of returns and the risk of different assets across investment horizons. This information forms the input for portfolio allocation. Following the Campbell–Viceira methodology, we consider a benchmark portfolio to be obtained by attributing optimal weights to bonds, stocks and T-bills. Therefore, in ${\bf x}_t^{{\rm Mkt}}$ we include excess returns on stocks and bonds, real returns on T-bills, while we include in ${\bf s}_t^{{\rm Mkt}}$ three factors commonly recognized as good predictors of these assets' returns. In particular, the predictors are the nominal short-term interest rate, the dividend–price ratio, and the yield spread between long-term and short-term bonds.

3.2 Data and VAR estimates

The information set includes the returns of three reference securities: (i) the real short rate, rtb, obtained as the difference between the log of the total return on a strategy investing in three-month German T-bills and the one-year CPI inflation rate; (ii) the excess return on stocks, xr, given by the difference between the log of the total return of an investment in MSCI Europe and the log-return on the German three-month T-bill; (iii) the excess return on bonds, xb, calculated as the difference between the log of the total return of an investment in 10-year German government bonds and the log-return on the German three-month T-bill. It also includes three conditioning state variables: (i) the German three-month T-bill rate, y; (ii) the log of the dividend–price ratio for a European benchmark securities portfolio, dp; (iii) the spread between the yield on the German 10-year government bond and the German three-month T-bill, spr.Footnote ⁶

In order to include in the estimation also the period when the nominal short rate hit the zero lower bound (i.e., the interval between 2012 and 2021), we modify the original CV procedure by using as predictor for the real short-term rate the shadow rateFootnote ⁷ as computed in De Rezende and Ristiniemi (Reference De Rezende and Ristiniemi2023). Notice that the resulting VAR is observationally equivalent to the original one. In fact, by definition, the nominal short-term rate is given by the maximum between 0 and the shadow rate. On the other hand, the use of the shadow rate circumvents the estimation problem determined by the short-term nominal rate being at the zero lower bound (see Carriero et al., Reference Carriero, Clark, Marcellino and Mertens2021).

The presence of the conditioning state variables is necessary to consider the modification of the long-term risk–return generated by asset return predictability. We define the vector ${\bf z}_t^{Mkt}$ as:

$${\bf z}_t^{Mkt} : = [ {rtb_t, \;xr_t, \;xb_t, \;y_t, \;dp_t, \;spr_t} ] .$$

Our simulation is based on a sample of annual observations covering the period between 1969 and 2021. This 53-year sample period includes several different scenarios for interest rates and inflation rates. Figure 1 shows the time series of the 10-year yield and 3-month rate of German government bonds and the annual inflation rate in Germany from January 1969 to December 2021.Footnote ⁸ We observe that the three-month rate reaches a level close to zero around the end of 2012, becomes negative in 2015, and then remains below zero until 2021. The 10-year yield shows a strong downward trend over the sample period, especially starting from 1992. The rate declines to 2% for the first time in 2012, then keeps decreasing and becomes significantly negative in 2019, with a minimum at −65 basis points. The inflation rate also follows a declining trend with large fluctuations in the 1969–1992 period and significantly less pronounced variations in the 1992–2020 period, and a jump to about 5% in 2021.

Figure 1. The evolution of interest rates and inflation in Germany. This figure plots monthly observations for the 10-year yield and 3-month rate on German government bonds and the annual inflation rate in Germany over the period January 1969 to December 2021.

In what follows, we analyze the performance of the minimum guarantee investment contracts (MG) and the life-cycle strategies (LC) over the 1969–2021 sample period.

We also run the VAR simulation using a truncated sample running from 1969 to 2012. This set of data therefore excludes the 2013–2021 extreme interest rate scenario driven by the Outright Monetary Transactions program carried out by the ECB since August 2012 and the subsequent Asset Purchase Programmes, that is, those non-standard monetary policy measures implemented by the ECB to support the monetary policy transmission mechanism (see, e.g., Eser et al., Reference Eser, Lemke, Nyholm, Radde and Vladu2019; Altavilla et al., Reference Altavilla, Carboni and Motto2021). A similar approach is adopted in Campbell et al. (Reference Campbell, Pflueger and Viceira2020) where the sample period ends in 2011 in order to avoid the issues related to the zero lower bound for nominal interest rates.

Table 1 reports the unconditional risk and return characteristics for the asset classes and the predictors in the estimation samples.

Table 1. Summary statistics

This table reports means and standard deviations for the three assets and the three predictors used in the estimation of the VAR model: rtb = ex-post real T-bill rate, xr = excess stock return, xb = excess bond return, dp = log dividend–price ratio, y = nominal T-bill yield, spr = yield spread. Panel A contains summary statistics for the sample 1969–2021, while panel B shows estimates for the truncated sample from 1969 to 2012.

We find that the main differences between the two samples regard the average level of the ex-post real T-bill rate and nominal T-bill yield, which are significantly higher for the 1969–2012 period excluding the zero lower bound regime, and the Sharpe ratio on stock returns, which is almost 30% higher for the longer sample period.

Figure 2 plots, for each sample, the term structure of volatilities and correlations, that is, the variation with the horizon of the riskiness, measured by the standard deviation of ex-post real rate of returns on a yearly basis, and of the correlation between the three asset classes under consideration. The estimated term structures of volatility are quite homogeneous across the two samples, while the term structures of correlation show some significant differences. In particular, the correlation between short-term and long-term bonds is relatively flat for long horizons in the truncated sample and instead increasing with the time horizon in the case of the full sample.

Figure 2. Term structure of volatility and correlations. This figure plots the term structure of volatilities and correlations, that is, the variation with the time horizon of the standard deviation of the ex-post annual real rate of returns and of the correlation between the three asset classes under consideration: short-term bonds, long-term bonds and stock market index. Panel A reports values for the full sample from 1969 to 2021, while Panel B shows values for the truncated sample from 1969 to 2012.

The estimated term structure of risk does not show relevant differences compared to the original CV results. As discussed by Campbell et al. (Reference Campbell, Sunderam and Viceira2009) and David and Veronesi (Reference David and Veronesi2013), the correlation between fixed income and equity investment returns may vary across different economic regimes and is difficult to capture within the present linear homoscedastic modeling approach. On the other hand, we use the VAR simply as a scenario generator, thus the heterogeneity of the correlation structure across different regimes could be used to test the robustness of the results with respect to different correlation structures.

Table 2 reports the estimated coefficients for the VAR specifications. Remarkably, the estimated coefficients show that the three state variables – the shadow rate, the dividend–price ratio, and the term spread – are all significant predictors for the three relevant asset classes in both samples. These predictors improve the accuracy of estimation of the long-term risk–return tradeoff. In fact, they capture the slow-moving trend and thus remove the volatility component that would be artificially induced by assuming a misspecified unconditional model with constant long-term mean that are still popular in the simulation analysis of pension products.

Table 2. Estimated coefficients of the VAR model

This table reports summary statistics for the estimated coefficients of the VAR specification. The variables included in the model are: rtb = ex-post real T-bill rate, xr = excess stock return, xb = excess bond return, dp = log dividend–price ratio, y = nominal T-bill yield, spr = yield spread. Panel A reports estimates based on the full sample from 1969 to 2021, while panel B shows estimates for the truncated sample from 1969 to 2012. t-statistics in parentheses.

The differences that emerge between the coefficients in the two samples reflect the effect of the inclusion of the negative interest rate regime and will be used in our discussion to evaluate the robustness of the results with respect to changing economic conditions. In other words, we rely on the use of the different regimes to assess the potential costs that could arise in different long-term savings strategies as the underlying scenarios vary.

4.1 The simulation approach

Contribution scheme. We fix a contribution (wealth accumulation) phase of 40 years, that is, the worker joins the defined-contribution (hereafter DC) pension plan at age 25 and leaves it after retirement at age 65. The following assumptions of pension contributions and wage evolution further underly our computations: (i) an annual initial wage of 18,000 euro, which corresponds approximately to the current euro area average net income; (ii) an annual wage growth rate of 2%, consistent with a deterministic wage growth rate compatible with euro area inflation and productivity growth rates;Footnote ¹⁰ (iii) a monthly wage contribution of 10% to the DC pension plan.

Under these assumptions, the worker/investor starts contributing 150 euro, that is, 10% of her initial monthly wage, to the DC pension plan. Every following month, the wage increases by 2% on an annual basis, implying a final annual salary at the retirement date of about 40,000 euro. Therefore, at the retirement date the fix percentage contributions of 10% of monthly salaries give rise to total contributions of about 110,000 euro for the DC pension plan.

We exemplify the performance of different allocation strategies by comparing the performance of different pension products using the same set of initial conditions and the same set of simulation scenarios.

Asset return dynamics. The first step to set up the simulation approach is common to both life-cycle and life-insurance investment analyses. In each simulation run, we use the VAR model to generate 5,000 possible scenarios for the random evolution of the market returns and the state variables. Our input data will be the annual rates of returns produced by the set of tradable instruments for a number of years sufficient to determine the final performance of the alternative strategies. The state variables determine information on the inflation rate, the dividend–price ratio, and the yield spread.

From each simulation, we compute the time series of real returns for each of the three tradable asset classes:

$$R_{t + 1}: = ( {r_{t + 1}^{EQ} ( \omega ) , \;r_{t + 1}^{SB} ( \omega ) , \;r_{t + 1}^{LB} ( \omega ) } ) , \;$$

where ω = 1, …,  NoPaths, t = 0, …,  T − 1 and EQ, SB, and LB denote Equity, Short-term Bonds, and Long-term Bonds, respectively.

This comparison will be performed in real terms in order to take into account the erosion of the portfolio value driven by inflation.

Portfolio strategies. An asset allocation strategy $\Pi _t = ( {\pi_t^{EQ} , \;\,\pi_t^{SB} , \;\,\pi_t^{LB} } )$ produces a return between time t and time t + 1 along the path ω, $R_{t + 1}^\Pi ( \omega )$, that is given by:

$$R_{t + 1}^\Pi ( \omega ) = \pi _t^{EQ} r_{t + 1}^{EQ} ( \omega ) + \pi _t^{SB} r_{t + 1}^{SB} ( \omega ) + \pi _t^{LB} r_{t + 1}^{LB} ( \omega ) , \;\quad t = 0, \;\ldots , \;T-1.$$

We assume that the account balance is regulated on a yearly basis, so that the return $R_{t + 1}^\Pi ( \omega )$ is capitalized in the account on a yearly basis. Administrative costs are charged up-front and correspond to 0.5% of the contributions and for each year the asset management fees are assumed to be equal to 0.5%.

4.2 Implementation of strategies

In what follows, we sketch the main steps that are necessary to compute the investment performance for each path of the asset returns and for each investment approach. First, we detail the asset-liability management procedure adopted by the insurance company to dynamically adjust the balance sheet in order to (i) service the policy holders, (ii) fulfill the regulatory requirements, and (iii) compensate the shareholders. In essence, this procedure aims at calculating the minimum guaranteed rates that an insurer can offer to comply with these three objectives. Then, we describe the life-cycle investment strategy adopted in the simulations, which is based on the ‘Poterba age-based scheme’ (see Poterba et al., Reference Poterba, Rauh, Venti and Wise2006).

4.2.1 Participating life insurance with capital guarantee

We quantify the performance of the minimum guarantee strategy implementing a traditional participating life insurance contract. We concentrate solely on the analysis of the financial risk, thus abstracting from the actuarial risk to avoid the need of using biometric data. The following procedure builds on the approach proposed by Graf et al. (Reference Graf, Kling and Ruß2011) and Eling and Holder (Reference Eling and Holder2013) and combines a risk-neutral and an historical-risk measure to simultaneously analyze the performance of the strategy from the point of view of the insurance company, the policyholder, and the regulator.

In order to compute the benefit distributed to the policyholder, L _T(ω), it is necessary to simulate a simplified model for the balance sheet of the insurer and for its evolution.

When a policy of this type is sold, the policyholder agrees to pay the annual contribution and in exchange the intermediary, that is, an insurance company, commits to credit the policyholder's account with a minimum nominal interest rate G each year. Since we intend to analyze the real return to the policyholder, the effective guaranteed real rate is the level G net of the ex-post inflation rate π _t measured during year t, .

Both policyholders and shareholders participate in the investment performance and receive a fraction of the annual surplus. To align the interests of shareholders to those of policyholders, the annual dividend d _t credited to the shareholders is usually assumed to be equal to a fraction of the surplus credited to the policyholders. The evolution of the single period asset performance is determined by the equation:

$$A_{t + 1}^-{ = } ( {1 + R_{t + 1}^\Pi ( \omega ) } ) A_t^ + , \;$$

where $A_t^-$ and $A_t^ +$ are the time t assets prior and posterior, respectively, to the distribution of the dividends and the investment of the collected premium:

$$A_t^ + { = } \max \left\{{\mathop A\limits_-^t -d_t, \;L_t} \right\} + P_t + c_t, \;$$

with P _t defined as the individual contribution at time t net of management fees and c _t as a capital injection. Notice that if the accumulated asset, net of dividends, falls short of the liabilities, the following capital injection is required

$$c_t = \max \{ {L_t-A_t^-{ + } d_t, \;0} \} $$

and the shareholders must be compensated for its provision.

The liabilities evolve as follows:

$$L_{t + 1} = ( {1 + G_t} ) L_t + sur_t, \;$$

where sur _t is an additional distribution, whose exact expression is detailed in Eling and Holder (Reference Eling and Holder2013), which implements the German legal requirements and the traditional surplus distribution policy described below.

The exact value credited by the insurance company to the policyholder is determined by a target rate z as long as the reserve quota x _t, $x_t: = ( ( A_t^-{-}d_t-L_t) /L_t)$, lies within a predetermined region x _t ∈ [a,  b]. If the reserve quota falls below a, the interest rate credited to the policyholder is the maximum between the guaranteed interest rate and the level of the participation rate that restores x _t to the minimum capital requirement value a. If crediting the target rate z leads to a reserve quota above b, then the company credits exactly the rate necessary to set x _t = b. In our analysis, we will assume a = 0.05 and b = 0.30, while in the simulations the target rate z is set in the range 3–5%.

According to the German legislation, at least δ = 90% of the earnings on book values must be distributed to the policyholders. Therefore, we assume that a fraction η = 0.7 of the market value $A_{t + 1}^-{-}A_t^ +$ is distributed to the policyholders and the final participation to the surplus is given by:

$$\delta \eta ( {A_{t + 1}^-{-}A_t^ + } ) , \;$$

while the dividend distributed to the shareholders is a fraction α = 0.05 of the surplus:

$$\alpha \delta \eta ( {A_{t + 1}^-{-}A_t^ + } ) .$$

Pricing fairness is guaranteed by ensuring that the risk-neutral present value of the dividend distributed to the shareholder d _t(ω) is sufficient to compensate the capital provisions c _t(ω), that is, it must be larger than (or equal to) the risk-neutral discounted value of future capital provisions net of the total change in capital reserves:

(13)$$E^{\rm {\mathbb Q}}\left[{\mathop \sum \limits_T^{t = 1} \displaystyle{{d_t-c_t} \over {B_t}}} \right] + \left\{{E^{\rm {\mathbb Q}}\left[{\displaystyle{{e_T} \over {B_T}}} \right]-e_0} \right\}\simeq 0, \;$$

where $e_t = A_t^-{-}L_t-d_t$ denotes the residual value of the reserve account at time t.

In order to compute the risk-neutral probability of the different scenarios, we assume a stochastic discount factor driven by the VAR innovations for the excess returns on the stock market index and the long-term bond:

$$\eqalign{-\log ( {m_{t + 1}} ) & = r_{0, t} + \displaystyle{1 \over 2}[ {\Lambda_{EQ}, \;\Lambda_{LB}} ] \Omega _{2x2}[ {\Lambda_{EQ}, \;\Lambda_{LB}} ] ^{\prime} \cr & \quad + [ {\Lambda_{EQ}, \;\Lambda_{LB}} ] [ {\varepsilon_{t + 1}^{xr} , \;\varepsilon_{t + 1}^{xb} } ] ^{\prime}, \;} $$

where r _0,t is the rate of return on the short-term bond as determined by the VAR dynamics; Λ_EQ and Λ_LB are the market prices of risk for stocks and for long-term bonds, respectively, which are assumed to be constant and set equal to the historical values in the sample used to estimate the VAR; Ω_2x2 is extracted from the variance–covariance matrix of the VAR residuals selecting the variances and covariances between equity and long-term bond excess returns.

The risk that the management strategy of the guarantee is unsuccessful and the account balance does not break even is then quantified considering two risk measures: the Probability of Shortfall (PS)

(14)$$PS: = {\rm {\mathbb P}}( {A_T < L_T} ) , \;$$

and the Expected Shortfall (ES)

(15)$$ES: = {\rm {\mathbb E}}^{\rm {\mathbb P}}[ {{\rm {\mathbb I}}_{\{ {A_T < L_T} \} }( {L_T-A_T} ) } ] .$$

To simplify the analysis, we will consider as acceptable a management strategy if the relative ES, that is, the ratio between the ES and the present value of future contributions, NPV _C, is smaller than 0.5%:

(16)$$\displaystyle{{ES} \over {NPV_C}} < 0.5\% .$$

The probability of shortfall is considered as a reference measure to assess the frequency of the paths where the insurance management of the policy does not break even.

The level of the minimum guarantee is computed imposing: (i) condition (13), that is, contract fairness and (ii) condition (16), that is, a solvency condition.

4.2.2 Life-cycle Poterba-style investment strategy

The Poterba age-based scheme dictates a percentage wealth allocation to stocks which decreases with the age of the investor. So, for example, if the wealth allocation to stocks at age 25 is 75%, the stock allocation at the retirement age of 65 is 35%, if we decrease the wealth allocation by 1% each year. In this investment strategy, the individual account A _t is updated following the same rule of total assets under management:

$$A_{t + 1} = ( {1 + R_{t + 1}^\Pi ( \omega ) -apf} ) ( {A_t + P_t} ) , \;$$

where P _t has been defined above as the individual contribution at time t net of the management fee and apf is a management fee on the annual performance which accounts for the costs of trading. We apply apf = 0.5%.

In order to reduce the disinvestment risk, we assume that the pensioner stops the contributions at the age of 65 and then may choose either to liquidate the investment, if the amount is higher than the money back amount, or wait until the age of 72. This simplified scheme is assumed to mimic the role of more sophisticated disinvestment options that are known in the literature.Footnote ¹¹

4.3 Asset allocation strategies

This section specifies the structure of (i) minimum guarantee investments, with a participation life insurance policy that can be offered at market prices, and (ii) Poterba-style life-cycle target-date investment funds.

We assume that the asset allocation of the minimum guarantee contracts is 5% in equity and 95% in long-term bonds:Footnote ¹²

(17)$$\pi ^{EQ} = 5\% {\rm ; \;}\quad \pi ^{LB} = 95\% , \;$$

where π ^EQ and π ^LB are constant percentage wealth allocations to equities and long-term bonds, respectively.

Note that in practice there are structural reasons that force an insurer who sells a guaranteed product to allocate most of the wealth in long-term bonds and thus to reduce the diversification of the asset allocation. First, the principle of duration matching, which implies that to lower the exposure of the balance sheet to interest rate volatility risk, the duration of assets must be close to the duration of liabilities, which is high due to the presence of the guarantee on the liability side. Second, capital requirements for equity are higher than those for bonds according to Solvency II regulation.

For comparison with the minimum guarantee investment strategy supported by the above allocation, we consider three Poterba-style life-cycle strategies with the following time-varying allocations to equities and long-term bonds only:

(18)$$\eqalign{& {\rm Low\ equity\colon }\;\pi _t^{EQ} = \displaystyle{{85-\tau } \over {100}}, \;\cr & {\rm Medium\ equity\colon }\;\pi _t^{EQ} = \displaystyle{{100-\tau } \over {100}}, \;\cr & {\rm High\ equity\colon }\;\pi _t^{EQ} = \displaystyle{{115-\tau } \over {100}}} $$

and $\pi _t^{LB} = 1-\pi _t^{EQ}$ in all cases. In equation (18), τ is the age of the life-cycle investor, ranging from an initial age of 25, when starting the life-cycle strategy, and an age of 65 at retirement. The choice to focus on zero allocations to cash also in the life-cycle strategy is taken in order to make the comparison more consistent with the implications of the allocation for minimum guarantee investment products. In this way, we can focus on a comparison with the long-term risk–return tradeoff resulting only from a time-varying allocation to equity and long-term bonds in the life-cycle strategy.

4.4 Computation of minimum guarantees

For simplicity, we assume an insurer's balance sheet that is regulated on a yearly basis. Therefore, the random yearly return $R_{t + 1}^\Pi$ on the insurer's asset portfolio is capitalized in the account with a yearly frequency, for each of the allocation scenarios introduced in the previous section. Moreover, a fee of 50 basis points is charged up-front on every policy contribution to cover the sales costs. Finally, the yearly asset return credited to policy holders is reduced by 50 basis points to cover the asset management fees.

The minimum guaranteed rate depends on market conditions and is determined using an iterative numerical procedure, which implies that the level of the minimum guarantee is lowered (raised) until the condition on contract fairness (13) and the solvency constraint (16) are simultaneously satisfied. We find that the minimum annual guaranteed rate affordable by the insurance company is equal to G ^F = 2.25% for the full sample period 1969–2021 and G ^T = 4.25% for the truncated sample 1969–2012.