The fraud triangle, fraud diamond, and fraud pentagon are three successive models that forensic accountants use to explain why occupational fraud happens. Each one keeps everything from the model before it and adds a single new element. The triangle (Donald Cressey, 1953) names pressure, opportunity, and rationalization. The diamond (Wolfe and Hermanson, 2004) adds capability. The pentagon (Jonathan Marks, Crowe Horwath) adds arrogance. Read in order, they move from explaining ordinary embezzlement to explaining large financial statement fraud at the top of an organization.
The Fraud Triangle
Criminologist Donald Cressey built the original framework from interviews with 133 convicted embezzlers in federal prisons in the late 1940s, publishing the results in 1953 in Other People’s Money: A Study in the Social Psychology of Embezzlement. His conclusion was that trusted employees become trust violators only when three conditions line up at the same time.1University of Portsmouth Research Portal. Deconstructing the Origins of Cressey’s Fraud Triangle Remove any one, and the theory holds that fraud becomes far less likely.2AGA. The Fraud Triangle
Cressey himself never used the phrase “fraud triangle” in his published work. The visual came later. His name is now inseparable from it, and the framework is referenced in over 8,500 academic papers and professional training programs worldwide.1University of Portsmouth Research Portal. Deconstructing the Origins of Cressey’s Fraud Triangle
Pressure
Cressey called this a “non-shareable financial problem.” The perpetrator faces a financial crisis they believe they cannot tell anyone about: medical debt, gambling losses, substance abuse, or the strain of maintaining a lifestyle their legitimate income can’t support. What makes the pressure non-shareable is the perceived cost of disclosure, usually status, a relationship, or a job. The key word is perceived. Outsiders might see obvious solutions, but the perpetrator has convinced themselves no legitimate option exists.
Opportunity
The second element is the perpetrator’s belief that they can commit fraud without being caught. This perception almost always traces back to weaknesses in internal controls: missing management reviews, no independent reconciliation of records, or a failure to separate incompatible duties. When one person controls both the recording and the custody of assets, they can misappropriate funds and cover their tracks in the same workflow.3University of Pennsylvania Office of Audit, Compliance and Privacy. Operational Internal Controls – Section: Segregation of Duties
The opportunity doesn’t need to be real. It needs to be real to the person considering fraud. A manager who knows they’re the only one reviewing a particular expense account perceives an opening even if an audit would eventually catch the discrepancy. That perceived window is enough.
Rationalization
The third element is the internal story the perpetrator tells themselves to justify crossing the line. This psychological bridge allows someone who considers themselves honest to commit a dishonest act. Common versions: they’re only “borrowing” and plan to pay it back, they deserve the money because they’re underpaid, or the organization has treated them badly enough to have it coming.
Rationalization is the hardest element for an organization to detect because it happens entirely inside someone’s head. It’s also the element most likely to collapse on its own, as guilt accumulates or the cognitive dissonance becomes unbearable.
The Fraud Diamond
The triangle explains the conditions under which someone becomes motivated to commit fraud, but it doesn’t account for why some people pull it off while others don’t. In 2004, David Wolfe and Dana Hermanson addressed that gap by proposing the fraud diamond, which adds a fourth element: capability.4The CPA Journal. The Fraud Diamond – Considering the Four Elements of Fraud
Capability refers to the personal traits and skills that determine whether someone can actually execute a scheme. Even with pressure, opportunity, and rationalization all present, not everyone has what it takes to exploit the situation. The diamond shifts the question from “why would someone commit fraud?” to “who is equipped to carry it out?”
What Capability Looks Like
Capability isn’t just technical knowledge. Wolfe and Hermanson identified a cluster of traits that combine to make certain individuals far more dangerous than others:
- Position and function that grant access to assets, systems, or financial reporting processes.
- Intelligence sufficient to understand the financial systems and identify control weaknesses.
- Confidence and ego to believe they can succeed without detection.
- Coercion skills to pressure or manipulate others into participating or staying silent.
- The ability to lie effectively and maintain a consistent false narrative under scrutiny.
- Stress tolerance to handle the ongoing anxiety of concealment without obvious behavioral changes.
The distinction matters for risk assessment. The employee who skims $50 from a cash register operates on pure opportunity. The executive who manipulates revenue recognition across multiple quarters for years has all three triangle elements plus the capability to build and sustain a sophisticated concealment strategy. Designing controls only around opportunity misses the reality that high-capability individuals can often override those controls.
The Fraud Pentagon
The most recent evolution is the fraud pentagon, developed by Jonathan Marks of Crowe Horwath (now Crowe LLP). It keeps all three triangle elements, incorporates the capability concept under the label “competence,” and adds a fifth element: arrogance.6International Journal of Management, Accounting and Economics. Testing the Crowes Pentagon Theory of Fraud on Financial Statement Fraud The five elements are pressure, opportunity, rationalization, competence, and arrogance.
Arrogance as the Fifth Element
Marks defines arrogance as “an attitude of superiority and entitlement or greed on the part of a person who believes that internal controls simply do not personally apply.” This isn’t ordinary overconfidence. It’s the mindset of executives who treat the organization’s assets as extensions of their own wealth and its rules as obstacles designed for lesser employees.7Association of Certified Fraud Examiners. The Mind Behind The Fraudsters Crime – Key Behavioral and Environmental Elements
The arrogant perpetrator often doesn’t need much rationalization. A mid-level employee might wrestle with guilt and construct elaborate justifications. The arrogant executive simply doesn’t believe the rules apply. That makes the pentagon particularly useful for analyzing financial statement fraud at the C-suite level, where schemes tend to be both larger and more brazen than asset misappropriation lower down.
How Arrogance Differs From Capability
The practical difference between the diamond and the pentagon is one of psychological profile. Capability asks whether someone can commit a sophisticated fraud. Arrogance asks whether someone believes they’re above the controls entirely. A capable fraudster might still worry about getting caught. An arrogant one doesn’t seriously consider the possibility. That distinction helps explain why some corporate fraud scandals grow so large before detection: the perpetrator at the top genuinely believed they were untouchable.
Which Framework Fits Which Fraud
The three models are complementary rather than competing. Which one is most useful depends on the kind of fraud being analyzed.
For most asset misappropriation (cash skimming, inventory theft, expense reimbursement schemes committed by lower-level employees), the triangle is usually enough. Pressure, opportunity, and rationalization explain the behavior, and the response is tighter controls and attention to employee circumstances.
The diamond becomes relevant when a scheme requires specialized knowledge or sustained concealment. A single bookkeeper who runs a multi-year fictitious vendor scheme has capability on top of the triangle elements, and prevention has to account for that; controls a capable person understands completely can eventually be worked around.
The pentagon is aimed squarely at executive-level fraud. When the person with the pressure and opportunity is also the person who sets the tone for whether controls apply at all, arrogance explains behavior the other models don’t quite capture. Large financial statement frauds tend to sit here.
A Note on Collusion
None of the three frameworks name collusion as an element, but it changes how each one plays out. When two or more people agree to work together, they can defeat controls that would stop any individual. Segregation of duties, which splits authorization, recording, and custody of assets across different employees so no one person can both commit and conceal fraud, is the clearest example. If the person recording transactions and the person reconciling accounts are working together, the control effectively disappears.3University of Pennsylvania Office of Audit, Compliance and Privacy. Operational Internal Controls – Section: Segregation of Duties
Collusive frauds are harder to detect and tend to cause larger losses. Sole perpetrators typically exploit a lack of controls, while collusive schemes are more often supported by a poor ethical tone set by leadership and the ability to override existing controls.8The CPA Journal. The Risks of Fraud Collusion
Using the Frameworks to Prevent Fraud
The real value of these models isn’t catching fraud after the fact. It’s designing an environment where fewer people reach the point of committing it. Each element points to a different kind of prevention.
Reducing Opportunity
Strong internal controls remain the most direct lever. The core principle is to separate authorization of transactions, recording of transactions, and custody of assets, so that no single employee controls more than one.3University of Pennsylvania Office of Audit, Compliance and Privacy. Operational Internal Controls – Section: Segregation of Duties Mandatory vacations force someone else to touch the processes an employee usually handles alone. Surprise audits, automated reconciliation, and rotation of personnel in sensitive roles reinforce the same principle: any control a single person understands completely is a control that person can eventually circumvent.
Addressing Pressure and Rationalization
Employee assistance programs, open-door management, and fair compensation reduce the financial pressure that starts the triangle turning. Attention to employee morale and financial wellness removes the first element for a meaningful portion of potential perpetrators.
Rationalization is harder to counter directly, but culture does the work. When leadership consistently models ethical behavior and enforces rules uniformly rather than exempting senior executives, it becomes harder for an employee to tell themselves that “everyone does it” or “they had it coming.”
Targeting Capability and Arrogance
For the risks the diamond and pentagon describe, governance matters most. Separating the CEO and board chair roles prevents one person from accumulating unchecked authority. Independent audit committees with direct access to external auditors create oversight that can’t easily be bypassed by a single executive. Background checks, ongoing monitoring in high-risk positions, and clear reporting lines all help identify situations where capability combines with arrogance to create serious exposure.