Retirement planning with ambiguous investment and mortality risks

Yang Shen∗, Jianxi Su†

April 27, 2018

Abstract

In this article, we study the strategic retirement planning problem for a wage earner facing stochastic

lifetime. The wage earner aims to decide on the optimal portfolio choice, consumption and insurance buying

rules over the pre-retirement phase, but meanwhile she concerns about the uncertainty of financial and

mortality models. In order to address the concern, the wage earner considers the optimal decisions under the

worst-case scenario selected from a set of plausible alternative models. We find that the investment ambiguity

and mortality ambiguity have substantially different impacts on the optimal decisions. Specifically, though

the worst-case investment scenario depends only on the financial environment, the design of the worst-case

mortality scenario is determined by the intricate interplays between the wage earner’s personal profile (e.g.,

health status, income dynamics, risk aversion, etc.) and the evolution of the financial market. What is more,

the study of mortality ambiguity is also closely related to the value of life expectancy which can be positive

and negative in general. Such a complicated theoretical structure underlying the risk of mortality ambiguity

can sometimes even overturn the direction of its impacts on the optimal decisions. Our paper highlights the

importance as well as the complexity for modeling ambiguity aversion in optimal retirement studies, which

desires more serious and critical treatments from the community of actuarial professionals.

JEL classification: D81, G11, J26

Key words and phrases: Mortality ambiguity, worst-case scenario, life insurance, retirement, annuitization.

∗Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada.†Department of Statistics, Purdue University, West Lafayette, IN, 47906, United States.

1

1 Introduction

The importance of effective retirement planning is self-evident. The need is further magnified due to the

recent historically low interest rate environment, the provision for flexible retirement, and the major shift

in retirement funding from defined benefit plans toward defined contribution plans. The increasing public

concerns with retirement funding inadequacy has spurred intensified research attempting to understand the

complex interdependencies among a broad spectrum of actuarial and economic factors at play. The research

on retirement planning falls within the rich literature of portfolio management, which has been growing in

popularity and importance. To name a few influential works that are of interest to actuaries, one of the earliest

studies in modern retirement risk management dates back to the work by Ramsey almost one century ago

(Ramsey, 1928). Later, the lifecycle model, serving as the building block for many modern retirement planning

frameworks, was established in Yaari (1965). The use of stochastic control theory to solve lifetime financial

planning problems was pioneered by Merton (1969). The model in Richard (1975) integrated uncertain lifetime

into a stochastic control environment and was a substantial leap along this line. More recently, via adopting

stochastic programming approach, Milevsky and Young (2007) examined the optimal annuitization, investment

and consumption strategy for retirees facing stochastic times of death; Zeng et al. (2015) studied the optimal

retirement problem for a wage earner with exponential utility functions; Moore and Young (2016) considered an

optimal investment strategy to minimize the probability of lifetime ruin when the model parameters are subject

to random shocks; Shen and Sherris (2018) solved an optimal investment-consumption-insurance problem with

systematic and unsystematic mortality risks.

Despite the long list of literature, there is a limited amount of research has considered the effects of model

uncertainty on dynamic retirement planning, which also plays a pivotal role in the general practice of actuarial

modeling. In behavioral economics, the investors’ fear of the uncertainty in the probability distributions of

stochastic actuarial and economic outcomes, is best referred to as the ambiguity aversion. A proliferation of

empirical studies have documented that investors are not only risk averse, but also ambiguity averse. The

study of ambiguity has led to a number of significant advances in our ability to rationalize the empirical

features of asset returns and portfolio decisions. Therefore, one of our major contributions in this current paper

revolves around model uncertainty in the context of dynamic retirement planning. In the field of financial

mathematics, a general recipe for studying investor’s ambiguity aversion is via the notion of robust control.

The development of robust control was first pioneered by Hansen and Sargent (1993) and Hansen et al. (2002)

in which they studied the discrete time asset pricing problems for rational decision-makers with fears of model

miss-specification. The work in Anderson et al. (2003) was a firm step forward along this research line by

extending the approach to a continuous-time setting, which was further popularized by Maenhout (2004). In

plain terms, the robust control approach deals with model uncertainty by taking alternative models into account.

The robust optimal strategies are then selected based on the so-called endogenous worst-case scenario. The

investor’s aversion toward model uncertainty is captured by a penalty term in the optimization procedure that

reflects the discrepancy between the reference model best representing the data-generating process, and the

alternative models. Recently, a relatively new yet promising area of actuarial research in which ambiguity

models have found fruitful applications is optimal reinsurance design (see, e.g., Li et al., 2018; Zeng et al.,

2016).

2

One major feature that sets our paper apart from the existing literature is that we simultaneously consider

the effects of investment ambiguity and mortality ambiguity within a holistic framework, which to the best of

our knowledge, is new of this kind. The motivations behind our investigation are two-fold. First, it is a widely

accepted concept that investors should take model uncertainty into account when making financial decisions.

Although modeling stock price dynamics has been a very well-established subject in financial mathematics,

there is still a lack of consensus on the precise magnitude of expected risk premium, and the dispersions can

be very high depending on the estimation methods. Second, at the heart of optimal retirement study is to

maximize the accumulated utility of one’s lifetime consumption. The investor’s subjective assessment of her

health status plays a decisive role in the financial decision-making process, and the change in the individual

mortality can pose a substantial influence to the magnitude of the lifetime discounted utility. Different from

the objective mortality which can be estimated from the population data and/or extracted from the actuarial

life tables, the micro-structure of the subjective mortality is extremely complicated and is closely related to the

investor’s occupation, wealth, life style, and other actuarial/economic determinants (Hurd and McGarry, 1995,

2002). The statistical assessment of individual mortality is notoriously hard, if it is not impossible. For this

reason, it natural for us to take the subjective mortality ambiguity into consideration and analyze the end results

on retirement planning under the worst-case mortality scenario, which fits the very conservative behaviors of

many soon-to-be retirees. All in all, we think the investment and mortality ambiguities are essential components

within the optimal retirement planning paradigm. By incorporating both kinds of ambiguities into the model, it

allows us to compare the relative importance on retirement planning with respect to investment and mortality

risks.

To briefly preview our model, in this article, we examine the optimal planning strategies including invest-

ment, consumption, and insurance purchase dynamics, for a utility maximizing individual receiving stochastic

labor incomes. The investor concerns with the model uncertainty in the investment return and subjective

mortality models. Thus, among a set of plausible models, the investor prefers the dynamic rules that can

perform reasonably well even in the worst-case scenario. In addition to the aforementioned dynamic rules, we

locate the optimal retirement time under the full annuitization post-retirement arrangement. Although this

all-or-nothing annuitization arrangement is rather restrictive (yet not uncommon), this assumption has been

widely used in the retirement planning literature (see, e.g., Chen et al., 2018; Milevsky and Young, 2007). The

expected present value of future incomes is named the human capital in the portfolio management literature.

The rate of depletion in human capital is clearly a driving factor behind retirement decisions, and we study its

impact on the optimal retirement timing. We consider an individual facing a stochastic time of death under

power utilities for consumption, pre-retirement bequest motive and retirement income. The dynamics in stock

prices and incomes are modeled by diffusion processes of which the evolution terms are perfectly correlated. We

openly admit that assuming the perfect dependence between stock and income dynamics may be unrealistic.

Nevertheless this simplifying assumption allows us to obtain relatively more visible solutions for the utility

maximization problem of interest, which in turn reveals more intuitive insights into the interplays of ambiguity

aversions and various actuarial/economic factors. This is exactly the main impetus for our article.

Our efforts in this current paper make several technical contributions. To the best of our knowledge,

our paper is the first attempt in the retirement planning literature to address the problem of investment and

mortality ambiguities at the same time within a unified optimization framework. Although investment ambiguity

3

has been relatively well studied, it is perhaps fair to state that mortality ambiguity is a rarely touched area when

it comes to the study of optimal retirement. Another recent work that considers the ambiguous mortality risk on

portfolio selection is Young and Zhang (2016). In our paper, we borrow the methodologies used to model credit

defaults ambiguity in mathematical finance, and translate them seamlessly to model the mortality ambiguity in

optimal retirement problems. Due to the correlation between the investment and income dynamics, our model

should also address the concern with the model uncertainty for the income process, as a byproduct. We derive

the optimal solution for the retirement problem in an analytical form, and rigorously show that the solution

always uniquely exists. The mathematics behind our retirement problem is interesting in its own right, and the

methodology we establish in this paper should open up new research directions for simultaneously incorporating

investment and mortality ambiguities in optimal retirement research.

Our paper also contains a number of economic contributions. First, by comparing the aforementioned two

kinds of ambiguities, we find that besides the risk and ambiguity aversion parameters the worst-case investment

scenario depends on the market parameters only, however the analysis for the worst-case mortality scenario

is much more complicated. The determinants of the worst-case mortality probability measure consist of both

the market and personal parameters such as mortality, income, utility preferences, etc. Second, the study of

mortality ambiguity highlights the negative value of life expectancy for sufficiently risk averse investors. When

the value of life expectancy is negative, we find that the worst-case mortality scenario takes place in the situation

when the individual lives longer than expected. That being said, the uncertainty about the mortality model can

be viewed as a mortality risk for less risk averse investors, but a longevity risk for more risk averse investors.

Third, our numerical study shows that both investment and mortality ambiguities have monotonic impacts

on the optimal strategies regardless of the level of risk aversion except for optimal consumption behavior. In

our retirement model, the impacts of ambiguity aversion on optimal consumption are reversed for investors

with varying level of risk aversion. This is because the mechanism behind optimal consumption behavior is

extremely complex, which involves the intricate interdependencies of investment, insurance demand, retirement

timing, time preference of money, etc. The simple percentage-based consumption rule oftentimes suggested by

financial planners is clearly insufficient for effective retirement planning. Our paper further advocates the need

for using sophisticated technique in the study of retirement planning especially when model uncertainty is a

major concern.

The reminder of this article proceeds as follows. Setting up general models for the actuarial and financial

environment in Section 2, we derive the optimal solutions for the utility maximization problem in Section 3.

In Section 4, we conduct a numerical study to illustrate the economic implications of our theoretical results.

Section 5 concludes the article. The appendix contains the mathematical details that are in addition to our

major actuarial and economic contributions.

2 Model description

In this section, we describe the actuarial and economic ingredients that play the fundamental roles in our

retirement planning problem. To facilitate our sequential discussion in a rigorous manner, we fix a probability

space (Ω,F ,P) equipped with a right-continuous, P-complete filtration F := Ft∈[0,T ], where T <∞. Also let

Wtt∈[0,T ] be a one-dimensional standard Brownian motion on (Ω,F ,F,P), which will be used to describe the

4

economic dynamics in our model.

2.1 The formulation of actuarial and economic environment

We first present the economic models, followed by the actuarial models. Our article is normative and assumes

a simple financial environment consisting of one risk-free asset (e.g., bank account), and one risky asset (e.g.,

listed common stock, stock index and market portfolio) which we term the stock hereafter. Through the risk-free

asset, one dollar of investment grows without uncertainty as

Bt = ert, ∀t ∈ [0, T ],

for some fixed risk-free interest rate r > 0. The stock price described by a stochastic process, Stt∈[0,T ], evolves

in accordance with a geometric Brownian motion (GBM) model:

dSt = St(µdt+ σdWt), S0 = s0 > 0, (1)

where µ > 0 and σ > 0 denote respectively the expected rate and the volatility of stock return.

We assume that the wage process itt∈[0,T ] satisfies the following GBM model

dit = it(αdt+ βdWt), (2)

where i0 > 0 is postulated to be a deterministic initial income level, α ∈ R is the income growth rate which can

be positive or negative, and β > 0 is the diffusion coefficient. In this article, we assume the income process and

the stock model share a common Brownian motion. Stated plainly, that is the shocks to the stock price and

labor income are perfectly correlated. This situation may occur when one’s income is primarily based on the

financial market performance (think, e.g., individuals working in the financial industry and/or receiving stock-

based compensation). Indeed, this assumption is made due to a technical standpoint and it has been widely

adopted in the portfolio management literature (see, e.g., Chen et al., 2018; Pirvu and Zhang, 2012; Shen and

Sherris, 2018, etc). Although oversimplified, it considerably facilitates the existence of analytic solutions to the

utility maximization problem, which in turn reveals more insights into the actuarial and economic factors at

play.

Next we proceed to discuss the actuarial assumptions. We consider an individual currently aged at y > 0,

and conventionally denote her by ‘(y)’. The remaining lifetime of the individual (y), succiently τy, is a non-

negative random variable defined on (Ω,F ,F,P) and stochastically independent of Wtt∈[0,T ]. The associated

instantaneous force of mortality curve, λy+s := λy(s) : R+ → R+, is generally a nondecreasing function in

s ∈ R+. The survival probability for the individual is

tpy := P[τy > t] = exp

(−∫ t

0λy+sds

), (3)

and the corresponding probability density function is

fτy(t) = λy+t exp

(−∫ t

0λy+sds

). (4)

5

As we mentioned earlier in the introduction section, survival probability can be assessed ‘subjectively’ by

the individual herself or ‘objectively’ by the annuity/insurance providers. Notationally, we separate these

two probabilities by putting a dot on top of the objective mortality (i.e.,.λy+t, t ∈ [0, T ]), and thus

.tpy :=

exp(−∫ t

0

.λy+sds

)is the objective survival probability. For exposition reasons, let us henceforth confine our-

selves to a simpler case in which the subjective and objective mortalities are discerned up to a scaling coefficient

δ > 0 (see, Milevsky and Young, 2007, for a similar setup). Namely,

λy+t = δ.λy+t, (5)

for all t ∈ [0, T ]. It is a simple matter to see that if δ < 1, then the individual is healthier than the population

in a coherent manner and vice versa. In the dragon of actuarial mathematics, the assumption in (5) is also

known as the proportional hazard distortion (Wang, 1996), which is similar to the transformation examined by

Johansson (1996) in the context of life economics.

Moreover, we introduce the subjective discount rate ρ > 0. Then the continuous annuity factors based on

the subjective and objective mortalities can be computed respectively via

a(ρ)y :=

∫ ∞0

sE(ρ)y ds and

.a(r)y :=

∫ ∞0

s

.E(r)y ds, (6)

where

sE(ρ)y := e−ρsspy and s

.E(r)y := e−rss

.py,

are the corresponding actuarial discount factors. As the subjective (resp. risk-free) rate is usually associated

with the subjective (resp. objective) mortality, henceforth we suppress the discount rate in the notations and

just simply write ay and.ay to denote the subjective and objective annuity factors. Stating plainly, the objective

factor.ay is the fair annuity price that insurance companies would charge before any expenses loadings, whereas

ay is the annuity price the individual thinks what it would have been based on her personal rate of time preference

(i.e., ρ) and her subjective mortality assessment. Throughout this article, we do not consider insurance loadings,

commissions, and profits, which in fact can be straightforwardly incorporated by using loaded force of mortality

curves (see, Huang and Milevsky, 2008; Huang et al., 2008).

2.2 The formulation of retirement planning

We now turn to describe the retirement problem of interest. Suppose that the individual (y), endowed with an

initial wealth x0 > 0 and having present income i0 > 0, is a utility maximizing investor. We work with two dates

of interest - the retirement date T > 0 and the time-to-death of individual τy. For ease of the presentation, we

divide the retirement planning problem into two phases, namely, the pre-retirement and post-retirement phases.

Over the pre-retirement phase, the individual receives a stochastic rate of labor income described by the model

(2). Meanwhile, the individual consumes at rate ct at time t ∈ [0, T ] and makes investment. Specifically, she

would allocate πt dollar amount of the total wealth to the stock market modeled by (1), and the rest to the

risk-free bank account.

In order to protect against accidental death, the individual chooses to purchase life insurance. With the life

insurance in place, if the individual dies during the pre-retirement phase, then an amount of bequest consisting

6

of the insurance benefit and the accumulated financial wealth will be inherited by the beneficiary. We assume the

individual can purchase any amount of infinitesimally short-term insurance which is guaranteed to be available

continuously in the insurance market and pays death benefit of 1/.λy+t per face value, if she dies at time t. The

life insurance demand at time t is denoted by kt, which can be positive and negative. When kt is negative,

the short-selling of life insurance can be understood as an annuity purchase, in which the individual receives

annuity payments and the beneficiary needs to pay back the premiums as a lump sum upon the individual’s

death before T (see, e.g., Pliska and Ye, 2007; Shen and Sherris, 2018, for a similar setup). To summarize the

aforementioned setup during the pre-retirement phase, let us report herein formally the individual’s financial

wealth trajectory Xtt∈[0,T ] associated with the triplet of financial strategies (π, c, k):

dXt = [rXt + πt(µ− r)− ct + it − kt]dt+ πtσdWt, X0 = x0 > 0. (7)

Upon retirement, we assume the individual will fully annuitize her accumulated financial wealth in exchange

for an annuity income flow, which will be fully consumed. This retirement arrangement is also known as the

institutional all-or-nothing annuitization, which is restricted but not so uncommon in practice. In fact, this

simple annuitization rule is quite attractive to many retirees from an actuarial perspective since the exposures

to longevity risk are fully eliminated, and thus it has been widely adopted in the retirement risk management

literature (e.g., Chen et al., 2018; Milevsky and Young, 2007; Pliska and Ye, 2007). On another line of reasoning,

Yaari (1965) established a theoretical model to show that a rational retiree having no bequest motive should

annuitize immediately her entire wealth. In our problem, since bequest motive is not included in the post-

retirement phase (see, Equation (8)), the full annuitization arrangement can be in fact justified theoretically.

Motivated by the seminal work of Yaari’s life-cycle model (Yaari, 1965), we assume the individual seeks to

maximize her expected utilities from the inter-temporal consumption over lifetime as well as the legacy left to

the beneficiary in the pre-retirement phase. It follows that the associated objective function is given by

J(t, x, i;π, c, k;T )

= Et,x,i[ ∫ T∧τy

te−ρ(u−t)U1(cu)du+ e−ρ(τy−t)U2

(Xτy +

kτy.λy+τy

)1τy<T +

∫ τy

Te−ρ(u−t)U3

(XT.ay+T

)du

](1)= Et,x,i

[ ∫ T

tu−tEy+t

[U1(cu) + λy+uU2

(Xu +

ku.λy+u

)]du+ T−tEy+t ay+T U3

(XT.ay+T

)], (8)

where Et,x,i denotes the conditional expectation given τy > t and (Xt, it) = (x, i) at t ∈ [0, T ], the functions U1,

U2, and U3 are increasing and concave utility functions for respectively the pre-retirement consumption, bequest

motive and post-retirement consumption. Here, equality ‘(1)=’ holds because of the independence assumption

between W and τy, and the annuity formula in (6).

Concerning the varying preferences toward the inter-temporal consumption and bequest motive, our article

operates under the Constant Relative Risk Aversion (CRRA) utilities. Namely, for some x > 0,

U1(x) =x1−γ

1− γ, U2(x) =

(ξ∗x)1−γ

1− γ, U3(x) =

(η∗x)1−γ

1− γ, (9)

in which γ ∈ (0, 1)∪ (1,∞) is the relative risk aversion parameter, and ξ∗ > 0 and η∗ > 0 are the relative utility

weights for accommodating the state-dependent preferences toward the bequest motive and post-retirement

7

consumption. In practice, since individuals typically enjoy more free time and leisures after retirement, the

value of η∗ should be greater than one in order to capture the leisure preference. For brevity, we set ξ := ξ(γ) =

(ξ∗)1−γ ∈ R+, η := η(γ) = (η∗)1−γ ∈ R+, and U := U1, which simply implies U2 = ξU and U3 = ηU . When

ξ = 0, then the individual assigns no utility weight toward the bequest motive. However, it is noteworthy

that the insurance demand may not be simply equal to zero in this case. In fact, the individual will short-sell

insurance (equivalently, purchase annuity) as much as possible because she no longer cares about the legacy

(Chen et al., 2018).

2.3 The formulation of ambiguity aversion

As mentioned in the introduction section, the individual (y) is concerned about the risk of miss-specifying

the investment and subjective mortality models, even through they are established in such a way that best

represents the data generating processes. She therefore considers a pool of alternative models. To formulate

the set of plausible alternative models, we follow the very active line of research in financial economics focusing

on ambiguity aversion and let θSt t∈[0,T ] and θλt t∈[0,T ] be two Ft-predictable processes valued in R and R+,

respectively. Then we define a family of equivalent (real-world) probability measures by setting

dQdP

∣∣∣∣FT

= Λ1(T ) · Λ2(T ),

where

Λ1(T ) := exp

− 1

2

∫ T

0(θSt )2dt−

∫ T

0θSt dWt

, (10)

and

Λ2(T ) := exp

∫ T∧τy

0

[θλt log(θλt )− θλt + 1

]λy+tdt+

∫ T

0log(θλt−)dZt

. (11)

Here the process Ztt∈[0,T ] is defined by Zt := Ht−∫ t

0 (1−Hs)λy+sds with Ht := 1τy≤t and is a P-martingale.

Denote by Θ the set of all pairs of R-valued and R+-valued Ft-predictable processes such that Q is a well-defined

equivalent probability measure. In the Radon-Nikodym derivative above, the first component (10) changes the

probability distribution of stock prices by adjusting the expected stock return, while the second component

(11) governs the distortion of the instantaneous morality. Precisely, under the alternative Q measure, it follows

from Girsanov’s theorem that the dynamics underlying the economic outcomes evolve according to

WQt := Wt +

∫ t

0θSudu,

which is a standard Brownian motion, and the subjective force of mortality becomes λQy+t := θλt λy+t. So, the

alternative models for the stock, income and wealth processes governed by the Q measure are respectively

dSt = St[(µ− σθSt )dt+ σdWQ

t

],

dit = it[(α− βθSt )dt+ βdWQ

t

],

8

and

dXt =[rXt + πt(µ− r − σθSt ) + it − ct − kt

]dt+ πtσdW

Qt .

Under Q, the individual’s survival probability and the corresponding probability density function are defined

similarly as (3) and (4), but with λy+s replaced by λQy+s. Our formulations for the alternative financial models

are consistent with much of the work in the financial economic literature (see, e.g., Escobar et al., 2015, 2018;

Maenhout, 2004, for similar specifications). The setup for the alternative mortality models is inspired by the

work of Bo and Capponi (2017) in which they used a similar approach to induce ambiguity aversion to the

default models of credit portfolio.

On a different note, in order to make our optimization problem well-posted, a penalty is incurred for

alternative models that deviate too far away from the reference models. The ‘distance’ between the alternative

and reference models is measured in terms of the relative entropy:

Γ(u) : =(θSu )2

2ΨS(u,Xu, iu)+

(1−Hu)g(θλu)λy+u

Ψλ(u,Xu, iu),

where u ∈ [0, T ] and

g(θλu) :=[θλu log(θλu)− θλu + 1

].

To simplify our following presentation, we denote by

Γ(u) : =(θSu )2

2ΨS(u,Xu, iu)+

g(θλu)λy+u

Ψλ(u,Xu, iu).

The practical justification behind this penalty term is that alternative models possessing low entropy are

difficult to distinguish statistically from the reference models and thus worth to be considered seriously (see,

also Maenhout, 2004, 2006). The state-dependent functions ΨS and Ψλ can be viewed as the stock and mortality

robustness preferences. Namely, the greater the value of ΨS (resp. Ψλ), the stronger the preference of robustness

becomes, the less the faith we place on the reference stock (resp. mortality) model. For analytic convenience,

throughout the rest of this article, we assume the robustness preference functions admit the following format:

Ψ(t, x, i) :=ψ

(1− γ)V (t, x, i), (12)

where ∈ (S, λ), ψS > 0 and ψλ > 0 are respectively the ambiguity aversion parameters on stock and subjective

mortality models. We refer the reader to Maenhout (2004) for a more detailed discussion on the choice of the

robustness preference function (12).

We are now in a position to specify the expected discounted utility for the ambiguity averse investor with

the set of plausible alternative models we have formulated in this current section. As a consequence of the full

annuitization arrangement upon retirement, the individual only has the freedom to control the financial strategy

and consumption behavior before her retirement. Therefore, we consider the effects of ambiguity aversion over

the pre-retirement phrase only. Note that

EQ[1−Hu|τy > t] = e−∫ ut λ

Qy+sds, ∀u ∈ [t, T ].

9

Incorporating the alternative Q measure governed by (θS , θλ) as well as the expected value of the accumulative,

discounted deviation penalty over [t, T ], the objective function (8) becomes

J∗(t, x, i;π, c, k, θS , θλ;T )

= EQt,x,i

[ ∫ T

tu−tE

Qy+t

[U1(cu) + λQy+uU2

(Xu +

ku.λy+u

)]du+ T−tE

Qy+t ay+T U3

(XT.ay+T

)+

∫ T

tu−tE

Qy+t Γ(u)du

],

where

u−tEQy+t := e−ρ(u−t)e−

∫ ut λ

Qy+sds, for u ∈ [t, T ],

is the actuarial discount factor under the alternative mortality model governed by the Q measure.

3 The robust optimal strategies

Section 2 formulated the actuarial and economic environment and set up a meaningful theoretical groundwork

for modeling ambiguity aversion. In this section, we are going to discuss the corresponding control strategies by

using the dynamic programming principle. Specifically, the individual is facing a utility maximization problem

about the intertemporal choices on investment, consumption, and insurance demand. We aim to identify the

optimal strategy over a set of admissible strategies which satisfy the following technical assumptions.

Definition 1. A triple (π, c, k) is called an admissible strategy if

1. π, c and k are progressively measurable processes, taking values in R, R+ and R, respectively;

2. π, c and k are almost surely integrable in the sense∫ T

0(|πt|2 + |ct|+ |kt|)dt <∞, P-a.s.;

3. the wealth equation (7) associated with (π, c, k) has a unique strong solution X such that X(t)+H(t, i(t)) ≥0, where H denotes the human capital function to be defined in (15);

4. the unique solution X is regular enough such that U(Xt)t∈[0,T ] is uniformly integrable.

The space of all admissible strategies is denoted by A.

The conditions in Definition 1 can be justified intuitively as follows. The first condition ensures the amount

of consumption to be positive and the overall strategies to be well-defined in the sense that they only rely on

the information up to present. The third condition states that the investor should not spend down the total

wealth in her personal balance sheet to zero before retirement. Speaking differently, that is every admissible

strategy should prevent bankruptcy to the individual at all cost, if there is no pension income to fall back on.

The second and fourth conditions are imposed for technical reasons, which are needed to verify the optimality

for the selected strategy (see, e.g., Shen and Sherris, 2018).

10

For a given retirement time T > 0, the value function for the ambiguity averse individual who seeks the

optimal investment, consumption and insurance rules (over the admissible set A) under the endogenous worst-

case scenario, which is chosen over Θ, can be formulated as

V (t, x, i;T ) := sup(π,c,k)∈A

inf(θS ,θλ)∈Θ

J∗(t, x, i;π, c, k, θS , θλ;T ). (13)

To solve the optimization problem in (13), let us first report the associated Hamilton-Jacobi-Bellman (HJB)

equation in what follows. For notational convenience, we suppress the retirement time in the notation of

the value function and simply write V (t, x, i) := V (t, x, i;T ). We also define a partial differential generator,

L(π,c,k,θS ,θλ), acting on the value function such that

L(π,c,k,θS ,θλ)[V (t, x, i)] = −(ρ+ θλλy+t)V + Vt + [rx+ π(µ− r − σθS) + i− c− k]Vx

+(α− βθS)iVi +1

2π2σ2Vxx +

1

2β2i2Vii + πσβiVxi.

The HJB equation for the retirement planning problem of interest can be derived as follows:

sup(π,c,k)

inf(θS ,θλ)

L(π,c,k,θS ,θλ)[V (t, x, i)] + U(c) + θλλy+tξU(x+ k/

.λy+t)

+(1− γ)V (t, x, i)

((θS)2

2ψS+g(θλ)λy+t

ψλ

)= 0 (14)

with terminal condition

V (T, x, i) = ηay+TU

(x

.ay+T

).

The concept of human capital plays a critical role in the study of the HJB equation (14). Stating bluntly,

human capital is the market price if an individual is willing to sell all her future incomes as a lump sum. Human

capital can take up a large portion of one’s personal balance sheet, and therefore it drives retirement decision

making. The market value of human capital at time t ∈ [0, T ] can be computed via

H(t, i) := H(t, i;T ) = Et,i[ ∫ T∧.τy

te−rs is ds

], (15)

where Et,i[·] denotes the conditional expectation under the risk-neutral measure P (see, Equation (20)), given

the current income level it = i, and.τy is the lifetime random variable of (y) following the objective mortality

law. The individual’s total wealth consisting of the present financial asset and human capital can be defined as

W (t, x, i) := x+H(t, i).

The next proposition formulates the human capital for the individual considered in our retirement problem.

Proposition 1. Assume that the stock and income dynamics are modeled by (1) and (2), and let r∗ := r−α+

(µ− r)β/σ. For the individual aged at (y + t), t ∈ [0, T ], with retirement age (y + T ) and present income level

it = i > 0, the human capital is given by

H(t, i) = i.a

(r∗)

y+t:T−t ,

where.a

(r∗)

y+t:T−t denotes the (T − t)-year term objective annuity factor.

11

In light of the formula reported in Proposition 1, the individual’s human capital is equivalent to the price

of i-unit of annuity contracts with the term of payments based on the time remaining in labor. The later the

retirement date (i.e., T ), the higher the value of the annuity factor, and thus the individual owns higher human

capital. The income growth rate manipulates the value of human capital in the same fashion. That is, the

higher the income growth rate, the smaller the discount factor becomes, thereby yielding a higher annuity factor.

On the other hand, a more volatile income process may harm the individual’s wealth of human capital since

the diffusion coefficient β is an add-on to the discount rate r∗. Interestingly, the human capital is independent

of the individual’s risk aversion and ambiguity aversion. This is indeed a very natural result since H(t, i) is

an objective measure that reflects the maximum amount of cash the individual can borrow from the financial

market when her future income acts as the collateral.

Denoting the worst-case probability measures by θS∗ and θλ∗, and the optimal investment, consumption,

and insurance rules by respectively π∗, c∗, and k∗, the next assertion studies the optimal strategies for the

ambiguity and risk averse investor under the actuarial/economic environment discussed in Section 2. For

notional convenience, for any (t,m) ∈ [0, T ]× R+, let us define

h(t,m) :=1− γγ

δ(1−γ)/γξ1/γ

[1− 1− γ

ψλlogm+

γ

ψλ

] [1− 1− γ

ψλlogm

]−2

,

A(t,m) : =(ξmλy+t)

1γ

(.λy+t)

1−γγ

,

B(t,m) : = (r +.λy+t)−

(ρ+mλy+t)

1− γ+

1

2

(µ− r)2

(ψS + γ)σ2+g(m)λy+t

ψλ,

and

F (m) :=(δ m)

1−γγ ξ

1γ

1− 1−γψλ

logm.

Proposition 2. For a fixed retirement time T > 0, the worst-case probability measures for the ambiguity and

risk averse investor are determined by

θS∗t =ψS

ψS + γ

µ− rσ

,

and θλ∗t such that it satisfies the ordinary differential equation (ODE):

(θλ∗t )′ +

B(t, θλ∗t )

[1 +

γ

ψλ − (1− γ) log(θλ∗t )

]−1

+

[1 +A(t, θλ∗t )

]h(t, θλ∗t )−1(θλ∗t )−(1−γ)/γ

θλ∗t = 0, (16)

with terminal value θλ∗T = D(T ), where D(T ) is the positive root of the algebraic equation

η1γ

(ay+T )1γ

(.ay+T )

1−γγ

= F (D(T )). (17)

12

The optimal strategies are given by

π∗(t, x, i) =µ− r

(ψS + γ)σ2W (t, x, i)− β

σH(t, i),

c∗(t, x, i) =W (t, x, i)

F (θλ∗t ),

and

k∗(t, x, i) = A(t, θλ∗t )W (t, x, i)

F (θλ∗t )− x

.λy+t.

The corresponding value function is

V (t, x, i) =W (t, x, i)1−γ

1− γ[F (θλ∗t )

]γ. (18)

A few qualitative findings stand out immediately from Proposition 2. First, recall that the expected stock

return under the worst-case scenario is given by µ∗t := µ − σθS∗t . Since θS∗ is an increasing function of the

ambiguity parameter ψS (when µ > r), a higher level of ambiguity aversion leads to a lower stock return in the

worst-case scenario. When ψS = 0, the investor has no concern with the stock model uncertainty, so θSt ≡ 0 for

all t ∈ [0, T ] and the investor would always stick with the reference model for stock. On a different note, the

risk aversion parameter γ manipulates the expected stock return in the worst-case scenario in a reverse manner.

A more risk averse investor is more reluctant to deviate from the reference model, and a relatively mild decline

in stock return becomes very severe. The worst-case probability measure for the mortality model consists of a

highly non-linear ODE which cannot be solved in a closed-form expression. Moreover, different from θS∗, it can

be seen from (16) that θλ∗ depends on not only ψλ but also ψS . This observation provides further evidence for

the complication in understanding the ambiguity effects on the mortality model and motivates us to carry out a

numerical study about θλ∗ in Section 4. Although there is no closed-form expression for θλ∗, one can still verify

that when ψλ goes to zero, then θλ∗ converges to 1. To see this, we know from Proposition 3 that (θλ∗t , F (θλ∗t ))

must be bounded and strictly positive for all t ∈ [0, T ]. Therefore, as ψλ → 0, the limit of θλ∗ must exist but it

cannot be a constant other than one.

Second, due to the perfect correlation between the stock and income dynamics, the presence of labor

income influences the speculative demand. In light of the formula for π∗, to determine the optimal investment

allocation, the individual first computes the speculative demand based on the total wealth of x+H(t, i), then

makes correction for the implicit market exposures inherited in the income process with a relative weight of

β/σ. It is simple to check that the greater the level of ambiguity and risk aversions, the smaller the allocation

to risky investment. Rewriting the optimal investment rule in terms of the relative portfolio weight can reveal

extra insight:π∗

x=

µ− r(ψS + γ)σ2

+i

x

[µ− r

(ψS + γ)σ2− β

σ

].a

(r∗)

y+t:T−t .

In the relative portfolio weight formula above, the first term on the right hand side is the well-known Merton

ratio/myopic demand (with ambiguity consideration) and the quantify i/x in the second term is known as the

13

wealth-to-income ratio in economics. If the investor is sufficiently ambiguity and risk averse, and/or the market

risk premium is low such that the coefficients satisfy

µ− r(ψS + γ)σ2

− β

σ< 0,

then we observe that high wealth-to-income ratio discourages the individual to gain speculative exposure. This

observation makes very intuitive sense to us.

Third, the optimal consumption rule for an individual with financial wealth x and stochastic income de-

scribed by (2) is identical to that for whom has financial wealth of x + H(t, i) but no future income. The

wealth-to-income ratio can again play a critical role in optimal consumption. Specifically, the propensity to

consumption out of the current financial wealth is given by

c∗

x=

[1 +

i

x

.a

(r∗)

y+t:T−t

] [F (θλ∗t )

]−1.

Since the function F (θλ∗t ) is always positive for all t ∈ [0, T ], the consumption propensity has an increasing

relationship with the wealth-to-income ratio.

Forth, since life insurance can be considered as a hedge against the loss of human capital, the level of

human capital has a positive effect on insurance demand and offers the incentive for purchasing insurance.

Depending on the utility coefficient toward bequest motive as well as the differences between the subjective and

objective mortalities, insurance demand can be positive and negative. As shown in the formula of k∗, unhealthy

individuals with higher subjective mortality tend to optimally purchase more insurance, and the alternative

mortality measure θλ∗ manipulates the optimal insurance demand in a similar manner. All the aforementioned

implications on optimal insurance demand coincide with our intuition.

Last, the value function (18) consists of two components W and F , wherein F is independent of the income

process. The value function of the current maximization problem is the same as that for another problem with

the initial financial wealth of x+H(t, i) but no labor income. More formally, that is

V (t, x, i) = V (t, x+H(t, i), 0).

This is because human capital can be viewed as the certainty equivalence of the investor’s future income,

though between these two problems, the optimal investment strategies are different. All in all, since the risk

profile of human capital is closely related to the retirement age as well as the characteristics of income process,

the interplays of these factors consequentially are of central importance in the course of optimal retirement

planning. In Section 4, we will conduct a more intensive numerical study to glean a higher level of insight into

the economic implications for the retirement model of interest.

We have so far obtained the optimal strategy for every fixed retirement time T > 0. Owing to the complexity

of the ODE (16), it is very difficult (if not impossible) to obtain an explicit expression for θλ∗ and F , thus for the

optimal strategy. One has to solve for θλ∗ and F numerically. Nevertheless we can still educe some fundamental

properties for the optimal strategy such as the existence as well as the uniqueness, which are by no means

trivial.

Proposition 3. The system of coupled equations (29) and (31) has a unique solution pair (θλ∗t , F (θλ∗t )) that

are bounded and strictly positive over [0, T ].

14

Proposition 3 ensures that the optimal strategy for our retirement problem always uniquely exist. Once

the numerical results for θλ∗ and F are obtained, we can calculate the optimal strategy and furthermore use

the corresponding value function to identify the optimal retirement timing, with which the individual’s utility

is maximized over all T ∈ [0, TM ] where TM > 0 is the maximal retirement age. Precisely, given x0 = x and

i0 = i, we define the optimal retirement time by

T ∗ := arg maxT∈[0,TM ]

V (0, x, i;T ).

If T ∗ = TM , then we claim the optimal retirement time does not exist and the individual must retire at TM . As

we mentioned earlier, we cannot obtain a closed-form expression for the value function, and thus the optimal

retirement time needs to be solved numerically.

It is noteworthy that the optimal retirement time we derive in this current article by no mean a discretionary

stopping time. Instead of planning for the optimal retirement time and financial strategies simultaneously, the

investor will first select the optimal investment, consumption and insurance rules for a fixed retirement time,

after which the optimal retirement time will be located to maximize the lifetime utility. From a technical

point of view, this kind of pre-committed retirement time arrangement can largely simplify the complexity

of our problem and allows us to consider essentially only a control problem (rather than a combined optimal

control and stopping problem). Our intentionally simple model has the merit of producing clean results that

are convenient to interpret. On a different line of reasoning, Chen et al. (2017) recently reported that in terms

of ‘welfare loss’, this type of pre-committed retirement decision indeed well approximates the sophisticated one

by means of optimal stopping time.

4 Numerical illustration

In this section, we present some numerical results to demonstrate the economic implications behind our theo-

retical results. At the outset, we recommend a realistic set of economic parameters: r = 2%, µ = 9%, σ = 0.2.

With regard to the income dynamics, in a recent report published by the National Bureau of Economic Re-

search, Guvenen et al. (2015) analyzed a large panel data set of individual incomes and concluded that, income

life cycle indeed exhibits a hump-shaped pattern and peaks at around age 50. More interestingly, the average

log income starts to decay after age 55 at an approximately constant rate (see, Figure 3 in Guvenen et al., 2015).

Similar empirical findings can be also found in, e.g., Cocco et al. (2005); Kolasa (2017); Munk and Sørensen

(2010), and so forth. We conjecture that this decreasing pattern over one’s post-peak income cycle is a major

economic factor that drives the optimal retirement timing. To gain insight into the aforementioned matter,

we hereby consider an 55-year-old individual having current income i = 50 thousand with income growth rate

α = −5% and income volatility β = 5%. Furthermore, we assume the maximal retirement age is 90 and the

financial wealth the individual presently owns is x = 100 thousand.

We assume the Gompertz law to model the individual’s force of mortality curve. The Gompertz law is one of

the arguably most well-known mortality models in the literature of actuarial mathematics and it has been used

intensively in the insurance industry nowadays. In our article, the Gompertz mortality function is formulated

as

λGMs := w1 exp(w2 s), (19)

15

for s, w1, w2 ≥ 0. To estimate the parameters, we fit the Gompertz mortality law against the 2010 - 2015 US

mortality table from the Human Mortality Database and obtain w1 = 6.08 and w2 = 0.08 for the objective

mortality curve. In the baseline scenario of the sensitivity analysis, we assume the subjective and objective

mortalities are identical (i.e., δ = 1).

Numerous studies in literature have suggested a wide range of estimation for the risk aversion coefficient.

However, the estimation of γ critically relies on the individual’s risk profile. In the absence of a widely accepted

point estimation, we conventionally consider and present our results under several choices of value for γ. Fur-

thermore, we assume the baseline utility coefficients for bequest motive and retirement leisure are ξ∗ = 1 and

η∗ = 2. Also the ambiguity aversion parameters for stock and subjective mortality are respectively ψS = 2 and

ψλ = 0.5. The subjective discount rate is set to be ρ = r. As some of the aforementioned choices of parameters

are artificial, we are going to carry out a detailed sensitivity analysis in what follows. Precisely, we are going to

change the value of one parameter at a time and assess the resulting impacts on the optimal retirement strate-

gies. Hereafter, to be consistent with how insurance is discussed in practice, we present the optimal insurance

demand in terms of the face value, i.e., κ∗t := k∗t /.λy+t. We are going to report the amount of optimal invest-

ment and consumption in terms of the propensity out of the total wealth (i.e., respectively π∗t /W (t, x, i;T ∗)

and c∗t /W (t, x, i;T ∗)). Since the optimal strategies are state-dependent and random, we routinely report these

quantities at t = 0.

4.1 Sensitivity study on ambiguity aversion

Our primary focus in this subsection is to study the impacts of ambiguity aversion on optimal retirement

planning. We first study the interconnection between the worst-case probability measures and various actu-

arial factors. The worst-case probability measure for stock model possesses a simple explicit expression (see,

Proposition 2), and as we have discussed in Section 3 its relationships with various parameters are relatively

straightforward to understand. Therefore, in this part of the numerical study, we concentrate on the mortality

ambiguity aversion.

When speaking of mortality ambiguity in the context of retirement planning, it is important for us to notice

that life expectancy may comprise negative economic value (see, Cordoba and Ripoll, 2017, for a wealth of

motivating discussions). This is particularly true for a utility maximizing investor with negative-valued utility

function. In our model, that corresponds to the case when γ > 1 (see, Figure 1, for a visualization of this

matter). What we can observe from Figure 1 is that as the subjective mortality multiplier δ increases, the

individual becomes less healthy, while the magnitude of the total utility may decrease or increase depending on

whether γ is less than or greater than one. For γ ∈ (0, 1), the value function is decreasing in δ, thus the value

of life expectancy is positive, and life insurance can be used to hedge the associated morality risk. In sharp

contrast, when γ > 1, the investor is more risk averse and she thinks the value of life expectancy is negative.

In this case, the investor can short-sell insurance, or equivalently purchase annuity, to hedge the longevity risk.

With the aforementioned insight about the value of life expectancy in place, we now study the sensitivities

of a few actuarial parameters that play a key role in determining the worst-case mortality probability measure.

The results of the sensitivity analysis are reported in Figure 2. Here is how we should interpret their economic

implications. First and foremost, θλ∗ is smaller (resp. greater) than one for γ > 1 (resp. γ ∈ (0, 1)). In other

16

0.7 0.8 0.9 1 1.1 1.2 1.3200

250

300

350

400

450

500

550

V

=0.6=0.7=0.8

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-14

-12

-10

-8

-6

-4

-2

V

=1.4=1.5=1.6

Figure 1: Comparison of the value function for a utility maximizing investor (without ambiguity aversion

consideration) with varying subjective mortality multiplier δ and the risk aversion γ.

words, when γ > 1, the worst-case mortality scenario occurs when the ambiguity averse individual lives longer

than expected. The reverse relationship holds for γ ∈ (0, 1). This is due to the difference in the sign of life

expectancy value between γ > 1 and γ ∈ (0, 1). In addition, the larger the value of ambiguity aversion, the

farther the worst-case probability deviates from the reference model. When there is no ambiguity aversion (i.e,

ψλ goes to zero), the worst-case model coincides with the reference model at θλ∗ = 1. This numerical result is

consistent with our theoretical investigation discussed in Section 3.

Different from the ambiguity aversion parameter, the change in the risk aversion parameter influences the

worst-case mortality measure in the same direction regardless of whether γ > 1 or γ < 1. This observation can

be explained from two distinct angles. First, when γ > 1, the increase in γ decreases the absolute magnitude

of the utility. Since the utility of the investor becomes more sensitive to the change in mortality, less deviation

from the reference model is needed to achieve the worst scenario. By contrast, the absolute magnitude increases

with γ if γ ∈ (0, 1), and hence farther deviation is needed in the worst-case scenario for larger value of γ ∈ (0, 1).

Another line of explanation is related to the optimal retirement timing. As we will see later on in the next

subsection, rational investors with higher risk aversion will choose to postpone retirement if γ > 1 and advance

retirement if γ ∈ (0, 1). Since we assume that ambiguity aversion only affects the pre-retirement decision, as

γ > 1 decreases or γ ∈ (0, 1) increases, the shorter duration of exposure to the ambiguity aversion makes the

investor consider the worst-case scenario in a more extreme manner at the initial stage and deviate more from

the reference mortality model.

Compared to the worst-case investment probability measure which is a simple function of the market pa-

rameters, the worst-case mortality probability measure depends on not only the personal parameters (e.g., ξ∗

and δ), but also the market parameters (e.g., µ, r, and σ). Regarding the impacts of the personal parameters,

as the bequest motivate becomes more preferable, death is less unfavorable and thus the individual lowers the

mortality assumption in the worst-case scenario. Although we do not intend to present the sensitivity plot for

δ herein, the same rationale can be also used to explain the relationship between θλ∗ and δ since unhealthy

individuals tend to purchase more insurance, thus lessen the downside of death. On another note, the increase

in the expected stock return intensifies the negative impact of death, and hence µ has a positive relationship

with the worst-case mortality measure. Because of space constraints, the sensitivity plot for σ is not included.

Our numerical study shows that σ has a negative relationship with θλ∗ since a higher stock volatility harms

17

risky investment return, but investment return and θλ∗ are positively correlated as we have already discussed.

0.0 0.5 1.0 1.5 2.00

0.2

0.4

0.6

0.8

1

1.5 2 2.50.3

0.4

0.5

0.6

0.7

0.8

0.5 0.75 1 1.25 1.50.625

0.63

0.635

0.64

0.06 0.08 0.09 0.11 0.120.628

0.629

0.63

0.631

0.0 0.5 1.0 1.5 2.00

5

10

15

0.3 0.4 0.5 0.6 0.71

2

3

4

5

0.5 0.75 1 1.25 1.52.4

2.45

2.5

2.55

2.6

2.65

0.06 0.08 0.09 0.11 0.122.48

2.5

2.52

2.54

2.56

Figure 2: Sensitivity plots of θλ∗ in response to the change of ψλ, γ, ξ∗, and µ, with γ > 1 in the first row and

γ ∈ (0, 1) in the second row.

Next, we proceed to study the impacts of ambiguity aversion on the optimal strategies. As the sign of

life expectancy value is different between γ > 1 and γ ∈ (0, 1), we separate our subsequent numerical study

into two cases. We first consider the case when γ > 1, which is assumed more commonly in the studies of

retirement planning. The first row of Figure 3 depicts the relationships between the stock ambiguity aversion

and the optimal strategies. If the investor has a higher level of stock ambiguity aversion, then the risky

investment becomes less attractive to the investor, and thus she rationally decreases the investment allocation

and considers to retire earlier in order to gain the additional utility over the post-retirement phase. In this

respect, we recall the reader that the post-retirement utility preference is assumed to be generally higher than

that over the pre-retirement phase (i.e., η∗ > 1). The change in stock ambiguity level also influences the optimal

consumption behavior. Generally, the analysis on optimal consumption behavior is rather complicated since it

depends on the interplays of multiple actuarial and economic factors. First, there are some factors that help to

boost the optimal consumption. As the stock investment is less appealing, the investor would shift more wealth

to consumption. Meanwhile, the earlier retirement may allow the investor to consume more in the initial stage

in order to achieve a desirable level of pre-retirement consumption. Second, there are also some factors that

can harm the optimal consumption and force the investor to be more frugal. The decreasing allocation to risky

investment lowers the potential return for supporting the original consumption level. The earlier retirement

timing also needs additional saving to support the investor’s retirement life. In our numerical example with

γ > 1, the investor is sufficiently risk averse such that the first effect is dominated by the second one, thus

the investor tends to consume less when ψS increases. For investors having a lower level of risk aversion such

that γ ∈ (0, 1), the first effect plays the dominating role, and thus there is a positive relationship between the

optimal consumption and stock ambiguity aversion (see, Table 1).

The change in the mortality ambiguity aversion mainly influences the optimal insurance demand and retire-

18

ment timing. Somewhat surprisingly, the increase in mortality ambiguity aversion lowers the demand for life

insurance. This is due to the fact that when γ > 1, the value of life expectancy is negative and the investor

tends to short-sell life insurance to hedge the risk of living longer than expected, namely the longevity risk. As

the investor expresses increasing concern about the mortality uncertainty, she raises the demand of hedging and

thereby purchasing more annuity. In regard to the optimal retirement timing, since we assume the investor has

no ambiguity aversion over the post-retirement phase wherein she has fully annuitized her financial wealth, the

investor rationally decides to retire earlier in order to eliminate the fear on model uncertainty. The mortality

ambiguity has also modest effects on the optimal consumption. Its impacts on optimal consumption are reversed

between the cases of γ ∈ (0, 1) and γ > 1 (see again, Table 1), due to the same reason as for the stock ambiguity

aversion.

1 1.5 2 2.5 3s

10

20

30

40

50

(%)

=1.5=2=2.5

1 1.5 2 2.5 3s

5.4

5.6

5.8

6

6.2

c(%

)

=1.5=2=2.5

1 1.5 2 2.5 3s

-90

-85

-80

-75

-70=1.5=2=2.5

1 1.5 2 2.5 3s

70

71

72

73

74

75

T

=1.5=2=2.5

0.5 1 1.5 2 2.5 315

20

25

30

35

(%)

=1.5=2=2.5

0.5 1 1.5 2 2.5 35.4

5.6

5.8

6

c(%

)

=1.5=2=2.5

0.5 1 1.5 2 2.5 3-100

-90

-80

-70=1.5=2=2.5

0.5 1 1.5 2 2.5 368

70

72

74

T

=1.5=2=2.5

Figure 3: Sensitivity plots on the optimal strategies in response to the change of ambiguity aversion parameters

when γ > 1 (from left to right): the optimal investment allocation proportion, consumption proportion, face

value of insurance demand, retirement age.

4.2 Sensitivity study on other actuarial/economic factors

This subsection is devoted to the sensitivity study of the other actuarial and economic parameters. To this end,

we keep all else being identical and vary the values of ξ∗, η∗, δ, α, and β one by one. Their marginal impacts

on the optimal strategies are displayed in Figure 4.

We interpret the trends shown in Figure 4 as follows. The utility coefficient for bequest motive ξ∗ = ξ1

1−γ

primarily affects the optimal insurance demand, and the post-retirement consumption utility coefficient η∗ =

η1

1−γ mainly influences the optimal retirement age and thus the optimal consumption (see the first and second

rows in Figure 4). The trend of change in optimal insurance demand is complex to analyze and it is related

to multiple determinants including the depletion of human capital, consumption, and bequest preference. On

the one hand, all else being equal, the increase of ξ∗ offers additional incentive for the investor to purchase

19

0.5 0.75 1 1.25 1.515

20

25

30

(%)

=1.5=2=2.5

0.5 0.75 1 1.25 1.55.5

5.6

5.7

5.8

5.9

6

c(%

)

=1.5=2=2.5

0.5 0.75 1 1.25 1.5-90

-80

-70

-60

-50=1.5=2=2.5

0.5 0.75 1 1.25 1.570

71

72

73

74

T

=1.5=2=2.5

1 1.5 2 2.5 315

20

25

30

35

(%)

=1.5=2=2.5

1 1.5 2 2.5 35

5.5

6

6.5

7

c(%

)=1.5=2=2.5

1 1.5 2 2.5 3-90

-85

-80

-75

-70=1.5=2=2.5

1 1.5 2 2.5 365

70

75

80

85

90

T

=1.5=2=2.5

0.8 0.9 1 1.1 1.215

20

25

30

35

(%)

=1.5=2=2.5

0.8 0.9 1 1.1 1.25.2

5.4

5.6

5.8

6

6.2

c(%

)

=1.5=2=2.5

0.8 0.9 1 1.1 1.2-90

-85

-80

-75

-70

-65=1.5=2=2.5

0.8 0.9 1 1.1 1.270

71

72

73

74

T

=1.5=2=2.5

-0.08 -0.06 -0.04 -0.02 0 0.0215

20

25

30

35

(%)

=1.5=2=2.5

-0.08 -0.06 -0.04 -0.02 0 0.025

5.5

6

6.5

c(%

)

=1.5=2=2.5

-0.08 -0.06 -0.04 -0.02 0 0.02-90

-80

-70

-60

-50

-40=1.5=2=2.5

-0.08 -0.06 -0.04 -0.02 0 0.0265

70

75

80

85

90

T

=1.5=2=2.5

0.02 0.04 0.06 0.080

10

20

30

40

50

(%)

=1.5=2=2.5

0.02 0.04 0.06 0.085.4

5.6

5.8

6

6.2

c(%

)

=1.5=2=2.5

0.02 0.04 0.06 0.08-90

-85

-80

-75

-70=1.5=2=2.5

0.02 0.04 0.06 0.0868

70

72

74

76

T

=1.5=2=2.5

Figure 4: Sensitivity plots on the optimal strategies in response to the change of mortality, income growth rate,

and income volatility parameters, when γ > 1.

more insurance. On the other hand, with a higher ξ∗, the desirable level of legacy is easier to achieve, and

hence the investor tends to lower the insurance purchase and shift more spending toward consumption. When

γ > 1, the latter effect is stronger than the former, and thus the optimal insurance demand exhibits a decreasing

20

trend in response to the change of ξ∗. The aforementioned relationship is reversed for γ ∈ (0, 1) (see, Table

1). Regarding the sensitivity of η∗, the increase in η∗ results in earlier retirement. The shorter length of the

pre-retirement phase turns out to allow the investor to consume more at the beginning.

Rows three to five in Figure 4 are about the sensitivities of the subjective mortality and income process.

All else held constant, investors who are less healthy incline to purchase more insurance and retire earlier. This

observation is of course in accordance with our intuition. The income growth rate α significantly influences

the optimal retirement time. The faster the income stream decreases, the earlier the retirement time becomes.

The decrease in income and early retirement together speed up the depletion of human capital. Since insurance

purchase is a hedge against losing human capital, the insurance demand decreases substantially as α becomes

smaller. Moreover, the dramatic drop in human capital forces the individual to allocate additional money

to risky investment in order to maintain a desirable level of speculation demand. The income volatility β

influences the retirement planning in the following manner. An increase in income volatility β leads to early

retirement and a drop in human capital, and thus the insurance demand declines. On a different note, owing

to the perfect correlation between the stock and income processes, the increase in income volatility provides

additional speculation exposures to the investor, and she therefore rationally decreases the risky investment.

By an already mentioned reason, for increasing δ, decreasing α and increasing β, the resulting early retirement

leads to the increase in current consumption.

We now turn to consider the case when γ ∈ (0, 1). For brevity, we do not intend to present the detailed

sensitivity plots for this case. Instead, we schematically summarize the sensitivity results in Table 1 and

highlight the differences in the direction of impact between the cases of γ ∈ (0, 1) and γ > 1. It is illuminating

to observe from the summary table that, between γ ∈ (0, 1) and γ > 1, the actuarial and income parameters

(i.e., from ψS to β in Table 1) have the same direction of impact on the optimal investment and the optimal

retirement timing. As we mentioned earlier, the sensitivities of various actuarial/economic factors on the

optimal consumption behavior are rather difficult to conclude, and they depend on the interactions of investment

allocation, retirement age, and the investor’s risk aversion. We observe from Table 1 that all the parameters

except the subjective mortality multiplier δ and the risk aversion γ, have reverse sensitivity trends on optimal

consumption between γ ∈ (0, 1) and γ > 1. The bequest motivate preference is the only parameter that has

a reverse relationship with the optimal insurance demand. According to our earlier discussion, this is because

the interplays of consumption, insurance purchase, and life expectancy value.

ψS ψλ ξ∗ η∗ δ α β γ

π∗(%) − + − + + − − −c∗(%) +? +? −? −? + +? −? +

κ∗ − + +? − + + − +

T ∗ − − + − − + − −?

Table 1: Summary of parameters sensitivities when γ ∈ (0, 1), with ‘+’ and ‘−’ indicate respectively an

increasing and decreasing relationship between the corresponding strategy and parameter, and ‘?’ indicates the

trend is reverse between γ ∈ (0, 1) and γ > 1.

21

5 Conclusions

In this paper, we studied an optimal retirement planning problem in which the investor is not only risk averse,

but also ambiguity averse. By translating the methodologies for modeling ambiguity aversion from mathemati-

cal finance to the study of optimal retirement, we were able to simultaneously study the ambiguous risk on both

investment and mortality models under a unified optimization framework. We obtained analytic solution for the

corresponding optimal financial strategy, and verified that the solution always uniquely exists. Via theoretical

and numerical analysis, our results reveal that while the study of stock ambiguity aversion is relatively straight-

forward, the interdependencies between mortality ambiguity and retirement planning are much more intricate.

Different from the worst-case investment scenario which is only sensitive to the change of economic environ-

ment, the worst-case mortality scenario is determined by the interplays between the individual’s personal risk

profile and the various economic factors. The study of mortality ambiguity is made even more challenging since

the value of life expectancy can be both positive and negative, depending on the risk aversion of the investor.

The change in risk aversion may turnover the direction of the impacts of mortality ambiguity on retirement

planning. These results highlight the role of subjective mortality model in making retirement decisions and the

importance for taking mortality uncertainty into account. In future research, it is interesting to investigate how

the individual’s concern about either mortality risk or longevity risk due to mortality ambiguity may contribute

to the understanding of annuitization puzzle.

References

Anderson, E. W., Hansen, L. P., and Sargent, T. J. (2003). A quartet of semigroups for model specification,

robustness, prices of risk, and model detection. Journal of the European Economic Association, 1(1):68–123.

Bo, L. and Capponi, A. (2017). Robust optimization of credit portfolios. Mathematics of Operations Research,

42(1):30–56.

Chen, A., Hentschel, F., and Steffensen, M. (2017). Precommitted, naive and sophisticated retirement time

decision making. SSRN Electronic Journal.

Chen, A., Hentschel, F., and Xu, X. (2018). Optimal retirement time under habit persistence: what makes

individuals retire early? Scandinavian Actuarial Journal, 2018(3):225–249.

Cocco, J. F., Gomes, F. J., and Maenhout, P. J. (2005). Consumption and portfolio choice over the life-cycle.

The Review of Financial Studies, 18:491–533.

Cordoba, J. C. and Ripoll, M. (2017). Risk aversion and the value of life. Review of Economic Studies,

84(4):1472–1509.

Escobar, M., Ferrando, S., and Rubtsov, A. (2015). Robust portfolio choice with derivative trading under

stochastic volatility. Journal of Banking and Finance, 61:142–157.

Escobar, M., Ferrando, S., and Rubtsov, A. (2018). Dynamic derivative strategies with stochastic interest rates

and model uncertainty. Journal of Economic Dynamics and Control, 86:49–71.

22

Guvenen, F., Karahan, F., Ozkan, S., and Song, J. (2015). What do data on millions of U.S. workers reveal

about life-cycle earnings risk? Technical report, National Bureau of Economic Research, Cambridge.

Hansen, L. P. and Sargent, T. J. (1993). Seasonality and approximation errors in rational expectations models.

Journal of Econometrics, 55(1-2):21–55.

Hansen, L. P., Sargent, T. J., and Wang, N. E. (2002). Robust permanent income and pricing with filtering.

Macroeconomic Dynamics, 6(1):40–84.

Huang, H. and Milevsky, M. A. (2008). Portfolio choice and mortality-contingent claims: The general HARA

case. Journal of Banking and Finance, 32(11):2444–2452.

Huang, H., Milevsky, M. A., and Wang, J. (2008). Portfolio choice and life insurance: The CRRA case. Journal

of Risk and Insurance, 75(4):847–872.

Hurd, M. D. and McGarry, K. (1995). Evaluation of the subjective probabilities of survival in the health and

retirement study. The Journal of Human Resources, 30:268–292.

Hurd, M. D. and McGarry, K. (2002). The predictive validity of subjective probabities of survival. Economic

Journal, 112(482):966–985.

Johansson, P.-O. (1996). On the value of changes in life expectancy. Journal of Health Economics, 15(1):105–

113.

Kolasa, A. (2017). Life cycle income and consumption patterns in transition. Central European Journal of

Economic Modelling and Econometrics, 9:137–172.

Li, D., Zeng, Y., and Yang, H. (2018). Robust optimal excess-of-loss reinsurance and investment strategy for

an insurer in a model with jumps. Scandinavian Actuarial Journal, 2018(2):145–171.

Maenhout, P. J. (2004). Robust portfolio rules and asset pricing. The Review of Financial Studies, 17:951–983.

Maenhout, P. J. (2006). Robust portfolio rules and detection-error probabilities for a mean-reverting risk

premium. Journal of Economic Theory, 128(1):136–163.

Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of

Economics and Statistics, 51(3):247–257.

Milevsky, M. A. and Young, V. R. (2007). Annuitization and asset allocation. Journal of Economic Dynamics

and Control, 31(9):3138–3177.

Moore, K. S. and Young, V. R. (2016). Minimizing the probability of lifetime ruin when shocks might occur:

Perturbation analysis. North American Actuarial Journal, 20(1):17–36.

Munk, C. and Sørensen, C. (2010). Dynamic asset allocation with stochastic income and interest rates. Journal

of Financial Economics, 96(3):433–462.

23

Pirvu, T. A. and Zhang, H. (2012). Optimal investment, consumption and life insurance under mean-reverting

returns: The complete market solution. Insurance: Mathematics and Economics, 51(2):303–309.

Pliska, S. R. and Ye, J. (2007). Optimal life insurance purchase and consumption/investment under uncertain

lifetime. Journal of Banking and Finance, 31(5):1307–1319.

Ramsey, F. P. (1928). A mathematical theory of saving. The Economic Journal, 38(152):543.

Richard, S. F. (1975). Optimal consumption, portfolio and life insurance rules for an uncertain lived individual

in a continuous time model. Journal of Financial Economics, 2(2):187–203.

Shen, Y. and Sherris, M. (2018). Lifetime asset allocation with idiosyncratic and systematic mortality risks.

Scandinavian Actuarial Journal, 2018(4):294–327.

Wang, S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26(1):71–92.

Yaari, M. E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer. The Review of Economic

Studies, 32(2):137–150.

Young, V. R. and Zhang, Y. (2016). Lifetime ruin under ambiguous hazard rate. Insurance: Mathematics and

Economics, 70:125–134.

Zeng, X., Wang, Y., and Carson, J. M. (2015). Dynamic portfolio choice with stochastic wage and life insurance.

North American Actuarial Journal, 19(4):256–272.

Zeng, Y., Li, D., and Gu, A. (2016). Robust equilibrium reinsurance-investment strategy for a meanvariance

insurer in a model with jumps. Insurance: Mathematics and Economics, 66:138–152.

Appendix A Technical proofs

Proof of Proposition 1. Let P be a probability measure equivalent to P such that

dPdP

∣∣∣∣Ft

= exp

− 1

2

(µ− rσ

)2

t−(µ− rσ

)Wt

. (20)

In fact, P is the risk-neutral measure since the discounted stock price process is a martingale under this measure.

The P-dynamics of the income process is

dit = it

[(α− (µ− r)β

σ

)dt+ βdWt

],

where

Wt := Wt +µ− rσ

t

is a standard Brownian motion under P. Therefore, we have

H(t, i) = Et,i[ ∫ T

ts−t

.Ey+t is ds

]24

= Et,i[ ∫ T

tis exp

−∫ s

t(r +

.λy+u)du

ds

]

= i

∫ T

texp

−∫ s

t

[(r − α+

(µ− r)βσ

)+.λy+u

]du

ds

= i.a

(r∗)

y+t:T−t .

This completes the proof.

Proof of Proposition 2. By the first order condition with respect to θS and θλ in (14), we obtain

−πσVx(t, x, i)− βiVi + (1− γ)V (t, x, i)θS/ψS = 0,

and

−λy+tV (t, x, i) + λy+tξU(x+ k/.λy+t) + (1− γ)V (t, x, i)g′(θλ)λy+t/ψ

λ = 0.

This leads to

θSt =ψS

1− γ

[πσ

Vx(t, x, i)

V (t, x, i)+ βi

Vi(t, x, i)

V (t, x, i)

], (21)

and

θλt (k) = exp

ψλ

1− γ

[1− ξU(x+ k/

.λy+t)

V (t, x, i)

]. (22)

Recall that we have assumed the objective and subjective mortalities are only different up to a constant

multiplier such that (see, Section 2)

λy+t.λy+t

= δ.

The first order condition with respect to π, c, k gives respectively

(µ− r − σθS)Vx + πσ2Vxx + σβiVxi = 0, (23)

−Vx(t, x, i) + U ′(c) = 0, (24)

and

−Vx(t, x, i) + ξ δ θλ(k)U ′(x+ k/.λy+t) = 0. (25)

We conjecture the following ansatz

V (t, x, i) =(x+ b(t, i))1−γ

1− γf(t)γ

25

with terminal conditions:

b(T, i) = 0, f(T ) = η1γ

(ay+T )1γ

(.ay+T )

1−γγ

,

is a solution to the HJB equation (14). Then, from the first order condition in (23)-(25), we obtain that the

optimal investment, consumption and insurance strategies must satisfy the following formats:

πt =(µ− r − σθS)

γσ2(x+ b(t, i))− β

σibi(t, i), (26)

ct =x+ b(t, i)

f(t), (27)

and

kt =.λy+t

[(ξ δ θλ(k)

) 1γ x+ b(t, i)

f(t)− x]. (28)

Combining formulas (21)-(22) and (26)-(28) gives the worst-case probability measures

θS∗t =ψS

ψS + γ

µ− rσ

,

θλ∗t = exp

ψλ

1− γ

[1− (δ θλ∗t )

1−γγ

ξ1γ

f(t)

], (29)

and the optimal investment and insurance strategies

π∗t =µ− r

(ψS + γ)σ2(x+ b(t, i))− β

σibi(t, i),

k∗t = A(t, θλ∗t )(x+ b(t, i))

f(t)− x

.λy+t.

We have thus far established the representations for the worst-case probabilities and the optimal strategies

in terms of b and f . We now turn to solve b and f . To this end, we substituting θS∗, θλ∗, π∗, c∗ and k∗ back

to the HJB equation and separate coefficients of (x+ b)−γfγ and (x+ b)1−γfγ−1. By doing so, we get

bt(t, i)− (r +.λy+t)b(t, i) +

(α− µ− r

σβ

)ibi(t, i) +

1

2β2i2bii(t, i) + i = 0, (30)

and

ft(t) +1− γγ

[(r +

.λy+t)−

1

1− γ(ρ+ θλ∗t λy+t) +

1

2

(µ− r)2

(ψS + γ)σ2+g(θλ∗t )λy+t

ψλ

]f(t) + 1 +A(t, θλ∗t ) = 0. (31)

It is a simple matter to check that (30) admits a unique solution satisfying

b(t, i) = Et,i[ ∫ T

tis exp

−∫ s

t(r +

.λy+u)du

ds

]26

= i.a

(r∗)

y+t:T−t ,

which matches the formulation of the human capital function H(t, i).

It only remains to solve f according to (29) and (31) which constitute a system of coupled, highly non-linear

algebraic and differential equations. To disentangle the dependence between θλ and f , we take logarithm on

both sides of (29) and write f in terms of θλ as

f(t) =(δ θλ∗t )

1−γγ ξ

1γ

1− 1−γψλ

log θλ∗t. (32)

We then have

ft(t) = h(t, θλ∗t )(θλ∗t )(1−2γ)/γ(θλ∗t )′. (33)

Substituting the derivative formula (33) into (31), we obtain an ODE satisfied by θ∗λt:

(θλ∗t )′ +

[(r +

.λy+t)−

1

1− γ(ρ+ θλ∗t λy+t) +

1

2

(µ− r)2

(ψS + γ)σ2+g(θλ∗t )λy+t

ψλ

][1 +

γ

ψλ − (1− γ) log(θλ∗t )

]−1

+[1 +A(t, θλ∗t )

]h(t, θλ∗t )−1(θλ∗t )−(1−γ)/γ

θλ∗t = 0, (34)

where the terminal value θλ∗T can be found by solving the following equation:

η1γ

(ay+T )1γ

(.ay+T )

1−γγ

=(δ θλ∗T )

1−γγ ξ

1γ

1− 1−γψλ

log θλ∗T.

We have now obtained the explicit expressions of b and f , and thus the optimal strategies. The proof for

the proposition is to be completed by showing the uniqueness and existence of a solution pair to the system of

coupled equations (29) and (31), or equivalently (32) and (34), which will be deferred to Proposition 3.

Proof of Proposition 3. First of all, it can be seen from the structure of (29) that θλ∗t ≥ 0. It follows from (31)

that

F (θλ∗t ) = f(t) = η1γ

(ay+T )1γ

(.ay+T )

1−γγ

e∫ Tt G(u,θλ∗u ))du +

∫ T

t[1 +A(s, θλ∗s )]e

∫ st G(u,θλ∗u ))duds, (35)

where

G(t, θλ∗t ) :=1− γγ

[(r +

.λy+t)−

1

1− γ(ρ+ θλ∗t λy+t) +

1

2

(µ− r)2

(ψS + γ)σ2+g(θλ∗t )λy+t

ψλ

].

So, it is clear f(t) ≥ 0. Next we show that θλ∗t must be bounded and strictly positive. By L’Hospital’s rule,

limm→0+

F (m) = limm→0+

1−γγ δ

1−γγ ξ

1γm

1−2γγ

−1−γψλ

1m

= −ψλγδ

1−γγ ξ

1γ limm→0+

m1−γγ

27

and

limm→+∞

F (m) = −ψλγδ

1−γγ ξ

1γ limm→+∞

m1−γγ .

If γ > 1 (resp. γ < 1), the first (resp. second) limit leads to that f(t) = F (θλ∗t ) converges to −∞ when θλ∗tgoes to 0+ (resp. +∞). This contradicts with f(t) ≥ 0. Therefore, we can find two positive constants θ and θ

such that θλ∗t ∈ [θ, θ]. From (35), we can see f(t) is also bounded and strictly positive.

Next we show the existence and uniqueness of the solution to (32) and (34), which are equivalent to (29)

and (31). The first-order derivative of F (θλ∗t ) is

F ′(θλ∗t ) =δ

1−γγ ξ

1γ (θλ∗t )

1−γγ1−γ

γ [1− 1−γψλ

log θλ∗t ] + 1−γψλ

[1− 1−γ

ψλlog θλ∗t

]2 . (36)

Recalling F (θλ∗t ) = f(t) > 0, we have 1 − 1−γψλ

log θλ∗t > 0 holds on [θ, θ]. Thus, if γ > 1, then F ′(θλ∗t ) < 0;

otherwise, F ′(θλ∗t ) > 0. In other words, F (θλ∗t ) is always monotone in θλ∗t . This implies that F (θλ∗t ) and θλ∗thave a one-to-one correspondence relationship. So, it suffices to consider the existence and uniqueness of the

solution θλ∗t to (34).

To this end, we consider the Banach space C([0, T ]; [θ, θ]) equipped with the sup norm, and rewrite (34) as

(θλ∗t )′ + Ψ(t, θλ∗t ) = 0, (37)

where

Ψ(t, θλ∗t ) :=

[(r +

.λy+t)−

1

1− γ(ρ+ θλ∗t λy+t) +

1

2

(µ− r)2

(ψS + γ)σ2+g(θλ∗t )λy+t

ψλ

][1 +

γψλ

1− 1−γψλ

log(θλ∗t )

]−1

+[1 +A(t, θλ∗t )

]h(t, θλ∗t )−1(θλ∗t )−(1−γ)/γ

θλ∗t .

Applying the mean-value theorem to Ψ(t, θλ∗t ), we easily obtain that Ψ(t, θλ∗t ) is Lipschtiz continuous in the

second argument over [θ, θ] uniformly for t ∈ [0, T ]. On the other hand, the monotonicity and range of F

guarantees that there exists a unique terminal value D(T ) ∈ [θ, θ] such that the algebraic equation (17) holds.

Following the standard procedure in the ODE theory and using the contraction mapping theorem, we can

conclude that (34) admits a unique solution on C([0, T ]; [θ, θ]). This completes the proof.

28