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Mathematics and Financial Economics

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24-08-2018

Turnpike property and convergence rate for an investment and consumption model

Authors: Baojun Bian, Harry Zheng

Published in: Mathematics and Financial Economics | Issue 2/2019

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Abstract

We discuss the turnpike property for optimal investment and consumption problems. We find there exists a threshold value that determines the turnpike property for investment policy. The threshold value only depends on the Sharpe ratio, the riskless interest rate and the discount rate. We show that if utilities behave asymptotically like power utilities and satisfy some simple relations with the threshold value, then the turnpike property for investment holds. There is in general no turnpike property for consumption policy. We also provide the rate of convergence and illustrate the main results with examples of power and non-HARA utilities and numerical tests.

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A utility function U and its dual function V are equivalent and can be recovered by each other from the relations $V(y)=\sup _{x>0}(U(x)-xy)$ and $U(x)=\inf _{y>0}(V(y)+xy)$. For example, if U is a power utility $U(x)=(1/p)x^p$ for $x>0$ and $p<1$, then its equivalent dual function is $V(y)=-(1/q)y^q$ for $y>0$ and $q=p/(p-1)$. Furthermore, $U_i(x)=V_i(y)+xy$ if and only if $U'_i(x)-y=0$. Therefore, from

$$\begin{aligned} {U_i'(x)\over x^{p_i-1}}= \left( {y^{q_i-1}\over -V_i'(y)}\right) ^{p_i-1} \end{aligned}$$

and $U_i'(\infty )=0$, we get the equivalence of conditions (1.3) and (1.4).

The result of Theorem 2.4 can be stated equivalently in terms of $p_i$ defined in (1.3), that is, if $p_1>p^*$ or $p_2\ge p^*$, where $p^*=q^*/(q^*-1)\ge 0$, then the turnpike property holds and the optimal amount of investment can be approximated by $ \lim _{T\rightarrow \infty }A(x,t)=(\theta /\sigma )(1-\max \{p_1,p_2\})^{-1}x$.

Note that $q^*$ only depends on the market price of risk $\theta $, the riskless interest rate r and the utility discount rate $\delta $.

Condition (1.9) is equivalent to the limits of the Arrow–Pratt coefficients of relative risk aversion of utilities $U_i$ being $1-p_i$, that is,

$$\begin{aligned} \lim _{x\rightarrow \infty }\left( -{xU_i''(x)\over U_i'(x)}\right) = 1-p_i,\quad i=1,2, \end{aligned}$$

(1.10)

and $p_1,p_2>p^*$. This is due to the dual relation of $U_i$ and $V_i$ defined in (1.5), which implies that if $y=U_i'(x)$ then $x=-V_i'(y)$ and $U_1''(x)=-1/V_1''(y)$, and therefore the equivalence of conditions (1.9) and (1.10).

The limits in conditions (1.4) and (2.6) are $-1$ which can be replaced by some constants. Specifically, if there exist $k_i>0$, such that $\lim _{y\rightarrow 0}\frac{V_i'(y)}{y^{q_i-1}}=-\,k_i$, then for $x\in R_+$, we still have the turnpike property (1.7). If, for some ${\hat{q}} <1$, $k\ge 0$, $\lim _{y\rightarrow \infty } \frac{V'_2(y)}{y^{{\hat{q}}-1}}=-\,k$, then, we have $\lim _{t\rightarrow \infty }\hat{R}(t)^{\frac{{\hat{q}}-1}{q_2-1}}C(x,t) =k x^{\frac{ {\hat{q}}-1}{\min \{q_1,q_2\}-1}}$, where $\hat{R}(t)=k_1e^{\lambda _1 t}+k_2\frac{e^{\lambda _2 t}-1}{\lambda _2}$ if $q_1=q_2$, $k_1^{\frac{q_2-1}{\hat{q}-1}}e^{\frac{q_2-1}{q_1-1}\lambda _1 t}$ if $q_1<q_2$, and $k_2\frac{e^{\lambda _2 t}-1}{\lambda _2}$ if $q_1>q_2$. The proof is the same as that of Theorem 2.4 with some obvious changes to include $k_i$ and k.

It is easy to verify that both (1.4) and (1.9) imply (2.8), so (2.8) is the weakest condition among three conditions. The function $V'(y)=y^{q-1}\ln y (y\le Y<1)$ is an example that satisfies (1.9), but not (1.4) (see [8], page 1346). The function $V'(y)=-y^{q-1}e^{hy\sin \frac{1}{y}}$, where $h<\frac{1-q}{2}$, is an example that satisfies (1.4), but not (1.9) (see Back et al. [1], page 178). However, if $\lim _{y\rightarrow 0} V_i''(y)/y^{q_i-2}$ exists, then both (1.4) and (1.9) are satisfied with L’Hospital’s Rule.

Assume f and g are well defined functions on $R_+\times R_+$ and $R_+$, respectively. We say f(x, t) converges to g(x) exponentially (or polynomially) as $t\rightarrow \infty $ if there exist constants $c>0$ and ${\bar{T}}>0$ and a well defined function D on $R_+$, such that $|f(x,t)-g(x)|\le D(x)e^{-ct}$ (or $|f(x,t)-g(x)|\le D(x)t^{-c}$) for all $x\in R_+$ and $t\ge {\bar{T}}$. Exponential convergence is much faster than polynomial convergence.

If $V_2\equiv 0$, we need only $\bar{q}\in (\max \{q_1, 0\}, 1]$.

We have ${\bar{q}}={1\over 2} (q+1)>q$ and ${\bar{q}}-1={1\over 2}(q-1)$. It is easy to verify that U is strictly increasing and strictly concave, $U(0)=0$, $U(\infty )=\infty $, $U'(0)=\infty $ and $U'(\infty )=0$. Therefore U is a utility function in the classical sense. This utility (for $p=3/4$) is used in Bian and Zheng [3] to illustrate the turnpike property and the convergence rate for a terminal wealth utility maximization problem.

If $\{e_n\}$ is a sequence of errors with exponential convergence, then there are positive constants M and c such that $|e_n|\le Me^{-cn}$. To find c, we may assume $|e_n|\approx Me^{-cn}$, which gives $ c\approx -\ln \left( |e_{n+1}|/|e_n|\right) $. In our numerical tests, we have chosen time horizon $t=1,2,5,10,25,50,100$, which does not have equally spaced intervals. An adjustment is needed to reflect this. Specifically, we estimate c by $ c_n:=- (1/m)\ln \left( |e_{n+m}|/ |e_n|\right) $, where m is an integer indicating the distance of indices n and $n+m$ for errors $e_n$ and $e_{n+m}$. For $n:=1,2,5,10,25,50,100$, the distance between adjacent points are $m=1,3,5,15,25,50$. We form a sequence $\{c_n\}$ to see if there is a limit which would indicate the approximate exponent of the exponential convergence rate.

Back, K., Dybvig, P.H., Rogers, L.C.G.: Portfolio turnpikes. Rev. Financial Stud. 12, 165–195 (1999)CrossRef

Bian, B., Miao, S., Zheng, H.: Smooth value functions for a class of nonsmooth utility maximization problems. SIAM J. Financial Math. 2, 727–747 (2011)MathSciNetCrossRefMATH

Bian, B., Zheng, H.: Turnpike property and convergence rate for an investment model with general utility functions. J. Econ. Dyn. Control 51, 28–49 (2015)MathSciNetCrossRefMATH

Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)CrossRefMATH

Cox, J., Huang, C.: A continuous time portfolio turnpike theorem. J. Econ. Dyn. Control 2, 491–507 (1992)MathSciNetCrossRefMATH

Guasoni, P., Kardaras, C., Robertson, S., Xing, H.: Abstract, classic, and explicit turnpikes. Finance Stoch. 18, 75–114 (2014)MathSciNetCrossRefMATH

Huang, C., Zariphopoulou, T.: Turnpike behaviour of long-term investments. Finance Stoch. 3, 15–34 (1999)MathSciNetCrossRef

Huberman, G., Ross, S.: Portfolio turnpike theorems, risk aversion, and regularly varying functions. Econometrica 51, 1345–1361 (1983)MathSciNetCrossRefMATH

Jin, X.: Consumption and portfolio turnpike theorems in a continuous-time finance model. J. Econ. Dyn. Control 22, 1001–1026 (1998)MathSciNetCrossRefMATH

10.

Robertson, S., Xing, H.: Long term optimal investment in matrix valued factor models. SIAM J. Financial Math. 8, 400–434 (2017)MathSciNetCrossRefMATH

Title: Turnpike property and convergence rate for an investment and consumption model
Authors: Baojun Bian
Harry Zheng
Publication date: 24-08-2018
Publisher: Springer Berlin Heidelberg
Published in: Mathematics and Financial Economics / Issue 2/2019
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-018-0226-3

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