by Dr Alex Tse, former Research Associate at Cambridge Endowment for Research in Finance and winner of the CERF Alumni Association (CERFAS) Best Paper Award
Portfolio managers in real life can be selfish, where they care about their own short-term periodic incentives rather than the long-term portfolio growth. How will their risk-taking behaviours change in such a case? Are there ways to combat undesirable risk-taking behaviours of such “myopic” portfolio managers? In the paper “Portfolio Selection, Periodic Evaluations and Risk Taking” that I co-authored with Harry Zheng, we explore a theoretical model to answer the above questions.
Classical optimal investment models typically consider utility derived from the terminal portfolio value at some fixed horizon as the optimisation criterion. But is it consistent with real life practices? For example, fund managers usually receive rewards at the end of a financial year based on their performance, which could include bonuses in monetary terms, gain or loss of professional reputation and other indirect payoffs such as change in client flow. Such rewards recur periodically over time, where each of them is individually linked to the portfolio performance over a short time horizon only. In a more general corporate setting beyond delegated portfolio management, staff appraisal tends to happen every year where employees will receive tangible or intangible benefits based on how well they have been performing in a given year. There are many practical contexts where an agent needs to make risky investment decision with objective driven by the periodic performance of the underlying portfolio instead of its terminal value at some arbitrary time point.
In this paper, we develop a continuous-time portfolio selection model where the agent’s utilities are derived from the portfolio periodic performance evaluated on a deterministic sequence of dates over an infinite horizon. The portfolio performance in a given period is measured by its ending value in excess of its starting value scaled by an exogenous performance benchmark parameter. Finally, the periodic performance measures are converted into agent’s utilities via an S-shaped function to capture limited liability protection, option-based compensation, or behavioural preferences.
Before introducing our theoretical results, it is useful to compare our framework against a standard model with a single terminal evaluation date only (see for example Berkelaar et al. (2004) and Carpenter (2000)). Due to the S-shaped utility function, the agent is insensitive to large losses (eg because they have limited liability protection) in the bad states of the world. Hence if they are falling behind, they opt to gamble by taking excessively large risky investment with the hope of getting out from the portfolio losses. In such a standard model, there is no incentive tied to the portfolio value beyond the single terminal evaluation date, so the agent does not care even if the portfolio becomes insolvent at the maturity date.
However, the incentive structure changes significantly once the performance criterion becomes periodic. The agent now has to balance the reward obtained from the current period (short-term component) as well as all the subsequent rewards in the future periods (long-term component). The agent now cares about portfolio insolvency because it leaves the agent no capital to be invested if the portfolio goes bust in the current period and then all the future rewards would be zero. The optimal risk-taking decision is determined via the trade-off between the short-term and long-term benefits, which is heavily influenced by the model parameters such as the exogeneous performance benchmark. In general, the future rewards component encourages value preservation. The existing literature on the “one-off models” therefore has the intuition that under a repeated, periodic setup, the risk-taking behaviours should be moderated. Nonetheless, such conjecture has not been formalised and examined in a proper theoretical model to date. Our results show that such intuition is incomplete at best.
In our model, it turns out that there are three main regimes of risk-taking behaviours which are primarily dependent on the performance benchmark parameter. The first regime is “risk-moderation”, where risky investment is bounded across all possible states of the world. The second regime is “risk-seeking”, where investment in the risky asset tends to infinity in the bad states of the world and there is a strictly positive chance of portfolio insolvency at the end of each evaluation period (this regime is the one consistent with the existing literature which considers a single evaluation date only). The final regime is “underinvestment”, where not only the risky investment becomes unbounded in the bad states of the world so insolvency is possible (as per the “risk-seeking” regime), but the investment level also tends to zero in the good states of the world and thus the agent is disinvesting when the portfolio performs well. The portfolio gross return then has an upper bound under this regime.
So why do the multiple regimes exists? Whether the repeated game nature of investment encourages value preservation (and in turn, moderates risk-taking behaviours) depends on the value of the future benefits relative to the one from the current period. When the performance benchmark is low or the market is favourable, the value of the investment game is high, and the agent has incentive to ensure solvency so they can perpetually extract periodic benefits by managing a healthy portfolio. This is achieved by adopting a bounded investment strategy across all possible states of the world and results in the “risk-moderation” regime. But for a moderately high value of performance benchmark, the long-term value of the investment game is not high enough to outweigh the short-term benefit from the current period. The agent will then put more decision weight on the short-term reward and consequently behave as in the standard “one-off model” where unboundedly high level of risk is taken in the bad states of the world – this is the “risk-seeking” regime.
The “underinvestment” regime is perhaps the more interesting one. When the performance target becomes excessively high, the agent understands that it will be difficult to outperform their yesterday’s self. They will hence intentionally underinvest to prevent the portfolio from growing too much. Otherwise, a realisation of good outcome in the current period will lead to a high starting portfolio value in the next period, which forms the basis of a much higher (i.e. demanding) benchmark for the performance evaluation in the next period. The intentional manipulation of performance benchmark via underinvesting can be viewed as a symptom of “underperformance-aversion”.
Clearly, the regimes “risk-taking” and “underinvestment” are not ideal from a welfare perspective since the portfolio managers’ myopia will lead to capped portfolio growth but more devastatingly the inevitable ruin of the portfolio in the long run due to unbounded risk-taking during downturns. Incentives should therefore be properly set up or “nudged” to encourage managers to act in accordance with the “risk-moderation” regime. Two important factors include reasonable performance target and avoidance of excessive punishment on underperforming agents.
In summary, periodic evaluation can be a useful mechanism to contain excessive risk-taking even if the agent exhibits S-shaped preference due to convex payoff scheme or behavioural biases. This suggests, for example, that cliquet-style contracts can be superior to standard instruments like long-dated call options within employee compensation package. The risk-taking moderation effect is not universal, however, as revealed by the different possible regimes within our theoretical model. It also raises a deeper question over how the short-term periodic interest of the agents can be better aligned with the long-term investment goal of the stakeholders. In the future, I am interested in understanding such issues better via a principal-agent extension of the current model.
Berkelaar, A.B., Kouwenberg, R. and Post, T. (2004) “Optimal portfolio choice under loss aversion.” Review of Economics and Statistics, 86(4): 973-987
Carpenter, J.N. (2000) “Does option compensation increase managerial risk appetite?” Journal of Finance, 55(5): 2311-2331