Rank-dependent predictable forward performance processes

Figure 1.  On left: Plots of six probability weighting functions: $ W(p):= p^{0.69} \Big(p^{0.69}+(1-p)^{0.69}\Big)^{-\frac{1}{0.69}} $ from [27] (the solid black curve); $ W(p)=0.65 p^{0.6} \Big(0.65 p^{0.6}+(1-p)^{0.6}\Big)^{-1} $ from [26] (the dashed blue curve); $ W(p)=\exp\left(-0.65\big(-\ln(p)\big)^{0.74}\right) $ from [21] (red dashed–dotted curve); $ W(p)=p $ (the cyan dotted line); a concave probability weighting function $ w(p)=p^{0.2} $ (the green dashdotdotted curve); and an extreme S-shaped probability weighting function $ W(p):= p^{0.3} \Big(p^{0.3}+(1-p)^{0.3}\Big)^{-\frac{1}{0.3}} $ (the purple long dashed curve). Parameter values for the first three are taken from various empirical studies, see page 9 of [9]. The last two probability weighting function are unrealistic as they highly overweight probabilities of extreme events. They are included to highlight the effect of probability weighting function on the RDPFPPs (See Figure 4). On right: The corresponding plots of the functions $ \Phi(\cdot) $ given by (7.3) for each probability weighting function and assuming the market Sharpe-ratio $ \lambda=0.4 $

Figure 2.  Each row corresponds to one probability weighting function in Figure 1. Left plots illustrate the $ \overline{\Phi}(\cdot) $, the concave envelope of each $ \Phi(\cdot) $ in Figure 1. Middle plots show the corresponding kernel $ k(1,\cdot) $ given by (7.4) and the corresponding resolvent kernel $ {k^*}(1,\cdot) := \sum_{i=1}^\infty k_i(1,\cdot) $, with the iterated kernels $ k_i(1,\cdot) $ given by (7.5). Right plots show the size of $ \sup_{0<\xi\le1} |k_i(1,\xi)| $ for different $ i $. Note that the vertical axis has a logarithmic scale. Since $ k_i(1,\cdot) $ vanish for sufficiently large $ i $, we can approximate $ {k^*}(1,\cdot) $ by summing a finite number of iterated kernels.

Figure 3.  Top left: Plots of three initial inverse marginal functions, namely, $ I_0(\cdot) $ given by (7.7) (red dashed-dotted curve), $ I_0(\cdot) $ given by (7.8) (the solid black curve), and $ I_0(\cdot) $ in (7.9) (the blue dashed curve). The first two belong to the special class of CMIM function with representation (6.4), while the third one is a CMIM function that does not belong to this class. Top right: The solutions $ I(y) $ of (7.2) for each of the three initial inverse marginals, the [27] probability weighting function (see Figure 1), and a market Sharpe-ratio $ \lambda=0.4 $. Each curve is computed numerically via (7.6), with $ q_0 $, $ \Phi'(\cdot) $, and $ {k^*}(1,\cdot) $ in the top row of Figure 2. For first two inverse marginals (namely, the red dashed-dotted curve and black solid curve), the solution $ I(y) $ can be found directly via the closed-form formula (6.6). These values are shown as data points (namely, the red circles and the black crosses), and match the numerical values calculated via the resolvent formula (7.6). There is no closed-form solution for the third inverse marginal (i.e. the blue dashed curve), as it does not belong the the special class of CMIM functions of the form (6.4). Note also that the third curve is less sensitive to the change of the probability weighting function. Bottom: The solutions $ I(y) $ of (7.2) for probability weighting functions in [26] and [21].

Figure 4.  Sensitivity of the solutions $ I(y) $ of (7.2) with respect to the probability weighting function $ W(\cdot) $ and the market Sharpe-ratio $ \lambda $. In both plots, the initial inverse marginal function $ I_0(\cdot) $ is given by (7.8) and is illustrated by the thin dotted green curve. On left: The solution $ I(y) $ of (7.2) corresponding to six different probability weighting functions from Figure 1 and assuming the market Sharpe-ratio $ \lambda=0.4 $. Note that $ I(y) $ corresponding to the identity weighting function (i.e. the expected utility criterion) is closest to the initial inverse marginal function $ I_0(\cdot) $, while the one corresponding to concave probability weighting function $ w(p)=p^{0.2} $ is the farthest away. The realistic probability weighting functions (i.e. those taken from [27, 26, 21]) yield solutions $ I(y) $ in between. Note also that that the solution corresponding to probability weighting functions from [27, 26] almost coincide. On right: Similar plot corresponding to a higher market Sharpe-ratio $ \lambda=1.5 $. Note also that the solutions I(y) corresponding to the more realistic probability weighting functions (those taken from [27, 26, 21] plus the identity) are more sensitive to the change of $\lambda $.