Deep Dive into Net Present Value (NPV): Part 3 - Pascal's Mugging

"Beware of false knowledge; it is more dangerous than ignorance." — George Bernard Shaw

-Source https://www.smbc-comics.com/comic/pascal39s-other-wager

Net Present Value (NPV) is widely regarded as the gold standard for financial decision-making, enabling businesses to compare options, weigh risks, and make decisions that maximize future returns. In this installment of our NPV series, we explore an intriguing edge case: Pascal’s Mugging, a thought experiment that highlights the challenges of NPV under extreme conditions.

What is Pascal’s Mugging?

Pascal’s Mugging originates from a philosophical problem that builds on Blaise Pascal’s famous wager. In Pascal’s Mugging, an individual encounters a situation where they are asked to act based on an extraordinarily high reward with an infinitesimally small probability of success. For example, imagine a stranger offering you the chance to save the universe in exchange for $10, but with odds so astronomically small as to be almost impossible. Despite the overwhelming improbability, the expected value calculated via NPV might seem to justify the bet due to the sheer magnitude of the reward.

The Problem of Extreme Values in NPV

NPV calculations rely on several key assumptions:

1. The ability to assign probabilities to outcomes.

2. A discount rate that reasonably reflects time value and risk.

3. Bounded inputs for risk and return.

In extreme scenarios, however, these assumptions become untenable. Consider the following issues:

1. Assigning Probabilities

In practical terms, assigning probabilities to astronomically improbable events is inherently speculative. When the probability is close to zero, even tiny misestimations can drastically skew the NPV calculation. For example, is the chance of saving the universe one in a billion or one in a trillion? The difference matters when the reward is enormous—but assigning either probability accurately is nearly impossible.

2. Discount Rates Under Extreme Scenarios

Discount rates are meant to reflect risk and opportunity costs, but under extreme conditions, they fail to capture the true uncertainty. For example, applying a standard 10% discount rate to a wager on saving the universe is nonsensical. There’s no market mechanism or comparable risk to guide a reasonable discount rate for such outcomes.

3. Bounded Inputs

Most real-world decisions occur within a range of bounded inputs. For businesses, assets, revenue, and lending capacities have natural upper limits. In Pascal’s Mugging-style scenarios, however, the lack of bounds leads to NPV results that are theoretically valid but practically meaningless.

Real-World Implications: When NPV Leads to Poor Decisions

The weaknesses exposed by Pascal’s Mugging have practical implications, especially in high-stakes financial decision-making. Consider lending decisions. An NPV-positive opportunity involving an extraordinarily high return but with a negligible probability of success might still “pencil out”—yet pursuing such opportunities would be disastrous for lenders who must operate within bounded capital and risk frameworks.

This issue is compounded by the sensitivity of NPV calculations to assumptions. Small changes in the probability, discount rate, or expected return can drastically alter the outcome, making the decision-making process overly reliant on precise and often speculative assumptions.

The Practical Lesson: Reasonable Scenarios Matter

Despite these challenges, NPV remains highly effective for more reasonable, bounded scenarios. The key lesson is that NPV works best when applied to situations with well-defined probabilities, manageable risks, and realistic outcomes. By ensuring inputs remain within a reasonable range, businesses can leverage NPV to make sound decisions without falling into the trap of overfitting assumptions to extreme conditions.

In the context of lending, for example, a reasonable lower bound for assets and operational capabilities ensures that NPV can provide actionable insights. Conversely, pushing the model to its extremes—whether through astronomical rewards, minuscule probabilities, or limitless growth assumptions—renders it unreliable.

Conclusion: Balancing Theory with Practicality

Pascal’s Mugging illustrates a critical weakness of NPV: its reliance on assumptions that break down under extreme conditions. While these edge cases are unlikely to arise in most business contexts, they serve as a reminder of the importance of bounded, realistic inputs in NPV calculations. NPV is an invaluable tool when used within reasonable bounds. For extreme scenarios, it’s worth supplementing NPV with other decision-making frameworks to ensure practical and logical outcomes.