What was Bernoulli's primary error in utility theory?

Bernoulli assumed that utility depends solely on absolute wealth levels, failing to realize that human beings evaluate outcomes as gains and losses from a reference point.

What are the shapes of the Prospect Theory Value Function?

The value function is S-shaped: concave for gains (risk aversion) and convex for losses (risk seeking), with a steeper slope in the loss domain.

What is the typical value of the Loss Aversion coefficient?

The loss aversion coefficient is empirically calibrated at lambda approximately equal to 2.25, meaning losses hurt roughly 2.25 times more than equal gains.

Home / Semester 4 / Behavioural Finance / Error in Bernoulli's Theory and Kahneman-Tversky Prospect Theory: Reference Dependence, S-Shaped Value Function, and Loss Aversion

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AKTU MBA Semester 4 • BMB FM 04 Unit 3: Prospect Theory, Framing & Inefficiencies Est. Reading Time: 10 Mins

Error in Bernoulli's Theory and Kahneman-Tversky Prospect Theory: Reference Dependence, S-Shaped Value Function, and Loss Aversion

MAuthored by mbanotes.in Academic Editorial Team

Verified Syllabus Alignment: AKTU MBA CoreTarget Score: 10/10 Marks (Section B/C)

⚡ 30-Second Revision Snapshot: Prospect Theory Foundations AKTU MBA BMB FM 04 | Unit 3 Core Definition The intellectual core of behavioral economics: Daniel Kahneman and Amos Tversky’s Prospect Theory, deconstructing Bernoulli’s wealth error and modeling loss aversion. Bernoulli’s Fatal Flaw Bernoulli assumed utility depends on total absolute wealth states, failing to recognize that humans evaluate outcomes as gains and losses from a reference point. Reference Dependence Carriers of value are changes in wealth (ΔW) relative to an arbitrary psychological baseline, not terminal net worth (W). S-Shaped Value Function Concave for gains (risk aversion for positive returns) and convex for losses (risk seeking to avoid losses), with diminishing sensitivity. x Loss Aversion Losses loom roughly 2.25 \times larger than equivalent gains: the psychological pain of losing INR 10,000 exceeds the pleasure of gaining INR 10,000.

Kahneman Tversky Prospect Theory S Shaped Value Function Loss Aversion Lambda 2.25 and Diminishing Sensitivity Infographic

Figure 3.1: Prospect Theory Foundations – Reference Dependence, S-Shaped Asymmetric Curve, and 2.25x Loss Aversion Slope.

1. Foundational Architecture: The Error in Bernoulli’s Expected Utility

In 1738, Swiss mathematician Daniel Bernoulli published Specimen Theoriae Novae de Metris Sortis, proposing that individuals evaluate risky prospects not by their mathematical expected monetary payoff, but by the expected utility of terminal wealth. For nearly 250 years, Bernoulli’s framework formed the unshakeable foundation of microeconomics and neoclassical finance.

However, in their monumental 1979 paper “Prospect Theory: An Analysis of Decision under Risk”, Daniel Kahneman and Amos Tversky uncovered Bernoulli’s fatal theoretical error:

“Bernoulli assumed that the utility that makes people happy or unhappy depends entirely on the absolute state of wealth. This assumption completely misses the psychological reality of adaptation: happiness depends on changes in wealth relative to an initial reference point, not total wealth.”

The Twin Billionaire Thought Experiment

Consider two corporate executives, Rajesh and Vikram, who both end the business day with identical net worths of INR 50 Crore:

Yesterday, Rajesh had INR 10 Crore, and made a spectacular INR 40 Crore profit today.
Yesterday, Vikram had INR 100 Crore, and suffered a devastating INR 50 Crore loss today.

Under Bernoulli’s expected utility theory, because Rajesh and Vikram possess the exact same terminal wealth of INR 50 Crore, their utility must be identical ($U(50\text{Cr}) = U(50\text{Cr})$). They should feel equally happy. In reality, Rajesh is ecstatic and euphoric, while Vikram is traumatized, clinically depressed, and in acute distress. Bernoulli’s theory cannot explain this because it lacks a reference point.

2. Kahneman & Tversky’s Prospect Theory: Value Function Properties

Prospect Theory replaces Bernoulli’s global utility function $U(W)$ with an S-shaped Value Function $v(x)$ defined over gains and losses relative to a psychological reference point ($x = W – W_{ref}$):

Value Function Property	Mathematical Formulation	Psychological & Market Implication
Reference Dependence	$v(0) = 0$; $x = W – W_{ref}$	Value is defined as gains ($x > 0$) or losses ($x < 0$) from a subjective baseline (usually purchase price).
Diminishing Sensitivity (Gains)	$v”(x) < 0$ for $x > 0$ (Concave)	The difference between gaining INR 10,000 and INR 20,000 feels much larger than between INR 1,10,000 and INR 1,20,000. Risk-averse for gains.
Diminishing Sensitivity (Losses)	$v”(x) > 0$ for $x < 0$ (Convex)	The psychological pain flattens as losses mount; investors become risk-seeking in the loss domain, gambling on double-down bets.
Loss Aversion ($lambda$)	$-v(-x) > v(x)$ for all $x > 0$; $lambda approx 2.25$	The slope of the value function is significantly steeper in the negative quadrant: losses hurt roughly 2.25 times more than equal gains feel good.

3. Fourfold Pattern of Risk & Loss Aversion in Capital Markets

Combining the S-shaped value function with the non-linear probability weighting function $w(p)$ generates Prospect Theory’s famous Fourfold Pattern of Risk Attitudes:

Domain	High Probability (Certainty Effect)	Low Probability (Possibility Effect)
Gains	Risk-Averse: Fear of disappointment leads investors to lock in paper gains quickly (selling winning stocks too early).	Risk-Seeking: Hope of large jackpot leads investors to overpay for IPOs, biotech penny stocks, and lottery tickets.
Losses	Risk-Seeking: Desperation to break even leads investors to hold losing stocks and double down on failing positions.	Risk-Averse: Fear of catastrophic loss drives individuals to purchase expensive insurance policies and protective put options.

4. Mathematical Model, Value Function Formulation & Worked Problem

We formalize the parametric Kahneman-Tversky (1992) Power Value Function and demonstrate its predictive power.

Mathematical Formulation: Kahneman-Tversky Value Function

\[ v(x) = \begin{cases} x^\alpha & \text{for } x \ge 0 \\ -\lambda (-x)^\beta & \text{for } x < 0 \end{cases} \] \[ V = \sum_{i=1}^n w(p_i) \cdot v(x_i) \]

Where:

$x$: Monetary deviation from the reference point ($x > 0$ for gains, $x < 0$ for losses).
$alpha, eta$: Diminishing sensitivity curvature parameters. In Kahneman & Tversky’s 1992 calibration: $\alpha = \beta = 0.88$.
$lambda$: Coefficient of Loss Aversion. Calibrated empirically at $\lambda = 2.25$.
$V$: Total subjective psychological prospect value.

Step-by-Step Worked Numerical Problem: Symmetrical Coin Flip Valuation

Problem Statement: An investor with substantial net worth is offered an actuarially fair 50-50 bet based on the flip of a fair coin ($p_1 = p_2 = 0.50$):

Heads ($x_1 = +\text{INR } 1,00,000$): Investor receives a tax-free cash gain of INR 1,00,000.
Tails ($x_2 = -\text{INR } 1,00,000$): Investor pays a penalty of INR 1,00,000.

Assume standard Prospect Theory parameters: $alpha = eta = 0.88$, loss aversion $lambda = 2.25$, and linear decision weights for 50-50 odds ($w(0.50) = 0.50$).

Required:

Calculate the mathematical expected monetary value $E(X)$ of the lottery.
Calculate the subjective value of the gain quadrant: $v(+1,00,000)$.
Calculate the subjective value of the loss quadrant: $v(-1,00,000)$.
Calculate the total psychological prospect value $V$ and determine whether the investor accepts or rejects the gamble.
Calculate the minimum positive winning payout $X_{win}^*$ required to make the investor willing to accept the risk of losing INR 1,00,000.

Step 1: Compute Mathematical Expected Value

\[ E(X) = (0.50 \times +1,00,000) + (0.50 \times -1,00,000) = +50,000 – 50,000 = \mathbf{\text{INR } 0.00} \]

Under neoclassical finance with zero risk aversion or infinitesimal stake relative to wealth, a risk-neutral agent is indifferent ($E(X) = 0$).

Step 2: Calculate Subjective Value of Gain $v(+1,00,000)$

\[ v(+1,00,000) = (1,00,000)^{0.88} \] \[ \ln(1,00,000) = 11.512925 \implies 11.512925 \times 0.88 = 10.131374 \] \[ v(+1,00,000) = e^{10.131374} = \mathbf{+25,118.86 \text{ utils}} \]

Step 3: Calculate Subjective Value of Loss $v(-1,00,000)$

\[ v(-1,00,000) = -\lambda \cdot |-1,00,000|^{0.88} = -2.25 \times (1,00,000)^{0.88} \] \[ v(-1,00,000) = -2.25 \times 25,118.86 = \mathbf{-56,517.44 \text{ utils}} \]

The psychological trauma of the INR 1,00,000 loss (-56,517 utils) is more than double the joy of the INR 1,00,000 gain (+25,119 utils)!

Step 4: Compute Total Prospect Value $V$

\[ V = 0.50 \times v(+1,00,000) + 0.50 \times v(-1,00,000) \] \[ V = 0.50(25,118.86) + 0.50(-56,517.44) = 12,559.43 – 28,258.72 = \mathbf{-15,699.29 \text{ utils}} \]

Since $V < 0$, the investor vehemently rejects the gamble, despite it having an expected monetary payoff of zero.

Step 5: Determine Minimum Break-Even Winning Payout $X_{win}^*$

For the investor to accept, $V = 0$:

\[ 0.50 \cdot v(X_{win}^*) + 0.50 \cdot v(-1,00,000) = 0 \implies v(X_{win}^*) = -v(-1,00,000) = 56,517.44 \] \[ (X_{win}^*)^{0.88} = 56,517.44 \implies \ln(X_{win}^*) = \frac{\ln(56,517.44)}{0.88} = \frac{10.942303}{0.88} = 12.434435 \] \[ X_{win}^* = e^{12.434435} = \mathbf{\text{INR } 2,51,930} \]

To risk losing INR 1,00,000 on a 50-50 coin flip, the investor demands a potential gain of at least INR 2,51,930 (a gain-to-loss ratio of $2.52:1$), perfectly reflecting loss aversion in real human psychology!

6. Corporate Case Vignette: Retail Investor Refusal to Cut Losses

Corporate Case Vignette: The Collapse of Yes Bank and Retail Averaging Down

Between 2018 and 2020, shares of private lender Yes Bank plummeted from an all-time high of INR 400 to below INR 15 as catastrophic non-performing loans (NPLs) and corporate governance scandals emerged. During this collapse, institutional mutual funds aggressively dumped their holdings, cutting exposure to zero.

Astoundingly, regulatory shareholding filings revealed that retail shareholder count surged from 3.5 Lakh investors to over 20 Lakh investors during the very same descent. Retail traders continuously “averaged down,” pouring fresh capital into a failing bank.

The Behavioral Diagnostic: This catastrophic capital allocation was driven by the convex loss region of Prospect Theory. Once investors were underwater relative to their purchase reference price (e.g., INR 250), cutting losses meant crystallizing a definitive emotional loss (experiencing the agonizing $lambda = 2.25$ pain penalty). By holding and buying more at INR 100 and INR 50, investors gambled in the risk-seeking loss domain, desperately chasing the slim probability of a miraculous turnaround to break even. Most lost over 95% of their capital.

7. AKTU Exam Drill: 2-Mark & 10-Mark University Questions

AKTU Pattern Section A (2 Marks Each)

Short-Answer Conceptual Checkpoints

Q1: What was Bernoulli’s primary error identified by Kahneman and Tversky?

Answer: Bernoulli assumed utility is derived from total absolute wealth states, overlooking that humans evaluate outcomes as gains and losses relative to a reference point.

Q2: State the two geometric shapes of the Prospect Theory Value Function.

Answer: It is concave in the domain of gains ($v” < 0$, risk-averse) and convex in the domain of losses ($v'' > 0$, risk-seeking).

Q3: What is the Loss Aversion coefficient ($lambda$) and its typical empirical value?

Answer: The loss aversion coefficient measures how much more painful a loss is compared to an equal-sized gain; empirically calibrated by Kahneman and Tversky at $lambda approx 2.25$.

Q4: Define ‘Diminishing Sensitivity’ in Prospect Theory.

Answer: The principle that the marginal psychological impact of an additional dollar decreases as the absolute distance from the reference point increases, both for gains and for losses.

Q5: What is the Fourfold Pattern of Risk?

Answer: The behavioral pattern predicting risk aversion for high-probability gains and low-probability losses, and risk seeking for low-probability gains (lotteries) and high-probability losses (gambling on bad investments).

AKTU Pattern Section B & C (10 Marks Each)

In-Depth Analytical & Long-Answer Questions

Q6: Detailed Critique of Bernoulli’s Utility Theory and Comprehensive Exposition of Prospect Theory. Contrast their mathematical formulations and economic implications.

Model Answer Outline:

Introduction: Review Bernoulli’s 1738 paper and its role in neoclassical economics. Explain the core assumption: $U(W)$.
The Fatal Flaw: Present Kahneman’s twin billionaire thought experiment proving that absolute wealth cannot explain emotional utility without a reference point.
Prospect Theory Architecture: Detail the three fundamental properties of the S-shaped Value Function: Reference Dependence, Diminishing Sensitivity, and Loss Aversion ($lambda = 2.25$).
Mathematical Derivation: Write down $v(x) = x^alpha$ for $x ge 0$ and $v(x) = -lambda(-x)^eta$ for $x < 0$.
The Fourfold Pattern of Risk: Draw the 2×2 matrix combining probability weighting $w(p)$ and the value function.
Capital Market Evidence: Discuss how loss aversion creates the Disposition Effect and retail averaging down in failing equities.

Q7: What is Loss Aversion? Solve a step-by-step mathematical problem demonstrating why a risk-averse investor rejects an actuarially fair 50-50 gamble, and derive the break-even winning payoff.