CURRENT TRENDS IN PRIVATE EQUITY

In asset allocation for robo-advising, it is desirable to elicit investors’ risk profile via several simple online questions and to provide advice consistent with conventional investment wisdom (e.g. rich people should invest more absolute amount of money in risky assets, and for long-term investment people should not short sell major stock indices whose returns are higher than the risk-free rate). We propose a dynamic portfolio choice model with the mean-variance criterion for portfolio logreturns that meets the two challenges. The model yields analytical and time-consistent optimal portfolio policies. INTRODUCTION Portfolio optimization has a long history in finance, dating back to celebrated works of single period mean-variance analysis by Markowitz (1952) and continuous-time expected utility maximization by Merton (1971). However, both the expected utility maximization model and the mean-variance model suffer from some disadvantages. For example, the way of estimating investors’ risk profile in a utility maximization model is not straightforward, despite that the optimal trading policy of an advanced utility maximization model conforms with conventional investment wisdom. The dynamic mean variance policies in the existing literature, on the contrary, can easily elicit investors’ risk profile but either are time inconsistent or have some conflicts with some conventional investment criteria. In Dai, Jin, Kou, & Xu (2018), we study a dynamic portfolio choice model with mean-variance criterion for portfolio log-returns (hereafter log-MV criterion, for short), instead of with the standard mean-variance criteria for terminal wealth in extant literature. Our model combines the advantages of both Markowitz’s and Merton’s models. More precisely, our contribution is threefold: (1) We complement the portfolio theory by proposing a dynamic log-MV criterion. As with Markowitz’s single-period model, our model offers a simple and intuitive way to identify the mean-variance preference parameter that measures the tradeoff between mean and variance of portfolio logreturns. In fact, for a complete market with constant market parameters, we obtain an explicit formula to characterize the relation between the expected target annual return and the mean-variance preference parameter. As a result, we can elicit the investors’ risk profile by asking investors to input their target annual returns. (2) By extending Basak and Chabakauri (2010) and Bjӧrk, Murgoci, and Zhou (2014), our model leads to an optimal time-consistent mean-variance policy. In a complete market, the policy is the same as the well-known Merton’s myopic strategy with constant relative risk aversion (CRRA) preferences. Moreover, our model is tractable even under incomplete market settings, and the resulting trading policy consists of familiar myopic and hedging components, as for CRRA preferences. Hence, our model possesses the merits of Merton’s model. (3) As two applications, we present analytical solutions for the dynamic log-MV model in two special cases, a general Heston’s stochastic volatility setting and a Gaussian mean return setting. Using the analytical solutions, we prove that the optimal trading VOL 4 l NOV 2018 PRIVATE EQUITY REVIEW 06 policies in both settings conform with conventional investment wisdom, such that (i) rich people should invest more dollar amount of money in risky assets; (ii) the longer the investment horizon, the more the proportional amount of money invested in risky assets; and (iii) for long-term investment, people should not short sell major stock indices whose returns are higher than the risk-free rate. Our model can be applied to the asset allocation part of robo-advising. Due to the very natural of general public, there are at least three distinctive requirements for the asset allocation part of robo-advising, compared to that of traditional financial advising. First, the computer asset allocation system needs to elicit investors’ risk profile automatically by asking some simple questions understandable to the general public. Second, the advice provided by the computer system must be time consistent in order to avoid confusing the general public. Third, to convince the general public, the advice provided by the computer system should be in line with conventional investment criteria. Though simple as they seem, the three requirements pose challenges to existing portfolio theories. The first requirement narrows the search for a desired model to the mean-variance framework. The second requirement rules out the pre-committed mean-variance strategies that are time inconsistent. The existing time-consistent meanvariance policies violate at least one of the investment criteria listed above. To our best knowledge, our paper is the first one focusing on dynamic portfolio choice models that can accommodate the three requirements simultaneously. ROBO-ADVISING In this section, we introduce the background of roboadvising and explain how to incorporate our model to robo-advising. Background of Robo-Advising Thanks to the rapid development of computer technology and mobile IT applications, there is a growing trend for major companies to start to build robo-advising systems and attempt to reproduce or beat the results from the traditional financial advising, as part of efforts to serve more customers effectively and to reduce costs.1 While traditional, person to person, financial advisors aim at 1 Robo-advising first appeared in 2008 (according to Investor's Business Daily, June 27, 2016), and has grown rapidly since then. As estimated by the US News & World Report (October 5, 2017), as of October 2017 robo-advisors had US$224 billion in assets under management. For example, Vanguard Group and Charles Schwab Corporation had about US$83 billion and US$19 billion in assets under management related to robo-advising in September 2017, respectively (Forbes Magazine, September 12, 2017). 2 We shall not study another related automatic investor service, namely trading stations, which help ordinary investors to design algorithms to trade assets automatically either in low or high frequencies. providing tailored and comprehensive (often including tax and estate planning and advising on insurance needs) financial advice to (perhaps mostly rich) investors, roboadvisors attempt to give automatic financial advice in cost efficient ways, targeting thousands of, if not millions of, ordinary investors.2 There are some commonalities between traditional financial advisors and robo-advisors. For example, both need some clients’ information, such as risk profile, investment horizon, investment amount, educational and retirement needs, estate planning, etc., before providing investment advice. Although our dynamic mean-variance analysis can be applied to the asset allocation part for both traditional financial advising and robo-advising, our analysis is more suitable for the latter pattern, due to three distinct requirements related to robo-advising. First, a roboadvisor must effectively identify clients’ risk profile based on simple inputs of clients. A robo-advisor does not have the luxury like a traditional financial advisor, who can talk to clients personally for an extended period (e.g. one hour) and can ask many questions to engage the clients to extract their risk profile. Furthermore, many times clients of traditional advisors may even have substantial financial knowledge, e.g., the meaning of utility functions. A typical robo-advisor can only ask clients to input answers (via the Internet) to some questions understandable to the general public. In this paper, for robo-advising we propose to ask clients several simple questions such as expected portfolio returns and investment time horizon. Second, to avoid confusing the general public, the asset allocation advice provided by robo-advisors should be time consistent. In fact, a robo-advising system should contain outputs that may be helpful to educate investors. For example, the outputs may give guidelines of (1) the portfolio strategy across time for a particular investor; (2) the cross-sectional portfolio strategies for investors with different investment horizons at time 0; (3) the changes in the portfolio strategy for a particular investor, should the estimated stock returns vary. For Items (1) and (2) it is very useful to consider time-consistent portfolio strategies, as for these strategies, ceteris paribus, Items (1) and (2) will be identical. However, for pre-committed mean-variance strategies, the two may not be identical at any time, thus creating potential confusions for the general public. The time-consistent strategies presented in this paper can avoid such confusions. PRIVATE EQUITY REVIEW NOV 2018 l VOL 4 07 Third, to convince the general public, a robo-advising system needs to provide recommendations that tally with the conventional investment wisdom, as clients can easily change their inputs on the Internet in order to test the effectiveness of the system. For example, suppose that the system suggests 60% of the total money in stocks and 40% in bonds when a client inputs 5 years investment horizon. However, if the client immediately changes the online input of the investment horizon to 15 years and the system adjusts the recommendation to 50% in stocks and 50% in bonds, then the client may lose trust in the system, as the conventional investment wisdom suggests that, to the contrary, people having longer investment horizon should invest more proportion of their wealth in stocks. Note that under incomplete market settings, as in the stochastic volatility model and in the Gaussian mean return model, the relation between the mean-variance preference parameter and the expected target annual return is not straightforward. For example, in the timevarying Gaussian mean return model, given a meanvariance preference parameter and reasonable market parameter values, the target annual return associat