OPTIMAL CONSUMPTION -INVESTMENT CHOICES UNDER WEALTH -DRIVEN RISK AVERSION Ruoxin Xiao

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OPTIMAL CONSUMPTION-INVESTMENT CHOICES UNDER

WEALTH-DRIVEN RISK AVERSION

Ruoxin Xiao

Department of Mathematics College of Sciences

Shanghai University

Shanghai, China

xiaoruoxin@shu.edu.cn

ABSTRACT

CRRA utility where the risk aversion coefﬁcient is a constant is commonly seen in various economics

models. But wealth-driven risk aversion rarely shows up in investor’s investment problems. This

paper mainly focus on numerical solutions to the optimal consumption-investment choices under

wealth-driven aversion done by neural network. A jump-diffusion model is used to simulate the

artiﬁcial data that is needed for the neural network training. The WDRA Model is set up for describing

the investment problem and there are two parameters that require to be optimized, which are the

investment rate of the wealth on the risky assets and the consumption during the investment time

horizon. Under this model, neural network LSTM with one objective function is implemented and

shows promising results.

Keywords

Investment Problem

Jump-Diffusion Model

Wealth-Driven Risk Aversion

CRRA

Neural Network

LSTM

1 Introduction

The theory of risk aversion is developed to cope with the measurement of uncertainty in economics. In canonical

theories, risk aversion is usually modeled by expected utility. The concept was ﬁrst tied to diminishing marginal utility

for wealth. In applications, economists derived speciﬁc functional forms to measure risk aversion. Two common

measures are the coefﬁcient of absolute risk aversion and the coefﬁcient of relative risk aversion, both deﬁned by Pratt

(1964) and Arrow (1965). The constant relative risk aversion (CRRA) utility functon is one of the most widely used

forms, in which risk aversion is modeled by a single constant parameter, ρ.

Though the model performs well due to its simplicity, it still puts serious constraints on individual preferences. Much

research have been done on modeling risk aversion in application. Risk aversion is studied in empirical analysis to be

affected by exogenous factors, mainly demographic, measuring the heterogeneity of investors. In Palacios-Huerta and

Santos (2004)[

], they modeled the degree of risk aversion endogenous to market arrangements. To take a step further,

this paper casts light on endogenous risk aversion model within the consumption-investment strategy problem.

Unlike the CRRA model where the risk aversion coefﬁcient is constant, the innovation we present in this paper

of consumption-investment optimization strategy is to take the inﬂuence of temporary wealth on risk aversion into

consideration. Risk aversion is no longer ﬁxed throughout the investment period, but a function that varies with the

changes in temporary wealth, the result of optimal consumption-investment strategy of each step. The ultimate goal is

to optimize the expected utility on consumption and terminal wealth through the risk-aversion-changing process.

The study of optimal consumption-investment problem can be traced back to Merton in 1969 (Merton, 1969, 1971).

Merton develops an explicit optimal investment strategy by using stochastic optimal theory. Currently, deep learning

skills have already been used in numbers of areas, including portfolio selection. Deep learning skills ﬁrst used to solve

optimal investment problem is done by Chen and Ge (2001).[2]

arXiv:2210.00950v1 [stat.ML] 3 Oct 2022

2 Jump-Diffusion Model and Wealth-driven Risk Aversion Model

2.1 Set up of the JD Model

We begin with a ﬁnancial market which operates continuously with a probability space (

Ω

) and a time horizon

0< t < T

is deﬁned as a ﬁltration reﬂecting all the available information at time t. We consider an investment

universe consisting of one risk-free asset

and one risky assets denoted by

. The price of assets

is governed by the

following stochastic differential equation[3]

dSt=St−(µdt +σdBt+dJt)(1)

Jt=

i=1

(eUi−1) (2)

where

is a standard Brownian Motion,

is a Poisson process with rate

and

is a sequence of independent

identically distributed (i.i.d) random variables such that has an asymmetric double exponential distribution, which refers

to Kou(2002)[4] with the density

fY(y) = pη2e−η1(y−a)1{y≥a}+qη2eη2(y−a)1{y≥a}(3)

η2>1, η2>0

where

p, q > 0

, and

p+q= 1

, representing the probabilities of upwards and downwards jumps compared with

y=α

Solving the SDE(1) above by Itˆo’s Lemma gives the dynamics of an asset price

St=S0exp((µ−σ2

2)t+σWt+

i=1

Ui)(4)

2.2 Set up of the WDRA Model

Consider a ﬁnancial market which consists of a riskless asset

and a risky asset

. An investor enters the market at

time 0 with initial wealth

. The price of the risk-free asset is governed by the following ordinary differential equation

dP0=rP0dt (5)

where

is assumed to be constant which represents the risk-free rate. The price of risky assets

is governed by the

following stochastic differential equation

dSt=St−(µdt +σdBt+dJt)(6)

We develop the binary utility function related to the state of wealth based on the Constant Relatively Risk Aversion

(CRRA) utility function

u(x, y) = x1−ρ(y)−1

1−ρ(y), ρ(y)6= 1 (7)

where

u(x, y) = log(x)

ρ(y)=1

and

indicates the level of relative risk aversion. This wealth-driven risk aversion

lets the coefﬁcient of risk aversion varies with the wealth.

We adopt the function form of wealth-driven risk aversion developed by Chu, Nie and Zhang(2014)[

] through their

empirical analysis.

ρ(W) = b0+b1·W+b2·W3(8)

where

is the ratio of individual wealth to the average wealth level and we set

b1= 0.13

b2=−0.45

, being

consistent with the empirical speciﬁcation in Chu (2014). We chose

such that the average risk aversion coefﬁcient is

3. The relationship between risk aversion and wealth is hump-shaped. Risk aversion ﬁrst increases with wealth and

then decreases with it, which suggests that the poorest group of people and the richest group of people are more of risk

takers than the those people in the middle.

The investor is endowed with wealth

at the beginning of the time horizon. And its objective is to maximize the the

expected utility by making consumption and investment choices during the time horizon. The expected utility during

the investment time horizon and the wealth process are given by the following formula

sup

θ,c

E[ζZT

e−ηtu(ct, wt)dt + (1 −ζ)e−ηtu(wT, wT)] (9)

dwt=θtwtS−1

tdSt+ (1 −θt)wtrdt −ctdt (10)

where

θt

represents the investment rate of the wealth on the risky asset

at time

(1−θt)

represents the investment rate

of the wealth on the riskless asset

at time

represents the consumption at time

represents the terminal wealth,

represents the subjective discount rate, and

represents the relative importance of the intermediate consumption and

the terminal wealth.

is the expectation of the whole stochastic process. The ﬁrst term and the second term in (9) is

used to measure the utility in terms of consumption and terminal wealth under the wealth-driven risk aversion.

2.3 Estimation of JD Model

By(1), the log return over a time interval ∆tis:

ln St+∆t

St=µ−σ2

2∆t+σ(Wt+∆t−Wt) +

Nt+∆t

i=Nt+1

Ui(11)

where we set the time interval is small(∆t=one day= 1/247 year) to approximate the log return as

ln St+ ∆t

St≈µ−σ2

2∆t+σ√∆tZ +BY (12)

where

is a standard normal random variables,

is a Bernoulli random variable with

P(B= 1) = λ∆t

and

P(B= 0) = 1 −λ∆t.

The density of (12) is given by the following formula

g(x) = 1−λ∆t

σ√∆tφ



t−µ−σ2

2∆t

σ√∆t



+λ∆tpη1eη2

1σ2∆t

2e−t−α−µ−σ2

2∆tη1Φ



t−α−µ−σ2

2∆t−η1σ2∆t

σ√∆t



+λ∆tqη2eη2

2σ2∆t

2et−α−µ−σ2

2∆tη2Φ

−

t−α−µ−σ2

2∆t+η2σ2∆t

σ√∆t



(13)

Firstly, to estimate an appropriate range of

, we count the number of data points which are outside the

3σ

(Gaussian estimated sigma) interval. Then Maximum Likelihood Estimation is used to estimate seven unknown

parameters(µ, σ, λ, p, η1, η2, α)in the model by empirical data. The maximum likelihood function is given as follow

max

para L(x1, x2, . . . ·xn;para) = max

para

i=0

ln g(xi|para)(14)

where

{xi}

is empirical data processed from the data of a stock within one year and

para = (µ, σ, λ, p, η1, η2, α)

During this process, Adam Optimizer is used to optimize the maximum likelihood function. As a result, the estimated

parameters

(µ, σ, λ, p, η1, η2, α)

(

-0.2438

,0.2579,2,0.0062,1.0879,0.2435,0.2)

. The density plot of g is shown in

Fig.1 along with the Gaussian kernel density estimation.

Figure 1: The ﬁtted density plot of g is compared with the Guassian kernel density estimation. The blue points stands

for the model and the blue line is used for the Guassian kernel density estimation by empirical data

2.4 Simulation of JD model

For the purpose of training neural networks, a set of data for the JD model is produced by simulation based on the

estimated parameters in the previous section. The initial price is set at

S0= 100

. We simulated the prices of a

hypothetical stock in

T= 247

days (one year) under the model for 100 times. The (12) could be divided into two part:

the diffusion part and the jump part.

•

The diffusion part: The diffusion part follows a normal distribution with mean

(µ−σ2

2)∆t

and standard

deviation σ√∆t.

•

The jump part:The jump part simulates the inter-jump time and the jump size. The inter-jump time follows the

exponential distribution

Exp(λ)

, which indicates that the time at which the jumps occur follows a Poisson

process with parameter

. Accordingly, we iteratively draw samples from Poisson distribution till the sum

of the samples exceeds 1 (since

T= 1

year). The sampled inter-jump times is stacked together to get the

jump times and ﬂoored for speciﬁc dates. For the jump size, it follows a double-exponential distribution. The

samples are draw from a double-exponential distribution for each time that the jump occurs and the number of

the samples at each time is the inter-jump time.

The simulation paths is shown in Figure 2. From the ﬁgure, it is clear that there are both small and steady ﬂuctuations

and occasional steep changes in the paths.

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摘要：

OPTIMALCONSUMPTION-INVESTMENTCHOICESUNDERWEALTH-DRIVENRISKAVERSIONRuoxinXiaoDepartmentofMathematicsCollegeofSciencesShanghaiUniversityShanghai,Chinaxiaoruoxin@shu.edu.cnABSTRACTCRRAutilitywheretheriskaversioncoefcientisaconstantiscommonlyseeninvariouseconomicsmodels.Butwealth-drivenriskaversionrare...

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