Stochastic Dominance: The Quantile

4.5 Summary

In this chapter, the FSD and SSD stochastic dominance rules were formulated in terms of the distribution quantiles. The advantage of the stochastic dominance rules based on the quantile formulation is that they can be extended relatively easily to the case where lending and borrowing at the riskless interest rate is allowed. A mix of the riskless asset with the risky asset, x_α, is given by x_α=(1−α) r+α x whose cumulative distribution is F_α. If there is an α>0 such that F_α dominates G by FSD, SSD or TSD, then we can safely conclude that for any mix of G with the riskless asset there will be a mix of F with the riskless asset which dominates it by these rules, respectively. Thus, if F_αDG for one positive value α, then the set F_α will dominate the set G_β. Such dominance is called Stochastic Dominance with a Riskless Asset (SDR). For U∈U₁, we developed FSD with a riskless asset denoted by FSDR. Similarly, for U∈U₂ we presented SSDR. Finally, we established stochastic dominance rules with a riskless asset in the more realistic case where the borrowing rate r_b is higher than the lending rate r_l. The quantile framework is also more convenient for developing an algorithm for FSD and SSD (as will be seen in the next chapter), but not allowed to be employed instead of the cumulative distributions for TSD and higher degree SD rules.