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Functional Clustering of Discount Functions for Behavioral Investor Profiling

This study employs Functional Data Analysis to reveal nuanced heterogeneity in temporal discounting behaviors across different investor temperaments, thereby refining behavioral investor profiling to better guide personalized financial strategies.

Original authors: Annamaria Porreca, Viviana Ventre, Roberta Martino, Salvador Cruz Rambaud, Fabrizio Maturo

Published 2026-05-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Annamaria Porreca, Viviana Ventre, Roberta Martino, Salvador Cruz Rambaud, Fabrizio Maturo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how people make decisions about money over time. Do they prefer $100 today or $110 next month? Do they get scared of the future, or do they plan for it?

For a long time, financial experts used a "one-size-fits-all" map to explain this. They assumed everyone thinks a little bit like a robot: if you wait, you get a little more, and that's it. But real life is messier. People get emotional, impulsive, or overly cautious.

This paper is like a team of detectives who decided to throw away the old, flat map and use a 3D scanner instead. Here is how they did it, explained simply:

1. The Old Map vs. The New Scanner

The Old Way (Traditional Models):
Think of traditional finance models as a black-and-white sketch. They try to draw a person's decision-making style with a few straight lines. They say, "You are a 'Guardian' type, so you are cautious," or "You are an 'Artisan' type, so you are impulsive." It's a simple label, like putting a sticker on a jar.

The New Way (Functional Data Analysis):
The authors used a tool called Functional Data Analysis (FDA). Imagine instead of a sketch, you have a live, moving video of how a person's mind works over time.

Instead of just asking, "Do you want money now or later?" they watched how the person's preference changed second by second, day by day, and month by month.
They turned the person's answers into a smooth, flowing curve (like a rollercoaster track) that shows exactly how much they value the future compared to the present.

2. The Four "Personality" Groups

The study started with a popular theory that divides investors into four personality types, based on how they act in the world:

Guardians (Preservers): The cautious planners who hate losing money.
Artisans (Accumulators): The thrill-seekers who want to make money fast and take risks.
Idealists (Followers): The dreamers who often follow the crowd.
Rationals (Independents): The logical thinkers who like to solve puzzles.

The researchers asked 170 people to fill out a survey about their money choices and their personality. Then, they used their "3D scanner" (FDA) to draw the unique "time-valuation curve" for every single person.

3. The Big Surprise: The "Hidden Sub-Groups"

Here is where the plot twist happens.

When the researchers looked at the old black-and-white sketches, the four groups looked distinct. But when they looked at the 3D videos (the curves), they saw something unexpected: The groups were messy.

The "Guardian" Group: The researchers expected all Guardians to look the same (cautious). But the curves showed that some Guardians were very steady, while others were actually quite jittery and changed their minds often. It was like a choir where everyone was supposed to sing the same note, but some were singing high, some low, and some were off-key.
The "Artisan" Group: They expected all Artisans to be impulsive. But the data showed some Artisans had curves that were surprisingly patient, while others were extremely frantic.

The Analogy: Imagine you have a bag of "Red Marbles" (Guardians). You expect them all to be the exact same shade of red. But when you shine a special light on them, you realize some are bright cherry red, some are dark burgundy, and some are pinkish-red. The old label "Red" wasn't specific enough.

4. Finding the Real Patterns

The researchers then used a smart computer program (Functional Clustering) to sort these curves into new groups, ignoring the old personality labels for a moment. They asked the computer: "If we just look at how people value time, who actually groups together?"

The computer found four new, distinct patterns that didn't match the original personality labels perfectly:

The "Grab It Now" Group: People who value today so much they almost ignore tomorrow.
The "Balanced" Group: People who weigh today and tomorrow fairly evenly.
The "Patient" Group: People who are very calm and value the future highly.
The "Switchers" Group: A unique group who start out wanting money now, but if they don't get it, they suddenly become very patient and wait for the future.

5. What This Means (According to the Paper)

The paper concludes that the old way of labeling investors (just saying "You are a Guardian") is too simple. It's like trying to describe a whole forest by just saying "It has trees."

By using this new "video" method, the authors found that:

People within the same personality type are actually very different from each other.
There are hidden sub-groups of investors that traditional models miss.
To truly understand how someone invests, you need to look at the shape and movement of their decision-making over time, not just a single label.

In short: The paper argues that we need to stop using static stickers to describe investors and start using dynamic movies to see the real, complex, and varied ways people handle money and time.

Technical Summary: Functional Clustering of Discount Functions for Behavioural Investor Profiling

Problem Statement
Classical finance models operate on the premise of rational investors who utilize all available information, a framework that often fails to capture anomalies in intertemporal choices and decision-making under uncertainty. Traditional parametric models struggle to account for individual differences in preferences, particularly how distinct personality temperaments influence the evaluation of trade-offs between present and future outcomes. While Pompian's model of four Behavioral Investor Types (BITs)—Accumulators, Preservers, Followers, and Independents—links Keirsey's temperament theory to financial decision-making, it relies on static classifications that may not fully capture the nuanced, dynamic nature of temporal discounting. The authors identify a gap in understanding how these temperaments specifically manifest in intertemporal decisions and whether significant heterogeneity exists within these broad categories.

Methodology
The study employs Functional Data Analysis (FDA) combined with functional clustering techniques to investigate temporal discounting behaviors. The research is grounded in the following methodological framework:

Data Collection: An online questionnaire was administered to 200 volunteers (170 valid responses) in February 2023. The instrument included intertemporal choice tasks designed to construct individual discount functions and a Keirsey Temperament Sorter II to classify participants into Artisan, Guardian, Idealist, and Rational temperaments. The intertemporal tasks utilized a 20-second timer to induce a sense of haste, aiming to reveal behavioral anomalies.
Construction of Discount Functions: Individual responses were interpolated to create continuous discount functions, $f(t)$ , representing the subjective perception of time and uncertainty. The study utilized the monotone smoothing method (Ramsay, 2005) to ensure the reconstructed functions satisfied the necessary constraints of a discount function: $f(0)=1$ , $f(t)>0$ , and strict monotonic decrease.
Functional Metrics: The analysis utilized $L_2$ -distances and semi-metrics based on the $r$ -order derivatives of the curves to measure proximity between investor profiles. This allowed for the quantification of within-group variability (heterogeneity within a temperament) and between-group variability (differences between temperaments).
Clustering Approach:
1. Conditional Clustering: Functional $k$ -means clustering was applied within each of the four pre-defined temperament groups to identify potential subgroups (sub-temperaments). The optimal number of subclusters was determined using an extension of the Elbow method applied to the Within-Cluster Sum of Squares (WCSS).
2. Unconditional Clustering: A separate functional $k$ -means analysis was performed on the entire dataset without prior knowledge of temperament labels to uncover latent investor profiles based solely on intertemporal choice patterns.

Key Results

Heterogeneity within Temperaments: Preliminary descriptive analysis and functional variance calculations revealed significant internal heterogeneity within each of the four Keirsey temperaments. For instance, Artisans and Idealists showed high variability in their discounting patterns, suggesting that a single "mean" profile for these groups is insufficient.
Subgroup Identification: The conditional functional clustering identified distinct subgroups within the original BITs. The Elbow method indicated that Artisans and Idealists required up to four subclusters, while Guardians and Rationals required three and four, respectively, to adequately capture internal variability.
New Investor Profiles: The unconditional functional $k$ $k$ -means analysis, ignoring pre-defined temperaments, converged on four distinct investor profiles based purely on the shape of their discount functions:
1. Green Cluster: Steeply declining curves, indicating strong impulsivity and a heavy undervaluation of future outcomes.
2. Cyan Cluster: Gradual decrease, representing a balanced approach with slight short-term preference but openness to long-term options.
3. Purple Cluster: Flat discount functions, indicating high patience and a stable valuation of future rewards.
4. Red Cluster: A dynamic pattern where rates drop sharply initially but stabilize, suggesting a shift from short-term to long-term focus depending on immediate returns.
Derivative Analysis: The analysis of first and second derivatives highlighted that the rate of change (velocity) and acceleration of discounting behaviors vary significantly across groups and time, further confirming that decision-making processes are dynamic rather than static.

Significance and Claims
The paper claims that the integration of FDA and functional clustering offers a methodological advancement over traditional parametric approaches in behavioral finance. The primary significance lies in demonstrating that classical Behavioral Investor Types (BITs) are too rigid to encompass the complexity of real-world investor behavior.

Refinement of Classification: The study challenges the adequacy of static categorizations (e.g., Pompian's BITs) by revealing that significant subpatterns exist within these broad categories. The findings suggest that investor profiles are far more diverse than previously thought.
Dynamic Understanding: By analyzing the functional derivatives, the authors argue that temporal preferences are not static but evolve, influenced by emotional drives and situational contexts.
Practical Implication: The refined classification provides a basis for financial advisors to move beyond rigid templates. Instead, they can tailor strategies to the specific temporal discounting behaviors and risk preferences of individual clients, acknowledging the diversity within traditional temperament groups.

The authors conclude that while their findings validate the importance of identifying subpatterns, future research is necessary to apply these methods to larger, more diverse samples to validate the generalizability of the identified subgroups.

1. The Old Map vs. The New Scanner

2. The Four "Personality" Groups

3. The Big Surprise: The "Hidden Sub-Groups"

4. Finding the Real Patterns

5. What This Means (According to the Paper)

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