Influence of risk tolerance on long-term investments: A Malliavin calculus approach

The risk–return trade-off is an important issue in finance. Investors assess returns against risk while considering investment strategies, and they choose strategies that maximize the returns while minimizing the risk. There are several ways to formulate the risk–return trade-off. One of the most commonly accepted forms is the utility function, which reflects the risk tolerance of an investor. We quantify the extent to which the expected utility is affected by small changes in the risk tolerance.

This study investigates long-term investment strategies. Under several market models, utility-maximizing portfolios and their expected utility are considered. For this purpose, we adopt the Malliavin calculus method and the Hansen–Scheinkman decomposition, through which the expected utility is expressed in terms of the eigenvalues and eigenfunctions of an operator. We conclude that the influence of risk tolerance on the long-term expected utility is determined by these eigenvalues and eigenfunctions.

The main objective of this study is to investigate the influence of small changes of the utility function on long-term investments. We recall the classical utility maximization problem

$$-u_{T}:=\max\mathbb{E}[U(\Pi_{T})]$$

over all admissible self-financing portfolios with a given initial capital for terminal time $T.$ The constant relative risk aversion (CRRA) utility function

$$U(x)=\frac{x^{\nu}}{\nu}$$

for $\nu<0$ is considered in this paper. The parameter $\nu$ represents how an investor measures his/her degree of risk tolerance. The optimal portfolio $\hat{\Pi}=\hat{\Pi}^{(\nu)}$ depends on the parameter $\nu,$ thus

$$-u_{T}=\mathbb{E}[U(\hat{\Pi}_{T})]=\mathbb{E}[\hat{\Pi}_{T}^{\nu}]/\nu\,.$$

We investigate the influence of small changes in the parameter $\nu$ on the expected utility over a long time horizon. The influence of small changes in the parameter $\nu$ in terms of its logarithmic value can be mathematically expressed as

$$\frac{\partial}{\partial\nu}\ln u_{T}\,.$$

We estimate the large-time behavior of this partial derivative as $T\to\infty.$

The main methodology for this analysis is a combination of the Hansen–Scheinkman decomposition and the Malliavin calculus technique presented in Sections 2 and 3. This approach is from Park (2018). First, we transform the expected utility into the expectation form

$$p_{T}=\mathbb{E}_{\xi}[e^{-\int_{0}^{T}q(X_{s})\,ds}]$$

for some Markov diffusion process $X=(X_{t})_{0\leq t\leq T}$ with $X_{0}=\xi$ and some measurable function $q.$ The process $X$ with killing rate $q$ induces an infinitesimal generator. Using the Hansen–Scheinkman decomposition, we can find an eigenvalue $\lambda$ and a positive eigenfunction $\phi$ of the generator as well as a measurable function $f$ such that the expectation is written as

$$p_{T}=\phi(\xi)e^{-\lambda T}f(T,\xi)\,.$$

The real number $\lambda$ and the functions $\phi$ and $f$ depend on the parameter $\nu.$ By differentiating with respect to $\nu,$ we have

$$\frac{\partial}{\partial\nu}\ln p_{T}=\frac{\partial}{\partial\nu}\ln\phi(\xi)-T\frac{\partial\lambda}{\partial\nu}+\frac{\partial}{\partial\nu}\ln f(T,\xi)\,.$$

Using the above-mentioned equation, we estimate the influence of risk tolerance on the long-term expected utility. If $\frac{\partial}{\partial\nu}\ln f(T,\xi)$ is bounded in $T$ on $[0,\infty),$ then

$$\Big{|}\frac{1}{T}\frac{\partial}{\partial\nu}\ln p_{T}+\frac{\partial\lambda}{\partial\nu}\Big{|}\leq\frac{c}{T}$$

for some positive constant $c.$ This implies that the influence of risk tolerance is asymptotically equal to the partial derivative of $-\lambda$ with respect to $\nu,$ which is the main conclusion of this study. The Malliavin calculus technique is used to verify that $\frac{1}{T}\frac{\partial}{\partial\nu}\ln f(T,\xi)$ is bounded in $T$ on $[0,\infty).$ We cover several market models, namely the Ornstein–Uhlenbeck process, the CIR process, the $3/2$ model, a quadratic drift model.

The remainder of this paper is organized as follows. The related literature is reviewed in Section 1.2. The utility maximization problem, the Hansen–Scheinkman decomposition and the Malliavin calculus method are explained as mathematical preliminaries in Section 2. The main ideas and arguments for investigating the influence of risk tolerance are discussed in Section 3. The influence of risk tolerance on utility-maximizing portfolios is illustrated in Section 4. Finally, our findings are summarized in Section 5. The technical details are presented in the appendices.

Many authors have studied the stability of the optimal investment strategy with respect to the utility function. Jouini and Napp (2004) studied in a general complete financial market the stability of the optimal investment-consumption strategy with respect to the choice of the utility function. More precisely, for a given sequence of utility functions that converges pointwise, they proved the almost sure as well as the $L^{p}$-convergence $(p\geq 1)$ of the optimal wealth and consumption at each date. Carassus and Rásonyi (2007) investigated the convergence of optimal strategies with respect to a sequence of utility functions. They also established the continuity of the utility indifference price with respect to changes in agents’ preferences. Nutz (2012) considered the economic problem of optimal consumption and investment with power utility. As the relative risk aversion tends to infinity or to one was proved, the convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models.

The dependence of the risk tolerance on the investment strategy has been studied many authors. Zariphopoulou and Zhou (2009) analyzed a portfolio choice problem when the local risk tolerance is time-dependent and asymptotically linear in wealth. This methodology allows the investment performance to be measured in terms of the risk tolerance and alternative market views. Mocha and Westray (2013) studied the sensitivity of the power utility maximization problem with respect to the investor’s relative risk aversion, the statistical probability measure, the investment constraints, and the market price of risk. Paravisini et al. (2017) estimated risk tolerance from investors’ financial decisions in a person-to-person lending platform. They developed a method that obtains a risk-tolerance parameter from each portfolio choice on the basis of the elasticity of risk tolerance to changes in wealth. Bi and Cai (2019) investigated the optimal investment–reinsurance strategies for an agent with state-dependent risk tolerance and value-at-risk constraints. They derived the closed-form expressions of the optimal strategies and discussed the impact of the risk tolerance. Delong (2019) considered agents whose risk tolerance consists of a constant risk tolerance and a small wealth-dependent risk tolerance. He investigated an exponential utility maximization problem for an agent who faces a stream of non-hedgeable claims.

Numerous studies have investigated the topic of long-term investment strategies. Fleming and Sheu (2000) considered an optimal investment model to maximize the long-term growth rate of the expected utility of wealth. The problem was reformulated as a risk‐sensitive control problem with an infinite time horizon. Hansen and Scheinkman (2009), Hansen (2012) and Hansen and Scheinkman (2012) exploited the Hansen–Scheinkman decomposition method and demonstrated a long-term risk–return trade-off. Guasoni and Robertson (2015) studied a class of static fund separation theorems that is valid for investors with a long time horizon and constant relative risk tolerance. Robertson and Xing (2015) investigated long-term portfolio choice problems by analyzing the large-time asymptotic behavior of solutions to semi-linear Cauchy problems.

Malliavin calculus has been studied in relation to various topics in quantitative finance. Fournié et al. (1999) investigated a probabilistic method for computations of Greeks in finance. This methodology is based on the integration-by-parts formula developed in Malliavin calculus. Benhamou (2003) showed that the Malliavin weight functions for Greeks must satisfy necessary and sufficient conditions expressed as conditional expectations. Alos et al. (2007) used Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be a diffusion or a Markov process. Borovička et al. (2014) studied the shock elasticity, which reflects the sensitivity with respect to a perturbation over a time instant. They proposed a Malliavin calculus method to compute the shock elasticity. Alòs and Shiraya (2019) studied the difference between the fair strike of a volatility swap and the at-the-money implied volatility of a European call option. They used the Malliavin calculus approach to derive an exact expression for this difference. Park and Sturm (2019) employed the Malliavin calculus method to investigate the sensitivities of the long-term expected utility of optimal portfolios under incomplete markets.

As a closely related article, Park (2018) conducted a sensitivity analysis of long-term cash flows. However, the perturbation form is different from this paper. He studied the extent to which the price of the cash flow is affected by small perturbations of the underlying Markov diffusion. He considered the drift and volatility perturbations in the underlying process

$$dX_{t}^{\epsilon}=b_{\epsilon}(X_{t}^{\epsilon})\,dt+\sigma_{\epsilon}(X_{t}^{\epsilon})\,dB_{t}\;,\,X_{0}^{\epsilon}=\xi$$

for the perturbation parameter $\epsilon$, and analyzed their influence to the cash flows. This paper, however, works with the investor’s risk tolerance and the perturbations in the underlying process is not considered.

We consider the classical utility maximization problem in financial markets. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\bf P})$ be a filtered probability space having a one-dimensional Brownian motion $(Z_{t})_{t\geq 0}$ and a $n$-dimensional Brownian motion $W$ with constant correlation $\rho=(\rho_{1},\cdots,\rho_{n})^{\top}=d\left\langle Z,\,W\right\rangle_{t}/dt.$ The measure ${\bf P}$ is referred to as the physical measure and the filtration $(\mathcal{F}_{t})_{t\geq 0}$ is the argumentation of the natural filtration of $Z$ and $W.$ We assume that the market has a state process $(X_{t})_{t\geq 0},$ which is a Markov diffusion satisfying

$$dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dB_{t}\,,\;X_{0}=\xi$$

(2.1)

There are $n+1$ assets $S^{(0)},S^{(1)},\ldots,S^{(n)}$ in the market, where $S^{(0)}$ is a risk-free asset and $S^{(1)},\ldots,S^{(n)}$ are risky assets. The assets dynamics are given as

		$\displaystyle\frac{dS^{(0)}_{t}}{S^{(0)}_{t}}=r(X_{t})\,dt\,,$		(2.2)
		$\displaystyle\dfrac{dS^{(i)}_{t}}{S^{(i)}_{t}}=(r(X_{t})+\mu_{i}(X_{t}))dt+\sum_{j=1}^{n}\upsilon_{ij}(Y_{t})\,dW^{(j)}_{t}\qquad 1\leq i\leq n\,.$

Here, $r$ is the short interest rate function, $\mu=(\mu_{1},\cdots,\mu_{n})^{\top}$ is the excess return function, and $\upsilon=(\upsilon_{ij})_{1\leq i,j\leq n}$ is the volatility matrix functions.

A portfolio is a $n$-dimensional process $\pi=(\pi_{t})_{t\geq 0}=(\pi^{(1)}_{t},\ldots,\pi^{(n)}_{t})_{t\geq 0}$, which represents the proportions of wealth in each risky asset. The wealth process $(\Pi_{t})_{t\geq 0}=(\Pi_{t}^{\pi})_{t\geq 0}$ of $\pi$ with positive initial capital $\omega$ satisfies

$$\frac{d\Pi_{t}}{\Pi_{t}}=(r(X_{t})+\pi_{t}\mu(X_{t}))\,dt+\pi_{t}\upsilon(X_{t})\,dW_{t},\quad\Pi_{0}=\omega>0.$$

(2.3)

We assume that the portfolio process $(\pi_{t})_{t\geq 0}$ is $(\mathcal{F}_{t})_{t\geq 0}$-adapted and the integrations in Eq.(2.3) are well-defined, that is, $\mathbb{E}\int_{0}^{t}|(r(X_{s})+\pi_{s}\mu(X_{s}))|+|\!|\pi_{s}\upsilon(X_{s})|\!|^{2}\,ds<\infty$ for all $t\geq 0,$ where $|\!|\cdot|\!|$ is the usual Euclidean norm. The wealth process $\Pi_{t}>0$ for all $t\geq 0$ a.s. since the initial wealth $\omega>0.$

In this market, we consider an agent who wants to maximize the expected utility of the terminal wealth over all possible portfolios. More precisely, we are interested in

$$\sup_{\pi}\mathbb{E}^{\bf P}[U(\Pi_{T}^{\pi})]$$

over all possible portfolios $\pi$ for given positive initial capital $\omega.$ The utility function is assumed to be a power function of the form

$$U(x)=x^{\nu}/\nu$$

(2.4)

for $\nu<0.$

This utility maximization problem can be solved by using stochastic control theory. Define the value function $u_{T}$ as

$$-u_{T}=-u_{T}(t,\omega,x)=\sup_{\pi}\mathbb{E}^{\bf P}[U(\Pi_{T}^{\pi})|\Pi_{t}=\omega,X_{t}=x]\,.$$

Following (Zariphopoulou, 2001, Proposition 2.1), we have

$$u_{T}(t,\omega,x)=-\frac{\omega^{\nu}}{\nu}(\mathbb{E}^{\mathbb{P}}[e^{\frac{\nu(1-\nu+\nu\rho^{\prime}\rho)}{1-\nu}\int_{t}^{T}(r+\frac{1}{2(1-\nu)}\theta^{\prime}\theta)(X_{u})\,du}|X_{t}=x])^{\frac{1-\nu}{1-\nu+\nu\rho^{\prime}\rho}}$$

where $\theta:=\upsilon^{-1}\mu,$ $b:=k+\frac{\nu}{1-\nu}\sigma\rho^{\prime}\theta$ and the $\mathbb{P}$-dynamics of $X$ is

$$dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dB_{t}$$

(2.5)

with a $\mathbb{P}$-Brownian motion $(B_{t})_{t\geq 0}.$ For $t=0,$ it follows that

$$u_{T}=u_{T}(0,\omega,\xi)=-\frac{\omega^{\nu}}{\nu}q_{T}^{\frac{1-\nu}{1-\nu+\nu\rho^{\prime}\rho}}$$

for

$$q_{T}:=\mathbb{E}^{\mathbb{P}}_{\xi}[e^{\frac{\nu(1-\nu+\nu\rho^{\prime}\rho)}{1-\nu}\int_{0}^{T}(r+\frac{1}{2(1-\nu)}\theta^{\prime}\theta)(X_{u})\,du}]\,.$$

(2.6)

Here, we have used the notation $\mathbb{E}^{\mathbb{P}}_{\xi}[\,\cdots]=\mathbb{E}^{\mathbb{P}}[\,\cdots|X_{0}=\xi].$ In particular, when the short rate is a constant $r,$

$$u_{T}=-\frac{\omega^{\nu}}{\nu}e^{r\nu T}p_{T}^{\frac{1-\nu}{1-\nu+\nu\rho^{\prime}\rho}}$$

(2.7)

where

$$p_{T}:=\mathbb{E}_{\xi}^{{\mathbb{P}}}[e^{-q\int_{t}^{T}(\theta^{\prime}\theta)(X_{u})\,du}]$$

(2.8)

and $q:=-\frac{\nu(1-\nu+\nu\rho^{\prime}\rho)}{2(1-\nu)^{2}}.$

In conclusion, the problem of analyzing the optimal expected utility boils down to the problem of analyzing the expectation $q_{T}$ in Eq.(2.6) (or $p_{T}$ in Eq.(2.8) if the short rate is constant) with the dynamics of $X$ given as Eq.(2.5). Later this will be used in Step I in Section 3.

We briefly review the Hansen–Scheinkman decomposition as a mathematical preliminary. Readers may refer to Hansen and Scheinkman (2009), Park (2018), and Qin and Linetsky (2016) for further details. Consider a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$ having a $d$-dimensional Brownian motion $B=(B_{t}^{(1)},\cdots,B_{t}^{(d)})_{t\geq 0}^{\top}.$ The family $(\mathcal{F}_{t})_{t\geq 0}$ is the completed filtration generated by $B.$

We begin with the state space $\mathcal{D}$ and four functions $b,\sigma,q,h.$ Let $\mathcal{D}$ be an open and connected subset of $\mathbb{R}^{d},$ and let $b:\mathcal{D}\to\mathbb{R}^{d}$ and $\sigma:\mathcal{D}\to\mathbb{R}^{d}\times\mathbb{R}^{d}$ be continuously differentiable functions. The matrix $\sigma$ is invertible. The function $q:\mathcal{D}\to\mathbb{R}$ is continuous and the function $h:\mathcal{D}\to\mathbb{R}$ is nonnegative continuously differentiable. For each $\xi\in\mathcal{D},$ assume that the SDE Eq.(2.1) has a unique strong solution $X$ on $\mathcal{D}.$

We can consider an infinitesimal generator and its eigenpair. Define the infinitesimal generator of the process $X$ with killing rate $q$ as

$$\mathcal{L}:=\frac{1}{2}\sum_{i,j=1}^{d}a_{ij}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}+\sum_{i=1}^{d}b_{i}\frac{\partial}{\partial x_{i}}-q,$$

(2.9)

where $a=\sigma\sigma^{\top}.$ For a real number $\lambda$ and a positive $C^{2}$-function $\phi:\mathcal{D}\to\mathbb{R},$ we say that a pair $(\lambda,\phi)$ is an eigenpair of $-\mathcal{L}$ if

$$\mathcal{L}\phi=-\lambda\phi\quad\textnormal{on }\mathcal{D}\,.$$

(2.10)

For $T>0,$ by applying the Ito formula, we can show that for each eigenpair $(\lambda,\phi),$ a positive process

$$M_{t}^{\phi}:=\frac{\phi(X_{t})}{\phi(\xi)}e^{\lambda t-\int_{0}^{t}q(X_{s})\,ds}\,,\;0\leq t\leq T$$

(2.11)

is a local martingale. Assume that this process is a martingale. We can define a measure $\hat{{\mathbb{P}}}_{t}^{\phi}$ on each $\mathcal{F}_{t}$ for $0\leq t\leq T$ by

$$\frac{d\hat{{\mathbb{P}}}_{t}^{\phi}}{d{\mathbb{P}}\;}=M_{t}^{\phi}\,.$$

The family $(\hat{{\mathbb{P}}}_{t}^{\phi})_{0\leq t\leq T}$ is consistent, i.e., $\hat{{\mathbb{P}}}_{t^{\prime}}^{\phi}|_{\mathcal{F}_{t}}=\hat{{\mathbb{P}}}_{t}^{\phi}$ for all $0\leq t\leq t^{\prime}\leq T.$ For fixed $T>0,$ we use the notation $\hat{\mathbb{P}}^{\phi}$ instead of $\hat{{\mathbb{P}}}_{T}^{\phi},$ suppressing $T.$ This measure on $\mathcal{F}_{T}$ is called the eigen-measure with respect to $\phi.$ The process

$$\hat{B}_{t}^{\phi}=-\int_{0}^{t}(\sigma^{\top}\nabla\phi/\phi)(X_{s})\,ds+B_{t}\,,\;t\geq 0$$

is a $\hat{{\mathbb{P}}}^{\phi}$-Brownian motion by the Girsanov theorem, and the process $X$ satisfies

$$dX_{t}=(b+\sigma\sigma^{\top}\nabla\phi/\phi)(X_{t})\,dt+\sigma(X_{t})\,d\hat{B}_{t}^{\phi}\,.$$

Under these circumstances, consider the decomposition of the discount factor

$$e^{-\int_{0}^{t}q(X_{s})\,ds}=M_{t}^{\phi}e^{-\lambda t}\frac{\phi(\xi)}{\phi(X_{t})}\,,\;t\geq 0,$$

(2.12)

which comes from Eq.(2.11). This expression is called the Hansen–Scheinkman decomposition. The expectation $p_{T}$ can be written as

	$\displaystyle p_{T}=\mathbb{E}_{\xi}^{\mathbb{P}}[e^{-\int_{0}^{T}q(X_{s})\,ds}]$	$\displaystyle=\phi(\xi)\,e^{-\lambda T}\,\mathbb{E}_{\xi}^{\mathbb{P}}\Big{[}\frac{M_{T}^{\phi}}{\phi(X_{T})}\Big{]}$		(2.13)
		$\displaystyle=\phi(\xi)\,e^{-\lambda T}\,\mathbb{E}_{\xi}^{\hat{\mathbb{P}}^{\phi}}\Big{[}\frac{1}{\phi(X_{T})}\Big{]}$
		$\displaystyle=\phi(\xi)e^{-\lambda T}f_{\phi}(T,\xi),$

where $f_{\phi}(t,x):=\mathbb{E}_{x}^{{\hat{\mathbb{P}}}^{\phi}}[\frac{1}{\phi(X_{t})}]$ for $t\geq 0$ and $x\in\mathcal{D}.$ The function $f_{\phi}$ is referred to as the remainder function. The decomposition in Eq.(2.13) is useful for the analysis of $p_{T},$ because the expectation $\mathbb{E}_{\xi}^{{\hat{\mathbb{P}}}^{\phi}}[\frac{1}{\phi(X_{T})}]$ depends on the final random variable $X_{T},$ whereas the expression

$$p_{T}=\mathbb{E}_{\xi}^{\mathbb{P}}[e^{-\int_{0}^{T}q(X_{s})\,ds}]$$

depends on the entire path of $(X_{t})_{0\leq t\leq T}.$ If we know the $\hat{\mathbb{P}}^{\phi}$-distribution of $X_{T}$, then we can analyze the expectation $\mathbb{E}_{\xi}^{\hat{\mathbb{P}}^{\phi}}[\frac{1}{\phi(X_{T})}]$ directly. For notational simplicity, when the eigenpair is specified, we use the notations $M,$ $\hat{{\mathbb{P}}},$ $(\hat{B}_{t})_{t\geq 0}$, and $f$ instead of $M^{\phi},$ $\hat{{\mathbb{P}}}^{\phi},$ $(\hat{B}_{t}^{\phi})_{t\geq 0}$, and $f_{\phi},$ respectively, suppressing $\phi.$

We summarize this section in the following proposition.

This section presents a brief review of Malliavin calculus. For further details, refer to Malliavin and Paul (2006) and Nualart and Nualart (2018). Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leq t\leq T},\mathbb{P})$ be a filtered probability space having a one-dimensional Brownian motion $B.$ The filtration $(\mathcal{F}_{t})_{0\leq t\leq T}$ is the natural filtration of $B.$ Define the set of all cylindrical random variables as

$$\mathcal{S}:=\Big{\{}f\Big{(}\int_{0}^{T}h^{(1)}(s)\,dB_{s},\cdots,\int_{0}^{T}h^{(n)}(s)\,dB_{s}\Big{)}:n\in\mathbb{N},h^{(1)},\cdots,h^{(n)}\in L^{2}[0,T],f\in C_{p}^{\infty}(\mathbb{R}^{n})\Big{\}}\,,$$

where $C_{p}^{\infty}(\mathbb{R}^{n})$ is the set of all $C^{\infty}$ functions such that $f$ and all its partial derivatives have polynomial growth. The Malliavin derivative of $F\in\mathcal{S}$ is defined as a stochastic process $DF=(D_{t}F)_{0\leq t\leq T}$ given by

$$D_{t}F=\sum_{i=1}^{n}h^{(i)}(t)\frac{\partial f}{\partial x_{i}}\Big{(}\int_{0}^{T}h^{(1)}(s)\,dB_{s},\cdots,\int_{0}^{T}h^{(n)}(s)\,dB_{s}\Big{)}\,.$$

Then, $D$ is a linear operator from $\mathcal{S}\subseteq L^{2}(\Omega)$ to $L^{2}([0,T]\times\Omega),$ and it is known that $D$ is closable. The closure is also denoted as $D.$ The domain of $D$ is the closure of $\mathcal{S}$ under the norm

$$|\!|F|\!|_{\mathbb{D}^{1,2}}:=|\!|F|\!|_{L^{2}(\Omega)}+|\!|DF|\!|_{L^{2}([0,T]\times\Omega)}$$

and is denoted as $\mathbb{D}^{1,2}.$

We will use the following propositions. Propositions 2.2, 2.3, and 2.5 are from (León et al., 2003, Lemma 2.1), (Protter, 2004, Theorem 39 on page 312), and (Park, 2018, Proposition A.1), respectively.

As a motivating example, consider a constant proportion portfolio when the underlying market follows the Black–Scholes model. Assume that the short rate is a constant $r\geq 0$ and the stock price follows

$$dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dZ_{t}$$

for $\mu\in\mathbb{R},$ $\sigma>0.$ With the initial capital $\omega>0,$ it is known that the optimal expected utility is

$\displaystyle-u_{T}:=\max_{\pi}\mathbb{E}^{\bf P}[U(\Pi_{T}^{\pi})]=\frac{\omega^{\nu}}{\nu}e^{(r+\frac{(\mu-r)^{2}}{2(1-\nu)\sigma^{2}})\nu T}.$

(4.1)

We aim to investigate the influence of the risk aversion on the long-term investment. We calculate the partial derivative

$$\frac{\partial}{\partial\nu}\ln u_{T}=\ln\omega-\frac{1}{\nu}+(r+\frac{(\mu-r)^{2}}{2(1-\nu)^{2}\sigma^{2}})T\,.$$

The partial derivative $\frac{\partial}{\partial\nu}\ln u_{T}$ grows linearly as $T\to\infty,$ and the linear growth rate is $r+\frac{(\mu-r)^{2}}{2(1-\nu)^{2}\sigma^{2}}.$

Under the physical measure ${\bf P},$ let the state process $X$ follow the OU process

$$dX_{t}=(b-kX_{t})\,dt+\sigma\,dZ_{t}\,,\;X_{0}=\xi$$

(4.2)

for $b,\xi\in\mathbb{R}$, $k,\sigma>0$ and assume that $\theta^{\prime}\theta(\cdot)=\cdot\,^{2}$ and $\rho^{\prime}\theta(\cdot)=\overline{\rho}\,\cdot$ for some constant $\overline{\rho}\in\mathbb{R}$ (this holds, for example, $d=1$ and the state process is the market price of risk). Then, the $\mathbb{P}$-dynamics of $X$ given in Eq.(2.5) is

$$dX_{t}=(b-aX_{t})\,dt+\sigma\,dB_{t}\,,\;X_{0}=\xi$$

(4.3)

and $p_{T}=\mathbb{E}^{{\mathbb{P}}}[e^{-q\int_{0}^{T}X_{u}^{2}\,du}]$, where $a=k-\frac{\nu\sigma\overline{\rho}}{1-\nu}$ and $q=-\frac{\nu(1-\nu+\nu\rho^{\prime}\rho)}{2(1-\nu)^{2}}.$

We now apply the Hansen–Scheinkman decomposition stated in Section 2.2. Eq.(A.3) gives

$\displaystyle p_{T}=f(T,\xi)e^{-\frac{1}{2}\eta\xi^{2}-\ell\xi}e^{-\lambda T},$

(4.4)

where $\alpha=\sqrt{a^{2}+2q\sigma^{2}},$ $\eta=\frac{\alpha-a}{\sigma^{2}},$ $\ell=\frac{b\eta}{\alpha},$ $\lambda=-\frac{1}{2}\sigma^{2}\ell^{2}+b\ell+\frac{1}{2}(\alpha-a)$, and

$$f(t,x)=\mathbb{E}_{x}^{\hat{\mathbb{P}}}[e^{\frac{1}{2}\eta X_{T}^{2}+\ell X_{T}}]\,,\;0\leq t\leq T\,,\;x\in\mathbb{R}$$

(4.5)

is the remainder function. Under the measure $\hat{\mathbb{P}},$ the process $X$ satisfies

$$dX_{t}=(\delta-\alpha X_{t})\,dt+\sigma\,d\hat{B}_{t}$$

for $\delta=\frac{b}{\alpha}.$ It follows that

$$\frac{\partial}{\partial\nu}\ln p_{T}=-\frac{1}{2}\xi^{2}\frac{\partial\eta}{\partial\nu}-\xi\frac{\partial\ell}{\partial\nu}-T\frac{\partial\lambda}{\partial\nu}+\frac{f_{\nu}(T,\xi)}{f(T,\xi)}\,.$$

(4.6)