# Intergenerational Risk Sharing in a Collective Defined-Contribution Pension
System: A Simulation Study with Bayesian Optimization

An Chen, Motonobu Kanagawa, and Fangyuan Zhang Institute of Insurance Science,
Ulm University, Germany<EMAIL_ADDRESS>Science Department, EURECOM,
France<EMAIL_ADDRESS>of Insurance Science, Ulm
University, Germany. (Corresponding<EMAIL_ADDRESS>
###### Abstract

Pension reform is a crucial societal problem in many countries, and
traditional pension schemes, such as Pay-As-You-Go and Defined-Benefit
schemes, are being replaced by more sustainable ones. One challenge for a
public pension system is the management of a systematic risk that affects all
individuals in one generation (e.g., that caused by a worse economic
situation). Such a risk cannot be diversified within one generation, but may
be reduced by sharing with other (younger and/or older) generations, i.e., by
intergenerational risk sharing (IRS). In this work, we investigate IRS in a
Collective Defined-Contribution (CDC) pension system. We consider a CDC
pension model with overlapping multiple generations, in which a funding-ratio-
liked declaration rate is used as a means of IRS. We perform an extensive
simulation study to investigate the mechanism of IRS. One of our main findings
is that the IRS works particularly effectively for protecting pension
participants in the worst scenarios of a tough financial market. Apart from
these economic contributions, we make a simulation-methodological contribution
for pension studies by employing Bayesian optimization, a modern machine
learning approach to black-box optimization, in systematically searching for
optimal parameters in our pension model.

##  1 Introduction

Pension reform is a crucial societal problem in many countries. Traditional
pension schemes, such as Pay-As-You-Go (PAYG) and Defined Benefit (DB) pension
systems, are becoming unsustainable because of the potential mismatch of
incoming and outgoing cash flows (i.e., contributions from working generations
and payments to retirees) driven by economic challenges such as demographic
change and long-term low interest rates. Many countries are addressing this
unsustainability, e.g., by “de-risking” DB plans (Lin et al.,, 2015; Kessler,,
2019) or by substituting the traditional schemes with fully or partially
funded pension schemes (e.g, Bodie et al., 1988; Poterba et al., 2007;
Broeders et al., 2013; Chen and Delong, 2015; Chen and Uzelac, 2015).

Defined Contribution (DC) pension plans are a representative example of funded
pension schemes. In a pure DC pension plan, retirement saving becomes a matter
of individual investment: Each participant chooses her investment plan for
retirement saving and pays a contribution to the plan on a regular basis; the
retirement benefits are determined by the performance of the invested plan and
the contributions. Examples of pure DC plans include the “Individual
Retirement Accounts” and the 401(k) plans in the United States (Copeland,,
2011).

However, the pure DC scheme has two critical problems as a public pension
system. One problem is that each individual needs to bear the investment risk.
Not everyone has the ability and time for making a good investment decision
(Benartzi and Thaler,, 2001). Another problem is that there may be an
unavoidable risk common for all individuals belonging to the same generation.
For instance, such a risk arises if the working period of this generation
largely overlaps with a worse economic period. Pure DC plan participants
cannot manage such a systematic risk.

###  1.1 Intergenerational Risk Sharing (IRS)

Collective Defined Contribution (CDC) pension schemes with multiple
generations can address the key problems of pure DC plans, and thus may be
more appropriate as a public pension system. In a CDC plan, each participant
pays a fixed contribution to the pension fund on a regular basis. Different
from pure DC plans, the CDC fund manages and invests the contributions from
all participants together. The retirement benefits of each participant are
determined by her accumulated contributions and the fund’s investment
performance. As the retirement benefits are essentially a redistribution of
accumulated contributions from all participants with different generations,
the CDC scheme can naturally implement intergenerational risk sharing (IRS),
i.e., risk sharing among different generations.

The core idea of IRS is as follows. Suppose that one generation (say
generation A) suffers more from a worse economic situation than other
generations, simply because this generation’s working period overlaps largely
with the worse economic period. Such a risk cannot be easily diversified
within one generation, but may be mitigated by sharing the risk with other
generations. For instance, there may be another generation (say generation B)
that benefits from a good economic situation more than other generations. By
sharing this surplus benefit of generation B with the unfortunate generation
A, the risk of this generation A can be reduced.111For classical analysis of
IRS, see e.g., Diamond, (1977); Gordon and Varian, (1988); Shiller, (1999).

In the literature on collective pension funds (including CDC funds), IRS has
been studied by many researchers. For example, Gollier, (2008) theoretically
proves that, under a certain assumption, a CDC fund can improve the welfare of
all generations (compared to the situation where each generation invests
individually and optimally) by enhancing the risk-taking ability of each
generation. Beetsma and Bovenberg, (2009) consider a two-period overlapping
generations model, and show that a collective DB scheme with an appropriate
investment policy can achieve the optimal IRS. A simulation study by Cui et
al., (2011) shows that a hybrid pension fund (in which both contributions and
retirement payments can be adjusted) can achieve the ex-ante Pareto welfare
improvement for all generations. Kurtbegu, (2018) studies a pension model
imitating a Dutch CDC pension fund for the time-horizon of 150 years, and
compares it with other pension systems by simulations, concluding that the CDC
pension system appears to be the best in terms of the IRS against demographic
shocks. The Dutch pension system is an example of CDC pension plans in
reality, and more analyses and details of it are available in, e.g., Bovenberg
et al., (2007); Nijman et al., (2014); Bovenberg and Nijman, (2019);
Westerhout, (2020).

###  1.2 Contributions

We contribute to the literature by investigating how intergenerational risk
sharing can be implemented in CDC pension systems. We consider a CDC pension
system with an overlapping generations model. Like pure DC funds, each
participant has an “individual benefit account” that keeps track of the asset
belonging to this participant. Each participant pays regular fixed
contributions to the pension fund, which are collected in the individual
benefit account. The pension fund invests its asset (originally the
contributions from the participants) in the financial market, and adjusts the
declaration rate, which determines the growth rate of individual benefit
accounts, according to the fund’s investment performance. Each participant’s
retirement benefits are determined by the terminal value of her benefit
account at the time of retirement.

For the adjustment of the declaration rate, we use the funding-ratio-linked
declaration rate studied in Goecke, (2013). This approach formulates the
declaration rate as a stochastic process depending on the expected log asset
return and the funding ratio — the ratio of the fund’s asset and liability.
For instance, given a good (resp. bad) fund’s investment performance, the
funding ratio becomes accordingly larger (resp. smaller), and the declaration
rate is increased (resp. decreased) so that each participant’s benefit account
grows more rapidly (resp. more slowly).

In this paper, we study the functionality of the funding-ratio-linked
declaration rate as a means of IRS in the CDC pension scheme. Suppose that a
worse period in the financial market has started at a certain point of time.
As the younger generations’ working periods may overlap with the worse period
much longer than the older generations, this results in higher risks for the
younger generations. In contrast, the worse economic period would result in
lower risks for the older generations, since their remaining working periods
are shorter and thus their assets are damaged less severely. Therefore in this
situation, the IRS should work for sharing the higher risks of the younger
generations with the older generations.

The funding-ratio-linked declaration rate automatically achieves this IRS.
Suppose that the worse economic situation makes the fund’s investment
performance worse. Then the declaration rate is automatically decreased,
letting each benefit account grow more slowly. This adjustment has a stronger
impact on the older working generations, since their retirement dates are
closer than for younger generations. It essentially results in pooling some
assets from the older generations to redistribute later to the younger
generations to reduce their risk. Similar reasoning can be made for a better
economic period, in which case younger generations share the risk of older
generations.

Our pension model has two parameters that need to be specified: one
determining the investment strategy and the other specifying the strength of
adjustment in the funding-ratio-linked declaration rate. We determine these
two parameters by formulating and solving a maximization problem of the
expected utility of a hypothetical social planner. We assume that this social
planner takes into account the retirement benefits of all generations
including those in the future. Since there is no analytic solution for the
maximization problem, we numerically search for the optimal parameters with
simulations using Bayesian optimization.

We perform an extensive simulation study covering several different settings
of i) the degree of relative risk aversion that decides the planner’s attitude
towards the risk, and ii) the market price of risk, given as the Sharpe ratio
of the expected market return divided by the volatility of the market. We
obtain the following observations regarding the functionality on the funding-
ratio-linked declaration rate as a mechanism for IRS, which we report in
Section 5:

  1. 1.

The evolution of each individual benefit account becomes much smoother than
the corresponding account in a pure DC with the same investment strategy. This
is due to the smoothing effect of IRS.

  2. 2.

IRS works particularly effectively in a tough market where the market price of
risk is low. In particular, in worst scenarios of the financial market, the
IRS protects the participants from losing too much, compared to the
corresponding participants in a pure DC plan.

These observations indicate that the CDC scheme with the funding-ratio-linked
declaration rate is promising as a public pension system.

Apart from the above economic contribution, we make a “meta” contribution in
terms of simulation methodology for studying pension systems. To numerically
optimize the two parameters in our pension model, we use Bayesian Optimization
(Shahriari et al.,, 2016), a modern machine learning approach to optimizing a
black-box objective function. The objective function in our case is the
mapping from the two parameters to the resulting expected utility. Since it
does not have an analytic expression, we numerically approximate the expected
utility by the Mote Carlo average over 10,000 simulations of the financial
market, where each simulation is over 100 years. As such, one evaluation of
our objective function for given candidate parameters is computationally
expensive.222With the authors’ computational environment, one evaluation of
the objective function requires a few minuets. Thus, if we evaluate 100
candidate pairs of the two parameters, in total we need more than one or two
hours to finish. For the optimization of such an objective function, brute-
forth approaches such as grid search have been used in the pension literature
[e.g., Cui et al., 2011; Bams et al., 2016]. However, grid search is
inefficient and requires a large number of function evaluations to obtain a
reliable solution. Bayesian optimization is much more efficient than grid
search, as it iteratively refines the search space by quantifying the
uncertainty of the response surface of the objective function based on
evaluated function values. Since Bayesian optimization can result in optimal
parameters with a smaller computational budget, this opens up many
possibilities for studying more sophisticated but more computationally
intensive models with many parameters. Thus, our “meta” contribution is to
show that Bayesian optimization is a useful tool for simulation studies of
pension systems in general.

The paper is organized as follows. Section 2 presents our pension model, and
Section 3 describes how Bayesian optimization can be used for selecting the
parameters in the pension model. Section 4 studies the optimal parameters
obtained for several different settings and the resulting simulation paths.
Section 5 investigates the intergenerational risk sharing in our pension model
by comparing it to the corresponding pure DC plan. Section 6 concludes.
Additional results are reported in the appendix.

###  1.3 Related work

Our work is closely related to a line of research on the IRS in CDC pension
models. Baumann and Müller, (2008) study the IRS in a CDC pension model using
a declaration rate based on the risk-free rate adjusted with the funding-ratio
process. They analyze the optimal investment strategy from the viewpoint of
one generation, and show that the Pareto improvement of the welfare can be
achieved, assuming a self-financing continuous model. Goecke, (2013) considers
a self-financing pension model similar to Baumann and Müller, (2008), but with
a declaration based on the expected asset return adjusted with the funding-
ratio process. This funding-ratio-linked declaration rate is what we study in
our work. Goecke, (2013) analyzes the properties of the resulting funding-
ratio process.

One key issue in pension reform is the mismatch of the incoming and outgoing
cash flows (i.e., the contributions from working generations and the payments
to retirees). Hence, the assumption that the pension fund is self-financing
(i.e., there is no mismatch between the cash flows) is not realistic in this
regard. There are a few works that extend the self-financing model of Goecke,
(2013) to more realistic CDC pension funds with multiple generations, e.g., by
explicitly modeling cash flows. Bams et al., (2016) consider a CDC pension
model with multiple generations similar to ours, in which the declaration rate
of Goecke, (2013) and a constant-mix investment strategy are used. Donnelly,
(2017) studies a CDC pension model with multiple generations, in which the
investment strategy is adjusted to achieve a target funding-ratio. She shows
that the resulting retirement benefits are more stable than those in a
corresponding pure DC plan in the sense that their variations become narrower.

We contribute to this line of research by providing novel insights about the
functionality of the funding-ratio-linked declaration rate as a means of IRS
in a CDC pension system. By thoroughly studying different settings for the
market price of risk of the financial market (captured by the Sharpe ratio)
and the risk attitude of the social planner (or the fund manager), we show
that the IRS works particularly effectively in worst scenarios of a tough
market for protecting the participants. This property, which is desirable for
a public pension system, is a new finding about the functionality of the IRS
in CDC pension systems. Moreover, our simulation experiments show that the
optimal investment strategy and the associated adjustment strength of the
declaration rate are highly dependent on the market price of risk of the
financial market and the risk attitude of the social planner.

Lastly, we emphasize that the use of Bayesian optimization has enabled our
extensive simulation study, as Bayesian optimization can drastically reduce
the computational times required. Without using Bayesian optimization, it
would have not been possible to obtain the above new findings, as the brute-
force approaches such as grid search are too time-consuming with the authors’
computational environment.333For grid search to match the accuracy of Bayesian
optimization, it usually requires a much larger number of function evaluations
and thus is more time-consuming. For example, for grid search, 100 function
evaluations amount to $10\times 10$ grids (for optimizing two parameters), by
which one can only obtain a rough estimate for the optimal solution. If one
chooses finer grids of size, say, $40\times 40=1600$, one needs 16 times more
computational costs. See Figure 3 in Section 3 for an illustration of how grid
search can be inefficient compared to Bayesian optimization. In this sense,
our work demonstrates the benefits of using Bayesian optimization, or machine
learning in general, in simulation studies of pension systems.

##  2 Model setup

###  2.1 The underlying financial market and pension asset dynamics

To introduce our stylized CDC pension model, we first define the setup. Let
$t\geq 0$ denote a time point, with the unit being one year (i.e., $t=1$ is
one year after the pension fund started at time $t=0$). For the financial
market, we consider the Black–Scholes model where there exist a risk-free
asset $F(t)\geq 0$ and a risky asset $S(t)\geq 0$, modelled as stochastic
processes whose dynamics are given by

$\displaystyle\mathrm{d}F(t)=$ $\displaystyle rF(t)\mathrm{d}t,\quad
F(0)=F_{0},$ $\displaystyle\mathrm{d}S(t)=$ $\displaystyle\mu
S(t)\mathrm{d}t+\sigma S(t)\mathrm{d}W(t),\quad S(0)=S_{0},$

where $r>0$ is the risk-free rate, $\mu>r$ and $\sigma>0$ are the constant
drift and volatility of the risky asset, respectively. We use $F_{0}>0$ to
denote the initial value of the risk-free asset, $S_{0}>0$ the initial value
of the risky asset, and $W(t)$ the standard Brownian motion under the real
world probability measure.

Denote by $A(t)\geq 0$ the asset of the pension fund at time $t\geq 0$. Assume
that, at time $t$, the fund invests the fraction $\pi(t)\geq 0$ of its asset
$A(t)$ into the risky asset $S(t)$, and the rest $1-\pi(t)$ into the risky-
free asset $F(t)$. Then, the dynamics of the asset $A(t)$ as a stochastic
process is given by

$\displaystyle\mathrm{d}A(t)=$
$\displaystyle\pi(t)A(t)\frac{\mathrm{d}S(t)}{S(t)}+(1-\pi(t))A(t)\frac{\mathrm{d}F(t)}{F(t)}$
$\displaystyle=$ $\displaystyle
A(t)\left[(\pi(t)(\mu-r)+r)\mathrm{d}t+\pi(t)\sigma\mathrm{d}W(t)\right],\quad
A(0)=a_{0},$ (2.1)

where $a_{0}>0$ is the initial asset of the fund. The constraint $\pi(t)\geq
0$ means that short selling ($\pi(t)<0$) is prohibited. This is consistent
with regulations on pension funds’ investment policies in reality.

We describe the basis scheme of our pension model. We consider an overlapping
generations model (OLG model) consisting of $N\in\mathbb{N}$ working
generations (e.g., $N=40$ generations of people of ages from 25 to 64). We
assume, for simplicity, that each generation consists of one hypothetical
participant444This essentially represents the situation where each generation
consists of about the same number of participants.. At the beginning of each
year $t=0,1,2,\dots$, the oldest working generation leaves the fund and a new
youngest generation joins it. As a consequence, there are always $N$ working
generations in the fund. The fund pays lump-sum benefits $B(t)>0$ to the
generation who has just retired, where $B(t)$ is determined by their benefit
accounts, introduced below. At the same time, each of $N$ working generations
pays a fixed amount $y>0$ to the fund as a contribution. Thus, the cash flows
(of the contributions from the $N$ working generations and the payments to the
retired generation) occur at discrete time points $t=0,1,2,...$ in our model,
while the asset $A(t)$ changes continuously. This implies that the asset
process $A(t)$ has a jump at the beginning of each year:

$A(t)_{+}=A(t)+\underbrace{Ny}_{\text{\scriptsize
Contributions}}-\underbrace{B(t)}_{\text{\scriptsize Payments}},\quad
t=0,1,2,...$ (2.2)

where $A(t)_{+}:=\lim_{\varepsilon\to+0}A(t+\varepsilon)$ denotes the right
continuous limit.

Figure 1: The OLG pension fund with one retired generation.

For ease of understanding, we illustrate the cash flows of the pension fund
with $N=40$ working generations in Figure 1.

###  2.2 Individual benefit accounts

We now introduce the individual benefit account for each generation. The
individual benefit account of one generation may be interpreted as the asset
that belongs to this generation, and its value at the retirement date becomes
the total amount of benefits to be paid to this generation after retirement.
As such, the individual benefit account is determined by the accumulated
contributions from the generation and the performance of the pension fund.

##### Generation indicators.

To describe this, we need to introduce some notations. Let
$\tau_{0}\in\mathbb{N}$ be the age of the youngest working generation (e.g.,
$\tau_{0}=25$) and $\tau_{R}\in\mathbb{N}$ be the age of the retired
generation (e.g., $\tau_{R}=65$). For the generation of age
$\tau\in\mathbb{N}$ at time $t$, we then define its unique indicator
$i=i(\tau,t)\in\mathbb{N}$ as

$i:=i(\tau,t):=\tau_{R}-\tau+[t],\quad\tau\in\mathbb{N},\quad t\geq 0,$ (2.3)

where $[t]$ denotes the integer part of $t$ (e.g., if $t=3.4$ then $[t]=3$).
The indicator $i$ represents the calendar year when this generation retires,
as can be seen by setting $\tau=\tau_{R}$ in (2.3).

Note that the generation $i$ of age $\tau$ at time $t$ is a working generation
if and only if $\tau_{0}\leq\tau<\tau_{R}$. Since $\tau=\tau_{R}+[t]-i$ by the
definition of $i$, this condition is equivalent to
$\tau_{0}\leq\tau_{R}+[t]-i<\tau_{R}$, or $i-(\tau_{R}-\tau_{0})\leq[t]<i$ by
rearranging. Thus, we can define the set of the indicators for all working
generations at time $t$ as

$I_{w}(t):=\\{i\in\mathbb{N}\mid i-(\tau_{R}-\tau_{0})\leq[t]<i\\}.$

It follows that the generation with indicator $i$ starts being a working
generation at time $t=i-(\tau_{R}-\tau_{0})$ and retires at time $t=i$, as
mentioned.

##### Individual benefit accounts.

We now define individual benefit accounts. Below let
$\tau_{i}:=\tau_{R}+[t]-i$ denote the age of generation $i$ at time $t$. For
generation $i$, let $B_{i}(t)\geq 0$ denote its individual benefit account at
time $t$, and we define its dynamics by

$\displaystyle B_{i}(i-(\tau_{R}-\tau_{0}))=0,$ (2.4)
$\displaystyle\mathrm{d}B_{i}(t)=B_{i}(t)\eta(t)\mathrm{d}t,\quad
t\in[i-(\tau_{R}-\tau_{0}),\ i],$ (2.5) $\displaystyle
B_{i}(t)_{+}=B_{i}(t)+y,\quad t\in\\{i-(\tau_{R}-\tau_{0}),\ \
i-(\tau_{R}-\tau_{0})+1,\ \dots,\ \ i-1\\},$ (2.6)

where $\eta(t)\in\mathbb{R}$ is the declaration rate at time $t$, introduced
below.

We first explain interpretations of the dynamics of the individual account.
The generation $i$ first becomes a working generation at time
$t=i-(\tau_{R}-\tau_{0})$, and at this point the individual benefit account
$B_{i}(t)$ starts from $0$ as in (2.4). The benefit account $B_{i}(t)$ evolves
continuously according to the declaration rate $\eta(t)$ as in (2.5), and
increases with its own contribution $y$ at the beginning of each year
$t=i-(\tau_{R}-\tau_{0}),\dots,i-1$, while the generation $i$ is in the
working generations as in (2.6). When the generation $i$ retires at time
$t=i$, the fund pays the amount of

$B(t)=B_{i}(t)=B_{i}(i),\quad t\in\mathbb{N}$ (2.7)

as a lump-sum retirement benefit.

##### Liability.

We define the liability $L(t)$ of the fund at time $t$ endogenously as the sum
of the individual benefit account values of all participants:

$L(t):=\sum_{i\in I_{w}(t)}B_{i}(t),\quad t>0,$

where $I_{w}(t)$ is the set of indicators for all the working generations at
time $t$. We define the liability in this way, since our fund is of the
defined-contribution type and the obligation of the fund is essentially its
benefit account values.

Due to the individual benefit accounts $B_{i}(t)$ in (2.5) and (2.6), the
dynamics of the liability $L(t)$ is then given by

$\displaystyle\mathrm{d}L(t)=L(t)\eta(t)\mathrm{d}t,\quad t\geq 0,\quad
L(0)=a_{0},$ (2.8) $\displaystyle L(t)_{+}=L(t)+Ny-B(t),\quad t\in\mathbb{N},$

where $B(t)$ is the payment to the retired generation $i=t$ at time $t$ in
(2.7). We set the initial value of the liability $L(0)$ as the initial value
of the asset $A(0)=a_{0}$, so that the funding ratio is $A(0)/L(0)=1$.

##### Declaration rate.

We now define the declaration rate $\eta(t)$ in (2.5). Let

$\mu(t):=\pi(t)(\mu-r)+r-\frac{1}{2}\pi(t)^{2}\sigma^{2}$

be the expected log-return value of the asset $A(t)$ at time $t$, which can be
obtained from (2.1). Then we define the declaration rate $\eta(t)$ at time $t$
as

$\displaystyle\eta(t)$ $\displaystyle=\mu(t)+\theta\ln(A(t)/L(t)),$ (2.9)

where $0\leq\theta\leq 1$ is an adjusting parameter. That is, the declaration
rate $\eta(t)$ is the expected log-return $\mu(t)$ adjusted by the logarithm
of the funding ratio $A(t)/L(t)$.

The adjustment by the funding ratio $A(t)/L(t)$ works as follows. Inserting
the definition of the declaration rate $\eta(t)$ in (2.5), the continuous
dynamics of the individual benefit account $B_{i}(t)$ of generation $i$ is
given as

$\displaystyle\mathrm{d}B_{i}(t)=B_{i}(t)\eta(t)\mathrm{d}t$
$\displaystyle=B_{i}(t)\left(\mu(t)+\theta\ln(A(t)/L(t))\right)\mathrm{d}t.$

Assume that, at time $t$, the asset $A(t)$ is larger than the liability
$L(t)$: $A(t)>L(t)$. Then we have $\ln(A(t)/L(t))>0$, and thus the declaration
rate $\eta(t)$ becomes larger than the expected log return $\mu(t)$ because of
the adjustment. Accordingly, the individual benefit account $B_{i}(t)$ grows
faster, and as a result the liability $L(t)$ also grows faster, as can be seen
from (2.8). On the other hand, assume that the asset $A(t)$ is smaller than
the liability $L(t)$, i.e., $A(t)<L(t)$. Then we have $\ln(A(t)/L(t))<0$, and
thus the declaration rate $\eta(t)$ is adjusted to be smaller than the
expected return $\mu(t)$. In this way, the individual benefit account
$B_{i}(t)$ grows more slowly, and the resulting liability $L(t)$ also grows
more slowly.

###  2.3 Optimization problem

Having introduced our model, we now define an optimization for finding the
investment strategy $\pi(t),~{}t\geq 0$ in (2.1) and the adjustment parameter
$\theta$ in the declaration rate in (2.9). We consider a hypothetical social
planner who runs the pension fund and cares about the benefits of all
generations, and define our objective function as the expected utility of this
social planner. To this end, let $U_{\gamma}:(0,\infty)\to(-\infty,\infty)$ be
a constant relative risk aversion (CRRA) utility function555See e.g. Chiappori
and Paiella, (2011) for an empirical study supporting the use of the CRRA
utility function. with relative risk aversion $\gamma\geq 0$:

$U_{\gamma}(x):=\frac{x^{1-\gamma}}{1-\gamma}\quad(\text{for
}\gamma\not=1),\quad U_{\gamma}(x):=\ln(x)\quad(\text{for }\gamma=1).$ (2.10)

We then define our objective function as the expected sum of the discounted
utilities (with a subjective discount factor $\beta\in(0,1)$) of the payments
to all the generations. The optimization problem is thus given as:

$\displaystyle\max\left\\{\mathbb{E}\left[\sum_{t=0}^{\infty}\beta^{t}U_{\gamma}(B(t))\right]\mid\pi(t)\geq
0,\ \ 0\leq\theta\leq 1,\ \ \text{subject to}\ \ A(t)>0\ \text{ for all}\
t\geq 0\right\\},$ (2.11)

where $B(t)$ is the pension payment to the retired generation $i=t$ at time
$t$ in (2.7).

In this work, for the investment strategy $\pi(t)$ in (2.11), we confine
ourselves to a constant mix strategy of the form

$\pi(t)=\pi\geq 0,\quad\forall t\geq 0.$

Thus, the optimization problem is with respect to this constant $\pi$ and the
adjustment parameter $\theta$. In general, a constant-mix strategy is easy to
implement, and hence is a prevailing candidate for a collective pension
design, see for example Cui et al., (2011), Goecke, (2013) and Bams et al.,
(2016).

As an analytic solution is not available for our optimization problem, we
numerically search for the optimal parameters using Bayesian optimization, a
modern machine learning approach to black-box optimization, as detailed in the
next section.

##### Relative risk aversion and investment regulations.

The relative risk aversion $\gamma$ in our model may be interpreted as
capturing a regulation on the pension fund’s investment policy, as $\gamma$
indirectly determines the investment strategy, as we will see in Section 4.
According to an annual survey of OECD, (2020), investment regulations on
pension funds differ among different countries. For example, in countries like
the U.S., the U.K., and the Netherlands, there is basically no limitation for
pension funds’ investment. On the other hand, in countries such as France and
Germany, there are relatively conservative upper-limits on the fraction of
pension funds’ assets to be invested in equity. Since in our model, a larger
(resp. smaller) $\gamma$ leads to a more conservative (resp. aggressive)
investment strategy, the choice of $\gamma$ may be interpreted as indirectly
representing the strength of a regulation.

##### Bankruptcy

In our pension model, we define bankruptcy to occur at time $t$ when the
pension’s asset becomes non-positive: $A(t)\leq 0$. Thus, the constraint
$A(t)>0$ for all $t>0$ in the optimization problem (2.11) is to require that
the fund does not go bankrupt. In practice, however, this hard constraint is
difficult to satisfy, because of the unboundedness of the Brownian motion
defining the financial market. In Section 3, we introduce a soft constraint
that encourages the numerically optimized parameters not to lead to
bankruptcy.

Note that bankruptcy is usually defined as when the pension’s asset is below
the pension’s liability. However, in our model, the liability is endogenously
defined as the sum of the benefit account values of working generations, and
is continuously adjusted according to the funding ratio. Thus, this definition
of bankruptcy is not appropriate for our pension model. This is why we define
bankruptcy as the pension’s asset $A(t)$ being not positive, i.e., when the
pension fund’s assets are depleted.

###  2.4 Entry cohorts

Here, we discuss the entry cohorts, i.e., the first generations who
participate in the fund at time $t=0$ when the pension fund starts. Before
this period, i.e., when $t<0$, we assume that these entry cohorts maintained
their individual benefit accounts $B_{i}(t)$ with a pure saving policy (where
$i$ is the indicator of one of the generations in the entry cohorts). In other
words, we assume for $t<0$ the individual benefit account $B_{i}(t)$ for
generation $i$ in the entry cohorts accumulated at the risk-free rate $r$:

$\mathrm{d}B_{i}(t)=rB_{i}(t)\mathrm{d}t,\quad t<0,$

and for $t\geq 0$ it evolves according to our design described in (2.5) (2.6).
Hence, we adopt the same way to deal with the entry cohorts as Bams et al.,
(2016).

##  3 Bayesian optimization (BO)

We explain here the details of Bayesian optimization (BO). As mentioned, we
use BO to maximize the expected utility in (2.11), to numerically search for
the optimal values of the investment strategy $\pi$ and adjustment parameter
$\theta$. BO is a modern machine learning approach to globally optimizing a
black-box objective function (Shahriari et al.,, 2016). For implementation, we
use the R package mlrMBO (Bischl et al.,, 2017).

BO has been widely used in applications where the objective function is
defined by simulations, and thus where the evaluation of the objective
function is computationally expensive. This is our case with the expected
utility (2.10), because we approximate the expectation with Monte Carlo
simulations. For the optimization of such a black-box objective function, grid
search is often used in practice. However, grid search is not efficient and
computationally demanding. BO is much more efficient than grid search, as it
iteratively refines the search space by taking into account the evaluated
function values.

This iterative refinement is enabled by using the so-called acquisition
function, which suggests the next point to evaluate. The acquisition function
is defined on the basis of the learning and uncertainty quantification of the
landscape of the objective function. This learning and uncertainty
quantification is done by a Bayesian statistical learning approach called
Gaussian process regression (or Kriging). This is why BO is called “Bayesian.”

We first describe the basic framework of BO in Section 3.1. We then explain
Gaussian process regression in Section 3.2 and the acquisition function in
Section 3.3. We demonstrate the application of BO in our simulation study in
Section 3.4. The readers who are only interested in the economic aspects of
the paper can directly move to the next section.

###  3.1 Basic framework

We first describe the basic framework of BO. Let $\Omega$ be a parameter set
and $f:\Omega\to\mathbb{R}$ be the objective function to be maximized. In our
problem, this parameter set is $\Omega=[0,3]\times[0,1]$ and each
$x:=(\pi,\theta)\in\Omega$ represents a pair of the investment strategy
$\pi\geq 0$ and adjustment parameter $\theta\geq 0$. We define the objective
function as the certainty equivalent (CE) of the expected utility in (2.11)
with input parameters $x=(\pi,\theta)$:

$\displaystyle f(x):=f(\pi,\theta):=CE(\pi,\theta),$ (3.1)
$\displaystyle\text{where}\quad CE(\pi,\theta)\geq 0\ \text{ is such that }\
U_{\gamma}(CE(\pi,\theta))=\mathbb{E}\left[\sum_{t=0}^{\infty}\beta^{t}U_{\gamma}(B(t))\right].$

Note that the expected utility is a function of $x=(\pi,\theta)$, as the
payment $B(t)$ depends on $\pi$ and $\theta$. Since the CRRA utility function
$U_{\gamma}$ is strictly monotonically increasing with respect to its
argument, the maximizer of the certainty equivalent is the same as the
maximizer of the expected utility:

$\arg\max_{(\pi,\theta)\in\Omega}CE(\pi,\theta)=\arg\max_{(\pi,\theta)\in\Omega}\mathbb{E}\left[\sum_{t=0}^{\infty}\beta^{t}U_{\gamma}(B(t))\right].$

Thus, the maximization of the expected utility can be equivalently formulated
as the maximization of the objective function (3.1).

In our simulation study, the expected utility is approximated by the Monte
Carlo average of 10,000 simulations of the asset process $A(t)$ (and thus the
resulting $B(t)$) for $t=0,1,\dots,T:=100$. Therefore, each evaluation of
$f(x)$ for a given $x=(\pi,\theta)$ involves 10,000 simulations over 100
years, which is computationally expensive.

##### Procedure of Bayesian optimization.

First, we generate initial design points $x_{1},\dots,x_{n_{\rm init}}$ for
some $n_{\rm init}\in\mathbb{N}$, and evaluate the function values
$f(x_{1}),\dots,f(x_{n_{\rm init}})$ on these points. One can generate these
initial points randomly (e.g, uniform sampling on $\Omega$) or
deterministically (e.g., grid points). In our study, we use the design given
by Latin hypercube sampling (McKay et al.,, 2000) on $\Omega$ with $n_{\rm
init}=10$.

Below we use the notation
$D_{n}:=\\{(x_{i},f(x_{i}))\\}_{i=1}^{n}\subset\Omega\times\mathbb{R}$ to
write the collection of points $x_{1},\dots,x_{n}$ and the resulting function
values $f(x_{1}),\dots,f(x_{n})$. $D_{n}$ can be understood as “data” or
“observations” about $f$ after $n$-time evaluations of the function. We also
denote by $\alpha(x;D_{n})$ the acquisition function, whose concrete form will
be introduced later in Section 3.3. The acquisition function $\alpha(x;D_{n})$
is a function of $x\in\Omega$ and defined from $D_{n}$.

BO iterates the following procedure for $n=n_{\rm init}+1,n_{\rm
init}+2,\dots,N$, where $N$ is the total number of function evaluations.

  1. 1.

Compute $x_{n+1}\in\arg\max_{x\in\Omega}\alpha(x;D_{n})$,

  2. 2.

Simulate $f(x_{n+1})$, and augment the data
$D_{n+1}:=D_{n}\cup\\{(x_{n+1},f(x_{n+1}))\\}$.

An estimate of the optimal parameters is then given as the maximizer from the
evaluated inputs $x_{1},\dots,x_{N}$:

$x^{*}\in\arg\max\left\\{f(x)\mid x\in\\{x_{1},\dots,x_{N}\\}\right\\}$

The acquisition function $\alpha(x;D_{n})$ determines the next point $x_{n+1}$
to evaluate the objective function $f$. Note that the computational cost of
solving $\max_{x\in\Omega}\alpha(x;D_{n})$ is negligible compared to the
computational cost of evaluating $f(x_{n+1})$, as $\alpha(x;D_{n})$ can be
evaluated cheaply.

The acquisition function is designed so as to balance the exploitation and
exploration. Exploitation is a strategy to search for in a region near the
current maximizer in $x_{n}^{*}:=\arg\max\\{f(x)\mid
x\in\\{x_{1},\dots,x_{n}\\}\\}$; exploration is to search for in a region far
from the evaluated points $x_{1},\dots,x_{n}$. This exploration-exploitation
trade-off is enabled by the learning and uncertainty quantification of the
response surface of $f$ from the data $D_{n}$. This is done by Gaussian
process regression, which we will explain next.

###  3.2 Gaussian process regression

Gaussian process regression (Rasmussen and Williams,, 2006) is a Bayesian non-
parametric method for learning (or approximating) an unknown function
$f:\Omega\to\mathbb{R}$ from its finite observations (data)
$D_{n}=\\{(x_{i},f(x_{i}))\\}_{i=1}^{n}$. Recall that Bayesian inference in
general proceeds as follows: a) define a prior distribution for the quantity
of interest, b) collect observations (data) related to that quantity, and c)
update the prior distribution to the posterior distribution using the observed
data, applying Bayes’ rule. In Gaussian process regression, the quantity of
interest is the unknown function $f$, and a’) one defines a prior distribution
of $f$ as a Gaussian process (or Gaussian random field), b’) collects data
$D_{n}=\\{(x_{i},f(x_{i}))\\}_{i=1}^{n}$, and c’) updates the prior Gaussian
process to the posterior Gaussian process, applying Bayes’ rule. See Figure 2
for illustrations of Gaussian process regression.

Figure 2: Illustrations of Gaussian process regression, adapted from Kanagawa
et al., (2018, Figure 1). Left: The green curves are 5 sample paths from the
prior Gaussian process (3.2) using the Matérn kernel (3.3) with $h=1$. The
thick black line is the prior mean function $m(x)=0$. Right: the three points
are noise-perturbed observations $(x_{i},y_{i})_{i=1}^{3}$, where
$y_{i}=f^{*}(x_{i})+\varepsilon_{i}$ with $f^{*}$ being the ground-truth
function and $\varepsilon_{i}$ being an independent zero-mean Gaussian noise
with variance $\sigma^{2}:=0.01$. The thick black curve is the posterior mean
function $m_{n}(x)$ in (3.5) and the two thin black curves are the posterior
standard deviation function $\sigma_{n}(x)=\sqrt{k_{n}(x,x)}$ in (3.7), in
which ${\bf K}_{n}^{-1}$ is replaced by $({\bf K}_{n}+\sigma^{2}{\bf I})^{-1}$
and ${\bf f}_{n}:=(y_{1},y_{2},y_{3})^{\top}$. (These modifications are
theoretically justified; see e.g. Kanagawa et al., (2018) and references
therein for details.) The green curves are 5 sample paths from the posterior
Gaussian process (3.4)

##### Prior Gaussian process.

A Gaussian process is completely specified by its mean function
$m:\Omega\to\mathbb{R}$ and covariance function $k:\Omega\times\Omega$. We
write $f\sim\mathcal{GP}(m,k)$ to mean that $f$ is a sample path of the
Gaussian process with mean function $\mu$ and covariance function $k$. Then we
have $\mu(x)=\mathbb{E}[f(x)]$, $x\in\Omega$ and
$k(x,x^{\prime})=\mathbb{E}[(f(x)-\mu(x))(f(x^{\prime})-\mu(x^{\prime}))]$,
$x,x^{\prime}\in\Omega$. By specifying $\mu$ and $k$, we implicitly specify
the corresponding Gaussian process.

For simplicity, we consider a Gaussian process with the zero-mean function
(i.e., $\mu(x)=0$, $\forall x\in\Omega$) for our prior distribution of the
objective function $f$:

$f\sim\mathcal{GP}(0,k).$ (3.2)

What we need is to specify the covariance function $k$. By doing so, we can
express our assumption or knowledge regarding key properties of the objective
function $f$, such as its smoothness and structure.

Popular choices of covariance kernels include square-exponential kernel
$k(x,x^{\prime})=\exp(-\|x-x^{\prime}\|^{2}/h)$ with $h>0$ and Matérn kernels.
In our study we use the so-called Matérn-$5/2$ kernel of the form

$k(x,x^{\prime})=\left(1+\frac{\sqrt{5}\|x-x^{\prime}\|}{h}+\frac{5\|x-x^{\prime}\|^{2}}{3h^{2}}\right)\exp\left(-\frac{\sqrt{5}\|x-x^{\prime}\|}{h}\right)$
(3.3)

where $h>0$ is a scale parameter.666In our simulation study, we use the
default value for $h$ of the mlrMBO package. Roughly, this kernel leads to
$f\sim\mathcal{GP}(0,k)$ that is almost surely twice differentiable (e.g.,
Kanagawa et al.,, 2018, Section 4.4). Thus, with this kernel we essentially
assume this degree of smoothness for the objective function, and this is our
prior assumption.

##### Posterior Gaussian process.

The use of a Gaussian process as a prior leads to an analytic expression of
the resulting posterior distribution. Given data
$D_{n}=\\{(x_{i},f(x_{i}))\\}_{i=1}^{n}$, the posterior distribution of $f$ is
also given as a Gaussian process

$f|D_{n}\sim\mathcal{GP}(m_{n},k_{n}),$ (3.4)

where $\mu_{n}:\Omega\to\mathbb{R}$ is the posterior mean function and
$k_{n}:\Omega\times\Omega\to\mathbb{R}$ is the posterior covariance function,
given by

$\displaystyle m_{n}(x)$ $\displaystyle=\mathbb{E}[f(x)|D_{n}]={\bf
f}_{n}^{\top}{\bf K}_{n}^{-1}{\bf k}_{n}(x),\quad x\in\Omega,$ (3.5)
$\displaystyle k_{n}(x,x^{\prime})$
$\displaystyle=\mathbb{E}[(f(x)-m_{n}(x))(f(x^{\prime})-m_{n}(x))|D_{n}]$
$\displaystyle=k(x,x^{\prime})-{\bf k}_{n}(x)^{\top}{\bf K}_{n}^{-1}{\bf
k}_{n}(x^{\prime}),\quad x,x^{\prime}\in\Omega,$ (3.6)

where ${\bf f}_{n}:=(f(x_{1}),\dots,f(x_{n}))^{\top}$, ${\bf
k}_{n}(x):=(k(x,x_{1}),\dots,k(x,x_{n}))^{\top}\in\mathbb{R}^{n}$ and ${\bf
K}_{n}:=(k(x_{i},x_{j}))_{i,j=1}^{n}\in\mathbb{R}^{n\times n}$. For the detail
of the above derivation, see Rasmussen and Williams, (2006).

The posterior mean function $m_{n}$ in (3.5) is an approximation of the
objective function $f$ based on the data $D_{n}$. It works as a
computationally cheaper surrogate model of $f$. On the other hand, the
posterior standard deviation

$\sigma_{n}(x):=\sqrt{k_{n}(x,x)}=\sqrt{\mathbb{E}[(f(x)-\mu_{n}(x))^{2}|D_{n}]}$
(3.7)

quantifies the uncertainty about the unknown function value $f(x)$. These
$\mu_{n}$ and $\sigma_{n}$ are the building blocks of the acquisition
function, as we will see next.

###  3.3 Acquisition function

We now introduce the concrete form acquisition function $\alpha(x;D_{n})$.
There are many acquisition functions proposed in the literature; see Shahriari
et al., (2016, Section IV). Most popular ones include the EI (Expected
Improvement), GP-UCB (Gaussian Process Upper Confidence Bound), and ES
(Entropy Search). In this paper, we use the EI acquisition function, which is
standard and theoretically well studied (Bull,, 2011). Let

$f_{n}^{*}:=\max_{i=1,\dots,n}f(x_{i}),\quad x_{n}^{*}\in\arg\max\\{f(x)\mid
x\in\\{x_{1},\dots,x_{n}\\}\\}.$

be the maximum and the maximizer of the objective function $f(x)$ over the
currently evaluated inputs $x_{1},\dots,x_{n}$. The EI acquisition function
$\alpha(x;D_{n})$ at $x$ is defined as the expected improvement of the
function value $f(x)$ over the current maximum $f^{*}_{n}$, where the
expectation is with respect to the posterior Gaussian process (3.4):

$\displaystyle a(x;D_{n})$
$\displaystyle:=\mathbb{E}_{f\sim\mathcal{GP}(\mu_{n},k_{n})}[\max(f(x)-f^{*}_{n},0)]$
$\displaystyle=\underbrace{\sigma_{n}(x)\phi\left(\frac{\mu_{n}(x)-f^{*}_{n}}{\sigma_{n}(x)}\right)}_{\rm
Exploration}+\underbrace{(\mu_{n}(x)-f^{*}_{n})\Phi\left(\frac{\mu_{n}(x)-f^{*}_{n}}{\sigma_{n}(x)}\right)}_{\rm
Exploitation},$ (3.8)

where $\phi:\mathbb{R}\to[0,\infty)$ is the probability density function of a
standard Gaussian random variable, and $\Phi:\mathbb{R}\to[0,1]$ is its
cumulative distribution function: $\Phi(y):=\int_{-\infty}^{y}\phi(s)ds$,
$y\in\mathbb{R}$.

The first term in (3.8) represents the exploration, as it becomes large when
$\sigma_{n}(x)$, which represents the uncertainty about the function value
$f(x)$, is large. This is typically the case when $x$ is far from already
evaluated locations $x_{1},\dots,x_{n}$. The second in (3.8) represents the
exploitation, as it becomes large when $\mu_{n}(x)-f_{n}^{*}$ is large and
$\sigma_{n}(x)$ is small. This is typically the case when $x$ is near the
current maximizer $x_{n}^{*}$. Thus, the EI acquisition function naturally
balances the exploration-exploitation trade-off, and the next point
$x_{n+1}\in\arg\max_{x\in\Omega}\alpha(x;D_{n})$ achieves such a balance.

##### Soft constraint for non-bankruptcy.

For the maximization of the expected utility (2.11), we have a constraint on
the parameters $x=(\pi,\theta)$ so that the bankruptcy, defined as $A(t)\leq
0$, does not occur for any time point $t>0$ and for any realization of the
financial market. This hard constraint is difficult to achieve in practice,
because of the unboundedness of the Brownian motion defining the financial
market. Therefore, in the simulation study we use the following soft
constraint: If $A(t)\leq 0$ happens at any $t>0$ for any of 10,000 simulations
of the financial market, we set the objective function value to its minimum:
i.e., $f(x)=0$. This encourages BO to avoid selecting parameters close to such
$x=(\pi,\theta)$.

###  3.4 Demonstration

Figure 3 shows an example of points $x=(\pi,\theta)$ evaluated by BO for
$\gamma=10$. The green points are the $n_{\rm init}=10$ initial design points
$x_{1},\dots,x_{n_{\rm init}}$ generated by Latin hypercube sampling. The
total number of evaluated points is $N=100$. The red point is the maximizer
$x_{N}^{*}$, and the blue points are the 10 other second-best parameters
(largely overlapping with the red point). For a comparison, we show $10\times
10$ grid points. Figure 2 also shows a heat map describing the landscape of
the objective function $f(x)=f(\pi,\theta)$ (red: higher; blue: lower).

Figure 3 shows that the parameters evaluated by BO concentrate in the region
around $\pi=0.25$, where the function values are high (as shown in the heat
map). In comparison to the $10\times 10$ grids, BO uses the same computational
budget of 100 evaluations more efficiently by focusing on the region where the
function values are high. This example describes the usefulness of BO in
simulation studies of a pension system in general, as BO can reduce
computational burdens substantially.

Figure 3: Demonstration of Bayesian optimization with $\gamma=10$ (see Section
3.4). Top: The points represent the values of $(\pi,\theta)$ evaluated by
Bayesian optimization. The green points are the initial 10 points given by
Latin hypercube sampling. The red point is the maximizer found by Bayesian
optimization after 100 evaluations, and the blue points (largely overlapping
the red point) are the 10 second-best points. Bottom A heat map representing
the response landscape of the objective function (red: higher, blue:lower).
Best seen in color.

##  4 Optimal parameters of the pension model

We study here the optimal values for the two parameters of the pension fund
(i.e., the investment strategy $\pi$ and the adjustment parameter $\theta$ of
the declaration rate) numerically obtained with Bayesian optimization, and how
these values depend on the relative risk aversion $\gamma$ of the social
planner (or the fund manager) and the market price of risk of the financial
market. In Section 4.1, we first describe the setup for our simulation study
also for the later sections on welfare analysis. In Section 4.2, we report the
optimal values for the two parameters of the pension model in different
settings, and the resulting trajectories of the funding-ratio process.

###  4.1 Simulation setup

We explain here a common setup for the following simulation studies.

##### Pension setting.

We set the annual contribution of a working generation to be the unit, $y=1$.
Each generation starts working at age $\tau_{0}=25$ and retires at age
$\tau_{R}=65$. Thus, one generation contributes to the pension fund for
$N=\tau_{R}-\tau_{0}=40$ years.

##### Settings of the financial market.

We first define the market price of risk. As defined in Section 2.1, we
consider a Black-Scholes market, and recall that $r>0$ is the rate of the
risk-free asset, and $\mu>r$ and $\sigma>0$ are the constant drift and
volatility of the risky asset, respectively. Then the the market price of risk
can be defined as

$\rho:=(\mu-r)/\sigma$

i.e., the excess return $\mu-r$ divided by the volatility $\sigma$ of the
risky asset. Intuitively, the market is tough if $\rho$ is small, and vice
versa.

We consider three settings of $\mu$, $r$ and $\sigma$ and write the resulting
market as $M1$, $M2$ and $M3$, corresponding to different levels of the market
price of risk $\rho$.

$\displaystyle M1:(\mu,r,\sigma)=(0.065,0.02,0.15),\quad\rho=0.3,$
$\displaystyle M2:(\mu,r,\sigma)=(0.065,0.01,0.25),\quad\rho=0.22,$
$\displaystyle M3:(\mu,r,\sigma)=(0.065,0.01,0.5),\quad\rho=0.11.$

Thus, $M1$ is the setting with the highest market price of risk among the
three settings, and $M3$ the lowest.

##### Expected utility.

We consider different values for the relative risk aversion $\gamma$ of the
CRRA utility function in the expected utility (2.11), and perform simulations
for each value. In this way, we can study the effects of $\gamma$, which
represents the risk attitude of the social planner, on the optimal investment
strategy $\pi$ and the adjustment parameter $\theta$ as well as on the
resulting operational characteristics of the pension fund. Specifically we
consider the set of values $\\{0.5,1,2,3,5,10\\}$ for $\gamma$, motivated by
existing studies; see e.g. Mehra and Prescott, (1985); Chiappori and Paiella,
(2011). We set the the subjective discount factor in the expected utility
(2.11) as $\beta=0.98$.

##### Range of the parameters.

As mentioned in Section 3.1, we specify the search space of the two parameters
(the investment strategy $\pi$ and the declaration rate adjustment parameter
$\theta$) as $(\pi,\theta)\in[0,3]\times[0,1]$. Note that $\pi>1$ can be
interpreted as the pension fund performing borrowing; we allow for such a
situation for our simulation experiments for completeness.

##### Simulation method and expected utility.

For the simulation of the asset process (2.1), we use a step size
$\Delta=1/12$ for a time-horizon of $T=100$ years (i.e., the simulation is run
from $t=0$ to $t=T=100$). This step size $\Delta=1/12$ implies that the asset
process and the declaration rate are updated on a monthly basis. The initial
values for the asset and liability are specified as $A(0)=L(0)=a_{0}$, so that
the initial funding-ratio becomes the unity: $A(0)/L(0)=1$. Note that $a_{0}$
is the sum of the assets of the entry cohorts at time $t=0$; see Section 2.4.
For a given pair of investment strategy $\pi$ and adjustment parameter
$\theta$, the expected utility in (2.11) is approximated as the empirical
average of the utilities over $10,000$ simulation paths.

##### Bayesian optimization.

As described in Section 3, we use Bayesian optimization to numerically search
for the optimal values of the two parameters. We set the number of initial
points as $n_{\rm init}=10$, and the total number of evaluations as $N=100$.
See Section 3 for details.

###  4.2 Optimal parameters and the funding-ratio process

Table 1 shows the optimal values for the investment strategy $\pi$ and the
adjustment parameter $\theta$ of the declaration rate, obtained with Bayesian
optimization for various settings of relative risk aversion $\gamma$ and the
financial market. We make the following observations.

Table 1: Optimal values for the investment strategy $\pi$ and the adjustment
parameter $\theta$ of the declaration rate obtained with Bayesian
optimization, for different settings of the financial market ($M1,M2,M3$) and
the relative risk aversion $\gamma$ of the social planner.

| $\gamma=0.5$ | $\gamma=1$ | $\gamma=2$ | $\gamma=3$ | $\gamma=5$ | $\gamma=10$
---|---|---|---|---|---|---
M1: | $\pi^{*}=2.822$ | $\pi^{*}=2.310$ | $\pi^{*}=1.244$ | $\pi^{*}=0.865$ | $\pi^{*}=0.560$ | $\pi^{*}=0.300$
| $\theta^{*}=0.882$ | $\theta^{*}=0.865$ | $\theta^{*}=0.598$ | $\theta^{*}=0.345$ | $\theta^{*}=0.235$ | $\theta^{*}=0.197$
M2: | $\pi^{*}=1.895$ | $\pi^{*}=1.088$ | $\pi^{*}=0.643$ | $\pi^{*}=0.442$ | $\pi^{*}=0.393$ | $\pi^{*}=0.158$
| $\theta^{*}=0.830$ | $\theta^{*}=0.533$ | $\theta^{*}=0.197$ | $\theta^{*}=0.125$ | $\theta^{*}=0.072$ | $\theta^{*}=0.146$
M3: | $\pi^{*}=0.544$ | $\pi^{*}=0.286$ | $\pi^{*}=0.179$ | $\pi^{*}=0.119$ | $\pi^{*}=0.069$ | $\pi^{*}=0.021$
| $\theta^{*}=0.470$ | $\theta^{*}=0.344$ | $\theta^{*}=0.135$ | $\theta^{*}=0.160$ | $\theta^{*}=0.209$ |  $\theta^{*}=1.080$x$10^{-5}$

##### Investment strategy.

Let us first study how the optimal investment strategy $\pi^{*}$, which is the
fraction of the pension asset invested in the risky asset, depends on the
relative risk aversion $\gamma$ and the setting of the financial market. For a
fixed financial market, the optimal investment strategy $\pi^{*}$ becomes
smaller as $\gamma$ increases (i.e., as the social planner becomes more risk-
averse), and vice versa. For instance, for the market $M2$, we have
$\pi^{*}=1.895$ for $\gamma=0.5$ and $\pi^{*}=0.158$ for $\gamma=10$.777Note
that $\pi^{*}=1.895$ can be interpreted as: the fund invests the full amount
of its asset in the risky asset, and also performs borrowing of the amount of
$89.5\%$ of its asset to invest in the risky asset. On the other hand, for a
fixed relative risk aversion $\gamma$, the optimal investment strategy
$\pi^{*}$ becomes smaller as the market price of risk decreases; recall that
$M3$ is the market with the lowest market price of risk $\rho=0.11$ and $M1$
the highest $\rho=0.3$. These observations regarding $\pi^{*}$ align with the
meanings of the relative risk aversion and the market price of risk.

##### Declaration rate adjustment parameter.

Let us now look at the optimal adjustment parameter $\theta^{*}$ of the
declaration rate. It tends to co-vary with $\pi^{*}$: for a large $\pi^{*}$,
the associated $\theta^{*}$ tends to be large (except the setting $M3$ with
$\gamma=2,3,5$, in which $\pi^{*}$ is not very large). This tendency may be
interpreted as: if $\pi^{*}$ is large, $\theta^{*}$ needs to be also large to
stabilize the funding-ratio process.

##### Funding-ratio process.

Figure 4 shows the funding-ratio process averaged over 10,000 simulations
using the optimal parameters $(\pi^{*},\theta^{*})$ from Table 1 for
$\gamma=3$ and each of the three settings of the financial market. We can
observe that the (averaged) funding-ratio process converges to $1$ or remains
around a value slightly above $1$. A similar pattern can be also observed with
the optimal parameters in other settings. However, this is not the case if
$\theta$ is too small; in our preliminary simulation experiments, we observe
that the pension fund can go easily bankrupt (i.e., $A(t)<0$ occurs at some
$t>0$) if the adjustment parameter $\theta$ is too small for a given
investment strategy $\pi$. Hence, we may understand the optimal adjustment
parameter $\theta^{*}$ as the one that enables the funding-ratio-linked
declaration rate to protect the fund from bankruptcy.

Note that the the funding-ratio increases sharply during the first 10 to 20
years of the pension fund. This phenomenon can be attributed to the rapid
increase of the benefit accounts of the entry cohorts during the early period
of the pension fund, because their assets before joining the pension fund
($t<0$) develop according to the risk-free rate (See Section 2.4).

Goecke, (2013) proves that, for a _self-financing_ pension fund using the same
funding-ratio-linked declaration rate and a constant mix strategy, the
logarithm of the funding ratio follows an Ornstein-Uhlenbeck process whose
expectation converges to $0$ (i.e., the funding ratio converges to $1$) as the
operation time tends to infinity. Since our pension design is an OLG model
involving discrete jumps of cash flows (i.e., annual contributions and
retirement payments), this result of Goecke, (2013) is not directly
applicable. However, the funding-ratio process shows a similar pattern in our
simulations. This suggests the wider applicability of the funding-ratio-linked
declaration rate than the self-financing fund considered in Goecke, (2013).

(a) $M1$

(b) $M2$

(c) $M3$

Figure 4: The funding-ratio process averaged over 10,000 simulations using the
optimal parameters in Table 1 for $\gamma=3$ and for each of the financial
market settings, $M1$, $M2$ and $M3$.

This section has studied the optimal parameters under various settings and the
resulting trajectories of the funding-ratio process. Note that these
parameters are optimal from the viewpoint of the social planner running the
fund, since they are the (numerical) maximizer of the expected utility of this
social planner. While this expected utility takes into account the benefits of
all the generations including those from the future, it does not directly
provide us with insights into the welfare of individual participants. In the
next section, we turn our attention to the resulting retirement benefits of
individual participants, and how intergenerational risk sharing works for
protecting the participants’ assets.

##  5 Welfare analysis and intergenerational risk sharing

This section analyses the welfare of each generation in the pension fund, with
a focus on how intergenerational risk sharing (IRS) works. We do this by
running simulations using the optimal parameters in Table 1 obtained for
various settings in the previous section. To understand the functionality of
IRS in our CDC pension model, we also consider the corresponding model for an
individual defined-contribution (IDC) plan in which IRS is not implemented. We
compare the benefit account of an individual participant in the CDC model and
that of the corresponding participant in the IDC model. By studying how they
differ, we can elucidate the mechanism of IRS.

Our focus is to understand the functionality of IRS implemented through the
funding-ratio-linked declaration rate in our CDC pension model. For this
purpose, we assume that an IDC participant under comparison uses the same
investment strategy $\pi$ as the CDC pension fund, so that the only difference
between the CDC and IDC benefit accounts is the existence of IRS. As for the
CDC participants, we use the number $i\in\mathbb{N}$ as an indicator of the
participant (or generation) in the IDC plan who starts working at time
$i-(\tau_{R}-\tau_{0})=i-40$ and retires at time $t=i$. Let $A_{i}(t)$ denote
the benefit account of the IDC participant $i$ at time
$t\in[i-(\tau_{R}-\tau_{0}),i]$, where $\tau_{R}=65$. Then its dynamics is
given by

$\displaystyle A_{i}(i+\tau_{R}-\tau_{0})=0,$
$\displaystyle\mathrm{d}A_{i}(t)=A_{i}(t)\left[(\pi(\mu-r)+r)\mathrm{d}t+\pi\sigma\mathrm{d}W(t)\right],\quad
t\in[i-(\tau_{R}-\tau_{0}),\ i],$ $\displaystyle A_{i}(t)_{+}=A_{i}(t)+y,\quad
t\in[i-(\tau_{R}-\tau_{0}),i-1]\cap\mathbb{N}.$

That is, the IDC participant $i$ accumulates the contribution $y$ annually in
her account for $N=40$ years (the third line), and the account evolves
according to the continuous dynamics determined by her constant mix investment
strategy $\pi\geq 0$ in the financial market (the second line). Her lump-sum
retirement benefit is given by her account value at the time of retirement
$t=i$, i.e., $A_{i}(i)$.

We make comparisons between the individual benefit accounts $B_{i}(t)$ in the
CDC pension model and the corresponding accounts $A_{i}(t)$ in the IDC model.
In Section 5.1, we first study the stability of the development of each
benefit account. In Section 5.2, we make comparisons regarding the retirement
benefits.

In the following, we may refer to our pension model as “CDC” (collective
defined-contribution) in contrast to the IDC model.

###  5.1 Stability of individual benefit accounts

We first study how each individual benefit account evolves over time by
simulating its path. As mentioned, the CDC uses the optimal parameters
$\pi^{*}$ and $\theta^{*}$ in Table 1, and the corresponding IDC plan uses the
same investment strategy $\pi^{*}$. In Figure 5, We show some examples of
simulated paths of the CDC account $B_{i}(t)$ and the corresponding IDC
$A_{i}(t)$ of generation $i=41$ (i.e., the generation who starts working at
time $t=1$ and retires at time $t=41$) for $\gamma=3$ and for the three
financial market settings. Note that each account lasts for $40$ years = $480$
months. More examples are given in Appendix A.

We can observe that, for each case, the CDC account evolves more smoothly than
the corresponding IDC account. This smoothness of the CDC account comes from
intergenerational risk sharing, since the only difference between the CDC and
IDC accounts is the declaration rate process in the CDC pension model. These
results show that the funding-ratio-linked declaration adds more stability to
the individual benefit accounts in the CDC pension system.

(a) $M1$

(b) $M2$

(c) $M3$

Figure 5: Examples of simulated trajectories of the CDC (black) and IDC (red)
accounts of generation $i=41$ for $\gamma=3$ and for the three settings of the
financial market ($M_{1}$, $M2$ and $M3$).

To quantitatively assess the smoothness of each path, we calculate the
increment-ratio-based roughness (IR-roughness), a measure of smoothness for a
stochastic process (Bardet et al.,, 2011).888One may consider calculating the
variance of each simulation path $g:[0,T]\to\mathbb{R}$ defined by
$\frac{1}{T}\int_{0}^{T}(g(t)-m_{g})^{2}\mathrm{d}t$ with the mean
$m_{g}=\frac{1}{T}\int g(t)\mathrm{d}t$ (or its discrete approximation), but
this variance does not provide an informative measure of smoothness. This is
because the value of the variance is mostly determined by the upward trend of
each path. For example, the mean $m_{g}$ of either CDC or IDC paths in Figure
5 (a) is around $100$, and the path increases from $0$ to around $200$.
Compared to this upward trend, the fluctuations of either path is relative
smaller and thus the resulting variance does not capture these fluctuations.
Therefore we use the IR-roughness as a measure of smoothness. To describe the
IR-roughness measure, we suppose that the path of each benefit account is
represented by a function $g:[0,T]\to\mathbb{R}$, where $T>0$ is its terminal
time. Discretizing the domain to $n-1\in\mathbb{N}$ intervals, the first-order
IR-roughness is defined as

$R^{1,n}(g):=\frac{1}{n-1}\sum_{j=0}^{n-2}\frac{\left|g(T\frac{j+1}{n})-g(T\frac{j}{n})+g(T\frac{j+2}{n})-g(T\frac{j+1}{n})\right|}{\left|g(T\frac{j+1}{n})-g(T\frac{j}{n})\right|+\left|g(T\frac{j+2}{n})-g(T\frac{j+1}{n})\right|}.$
(5.1)

By the triangle inequality, the numerator in the sum is less than or equal to
the denominator, and thus $R^{1,n}(g)$ takes values between $0$ and $1$. When
the signs of the two increments $g(T(j+1)/n)-g(Tj/n)$ and
$g(T(j+2)/n)-g(T(j+1)/n)$ are the same, the numerator equals the denominator,
while when those signs are different, the numerator is smaller than the
denominator. As such, $R^{1,n}(g)$ reflects the sign changes of the function
$g$ and thus quantifies its roughness. Intuitively, $R^{1,n}(g)$ is close to
$0$ when $g$ is rough, and is close to $1$ when $g$ is smooth. In fact, Bardet
et al., (2011) show that, for a sufficiently smooth $g$, $R^{1,n}(g)$
converges to $1$ as $n\to\infty$.

Tables 2 reports the IR-roughness of the CDC and IDC accounts of generation
$i=41$, averaged over 10,000 simulation paths, for different settings of the
financial market and relative risk aversion $\gamma$. For each setting, the
IR-roughness of the CDC is larger than the IDC, and this implies that the CDC
account evolves more smoothly than the corresponding IDC account on average.

Table 2: IR roughness measure for the CDC and IDC accounts of generation $i=41$, averaged over 10,000 paths. |  | $\gamma=0.5$ | $\gamma=1$ | $\gamma=2$ | $\gamma=3$ | $\gamma=5$ | $\gamma=10$
---|---|---|---|---|---|---|---
M1 | CDC | 0.933 | 0.934 | 0.950 | 0.972 | 0.988 | 1.000
| IDC | 0.723 | 0.724 | 0.728 | 0.731 | 0.737 | 0.750
M2 | CDC | 0.927 | 0.942 | 0.974 | 0.987 | 0.994 | 0.994
| IDC | 0.722 | 0.724 | 0.727 | 0.731 | 0.732 | 0.747
M3 | CDC | 0.941 | 0.957 | 0.982 | 0.984 | 0.987 | 1.000
| IDC | 0.723 | 0.727 | 0.732 | 0.738 | 0.748 | 0.777

###  5.2 Comparison of retirement benefits

We now compare the retirement benefits of the CDC participants and those of
the corresponding IDC participants, to investigate how IRS affects the
retirement benefits. As we simulate our CDC pension model for 100 years, there
are 61 generations whose working periods are contained in the 100 operating
years of the pension fund (i.e., generations $i=40,\dots,100$; recall that
generation $i$ is those who retire at time $t=i$). We study the retirement
benefits of these generations.

####  5.2.1 Quantile values of retirement benefits

Figure 6 shows the median values of the retirement benefits over 10,000
simulations for generation $i=40,41,\dots,100$, for each of the CDC and IDC
plans, under different settings of the financial market and relative risk
aversion $\gamma$. (Other settings of $\gamma$ are shown in Appendix B.) In
most cases, the retirement benefits of the IDC are higher than the CDC. The
difference between the IDC and CDC is relatively larger for more risky
investment strategies ($\gamma=0.5$), but the difference is smaller for less
risky investment strategies $\gamma=3,10$ and in particular for the tough
market $M3$.

Figure 7 describes the corresponding lower $1\%$ quantile values of the
retirement benefits over 10,000 simulations (other cases of $\gamma$ are shown
in Appendix C). These are the worst scenarios, and the trend is different from
Figure 6: In many cases, in particular in relatively tough markets ($M2$ and
$M3$) with low market prices of risk, the CDC retirement benefits are higher
than the corresponding IDC retirement benefits. These results indicate that
intergenerational risk sharing with the funding-ratio-linked declaration rate
works well for protecting the CDC participants from losing too much, in the
worst scenarios of a tough market. (Recall that the only difference between
the CDC and IDC models here is the existence of the declaration rate process
in the CDC model).

(a) M1: $\gamma=0.5$

(b) M2: $\gamma=0.5$

(c) M3: $\gamma=0.5$

(d) M1: $\gamma=3$

(e) M2: $\gamma=3$

(f) M3: $\gamma=3$

(g) M1: $\gamma=10$

(h) M2: $\gamma=10$

(i) M3: $\gamma=10$

Figure 6: Median values of retirement benefits over 10,000 simulations (Black:
CDC; Red: IDC). The subfigures show results for different settings of the
financial market ($M1$, $M2$, $M3$) and the relative risk aversion
$\gamma=0.5,3,10$ of the social planner running the pension fund. In each
subfigure, the horizontal axis indicates different generations
$i=40,41,\dots,100$, and the vertical axis shows their retirement benefits.

(a) M1: $\gamma=0.5$

(b) M2: $\gamma=0.5$

(c) M3: $\gamma=0.5$

(d) M1: $\gamma=3$

(e) M2: $\gamma=3$

(f) M3: $\gamma=3$

(g) M1: $\gamma=10$

(h) M2: $\gamma=10$

(i) M3: $\gamma=10$

Figure 7:  Lower $1\%$ quantile values of retirement benefits over 10,000
simulations (Black: CDC; Red: IDC). The subfigures show results for different
settings of the financial market ($M1$, $M2$, $M3$) and the relative risk
aversion $\gamma=0.5,3,10$ of the social planner running the pension fund. In
each subfigure, the horizontal axis indicates different generations
$i=40,41,\dots,100$, and the vertical axis shows their retirement benefits.

To support the above findings, we show in Figure 8 the left tail region of the
(empirical) cumulative distribution function of the retirement benefits of
generation $i=41$ over 10,000 simulations, for the three settings of the
financial market and relative risk aversion $\gamma=3$. (Results for other
settings are given in Appendix D). For the relatively tough financial markets
$M2$ and $M3$, the cumulative distribution function for the IDC is higher than
the CDC in the left tail region. This implies that there exist many more cases
in 10,000 simulations where the IDC retirement benefits are lower than the CDC
in the worst scenarios. Hence, the CDC participants are better protected in
the worst scenarios of the tough markets with low market prices of risk. The
trend is the opposite in the financial market $M1$, where the market price of
risk is higher.

(a) M1

(b) M2

(c) M3

Figure 8: Left tail region of the (empirical) cumulative distribution function
of the retirement benefits over 10,000 simulations of generation $i=41$, for
the three settings of the financial market ($M1$, $M2$ and $M3$) and relative
risk aversion $\gamma=3$ (Black: CDC; Red: IDC). The horizontal axis indicates
the amount of retirement benefits (payoffs), and the vertical axis is the
value of the cumulative distribution function.

To summarize, our main finding is: intergenerational risk sharing via the
declaration rate adjustment works for protecting the CDC pension participants
from losing too much in the worst scenarios of a tough market. We argue that
this functionality of intergenerational risk sharing is important, since it
makes the pension system robust against the worst-case scenarios of the
financial market. Such robustness is particularly desirable for public pension
systems.

####  5.2.2 Certainty equivalents

We study here the welfare of pension participants in terms of the certainty
equivalents of their expected utility levels. For simplicity, we assume that
each participant has the same CRRA utility function $U_{\gamma}$ in (2.10) as
that of the social planner managing the pension fund. While this is not
necessarily true in reality, we make this assumption to focus on the effects
of intergenerational risk sharing.

For each generation $i=40,41,\dots,100$, recall that $B_{i}(i)$ and $A_{i}(i)$
are the retirement benefits when participating the CDC and IDC pension plans,
respectively; see Section 2.2 and the begging of Section 5. Then, the
corresponding certainty equivalents with the CDC and IDC plans, denoted
respectively by $CE_{i}^{(CDC)}\geq 0$ and $CE_{i}^{(IDC)}\geq 0$, are defined
by

$U_{\gamma}(CE_{i}^{(CDC)})=\mathbb{E}[U_{\gamma}(B_{i}(i))],\quad
U_{\gamma}(CE_{i}^{(IDC)})=\mathbb{E}[U_{\gamma}(A_{i}(i))],$

where, as before, we approximate each expectation by 10,000 simulations.

Figure 9 shows the computed certainty equivalents for different settings of
the financial market and relative risk aversion $\gamma$. (Results for other
settings are given in Appendix E). Consistent with our findings so far, the
certainty equivalents of the CDC participants are higher than the
corresponding IDC in tough markets ($M2$ and $M3$) with low market prices of
risk, and the opposite trend is observed for the market with a high market
price of risk ($M1$). Since the only difference between the CDC and IDC plans
is the declaration rate process in the CDC, these results suggest that
intergenerational risk sharing of the CDC via the declaration rate adjustment
works for improving the participants’ welfare in a tough market.

(a) M1: $\gamma=0.5$

(b) M2: $\gamma=0.5$

(c) M3: $\gamma=0.5$

(d) M1: $\gamma=3$

(e) M2: $\gamma=3$

(f) M3: $\gamma=3$

(g) M1: $\gamma=10$

(h) M2: $\gamma=10$

(i) M3: $\gamma=10$

Figure 9: Certainty equivalents of the CDC and IDC participants for
generations $i=40,41,\dots,100$ for different settings of the financial market
($M1$, $M2$ and $M3$) and the relative risk aversion $\gamma$ (Black: CDC;
Red: IDC). The horizontal axis indicates different generations, and the
vertical axis shows the values of their certainty equivalents.

##  6 Conclusions

We investigate the functionality of intergenerational risk sharing (IRS) in a
CDC pension model, implemented through the funding-ratio-linked declaration
rate process. Our main findings, obtained by comparisons to the corresponding
IDC pension model, are that 1) IRS makes the development of each individual
benefit account more stable, and that 2) IRS protects the participants from
losing too much in the worst scenarios of a tough financial market. These
characteristics of IRS are promising for public pension systems, since they
make the pension fund robust against possible worst-case scenarios.

There are several possible future directions. Since our work is based on
simulations, it is straightforward to extend our stylized model to a more
realistic one. For instance, one may consider a more detailed realistic model
for the entry cohorts, which is important for studying the early phase of the
pension fund. Similarly, one can study the pension system in more realistic
models for the financial market other than the Black-Scholes model, e.g.,
those with fat tails.

Our extensive simulation study is enabled by the use of Bayesian optimization
for numerically searching for the optimal parameters in the pension model.
Bayesian optimization is much more efficient than traditional brute-force
approaches such as grid search, and has been widely used in many areas of
science and engineering. In this work, we demonstrate the usefulness of
Bayesian optimization in simulation studies of a pension system. We believe
that our work can encourage the use of Bayesian optimization, or more
generally machine learning techniques, in pension research, opening up many
possibilities for studying more realistic but computationally intensive
pension models.

##  Appendix A Simulated paths of individual benefit accounts

Figure 10 describes supplementary results for the simulation experiments in
Section 5.1 (Figure 5).

(a) $M1:\gamma=0.5$

(b) $M2:\gamma=0.5$

(c) $M3:\gamma=0.5$

(d) $M1:\gamma=1$

(e) $M2:\gamma=1$

(f) $M3:\gamma=1$

(g) $M1:\gamma=2$

(h) $M2:\gamma=2$

(i) $M3:\gamma=2$

(j) $M1:\gamma=5$

(k) $M2:\gamma=5$

(l) $M3:\gamma=5$

(m) $M1:\gamma=10$

(n) $M2:\gamma=10$

(o) $M3:\gamma=10$

Figure 10: Examples of simulated trajectories of the CDC (black) and IDC (red)
accounts of generation $i=41$ for relative risk aversion
$\gamma\in\\{0.5,1,2,5,10\\}$ and for the three settings of the financial
market ($M_{1}$, $M2$ and $M3$).

##  Appendix B Median values of retirement benefits

Figure 11 describes supplementary results for the simulation experiments in
Section 5.2.1 (Figure 6).

(a) M1: $\gamma=1$

(b) M2: $\gamma=1$

(c) M3: $\gamma=1$

(d) M1: $\gamma=2$

(e) M2: $\gamma=2$

(f) M3: $\gamma=2$

(g) M1: $\gamma=5$

(h) M2: $\gamma=5$

(i) M3: $\gamma=5$

Figure 11: Median values of retirement benefits over 10,000 simulations
(Black: CDC; Red: IDC). The subfigures show results for different settings of
the financial market ($M1$, $M2$, $M3$) and the relative risk aversion
$\gamma\in\\{1,2,5\\}$ of the social planner running the pension fund. In each
subfigure, the horizontal axis indicates different generations
$i=40,41,\dots,100$, and the vertical axis shows their retirement benefits.

##  Appendix C Lower 1% quantile values of retirement benefits

Figure 12 describes supplementary results for the simulation experiments in
Section 5.2.1 (Figure 7).

(a) M1: $\gamma=1$

(b) M2: $\gamma=1$

(c) M3: $\gamma=1$

(d) M1: $\gamma=2$

(e) M2: $\gamma=2$

(f) M3: $\gamma=2$

(g) M1: $\gamma=5$

(h) M2: $\gamma=5$

(i) M3: $\gamma=5$

Figure 12:  Lower $1\%$ quantile values of retirement benefits over 10,000
simulations (Black: CDC; Red: IDC). The subfigures show results for different
settings of the financial market ($M1$, $M2$, $M3$) and the relative risk
aversion $\gamma\in\\{1,2,5\\}$ of the social planner running the pension
fund. In each subfigure, the horizontal axis indicates different generations
$i=40,41,\dots,100$, and the vertical axis shows their retirement benefits.

##  Appendix D Left tail region of the cumulative distribution of retirement
benefits

Figure 13 describes supplementary results for the simulation experiments in
Section 5.2.1 (Figure 8).

(a) M1: $\gamma=0.5$

(b) M2: $\gamma=0.5$

(c) M3: $\gamma=0.5$

(d) M1: $\gamma=1$

(e) M2: $\gamma=1$

(f) M3: $\gamma=1$

(g) M1: $\gamma=2$

(h) M2: $\gamma=2$

(i) M3: $\gamma=2$

(j) M1: $\gamma=5$

(k) M2: $\gamma=5$

(l) M3: $\gamma=5$

(m) M1: $\gamma=10$

(n) M2: $\gamma=10$

(o) M3: $\gamma=10$

Figure 13: Left tail region of the (empirical) cumulative distribution
function of the retirement benefits over 10,000 simulations of generation
$i=41$, for the three settings of the financial market ($M1$, $M2$ and $M3$)
and relative risk aversion $\gamma\in\\{0.5,1,2,5,10\\}$ (Black: CDC; Red:
IDC). The horizontal axis indicates the amount of retirement benefits
(payoffs), and the vertical axis is the value of the cumulative distribution
function.

##  Appendix E Certainty equivalents of pension participants

Figure 14 describes supplementary results for the simulation experiments in
Section 5.2.2 (Figure 9).

(a) M1: $\gamma=1$

(b) M2: $\gamma=1$

(c) M3: $\gamma=1$

(d) M1: $\gamma=2$

(e) M2: $\gamma=2$

(f) M3: $\gamma=2$

(g) M1: $\gamma=5$

(h) M2: $\gamma=5$

(i) M3: $\gamma=5$

Figure 14: Certainty equivalents of the CDC and IDC participants for
generations $i=40,41,\dots,100$ for different settings of the financial market
($M1$, $M2$ and $M3$) and the relative risk aversion $\gamma\in\\{1,2,5\\}$
(Black: CDC; Red: IDC). The horizontal axis indicates different generations,
and the vertical axis shows the values of their certainty equivalents.

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