Increases in risk aversion and the distribution of portfolio payoffs

Oliver Hart proved the impossibility of deriving general comparative static properties in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs: A is less risk averse than B iff Aʼs payoff is always distributed as Bʼs payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributions over time (but not at every point in time).