Stochastic dominance is a concept used in decision-making under uncertainty and portfolio optimization to compare two probability distributions representing uncertain outcomes (such as returns on investments). It helps us determine which investment option is more preferable based on the preference of the decision maker. There are three types of stochastic dominance: first-order, second-order, and third-order. Below, we will focus on the first two, as they are the most widely used in finance.
1. **First-order stochastic dominance (FSD):**
Let X and Y be two random variables representing the outcomes of two investment opportunities. Denote their cumulative distribution functions (CDFs) as FX(x) and FY(y). X is said to first-order stochastically dominate Y, denoted as X≻1Y, if:
FX(x) ≤ FY(x) ∀x
This means that the probability of receiving a return less than or equal to x is always lower for the investment X than for the investment Y. In terms of utility, this implies that for any risk-averse investor (with monotonically increasing utility function), X is a preferred investment over Y.
2. **Second-order stochastic dominance (SSD):**
Let X and Y be two random variables representing the outcomes of two investment opportunities. Denote their CDFs as FX(x) and FY(y). X is said to second-order stochastically dominate Y, denoted as X≻2Y, if:
∫ − ∞xFX(u)du ≤ ∫ − ∞xFY(u)du ∀x
This condition implies that the area under the CDF curve of X (i.e., the probability-weighted outcome) is always less than or equal to that of the CDF curve of Y up to a certain return level x. This means that the average investor with a concave (risk-averse) utility function would prefer investment X over investment Y.
In portfolio optimization, stochastic dominance can be used as a criterion for selecting the best portfolio, given the investor’s preference for risk. Instead of relying solely on expected return and volatility, stochastic dominance takes the entire distribution of returns into account, providing a more robust decision-making framework.
For example, suppose there are two portfolios, A and B, with the following return probability distributions:
- Portfolio A: P( − 10%) = 0.2, P(10%) = 0.6, P(30%) = 0.2
- Portfolio B: P( − 20%) = 0.1, P(10%) = 0.7, P(40%) = 0.2
If we compare their CDFs, we can determine which portfolio first- or second-order stochastically dominates the other. This comparison can help the investor make better investment decisions under uncertainty, based on their risk preferences.
To summarize, stochastic dominance is a powerful tool for decision-making under uncertainty and portfolio optimization. By taking into account the entire probability distribution of returns, it allows investors to choose the most preferable investment according to their risk preferences.