Foundations for Wash Sales

TL;DR

This paper models wash sales using variations of the birthday problem and explores their implications for tax advantages, providing probabilistic insights into wash sale occurrences in unmanaged portfolios.

Contribution

It introduces a novel probabilistic framework for understanding wash sales by applying birthday problem variations and connects wash sales to the Littlewood-Offord problem.

Findings

Necessary conditions for wash sales based on birthday problem variants

Probabilistic estimates of wash sale likelihood in random portfolios

Insights into capital gains and losses related to wash sales

Abstract

Consider an ephemeral sale-and-repurchase of a security resulting in the same position before the sale and after the repurchase. A sale-and-repurchase is a wash sale if these transactions result in a loss within $\pm 30$ calendar days. Since a portfolio is essentially the same after a wash sale, any tax advantage from such a loss is not allowed. That is, after a wash sale a portfolio is unchanged so any loss captured by the wash sale is deemed to be solely for tax advantage and not investment purposes. This paper starts by exploring variations of the birthday problem to model wash sales. The birthday problem is: Determine the number of independent and identically distributed random variables required so there is a probability of at least 1/2 that two or more of these random variables share the same outcome. This paper gives necessary conditions for wash sales based on variations on the birthday problem. This allows us to answer questions such as: What is the likelihood of a wash sale in an unmanaged portfolio where purchases and sales are independent, uniform, and random? This paper ends by exploring the Littlewood-Offord problem as it relates capital gains and losses with wash sales.