Let's play ... "Guess the y-axis!"

#1
A financial-services company needs to understand its risk in underwriting obligations that have considerable uncertainty, often gyrating between zero and 100% projected probability of occurrence (0.00-1.00).

The company assembles a team of industry experts to determine if they are well-hedged across the range of possible occurrences. The task force consults industry information sources that are recognized standards, utilized by all the competitive financial-services companies, and the team constructs a model that shows that business is projected to have positive returns on all possible outcome probabilities (see below).

Linear.HOUSE.return,9-12,2.jpg

The company sees that while the data is very noisy, except when oddly symmetrically-ordered around 0.50, the metric stays positive across the range of probabilities (the x-axis is zero on the y-axis), so they are reassured.

The company works in an industry where the outcomes are often wildly unpredictable anyway, and they need to be comfortable with uncertainty, and so they incorporate a safety layer of margin and commission into their contracts to buffer against risk.

All their competitors operate in the same way, and the growing profitability and dynamism of the industry segment is very reassuring to both the company and their shareholders.

As the industry’s growth attracts other entrants and investment into this financial-service company’s market, competition becomes fierce, and margins tighten.

Sometime later, an upstart consulting firm approaches the financial-services company and maintains that they have developed a new way to analyze the company’s data for projections of the critical metric. At first, the financial-services company is a bit skeptical, since the consulting firm has no established track record in this industry, which they’ve been operating in for many years and are considered leaders, but is impressed that the consultants seem to understand the cloudy uncertainty in that specific financial market and claim to know how best to understand and hedge risks to maximize returns.

The financial-services company contracts the proprietary technology and services of the consulting firm.

Using the same set of data from the original in-house study, the consulting firm shows that the critical metric is actually very ordered when projecting from its implied probability of occurrence (see below).

Nonlinear.HOUSE.return,9-12.jpg

The consultants astutely note that this dependent-variable data is a projection derived from an implied probability determined from an ever-changing abstraction, which is what the company’s financial-derivative product is. They stress that inevitably there will be significant scatter around this projection when compared against actual results, but this projection will act as a consistent metric of expectation.

The company is comforted by this assessment of inherent uncertainty in their business, as they do already see lots of scatter within their existing already-noisy projections of the metric (see first chart, above), and much much more when actual real-world results are evaluated in comparison.

With this new methodology for determining their critical metric, the financial-service company can better monitor and understand performance, both projected and realized, as well as its inherent uncertainty. This leads to a virtuous cycle of less risk volatility, a better-priced product for its customers, and therefore more business in a market that is crowded and undifferentiated.

Q: What is the MYSTERY METRIC?
 
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#2
at high risk of just shown some huge ignorance in general, but i see that some values have 2 probabilities.
for example -1 has 0.82 ok; but -0.5 have 0.7 and 0.9