Comparison with CRRA

We conclude the case study section by exploring the impact of minimum consumption and wealth floors on calibration and optimal controls. For this sake, we fit the model Maptq,bptqto the very same parameters and target curves as before, but now enforce ¯cptq ” 0 and F ” 0. This CRRA model is referred to as M_aptq,bptq^CRRA. Table 4.4 provides the estimated parameters and the sum of the squared relative residuals. In terms of this sum, it is clear that model Maptq,bptqprovides a more adequate fit than model M_aptq,bptq^CRRA, its sum is only 4.82% of the sum which corresponds to M_aptq,bptq^CRRA. Going even further, all three benchmark models Ma,bptq, Maptq,b and Ma,b from the previous subsection, which all consider minimum levels for consumption and wealth, provide a more precise fit than M_aptq,bptq^CRRA in view of the sum of squared relative residuals. This shows that the introduction of floors for consumption and wealth in the model is essential.

Figure 4.7 visualizes the estimated input functions, Figure 4.8 provides the graphics about the fitted consumption and relative risky portfolio process with the expected wealth and stock price path. Besides a larger sum of the squared relative distances for model M_aptq,bptq^CRRA, especially the fitted risky investments ˆπ^‹pt; v0qin Figure 4.8 show that zero floors for consumption and wealth (¯cptq ” 0 and F ” 0) leads to an imprecise calibration and a large deviation from its given target curve due to a drop in model flexibility. Table 4.4 suggests that this drop in flexibility is attempted to be compensated by a higher risk aversion in terms of more negative estimated values for ˆb and bptq, see also Figure 4.7.

Sum of squared relative distances

ˆb aptq bptq

M_aptq,bptq 6.0425 ´0.9849 a₀ “5.2864 ˆ 10⁷,

λa“ ´0.6673

b₀ “ ´4.9731, λb “ ´0.0340 M_aptq,bptq^CRRA 125.3497 ´4.4867 a0 “0.6238 ˆ 10⁷,

λ_a“ ´0.8689

b0 “ ´9.7397, λ_b “ ´0.0192 Table 4.4: Calibrated parameters and sum of squared relative residuals for CRRA.

40 35 30 25 20 15 10 5 0

0 1 2 3 4 5 6 10⁷

(a) aptq.

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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

(b) bptq.

Figure 4.7: Estimated preference functions aptq and bptq for CRRA.

40 35 30 25 20 15 10 5 0 2

2.5 3 3.5 4 4.5 10⁴

(a) Fitted consumption rate.

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0.2 0.4 0.6 0.8 1 1.2

(b) Fitted relative risky investment.

40 35 30 25 20 15 10 5 0

4 6 8 10 12

10⁵

(c) ErV^‹pt; v0qs.

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100 200 300 400 500 600 700 800

(d) ErP ptqs “ p1e^µt.

Figure 4.8: Fitted expected consumption rate c^‹pt; v0q and relative risky investment ˆπ^‹pt; v0q, ex-pected wealth process ErV^‹pt; v0qs and stock price process ErP ptqs for CRRA.

“Nahles-Rente”/“Sozialpartnermodell”

An optimal policy has the property that whatever the initial state and ini-tial decision are, the remaining deci-sions must constitute an optimal pol-icy with regard to the state resulting from the first decision.

Richard Bellman

In the recent low interest rate environment, traditional pension products allocate a high fraction of wealth to defensive assets and thus offer only a relatively small expected return on the investments due to promised guarantees. By this, the pension fund wealth of a client grows at a very small rate and thus the future pension payments will be rather low. Generally, clients seek for and desire a stable evolution of their reported wealth (and their pension) at a high expected return and with a limited downside. To allow for a performance or return seeking characteristic, the new “Nahles-Rente” pension product basically comes with no pension cash flow guarantee at all. The wealth accumulation phase shows similarities to a defined contribution (DC) plan, but with an additional smoothing process to stabilize the reported wealth evolution. The wealth decumulation phase can be regarded as a generalization of a defined benefit (DB) plan, where the pensions stay constant as long as the wealth remains inside a pre-defined corridor. Some more details, information and current status can be found in aba and IVS (2017) and Pohl (2019). For a proposal and some discussion of an alternative model formulation that designs a pension product without guarantees we refer to Boado-Penas et al. (2020), where the modeling is related to but differs from our approach both on the accumulation and decumulation part. Within the accumulation phase prior to retirement, the “Nahles-Rente” thus provides a flexible setup that allows seeking for higher returns compared to products with guarantees and managing the total fund’s wealth with two accounts: a reported wealth balance (primary account) and a buffer balance (secondary account). The latter aims to smooth the reported wealth and to accumulate some buffer amount that can be used during the post-retirement phase to decrease the probability of pension shortenings as clients generally fear reductions in pension payments. As this new pension product is currently in a development stage and is being built up, we study the impact of the associated model. This chapter is a reproduction of Lichtenstern and Zagst (2020) with minor changes. In Section 5.1 the portfolio problem associated with the accumulation phase is studied, whereas Section 5.2 deals with the decumulation phase.

Generally we refer to Chapter 4 for a detailed literature overview on optimal investment management over the entire life-cycle. Additional specific publications for this chapter are cited at the relevant places.

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5.1 Accumulation phase

In this section we present an innovative optimal investment strategy for the accumulation phase of a private pension insurance plan or pension saving scheme under a portfolio smoothing mechanism.

The pension fund consists of two accounts: the primary account or investment portfolio is reported to the customers, the secondary account or buffer account acts in the background. Reporting a smoothed, but steadily increasing portfolio wealth or primary account balance is quite important for insurance companies and their clients, for instance due to planning stability and to avoid sur-prising drops in the level of future expected annuity payments. In general, it is unclear how an optimal portfolio and investment strategy look for a given buffer scheme. The main findings and contributions comprise the closed-form solution to the continuous-time optimal investment problem under a wealth-dependent buffer scheme, the derivation of risk and return figures and the presenta-tion of an explicit parametric setting such that the terminal buffer balance is always non-negative.

Finally, within a numerical case study, we illustrate the optimal portfolio policy and the stochastic accumulation of a buffer account and its benefits compared to asset allocation strategies without a buffer rule.

The accumulation phase can be modeled on a single client or on a cohort basis with individual buffer accounts. We assume that customers belonging to the same age-cohort can be managed as a group and that the group specific collective portfolios and buffer processes can be managed separately. A cohort-specific account or portfolio and a cohort-specific buffer account is assigned to each cohort. This allows to apply a dynamic, age-dependent (besides a market-dependent) investment strategy, that can invest the pension fund’s asset into the N ` 1 assets, which may add value to the clients wealth compared to a constant capital allocation mix. The number of cohorts m is specified as the oldest age cohort of the participants and mj is the number of people in age cohort-j for j “ 1, 2, ..., m. We assume that plan members in cohort-j are endowed with a salary process yjptq which has deterministic dynamics. A certain proportion λ of the salary is paid into the plan continuously restricting us to the inflow model λyjptq(employee) and ηpλyjptqq(employer), i.e. the total contribution rate is given by p1 ` ηqλyjptq. The standard example is λ “ 4% and η “ 15% for all ages during pre-retirement. It is important to note that intertemporal changes to other (“Nahles-Rente”) pension products are excluded in this work for the sake of simplicity. Within our model we implicitly assume all customers to survive until retirement entry time T . Another reason for not considering mortality in the accumulation phase is the specific product design: if a client dies prior to retirement time T , the fund does not pay out any bequest payments. The contributions are then invested in the financial market following custom-tailored dynamic investment strategies for each cohort. The specific investment strategy is obtained by solving the expected utility maximization of the investment portfolio accumulated until retirement. This problem is explained and solved in what follows. The problem needs to be solved for every cohort. Because of keeping the notation simple and understandable we use a quite general notation. For instance, the inflow process (premium or contribution to the pension fund) of one cohort is simply denoted by yptq instead of p1 ` ηqλyjptq for cohort j.

The part on the accumulation phase is structured as follows: Section 5.1.1 introduces the considered financial market model, formulates the pension fund dynamics of this new German pension product and states the associated portfolio optimization problem under a general buffer rule. In Section 5.1.2 we propose a specific wealth-dependent buffer mechanism and determine the corresponding optimal asset allocation policy, the optimal reported and total wealth accounts and buffer balance in closed-form. As the applied wealth-dependent buffer system can lead to a negative total buffer

amount, we further elaborate on a flexible parametric setup that allows for a smoothing benefit besides a positive terminal buffer balance. Moreover, analytic formulas for risk and return measures are derived. The obtained results for the optimal strategy and balances are visualized and analyzed in a numerical case study. All proofs are stored in Appendix C.