Parameterization of prospect theory.
1.0 Abstract
This study examines a critical evaluation of the parameterization of Prospect theory in section II. A of Kaustia (2010, pp. 794-796). A structural econometric model of decision making under risk for investors will show their propensity towards buying or selling of a stock does not increase in either direction. Trading analysis, on the other hand, shows that this propensity to sell jumps at zero return, but it is approximately constant over a wide range of losses and increasing or constant over a wide range of gains.
A fundamental theory under prospect theory is that; human beings do not always behave rationally. This is true as long as there are persistent biases motivated by psychological factors that influence people’s choices under periods of uncertainty (Baker et al., 2010). Prospect theory explains three biases that affect people’s preferences when making decisions:
- Certainty effect: a tendency for people to overweight options that are certain and be risk-averse for gains.
- Isolation effect: people’s tendency to act on information that stands out and differs from the rest. Isolation can is described through the Framing effect andthe Bandwagon effect. The Framing effect occurs when someone reacts to a particular choice in different ways, depending on how it was presented. We tend to avoid risk when a positive frame is given but seek risks when a negative framework is presented. The Bandwagon effect is a tendency to do or believe things because other people do or feel the same.
- Loss aversion: where people prefer to avoid losses to acquiring equivalent gains. The pain of losing something is twice as powerful as the pleasure of gaining.
2.0 Introduction
To explain prospect theory correctly, expected utility must first be described. Expected utility is a normative claim from many economists who believe that we are rational beings and make consistent decisions that allow us to be economically better off. They have even coined the term ‘homo economicus’ to describe such rational behavior. Those who act under the framework of expected utility calculate the probability of an event occurring, and multiply the probability by the utility or pleasure they expect to get out of the event. With the goal of maximizing expected utility in mind, they consistently make rational decisions based on expected utility calculations. Prospect theory considers preferences as a function of “decision weights,” and it assumes that these weights do not always match with probabilities (Haliassos, 2015). Specifically. Prospect theory suggests that decision weights tend to overweight small probabilities and under-weight moderate and high probabilities.
However, prospect theory (developed by Tversky and Kahneman) differs from expected utility theory in that it describes how people behave in real life, not how they should behave according to economic standards. Prospect theorists believe that humans are irrational and make systematic errors in judgment. Prospect theory claims that people are more risk-averse when it comes to gains, and become risk-seeking when facing a loss. Both behaviors can be explained by decreasing sensitivity to gains and losses, and the overwhelming desire for the status quo to be maintained. Another point that prospect theory covers is of loss aversion; it posits that losses loom larger than gains. That is, the disutility from incurring a loss is greater than utility obtained from a size- gain.
3.0 Prospect theory and the Sell Versus Hold Decision
3.1 Base Case Parameterization
The Prospect theory value function has been used by financial economists modeling asset allocation and pricing. The general form of the prospect theory value function is
Where x is the gain with respect to the reference point. is the coefficient for loss aversion, and and are coefficients of risk aversion and risk seeking, respectively on the experiments. Tversky and Kahneman (1992) estimate θ to be 2.25, and both and to be equal to 0.88. In addition to the value function, prospect theory contains other elements, such that framing and probability distortions, but have received less needed attention in quantitative analysis.
The prospects theory value function coupled with additional assumptions results in predictions for investors’ selling behavior. First, assume that investors reflect past and anticipated gains and losses relative to the purchase price of the stock. Second, assume that investors do not integrate returns across stocks (i.e., they consider each stock separately). Given these assumptions, the investor is risk seeking in the domain of losses, and risk averse in the domain of gains, for each stock. These assumptions are made, either explicitly or implicitly, in the finance literature linking empirical evidence on the disposition effect with prospect theory (Venezia, n.d.).
In what follows, a demonstration of the implications of prospect theory preferences for an investor’s motivation to enter and exit from a stock position. Consider an asset with a normally distributed return having an expected value of 12% and a standard deviation of 25%. Now consider an investor with preferences given by expression of the prospect theory function, such that x, the gain with respect to the reference point is specified as the percentage rate of return. The parameters of expression in the prospect function are these estimated by Tversky and Kahneman. To obtain the prospect value the risky asset, one takes the expectation of the value function over the distribution of the asset return to the reference point, that is,
Prospect value
Where f is a probability density function of the return with respect to the reference point at the end of the investment horizon. In this example, numerical integration will yield a positive value of 3.37 for the risky asset.
Now assume that the investor buys into the position, and consider what happens to the prospect value as the price of the asset changes. If the value of the asset falls by 25, the distribution of x shifts to the left by 2%, the distribution of x shifts to the left by 2% and the prospect value of holding the position falls to 1.5. However, the value of liquidating the position , that is, a certain return of 2%, has a prospect value of -4.1, and the investor is thus better off holding the asset. If the asset price increases by 2%, the prospect value of the position will be 5.2 and the liquidation value will be 1.8, in which case the investor will hold the position.
To graph the prospect theory value function with the canonical parameterization ( = 0.88, =0.88, =2.25) it will show that the prospects value of selling the asset minus the value of holding, for a range of asset prices. This difference is negative for all asset prices, and the investor is thus always better off sticking with the risky investment. To motivate a sale, the risky investment must be less favorable than in the example above. However, if it is initially valued at less than the risk-free asset, the investor will not take the position. This point is also made by Gomes (2005).
Now consider what happens if other factors (e.g., liquidity needs) cause a need to sell stocks. The difference between the value of selling and the value of holding as a function of asset price may then become important. This difference can be thought of as the propensity to sell the position; there is a local maximum close to 0 (at 0.85% return). Although the propensity to sell decreases in both directions, it decreases at a faster rate in the loss domain.
In sum, for conventional value function parameters, prospect theory predicts no realizations of gains up to 30% return. An exogenous factor is needed to induce a sale. Considering conventional return ranges, the likelihood of a sale is greatest when the price is close to the purchase price, and it declines in both directions. The likelihood of a sale is lower for moderate losses than for moderate gains, which is consistent with the disposition effect.
3.2 Data and Estimation
Because our values in the data set are binary and are values between 0 and 1 we shall use probit model. The right side will be modelled as a probability of each gambling outcome in the lottery. Probit has the advantage of generating fitted values which can be interpreted as probabilities. The equation is nonlinear, and does not generate well-defined residuals which can be minimized. Least squares estimation is therefore not possible.
The iteration history is a listing of the log likelihood of each iteration for the probit model. Probit regression uses maximum likelihood estimation, which is an iterative procedure. The first iteration (called iteration 0) is the loglikelihood of the “null” or “empty” model; that is; model with no predictors. At the next iteration (called iteration 1), the specified predictors are included in the model. In this case the predictors are choice_a, probH_a, payH_a, payL_a, payH_b, payL_b, frame, male and age. At each iteration, the log likelihood increases because the goal is to maximize the loglikelihood. When the difference between successive iterations is very small, the model is said to have converged and iterating stops.
In the Model Summary, the loglikelihood of the fitted model is used in the Likelihood Ratio Chi-Squared test of whether all predictors’ regression coefficients in the model are simultaneously zero. The Number of observations in our probit model are 6900; for which all of the response and predictor variables are non-missing. LR chi2 (3) is the likelihood Ratio (LR) Chi-Square test that at least one of the predictors’ regression coefficient is not equal to zero. The number in the parentheses indicates the degrees of freedom of the Chi-Square distribution used to test the LR Chi-Square statistic and is defined by the number of predictors in the model(3).
Prob> chi2 is the probability of getting a LR test statistic as extreme as, or more so, than the observed statistic under the null hypothesis is that all of the regression coefficient are simultaneously equal to zero. In other words, this is the probability of obtaining the chi-square statistic (485.55) or one more extreme if there is in fact no effect of the predictor variables. This p-value is compared to a specific alpha level, our willingness to accept a type I error, which is typically set at 0.00. The small p-value from the LR test, 0.0001, would lead us to conclude that at least one of the regression coefficients in the model is not equal to zero. The parameter of the chi-square distribution used to test the null hypothesis is defined by the degrees of freedom in the prior line, Chi2 (3). Pseudo R2 is the McFadden’s pseudo R-squared. It is the proportion of variance of the response variable explained by the predictors.
For the parameter estimates admit is the binary response variable predicted by the model. Coef. Are the regression coefficients. The predicted probability of a game would be
F (23.01222 – 0.5824407*choice_a + 0.2850547*probH_a – 0.2432696*probH_b + 0.015503*payH_a + 0.0377925 payL_a – 0.0154382* payH_b -0.0399125* payL_b – 0.028443* frame – 3.671501* male + 0.1861389*age)
Where F is the cumulative distribution function of the standard normal. The increase in probability attributed to a one unit increase in a given predictor is dependent both on the values of the other predictors and starting value of the given predictors. For example, if we hold and constant are zero, the one unit increase in payH_a from 2 to 3 has a different effect than the one unit increase from 3 to 4. A positive coefficient means that an increase in the predictor leads to an increase in the predicted probability. A negative coefficient means an increase in the predictor leads to a decrease in the predicted probability.
Pearson’s chi-squared test is a goodness of fit test that examines the sum of the squared differences between the observed and expected number of cases per covariate pattern divided by its standard error. In the dataset 2541 have been correctly classified and have predicted the outcome of grater or equal to 0.5. In total, 3990 have been correctly classified out of the 6900 observations. That makes a 62.54% correct classification by the fitted model. The Confidence Interval used is 95%, for a given predictor with a level of 95% confidence, we’d say that we are 95% confident that the true coefficient lies between the lower and upper limit of the interval. It is calculated as Coef. ()*(Std.Err.), where is a critical value on the standard normal distribution. The CI is equivalent to the z test statistic: If the CI includes zero, we’d fail to reject the null hypothesis that a particular regression coefficient is zero given the other predictors are in the model. An advantage of the confidence interval is that it provides a range where the “true” parameter may lie.
4.0 Conclusion
Prospect theory can be used to explain a few illogical finance behaviors. For example, there are people who do not wish to put their money in the bank to earn interest or who refuse to work overtime because they fear to pay taxes. Although these people would benefit from after tax income, prospect theory suggests that the benefit (or utility gained) from the extra money is not enough to overcome the feelings of loss incurred by paying taxes.
Prospect theory also explains the occurrence of the disposition effect, which is the tendency for investors to hold on to losing stocks for too long and sell wining stocks too soon (HE, 2017). The most logical course of action would be to hold on to winning stocks in order to prevent escalating losses. The prospect theory does have a “break-even effect”, its rationale is a combination of the scarcity principal and a natural tendency to want to avoid losses (Kahneman). People generally feel like its better not to lose $5 than to find $5. In a gambling casino, at the end of the day you know that you only have a few chances left to win back your money (either limited by time or cash flow) and when something is scarce we become less rational and it affects our behavior, this coupled with a desire to not lose promotes risk seeking behavior, as risk aversion seems less useful when you’re already losing.
References
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Haliassos, M. (2015). Household finance. Edward Elgar Pub. Ltd.