Shortfall aversion

Abstract

Shortfall aversion reflects the higher utility loss of spending cuts from a reference than the utility gain from similar spending increases. Inspired by Prospect Theory's loss aversion and the peak‐end rule, this paper posits a model of utility from spending scaled by past peak spending. In contrast to traditional models, which call for spending rates proportional to wealth, the optimal policy in this model implies a constant spending rate equal to the historical peak when wealth is relatively large. The spending rate increases when wealth reaches a model‐determined multiple of peak spending. In 1926–2015, shortfall‐averse spending is smooth and typically increasing.

Optimal consumption with reference to past spending maximum

“…The nonnegativity constraint needs to be taken care of in solving the HJB equation. As our utility differs from [ 2 ] and [ 12 ], the feedback optimal consumption is naturally distinct from their results. But we can compare the optimal consumption behaviour when the wealth level becomes extremely low and extremely high.…”

“…Given the feedback optimal controls in ( 3.30 ) and ( 3.31 ), we briefly present here some comparison results with Arun [ 2 ] and Guasoni et al [ 12 ] on how the optimal consumption is affected by the past spending maximum. We stress that [ 2 ] considers optimal consumption under a standard time separable power utility, i.e., , while the drawdown constraint , , is only imposed in the set of admissible controls.…”

“…A similar optimal dividend control problem with a drawdown constraint is also formulated and studied in Angoshtari et al [ 1 ]. On the other hand, the authors of [ 12 ] study the optimal consumption under a Cobb–Douglas utility, i.e., , where the utility is defined on the ratio of the consumption rate and the consumption running maximum. By virtue of the power utility on the consumption rate, the optimal consumption in [ 2 ] and [ 12 ] automatically satisfies .…”

“…On the other hand, the authors of [ 12 ] study the optimal consumption under a Cobb–Douglas utility, i.e., , where the utility is defined on the ratio of the consumption rate and the consumption running maximum. By virtue of the power utility on the consumption rate, the optimal consumption in [ 2 ] and [ 12 ] automatically satisfies . On the other hand, we are interested in exponential utility that measures the difference between the consumption rate and the consumption running maximum.…”

“…In addition, in Arun [ 2 ] and Guasoni et al [ 12 ] (see also Angoshtari et al [ 1 ]), one can change variables and focus on the new state process to reduce the dimension. As a consequence, all thresholds for the wealth variable separating different regions for the piecewise optimal control in these works are simply linear functions of .…”