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Rejecting Monotonicity in Risk Models

Let’s unpack and challenge the assumption that in most risk models, the relationship between ϵ (typically an error term or random variable) and y(ϵ) (the outcome variable) is monotonic—meaning that if ϵ increases (or decreases), y(ϵ) does too, and similarly, if the variance of ϵ increases, the variance of y(ϵ) follows suit. The examples given are "if prices increase, so does income" and "if the variance of ϵ increases, so does the variance of y(ϵ)." I’ll argue why this isn’t universally true and provide counterexamples to reject the assumption.

First, the claim hinges on a specific interpretation of "most risk models" and assumes a direct, positive, and monotonic linkage between ϵ and y(ϵ). In many risk models, ϵ represents stochastic noise or an exogenous shock, and y(ϵ) is the modeled outcome, often a function of ϵ and other variables. Monotonicity implies that the function y(ϵ) consistently increases (or decreases) as ϵ increases, without reversing direction. However, this doesn’t hold across all risk models, as the relationship depends heavily on the model’s structure and purpose.

Consider a simple linear risk model: y(ϵ) = a + bϵ, where b is a coefficient. If b > 0, then yes, as ϵ increases, y(ϵ) increases monotonically. But if b < 0, the relationship flips—ϵ goes up, y(ϵ) goes down. For example, in financial risk models, ϵ might represent a market shock, and y(ϵ) could be portfolio value. A positive shock (ϵ > 0) might reduce value if the portfolio is short on an asset, making the relationship non-monotonic in the assumed direction. Already, the sign of b shows that monotonicity isn’t guaranteed.

Now, let’s reject the specific example "if prices increase, so does income." This assumes ϵ is a price variable and y(ϵ) is income. In economic risk models, this isn’t always true. If prices of goods rise (ϵ increases), income doesn’t automatically follow. For a worker with fixed wages, real income might decrease due to inflation, even if nominal income stays constant. In a supply chain risk model, higher input prices (ϵ) could squeeze profit margins, reducing income for a firm. The relationship can be negative or non-monotonic, depending on elasticity, substitution effects, or market power—none of which are addressed in the blanket assumption.

Next, consider non-linear models, which are common in risk analysis. Take y(ϵ) = ϵ², a quadratic function. If ϵ increases from -1 to 1, y(ϵ) goes from 1 to 0 to 1—not monotonic, as it decreases then increases. In risk models for extreme events (e.g., natural disasters), outcomes like damage costs (y(ϵ)) might peak at a certain shock magnitude (ϵ) and then plateau or drop due to saturation effects (e.g., everything’s already destroyed). Monotonicity fails here entirely.

On the variance point: "if the variance of ϵ increases, so does the variance of y(ϵ)." This sounds plausible but isn’t universal. In a linear model y(ϵ) = a + bϵ, Var(y) = b²Var(ϵ), so yes, higher Var(ϵ) increases Var(y). But in non-linear or constrained models, it’s not so simple. Suppose y(ϵ) = sin(ϵ), a bounded function. As Var(ϵ) increases, ϵ takes larger swings, but Var(y) doesn’t scale proportionally—it’s capped because sin(ϵ) oscillates between -1 and 1. In risk models with thresholds (e.g., y(ϵ) = 1 if ϵ > 0, else 0), variance of y(ϵ) depends on the distribution of ϵ crossing the threshold, not just its spread. A higher Var(ϵ) might not shift Var(y) much if the threshold isn’t breached more often.

Finally, "most risk models" is too broad. In portfolio risk (e.g., VaR), operational risk, or ecological models, relationships between inputs and outcomes vary widely—linear, non-linear, monotonic, or not. Monotonicity might hold in stylized cases (e.g., basic asset pricing), but exceptions abound in practice, especially when feedback loops, constraints, or transformations are involved.

The assumption overgeneralizes. Monotonicity between ϵ and y(ϵ), or their variances, isn’t a default property of most risk models—it’s a special case, not the rule. Counterexamples like negative coefficients, non-linear functions, and bounded outcomes dismantle it. Thus, I reject it as a universal claim.

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