First order stochastic dominance (FSD) is a fundamental concept in decision theory and economics. It compares two probability distributions and quantifies the preference for one distribution over the other. FSD is closely related to the concepts of expected utility, risk aversion, and the Lorenz curve. FSD is a useful tool for analyzing investment decisions, ranking portfolios, and assessing the welfare implications of income and wealth distributions.
First Order Stochastic Dominance
The first order stochastic dominance (FSD) comes when the cumulative distribution function (CDF) of one random variable is always greater than or equal to the CDF of another random variable for all possible values of the random variable. In other words, one random variable is said to dominate the other in the FSD when it is always at least as good as, and sometimes better than, the other random variable.
There are three different ways to express FSD:
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Weak FSD: A random variable X weakly first-order stochastically dominates a random variable Y if and only if the CDF of X is greater than or equal to the CDF of Y for all possible values of the random variable. Mathematically, this can be expressed as:
F_X(x) ≥ F_Y(x) for all x -
Strict FSD: A random variable X strictly first-order stochastically dominates a random variable Y if and only if the CDF of X is greater than the CDF of Y for all possible values of the random variable, except possibly at one point, where they may be equal. Mathematically, this can be expressed as:
F_X(x) > F_Y(x) for all x except possibly at one point -
Mean-Preserving Spread: A random variable X is said to have a mean-preserving spread over a random variable Y if X can be obtained from Y by a spread preserving transformation. This means that the mean of X and Y is the same, but the variance of X is greater than the variance of Y.
The FSD is a useful tool for comparing the riskiness of two random variables. A random variable that dominates another random variable in the FSD is said to be less risky than the other random variable. This is because the FSD implies that the probability of a loss is lower for the dominating random variable than it is for the dominated random variable.
Example
Suppose we have to invest in two different stocks, A and B. Stock A has a 50% chance of returning 10% and a 50% chance of returning -10%. Stock B has a 25% chance of returning 20%, a 50% chance of returning 0%, and a 25% chance of returning -20%. The CDFs of the two stocks are shown in the table below:
| Return | CDF of Stock A | CDF of Stock B |
|---|---|---|
| -20% | 0 | 0.25 |
| -10% | 0.5 | 0.75 |
| 0% | 1 | 1 |
| 10% | 1 | 1 |
| 20% | 1 | 1 |
As can be seen from the table, the CDF of Stock A is always greater than or equal to the CDF of Stock B. This means that Stock A dominates Stock B in the weak FSD. This implies that Stock A is less risky than Stock B, as it is always at least as good as Stock B, and sometimes better.
Question 1:
What is the definition of first order stochastic dominance (FSD)?
Answer:
First order stochastic dominance (FSD) is a measure of the ranking of two probability distributions based on their cumulative distribution functions (CDFs). It indicates whether one distribution is more likely to produce higher outcomes than another.
Question 2:
Explain the concept of the expected value in relation to FSD.
Answer:
The expected value of a distribution is the average value that is expected to occur. FSD is related to the expected value because a higher expected value indicates a higher probability of obtaining higher outcomes, which is the basis of FSD.
Question 3:
How is FSD used in financial decision-making?
Answer:
FSD is used in financial decision-making to compare the risk and return profiles of different investment options. By calculating the FSD between different distributions of potential returns, investors can determine which option has a higher probability of providing higher returns.
Welp, there you have it – a basic overview of first order stochastic dominance. I know it’s a bit of a mouthful, but it’s a useful concept to get your head around if you’re interested in making financial decisions under uncertainty. Thanks for sticking with me through this brief exploration! If you have any questions or want to dive deeper, feel free to give me a holler again. Until next time, keep exploring the financial landscape!