First and second order stochastic dominance given two asset payoffs

Question

Assume that there are $3$ equally likely states of the world. There are two assets, $Z$ and $Y$, with the payoffs $Z = \{10,0,10\}$ and $Y = \{0,10,20\}$. Determine whether there is state-by-state, FSD and/or SSD between these two investments.

Attempted solution:

$$E[Z] = \frac{1}{3}\times 10 + \frac{1}{3}\times 0 + \frac{1}{3}\times 10 = \frac{20}{3}$$ and $$E[Y] = \frac{1}{3}\times 0 + \frac{1}{3}\times 10 + \frac{1}{3}\times 20 = 10$$

So, $E[Z] < E[Y]$. Given these payoffs I don't see how we can determine whether we FSD or SSD since we do not have their associated PDF's.

Answer 1

FSD

Start with definitions. Z FSD Y means $F_Z(x) \leq F_Y(x)$ for all $x$. Let's look at the CDFs.

$$F_Z(x) = \begin{cases} 0 & \text{ if } x < 0 \\ \frac{1}{3} & \text{ if } 0 \leq x < 10 \\ 1 & \text{ if } x \geq 10 \end{cases} $$

$$F_Y(x) = \begin{cases} 0 & \text{ if } x < 0 \\ \frac{1}{3} & \text{ if } 0 \leq x < 10 \\ \frac{2}{3} & \text{ if } 10 \leq x < 20 \\ 1 & \text{ if } x \geq 20 \end{cases} $$

From these we can see that $F_Z(x) = F_Y(x)$ for $x < 10$ and $x \geq 20$, but $F_Z(x) > F_Y(x)$ for $10 \leq x < 20$

So we see that Y FSD Z.

SSD

Y FSD Z implies Y SSD Z.