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 ±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. Suitable variations of the birthday problem are new to this paper. 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? Portfolios containing options may lead to wash sales resembling these characteristics. This paper ends by exploring the Littlewood-Offord problem as it relates capital gains and losses with wash sales.
US Internal Revenue Service, Investment Income and Expenses (including Capital Gains and Losses) (2015) IRS Publication 550, Cat. No. 15093R for 2014 Tax Returns. See Pages 59+. https://www.irs.gov/pub/irs-pdf/p550.pdf
Constantinides, G.M. (1984) Optimal Stock Trading with Personal Taxes: Implications for Prices and the Abnormal January Returns. Journal of Financial Economics, 13, 65-89.
DasGupta, A. (2005) The Matching, Birthday and the Strong Birthday Problem: A Contemporary Review. Journal of Statistical Planning and Inference, 130, 377-389.
Bradford, P., Perevalova, I., Smid, M. and Ward, C. (2006) Indicator Random Variables in Traffic Analysis and the Birthday Problem. 31st Annual IEEE Conference on Local Computer Networks (LCN 2006), Tampa, 14-16 November 2006, 1016-1023.