Prospect Theory

Prospect Theory is a behavioral economic theory developed by Daniel Kahneman and Amos Tversky in 1979. It describes how individuals assess their potential losses and gains when making decisions under uncertainty. Unlike traditional economic theories that assume rational behavior, Prospect Theory accounts for the psychological nuances involved in decision-making. This makes it particularly relevant to fields like finance, trading, and investing, where real-world decisions often deviate from purely rational models.

Key Components

1. Value Function

The value function is central to Prospect Theory and it describes how people value potential gains and losses. Unlike traditional utility theory which considers the absolute value of an outcome, the value function in Prospect Theory is:

Reference-dependent: The value of an outcome is determined relative to a reference point (usually the status quo or the current state).
Loss Aversion: Losses loom larger than gains. The disutility of losing $100 is greater than the utility of gaining $100.
Diminishing Sensitivity: The value function is concave for gains and convex for losses. This means that the marginal value of gains decreases with size, and similarly, the marginal disutility of losses decreases with size.

Mathematically, the value function (v) can be represented as: [ v(x) = \begin{cases} (x - [lambda](../l/lambda.html))^[alpha](../a/alpha.html) & \text{if } x \geq [lambda](../l/lambda.html)
-[kappa](../k/kappa.html) ([lambda](../l/lambda.html) - x)^[beta](../b/beta.html) & \text{if } x < [lambda](../l/lambda.html) \end{cases} ] Where:

(x) is the outcome.
([lambda](../l/lambda.html)) is the reference point.
([alpha](../a/alpha.html)) and ([beta](../b/beta.html)) are parameters reflecting diminishing sensitivity.
([kappa](../k/kappa.html)) reflects loss aversion.

2. Probability Weighting Function

The probability weighting function describes how people perceive the probability of outcomes. People tend to overweigh small probabilities and underweigh large probabilities. This is contrary to the expected utility theory where probabilities are considered linearly.

This function can be represented as: [ \pi(p) = \frac{p^[gamma](../g/gamma.html)}{(p^[gamma](../g/gamma.html) + (1-p)^[gamma](../g/gamma.html))^{1/[gamma](../g/gamma.html)}} ] Where:

(p) is the probability of the outcome.
(\pi(p)) is the decision weight.
([gamma](../g/gamma.html)) is an exponent that determines the curvature of the weighting function.

How It Works

Decision-Making Process

Prospect Theory proposes a two-stage process for decision-making under risk:

Editing Phase: Potential outcomes and probabilities are simplified using a series of cognitive operations, such as coding (setting reference points), combination (aggregation of probabilities), segregation (separating riskless from risky components), and cancellation (canceling out common attributes).
Evaluation Phase: Each edited prospect is evaluated by combining the value of outcomes and decisions weights. The prospect with the highest value is chosen.

Cumulative Prospect Theory

An advancement over the original model is Cumulative Prospect Theory (CPT), which extends the theory to account for multiple or compound outcomes. In CPT, outcomes are ordered and cumulative probabilities are used in the weighting function, allowing for a more comprehensive model that includes scenarios like portfolio optimization.

Examples in Real Life

Financial Markets

Investors often deviate from the rational behavior predicted by traditional financial theories. Examples include:

Loss Aversion: Investors are more likely to hold on to losing stocks, hoping they will rebound, rather than selling them at a loss.
Over-weighing Small Probabilities: Investors might over-invest in highly speculative stocks with small chances of huge gains, akin to buying lottery tickets.
Risk Behavior in Gains vs. Losses: Investors tend to be risk-averse when facing gains but become risk-seeking when facing potential losses.

Behavioral Biases

Behavioral biases explained by Prospect Theory include:

Endowment Effect: People value what they own more than identical items they do not own.
Status Quo Bias: Preference for the current state of affairs, leading to under-reaction in trading.
Disposition Effect: The tendency to sell winning investments and hold onto losing investments.

Insurance and Gambling

Prospect Theory explains why people purchase insurance and gamble:

Insurance: Even though the expected value of insurance is negative, people overweigh the small probability of a catastrophe.
Gambling: Similarly, gamblers overweigh small probabilities of large wins, resulting in higher stakes in lotteries and casinos.

Implications for Algorithmic Trading

Algorithmic trading relies heavily on models that predict market movements and make trading decisions. Understanding Prospect Theory allows developers to:

Incorporate human decision-making biases into algorithms, improving prediction models.
Design algorithms that exploit behavioral biases in the market (e.g., exploiting the disposition effect).
Enhance risk management strategies by considering how traders perceive risk differently from traditional models.

Conclusion

Prospect Theory provides a more realistic framework for understanding decision-making under risk than traditional rational models. Its insights into how people actually perceive gains, losses, and probabilities make it invaluable in fields like trading and finance. By integrating these behavioral insights, financial professionals and algorithm developers can create better strategies, mitigate risk, and ultimately improve financial outcomes.

For further reading and a comprehensive dive into this theory, you may refer to the following resources: