Pessimistic Portfolio Choice with One Safe and One Risky Asset and Right Monotone Probability Difference Order

Table 1

Example of distorted excess return with distortion function 𝑔(𝑥)=𝑥^𝛽.

Asset	Probability				𝑟	𝐸[𝑋]	𝐸𝑔[𝑋]
	0.25	0.25	0.25	0.25			𝑥^1.01	𝑥^0.95	𝑥^0.90	𝑥^0.80	𝑥^0.10	𝑥^0.01
𝑋1	0.01	0.03	0.04	0.08	0.04	0.04	0.040	0.039	0.038	0.036	0.014	0.011
𝑋2	0.01	0.04	0.05	0.10	0.04	0.05	0.050	0.049	0.047	0.045	0.017	0.011
𝑋3	0.03	0.04	0.06	0.15	0.04	0.07	0.070	0.069	0.067	0.092	0.035	0.031
𝑋4	0.05	0.07	0.08	0.20	0.04	0.10	0.100	0.098	0.096	0.092	0.057	0.051

Let X denote a lottery (𝑥₁, 𝑝₁; 𝑥₂, 𝑝₂; 𝑥₃, 𝑝₃; 𝑥₄, 𝑝₄), where the 𝐸_𝑔[𝑋] can be calculated via the formula: 𝐸_𝑔[𝑋]=∑⁴_𝑖=1𝑥_𝑖𝑤(𝑝_𝑖), where 𝑤(𝑝_𝑖)=[𝑔(𝑃_𝑖)−𝑔(𝑃_𝑖−1)], 𝑃_𝑖 being the cumulative probability and 𝑃₀=0.