Optimal investment strategy for DC pension plan with inflation risk under the hybrid stochastic volatility model

In this paper, we investigate an optimal investment strategy for defined-contribution (DC) pension plan under hybrid stochastic volatility (Heston–Hull–White) model, taking account of the inflation risk and the stochastic salary. The fund wealth is invested in financial market consisting of a risk-free asset, an inflation-indexed bond and a stock with hybrid Heston–Hull–White model. The goal of the pension fund manager is to maximize the expected utility of the terminal real wealth. We derive the Hamilton–Jacobi–Bellman (HJB) equation through the dynamic programming principle, under the constant relative risk aversion (CRRA) utility function, the optimal investment strategy is obtained. Finally, a numerical example is presented to characterize the impacts of financial parameters on the optimal investment strategy.


Introduction
As a continuous increase in the ageing population, the optimal investment strategy for DC pension plan has recently become a popular significant subject.There are two major types to manage pension fund: the defined benefit (DB) pension plan and the defined contribution (DC) pension plan.In the DB pension plan, the benefits at retirement are defined in advance by the sponsor and the contributions from the pension plan member also are set and adjusted to maintain the fund in balance.In the DC pension plan, the contributions are fixed and the benefits provided by the plan depend on the investment returns of the fund's portfolio.Compared with the DB pension plan, the DC pension plan relieves the financial pressure by transferring the investment and longevity risk from the sponsor to the pension plan member.Consequently, the optimal investment strategy for DC pension plan, which is also the topic of this paper, has attracted much attention in pension fund management.
In past decades, many scholars have been devoted to the study of the optimal asset allocation strategy.Some researchers start with the dynamic programming principle, which was first used by Merton (1969Merton ( , 1971)).Fei et al. (2021) researched an optimal intertemporal asset allocation strategy of a multinational corporation with exchange rate risk and derived the optimal investment strategy by using dynamic programming principle.Li et al. (2021) extended the research on Merton's optimal CONTACT Dengfeng Xia dengfengxia@ahpu.edu.cnconsumption and portfolio selection under the framework of model uncertainty.Boulier et al. (2001), Vigna and Haberman (2001), Haberman and Vigna (2002) and Deelstra et al. (2003) studied the management of DC pension plan by the optimal control theory.These studies have assumed that the risky asset price dynamics driven by a geometric Brownian motion (GBM) which implies that the volatility of risky asset price is a constant or a deterministic function.However, the volatility of risky asset price is time change on the market.The assumption of constant volatility of risky asset price contradicts with empirical evidence of fluctuating volatility.So it is realistic to incorporate stochastic volatility into the investment problem.
Recently, various researchers studied the optimal investment strategy for DC pension plan with stochastic volatility.Some researchers have used a constant elasticity of variance (CEV) model to describe the price of risky assets.Gao (2009) used the stochastic optimal control, power transform and variable separation to derive the explicit solutions of the optimal investment strategy for DC pension plan under the CEV model.Li et al. (2017) explored the DC pension investment with default risk under the CEV model and derived the equilibrium investment strategies and the corresponding equilibrium value functions via a game theoretic framework.Yong et al. (2021) investigated the optimal investment for DC pension plan under a CEV model with power utility.
Apart from the CEV model, the Heston's stochastic volatility (SV) model was also used to describe the price of the risky asset.Li et al. (2016), Guan and Liang (2016), Li et al. (2012) and Ma et al. (2020) all considered the optimal investment strategy for insurers with the stock price given by the Heston's SV model under mean variance criterion or the criterion of utility maximization.
However, because the investment horizon for DC pension plan often takes a long period, generally from 20 to 40 years.It is therefore crucial to take the risk of interest rate into account in the market.In the literature on optimal portfolio strategies, the dynamics of interest rate are modelled by Vasicek model, Cox-Ingersoll-Ross (CIR) model and affine interest rate models (including the Vasicek model and the CIR model).These approaches have been widely used in practice.Boulier et al. (2001) used Vasicek model to describe the dynamics of interest rate and obtained the optimal investment strategy to maximize the CRRA utility of terminal value.Battocchio and Menoncin (2004), Han and Hung (2017) and Wang et al. (2021) also modelled the interest rate by Vasicek model.Han and Hung (2012) described the dynamics of interest rate by CIR model and obtained the optimal investment strategy for DC pension plan with downside protection under stochastic inflation.In the works of Zhang and Rong (2013), Guan andLiang (2014, 2015) and Zhang et al. (2020), the interest rate modelled by affine structure, which combine both Vasicek model and CIR model.However, these models are built upon a particular diffusion process with constant coefficients, which is not suitable to solve the problems related to pension fund management.Therefore, to avoid the above problem, we use the Hull-White model to describe the dynamics of interest rate in this paper.
For a long-term investment, not only financial risk but also stochastic inflation and salary must be considered.The accumulated inflation in long time horizon investment could severely damage the real payoff of the investor.Fei et al. (2021) investigated the principalagent problem with the moral hazard under inflation risk, they found that the inflation risk affects the agent's optimal compensation.Battocchio and Menoncin (2004) first took into account the inflation risk in the DC pension plan and derived an optimal investment strategy under the constant absolute risk aversion (CARA) utility.Han and Hung (2017) showed that the inclusion of indexed bond in the economy helps us to hedge against the inflation risk.Chen et al. (2017) considered the optimal investment strategy for DC pension plan with inflation risk, salary risk and longevity risk under loss aversion and minimum performance constraint.Wang et al. (2021) recently investigated the optimal investment strategy for DC pension plan with stochastic inflation and salary and modelled the risky asset by Ornstein-Uhlenbeck (O-U) process.Similarly, Yan et al. (2021) incorporated the stochastic inflation and salary into a robust optimal investment problem for DC pension plan with a return of premiums clause and derived the robust optimal investment strategy by adopting a new state variable which represents the accumulated premiums.In this paper, we incorporate the stochastic inflation and salary into our model and introduce an index bond to hedge against the inflation risk.
In most previous literature, the stochastic interest rate and stochastic volatility have been treated separately.However, it is more practical to take these two risks into account in DC pension plan concurrently.Due to the fact that the Heston's SV model is unable to generate enough skews or smiles in the implied volatility as market required in many cases, especially for a short maturity.Grzelak and Oosterlee (2011); Grzelak et al. (2012) extended the Heston's SV model by introducing a Hull-White stochastic interest rate which is the hybrid Heston-Hull-White model where the correlations between the underlying processes are assumed to be non-zero.Later, Mwanakatwe et al. (2019) used the hybrid Heston-Hull-White to model the dynamics of risky assets price and solved the optimal investment and benefit payments strategy for DC pension plan with an income drawdown option.Furthermore, Kim et al. (2014) assumed that the stochastic interest rate is driven by a Hull-White process and incorporates it into a stochastic volatility model to evaluate the sensitivity of option price to changes in interest rate.Guan and Liang (2014) introduced the optimal investment and consumption problem for DC pension plan with stochastic affine interest rate and Heston's SV model.They derived the explicit solution for the optimal investment consumption strategy by applying the stochastic dynamic programming under the power utility and logarithmic utility.Similarly, Wang and Li (2018) considered a robust optimal investment problem for an ambiguity-averse member (AAM) of DC pension plan which the stochastic interest rate and stochastic volatility are described by an affine model and Heston's SV model, respectively.
With the above in mind, in this paper, we introduce stochastic inflation, stochastic salary and hybrid Heston-Hull-White model together into a DC pension investment problem.We assume that the DC plan member is a risk-neutral investor and has access to a financial market consisting of riskless asset, a stock and a bond, which can be bought and sold without incurring any transaction costs or restriction on short sales.Specifically, the stochastic interest rate follows Hull-White model, meanwhile, the stock price is described by the hybrid Heston-Hull-White model.The salary is also stochastic and driven by a GBM.Based on the above settings, we establish an optimal investment problem for DC pension plan under the CRRA utility.Using the dynamic programming approach, we derive analytical expressions of the optimal investment strategy, as well as the corresponding value function.Finally, the economic implications of our theoretical results analysed by using a numerical example.
The main contributions of this paper are as follows: (i) In an optimal investment problem for DC pension plan, we introduce stochastic inflation, stochastic salary and hybrid Heston-Hull-White model simultaneously, which is not considered in the aforementioned literature.(ii) We derive the explicit solutions of the optimal investment strategy and the value function for DC pension plan by a dynamic programming principle.(iii) We show the impacts of the parameters on the inflation, the stock's volatility and the salary on the optimal investment strategy, respectively.We find that the inflation, the stock's volatility and the salary affect the optimal investment strategy significantly.
The structure of this paper is as follows.Section 2 introduces the financial markets, which include a risk-free asset, an inflation-indexed bond and a stock.Section 3 deduces the real wealth process and optimization problem of DC pension plan.Section 4 drives explicit expressions of the optimal investment strategy and the corresponding value function.Numerical analysis is given in Section 5. Section 6 concludes this paper.

Mathematical model
In this section, we introduce the financial market and pension wealth process.Let ( , F, {F t } t≥0 , P) be a complete filtering probability space, where {F t } t≥0 is rightcontinuous and P-complete, and F t represents the information available until time t, the accumulation period of a DC pension plan is [0, T].Suppose that all the stochastic processes are defined on the probability space ( , F, {F t } t≥0 , P) and adapted to the filtration {F t } t≥0 .In addition, we assume that trading is continuous and there are no transaction costs or taxes in the financial market, and short selling is permitted.
Since the investment horizon for DC pension plan often takes a long period, the inflation risk is a dangerous factor that can severely damage the real payoff of the pension plan.Therefore, in this paper, we incorporate the inflation risk into our model.We assume that the commodity price index in this paper follows the diffusion process : = μ P dt + σ P dW P (t), where P(0) = p 0 > 0 is the initial condition, the constant μ P represents the expected rate of inflation, the constant σ P > 0 is the volatility of inflation rate, and W P is a one-dimensional standard Brownian motion.
In this paper, we suppose that the financial market consists of three financial assets: a risk-free asset, an inflation-indexed bond and a stock.
(i) The price of the risk-free asset follows the ordinary differential equation (ODE): where r(t) is the nominal interest rate.
(ii) The risky asset in the market is a stock, whose price process is described by the hybrid Heston-Hull-White model: where S(t), V(t), and r(t) are three random variables to represent the stock price, the volatility of an asset and the risk neutral interest rate, respectively.The parameters λ S , σ S , κ, ψ, σ V , λ r are positive constants and κ is the mean reversion rate in volatility, σ V is the volatility parameter, ψ denotes the long-term variance level, the long-run mean of interest rate which is time dependent is provided by the function θ(t), σ r is the interest rate volatility, the constant λ r determines the speed of mean reversion, and λ S √ V(t) and λ P represent the market price of the risk sources W S (t) and W P (t), respectively.Also, W S (t), W V (t) and W r (t) are the standard Brownian motions under a risk-neutral measure.We assume that W P (t) is independent of W V (t) and W S (t), W S (t) is independent of W r (t), while W r (t) and W P (t) are dependent with dW r dW P = ρ rP dt, W S and W V are dependent with dW S dW V = ρ SV dt, where (iii) We introduce an inflation-indexed bond into our portfolio model to hedge against the inflation risk.The price of the inflation-indexed bond is modelled by the diffusion process: (4) where R(t) represents the real interest rate and R(t) + μ P is the appreciation rate of the inflation-indexed bond.
Remark 2.1: Under the risk neutral measure, by the pricing theory of the derivative (to avoid arbitrage), we can get the relationship as follows: R(t) + μ P − λ P σ P = r(t).
( 5 ) Due to the income directly impacting the investment, we assume that the income is stochastic and driven by the source of uncertainty from inflation, the nominal salary L N (t) of the pension plan member is described by where μ L > 0 is the average growth rate of the income and σ L is the volatility rate.

Model formulation
In this paper, we consider the optimal portfolio problem of a DC pension plan member before retirement time T.
We assume that the DC pension plan member continuously contributes a constant proportion c (c ∈ [0, 1]) of his/her salary to the pension fund account before retirement.Let π S (t) and π B (t) denote the investment proportion in the stock and the inflation-indexed bond, respectively.Then π 0 (t) = 1 − π S (t) − π B (t) is the investment proportion in the risk-free asset.We use the stochastic process π(t) = (π B (t), π S (t)) to represent an investment strategy.Let X N (t) denote the nominal wealth process at time t ∈ [0, T], Then, the nominal wealth process X N (t) satisfies the following stochastic differential equation (SDE): The investment horizon for DC pension plan often takes a long period, the effect of inflation becomes noticeable for investors.Since the purchasing power of the nominal wealth is reduced by the inflation, the investors are more concerned about the utility of the real wealth, which is defined as the ratio between the nominal wealth and the price index.Now denote dX(t) = d(X N (t)/P(t)), dL(t) = d(L N (t)/P(t)).The following real wealth process X(t) and real salary L(t) can be obtained by using Itô's formula: Definition 3.1 (Define(Admissible Strategy)): A strategy π = (π B (t), π S (t)) is said to be admissible if: ( (3) The SDE (8) has a unique solution.Denote = {π(t)|π(t) = (π B (t), π S (t)), 0 t T} as the set of all admissible strategies.We can rewrite the evolution of real wealth process X(t) as follows: where The pension manager invests the wealth in the financial market and aims to find an optimal investment strategy to maximize the expected utility of wealth at time T, the maximization problem is described by max π∈

E(U(X(T))).
(10) We assume that the pension manager has CRRA preference and the corresponding utility function is as follows: where γ > 1 is the relative risk aversion coefficient.

The optimal investment strategy
This section is dedicated to deriving the optimal investment strategy for problem (10).For an admissible strategy π(t) ∈ , the value function V is defined as with the boundary condition V(T, x, r, v, l) = U(x).We define the controlled infinitesimal generator A and where , x, r, v, l) with respect to the variables t, x, r, v, l, respectively.According to the principle of stochastic dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation can be derived as with the boundary condition V(T, x, r, v, l) = U(x).
We assume that Based on the above setting, we derive a solution to the HJB Equation ( 14), and the associated results are presented in the following theorem.
Theorem 4.1: Let > 0, for the optimal investment problem Equation (10) with wealth process Equation (9) and the power utility function, a solution V (t, x, r, v, l) to Equation ( 14) with boundary condition V (T, x, r, v, l) and the corresponding optimal investment strategy is obtained as A Proof: See Appendix.

A numerical example
In this section, we provide a numerical example to illustrate the impacts of model parameters on the optimal investment strategy.We mainly focus on the impacts of the risk aversion coefficient γ , the salary parameters μ L , σ L , the stock volatility parameters κ, σ V , and the inflation parameters μ P , σ P .For simplicity and generality, we focus on analysing the initial investment time is t = 0. Throughout the analysis, the basic values of parameters, unless otherwise stated, are shown in Table 1.
Figure 1 displays the effects of the risk aversion coefficient γ on the optimal investment strategy.With the increasing of γ , the investment proportion in the stock decreases, but the investment proportion in the inflationindexed bond increases.This is mainly because higher risk aversion requires us to hedge more risks in the market.The risk from the inflation can be hedged by inflation index bonds, while stock risk is non-hedgeable.Therefore, the fund manager is more inclined to reduce the investment proportion in the stock and increase the investment proportion in the inflation-index bond, which is consistent with the actual market conditions and further illustrates the validity of our model.
Figures 2 and 3 illustrate the effects of the appreciation rate μ L and volatility rate σ L of the salary on the optimal investment strategy.Along with the growth of μ L , the investment proportion in the stock increases, whereas the investment proportion in the inflation-indexed bond decreases due to the substitution effect.The growth of μ L brings more income to the member and the risk resistance is strong.Therefore, the fund manager prefers to increase the investment proportion in the stock to obtain higher returns.However, as σ L increases, the salary has more uncertainty caused, and hence the fund manager prefers to reduce the investment proportion in the stock and the inflation-indexed bond to avoid the risk.Figures 4 and 5 show the effects of the mean-reversion speed κ and the volatility of variance σ V of the stock on the optimal investment strategy.With the increasing of κ, the stock return becomes more predictable, and hence the fund manager prefers to increase the investment proportion in the stock, whereas the investment proportion in   the inflation-indexed bond decreases due to the substitution effect.A larger σ V usually implies a higher volatility risk of variance of the stock, so the fund manager prefers to reduce the investment proportion in the stock to avoid the risk, while the investment proportion in the inflationindexed bond should increase due to the substitution effect.Furthermore, comparing with Figures 2 and 3, we find that under the Heston-Hull-White model, the volatility rate σ L of the salary has a greater impact on the optimal investment strategy than the volatility of variance σ V of the stock.Figures 6 and 7 demonstrate the effects of the expected rate μ P and volatility rate σ P of the inflation on the optimal investment strategy.When μ P becomes larger or σ P becomes smaller, the investment proportion in the stock decreases, while the investment proportion in the inflation-indexed bond increases along with μ P and σ P .This can be attributed to the fact that, with the growth of μ P and σ P , the fund manager prefers to increase the investment proportion in the inflation-indexed bond to hedge against the inflation risk.When σ P increases within a controllable range, increasing the investment proportion in the stock can maintain and increase the value of assets and avoid the adverse situation of the decline of wealth purchasing power caused by inflation to a certain extent.So the fund manager prefers to increase the investment proportion in the stock.

Conclusion
This paper investigates an optimal investment strategy for DC pension plan before retirement.The inflation risk and stochastic salary are considered in our model.
To hedge against the inflation risk, an inflation-indexed bond is introduced into the financial model.Moreover, the financial market also contains a risk-free asset and a stock whose price dynamics is assumed to driven by the hybrid Heston-Hull-White model.In this model, both the stochastic volatility and interest rate processes influence the optimal investment strategy.We derived the HJB equation and the explicit solutions of the optimal investment strategy and the value function for DC pension plan under the CRRA utility are derived via the dynamic programming principle and separation variables method.Finally, a numerical example is used to illustrate the effects of financial parameters on the optimal investment strategy.
In the numerical example, we have shown that the higher risk aversion coefficient is, the less (more) investment proportion in the stock (the inflation-indexed bond) would be taken.Also, the appreciation rate of the salary is positively correlated with the investment proportion in the stock, while the volatility rate is negatively correlated.Furthermore, mean-reversion speed is positively correlated with the investment proportion in the stock while the volatility of variance of the stock displayed inversely proportional relationship.Finally, inflation risk will also have significant impacts on the optimal investment strategy.Smaller volatility rate or larger expected rate of the inflation will lead to less investment proportion in the stock.

Figure 3 .
Figure 3.The impacts of μ L and σ L on π * B .

Figure 4 .
Figure 4.The impacts of κ and σ V on π * S .

Figure 7 .
Figure 7.The impacts of μ P and σ P on π * B .