Ambiguity premium and transaction costs☆
Introduction
Since the seminal references of Merton, 1969, Merton, 1971, there has been a very active line of research focusing on optimal consumption and portfolio choice by relaxing Merton’s restrictive assumptions such as independent and identical distributed of returns, the absence of uncertainty about returns, frictionless markets, and the absence of labor income etc. Our work sits squarely within the bulk of this research particularly by considering ambiguity (or model uncertainty) in accordance with Maenhout (2004) proposing homothetic robustness with wealth independence, and concerning the friction markets with transaction costs (Constantinides, 1986, Liu and Loewenstein, 2002).
Maenhout (2004) claims “Robustness amounts therefore to an increase in effective risk aversion, at least within the confines of the environment studied here”. This is precisely the direction we take on the paper. In the simplest possible setup in every other dimension, we isolate and very closely investigate the new issues introduced by ambiguity aversion on portfolio choice especially with transaction costs. In the standard (Merton, 1969, Merton, 1971) framework, we generalize the optimal investment model of an ambiguity averse investor with transaction costs. That is, this paper demonstrates the joint presence of ambiguity aversion (Maenhout, 2004) and transaction costs (Liu and Loewenstein, 2002).
Along the lines of Maenhout (2004), we first show that ambiguity leads to an increase in effective risk aversion by ambiguity aversion even with transaction costs. In order to address the importance of ambiguity aversion for portfolio choice in the friction markets, we compute the utility cost (measured in certainty equivalent wealth units) associated with suboptimal investment decisions, which is the so-called ambiguity premium. More specifically, we compute the utility loss incurred by investors who ignore ambiguity aversion with and without transaction costs. Overall, ignoring ambiguity aversion with and without transaction costs generates large ambiguity premia when ambiguity aversion is moderate. The premia are quite substantial, generating as high as 6% of wealth for moderate ambiguity aversion. Further, the cost of ignoring ambiguity aversion becomes larger with higher ambiguity aversion. The premia with transaction costs can be higher than 12% of wealth for high ambiguity aversion. This would, thus, still support the importance of ambiguity aversion for portfolio choice, even concerning the friction markets.
Section snippets
The basic model
There are two assets in the market: a risk-free bond and a risky stock. The bond has a return and the stock price follows a geometric Brownian motion with its expected return and the volatility . Trading stocks entails transaction costs of which the ask price is , and the bid price is , where and denote proportional transaction costs. We denote by a standard one-dimensional Brownian motion under a well-defined probability measure. Let and
Solution
We now solve the HJB equation (4) for the ambiguity averse investor’s investment problem with transaction costs (3). Following Maenhout (2004), we decide to use homothetic robustness removing wealth effects and allowing for analytical tractability. Specifically, we alter the investor’s state-dependent ambiguity aversion by replacing it with constant divided by : where represents the investor’s constant ambiguity aversion instead of her
Conclusion
In this paper, we develop a tractable workhorse investment model of an ambiguity averse investor with transaction costs. Our analysis demonstrates that ambiguity aversion does matter for portfolio choice, even concerning the friction markets. We show the economic significance of correctly considering ambiguity aversion in the optimal investment with transaction costs. If an investor neglects ambiguity aversion in her routine investment decision with and without transaction costs, the utility
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We would like to thank Max Croce, Alexander Carol, Tony He, John Doukas, Murali Jagannathan, Min Dai, Alain Bensoussan, Emanuele Borgonovo, Jang-Koo Kang, Hyeng Keun Koo, Yong Hyun Shin, and seminar participants at Ajou University, 2013 Co-conference in Finance of Korea, 2013 Korean Operations Research and Management Science Society Conference, 2014 Asia Pacific Industrial Engineering and Management Systems Conference, 2015 Quantitative Methods in Finance Conference, and 2019 Korean Association of Financial Engineering Conference for helpful discussions and insightful comments. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF2020R1A2B5B0100172111).