The paper's tradable conclusion is an unbounded multiplier. Liu, Liu and Shen find that, in favorable states, inverse S-shaped probability weighting pushes an insurer's risky fraction toward (theta/sigma) * (a + sigma_theta) / ((1-p) sigma_theta). Merton gives theta/(sigma(1-p)). The first expression exceeds the second by (a + sigma_theta)/sigma_theta, a ratio that grows without limit as the distortion parameter a increases. Using the plotted values (a = 1.17, sigma_theta = 0.40), the ratio reaches 3.93.
Their asset follows a stylized Black-Scholes diffusion, leaving us without a market against which to trade the result. Our nearest implementation would spread exposure across liquid US equity, Treasury, credit and commodity ETFs. It would retain one mechanism: maximize distorted utility of terminal wealth while imposing a minimum probability of clearing a contractual threshold. The closed forms disappear after that substitution. They rely on deterministic parameters, continuous trading and exact replication; an ETF portfolio rebalances at discrete intervals. We also lack real participating liabilities, guarantees and capital inputs. Any contract payoff or aspiration threshold would therefore be hypothetical rather than calibrated, while the distortion parameters cannot be observed in prices. The exercise below neither tests nor replicates the paper's method.
The contract creates the nonconcavity
An insurer sells a participating (profit-sharing) contract, invests the premium in its general account, and earns through risky returns. In the illustration, mu = 0.07 and r = 0.01, giving a 6-point premium at sigma = 0.15. At maturity, the policyholder receives a guarantee and a share of surplus above it. The insurer retains what remains.
The policyholder payoff is Theta(X_T) = L_T^g + delta(eta X_T - L_T^g)^+ - (L_T^g - X_T)^+; the insurer receives Psi = X_T - Theta. That residual pays nothing below the guarantee, keeps the excess through the middle band, then becomes (1 - delta eta)X_T - (1-delta)L_T^g after the fund exceeds L_T^g/eta. The authors combine this piecewise-affine payoff with CPT utility, (x-B)^p above the benchmark and -k(B-x)^q below it. Once p = q = gamma, the effective objective becomes nonconcave and develops kinks at L_T^g and L_T^g/eta. The guarantee and surplus split produce the nonconcavity.
A rank-dependent distortion w comes next, taken from the He and Zhou (2016) normal-shift family. The aspiration constraint P(X_T >= L) >= alpha serves as a solvency target. Because the distortion breaks the tower property, the authors use quantiles instead of dynamic programming. They transform variables through y = 1 - w^{-1}(1-x), form concave envelopes for the utility and budget function phi, then maximize the Lagrangian point by point. This yields closed-form terminal wealth, a replicating wealth process and the precommitted portfolio pi* = -(sigma sigma')^{-1}(mu - r 1) xi_t dX*/d xi_t. The construction covers complete markets and incomplete ones through the minimal market price of risk.
The paper contains no data.
Its illustrations have one risky asset and use mu = 0.07, r = 0.01, sigma = 0.15, T = 1, guarantee L_T^g = 1, p = 0.4, eta = 0.6, x_0 = 1, and distortion parameters a = 1.17, b = 0.86, z-bar = 0.33 (all three become zero for the undistorted benchmark). Four results emerge. Whenever a > 0, the distorted allocation approaches a line strictly above Merton. The risky percentage traces a valley and jumps sharply at the envelope's non-differentiable point. Undistorted curves for t = 0, 0.5, 0.7, 0.9 with T = 1 coincide; distorted curves separate. Initial capital alone also reverses the policy, from aggressive at x_0 = 0.5 to almost wholly invested in the risk-free asset at x_0 = 3.
The free dial is 1.17
The authors say directly that they never estimate it. They choose the distortion family because normal shifts preserve a Gaussian pricing kernel and leave the duality integrals explicit. Their concluding section assigns calibration of distortion parameters from institutional portfolio data to future work. Section 3 states the requirement plainly: the closed form "requires a parametric family that preserves analytical tractability under the quantile transformation." The headline multiplier therefore depends on a free parameter, made free by the same tractability choice that delivers the formula. Their first proposed extension, an optimization framework addressing ambiguity in behavioral parameters and model misspecification, carries the main empirical burden.
The arithmetic is stark. Given mu - r = 0.06 and sigma = 0.15, theta = 0.40; with T = 1, sigma_theta = 0.40. The Merton allocation is 0.40/(0.15 x 0.6) = 4.44 times wealth. The distorted limit comes to 2.667 x 1.57/0.24 = 17.4 times wealth. Under the normal-shift family, choosing a distortion parameter of 1.17 at sigma_theta = 0.40 multiplies the Merton weight by 3.93. At this volatility and equity premium, a 444% gross position becomes a 1,740% position without reference to any observed portfolio holding.
Divergence in the ratio occurs in one regime. As xi_t -> infinity with lambda phi'(c) > gamma^+, wealth approaches zero and pi*/X* explodes. At the opposite end, the ratio converges to 17.4 times wealth as the dollar position diverges. The model includes no transaction costs, short-sale constraints or leverage limits. Default is measured only at maturity, and the policyholder absorbs any shortfall below L_T^g.
Does the solvency floor bind?
The aspiration constraint supplies the insurance hook and enters the mathematics directly. Its terms appear through Q(1 - w(alpha)) >= L, with a correction that lifts terminal wealth to L across the binding quantile region. Yet we did not find a numerical value for alpha or L in any illustration. The figures print mu, r, sigma, T, a, b, z-bar, L_T^g, p, eta and x_0. They also omit the surplus-sharing rate delta, which determines the bad-state limit (1-delta)L_T^g/(1-delta eta).
The pictured regime switches attributed to "regulatory thresholds" therefore come from the envelope kink and the critical Lagrange multiplier. In the worked case, the kink lies at a_1 = c_U = L_T^g/eta, or 1.667 with eta = 0.6 and L_T^g = 1, and is interpreted as a reserve threshold. The formulas contain the probability floor; the numerical work leaves it unspecified. A reader seeking the risky-weight cost of a 99% one-year solvency target will not find that answer here.
Two arithmetic disagreements remain
Proposition 2 places (1-p) sigma_theta in the denominator of the favorable-state limit. The proof instead produces p sigma_theta in equations (57) and (58). With p = 0.4, those expressions give 17.4 and 26.2 times wealth. The paper therefore leaves its headline level unsettled.
Second, Section 2 links initial capital to the liability through x_0 = L_0/eta, while L_T^g = L_0 e^{gT} and g lies in (0, r). Setting L_T^g = 1, T = 1, r = 0.01 and eta = 0.6 places x_0 between about 1.65 and 1.67. The illustrations instead set x_0 = 1, while Figure 4 uses 0.5 and 3. Holding the guarantee fixed while changing initial capital alters the liability-to-asset ratio. The initial-capital regime switch consequently compares contracts as well as balance sheets.
Case (d) remains a gap acknowledged by the authors. If x_0 + H(0) lies within the union of intervals I_k, Theorem 1(d) says "our method fails to provide an optimal solution". Proposition 1 describes the same gap as a Lagrange construction that cannot identify an attainable optimizer. The worked example has one kink (a_0 = 0, a_1 = c_U = L_T^g/eta, a_2 = infinity). Because the final utility segment is nonlinear, the paper's bound |Lambda_n| <= n applies. With n = 1, at most one interval of initial capital can have this property. We did not find its location for the plotted parameters.
Precommitment is what the paper delivers
The abstract openly concedes time inconsistency. The conclusion converts it into a prescription, saying that inverse S-shaped distortion "induces excessive risk-taking and time inconsistency, necessitating pre-commitment strategies to align ex ante and interim optimality." The mathematics supports that reading. Under unchanged market dynamics, the policy that must be fixed in advance shifts from aggressive at x_0 = 0.5 to almost entirely risk-free at x_0 = 3. It also jumps at the envelope kink L_T^g/eta, equal to 1.667 under their parameters. Later re-optimization produces another answer, as Figure 3 demonstrates (t = 0 with T = 0.1, 0.3, 0.5, 1, remaining horizon held fixed).
As a stress-test generator, the framework has value. It prices the behavior of a distortion-prone insurance manager around a reserve threshold. The introduction also identifies convex compensation and participation components as another source of the same nonconcavity. Its directional result matters: after the guarantee has been cleared, distortion weakens lock-in, reversing the classic S-shaped result. A risk function fitted to lock-in behavior could therefore understate exposure in those states.
What survives our market substitution
We would sweep a and b because we cannot estimate them. That sweep contains the result: with a = 0, the limiting fraction equals Merton; with a = 1.17, it reaches 3.93 times Merton.
One number would change my view: an estimate of a with a standard error, derived from portfolio data, for which the distorted line remains above Merton by a factor an insurer could finance. Until such an estimate exists, 3.93 times Merton remains a consequence of choosing a = 1.17.