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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.

Wash sales occur when a security is sold and quickly bought back with the sole intent to capture a tax loss from the sale. Wash sales impact a portfolio’s tax liabilities. Deter- mining the likelihood of wash sales is also important for understanding investment strategies and for comparing actively and passively managed portfolios. Wash sales apply to investors, but not to market makers.

Taxes play a significant role in economics and finance. Taxes influence behavior, shape the engineering of financial transactions, and sometimes have unintended consequences. Therefore, thoughtful analysis is imperative for taxes. This paper adds firm mathematical foundations to aid the understanding of wash sale taxes.

The main goal of this paper is: To provide foundations for certain wash sales-in cases when they may occur as well as the capital gain implications. This may also help differentiate managed funds and unmanaged index funds in terms of wash sales.

Wash sales are sometimes created by the exercise of options, thus a portfolio manager may not be able to avoid a wash sale in some contexts. For example, suppose an in-the-money American-style put option is written in a portfolio. Provided this option remains in-the-money, it may be exercised by its holder^{1} at anytime up to its expiry. If the exercise of this put option replaces shares sold at a loss in the prior 30 days, then this is a wash sale. This option’s exercise is beyond control of the portfolio manager.

The foundations given here start with variations of the classical birthday problem from probability theory [

For convenience, let

Suppose a security is sold at a loss on day

Definition 1 (US wash sale [

1) The loss

2) The cost-basis of the shares repurchased on

Short positions may also be wash sales. For example, consider holding a short position of 100 shares of a security starting on date

Consider a wash sale as described by Definition 1, where

This means such a wash sale gives

Wash sales may be avoided by restricting each security in a portfolio to be either purchased or sold only every 31 calendar days. This restriction may not be suitable for many portfolios. In a portfolio containing options, it may be impossible to maintain this restriction.

It has also been suggested, e.g. [

Historically many securities are assumed to only trade on about

There has not been much research on wash sales, e.g., [

The birthday problem is classical.

Definition 2 (Birthday-Collision) Given two random variables

To model random wash sales, this paper assumes independent identically distributed random variables. A common statement of the birthday problem is:

Definition 3 (Birthday Problem) Consider n days in a year and k independent identically distributed (iid) uniform random variables whose range is

According to a blog post by Pat B [

Bounds of day counts for the birthday problems include [

The birthday problem applied to boys and girls (random variables with different labels) are discussed in [

Tight bounded Poisson approximations for birthday problems are given by [

Applications of the birthday problem include: computer security [

Results on the expectation for getting j different letter k-collisions are given by [

The Littlewood-Offord problem hails from complex analysis [

Section 2 reviews variants the birthday problem applied here. First the classical birthday problem is discussed. Next this section progresses through the

Subsection 2.1 gives an example of wash sales based on boy-girl birthday collisions of a single day.

Section 3 generalizes results of the previous sections. In particular, it shows how to compute

Subsection 3.1 gives an example of wash sales based on boy-girl birthday collisions over a range of

Finally, Section 4 explores how wash sales impact capital gains and losses. Since wash sales are capital losses, they may offset capital gains. Several results, including the Littlewood-Offord problem, are applied to capital gains and losses as they may be impacted by wash sales.

The birthday problem is often applied to finding the probability of coincidences. So there is a rich literature on variations of the birthday problem [

A key question is: Over n consecutive days for what integer k does

given n days, what is the least k iid uniform random variables so that

Solutions to this basic variation of the birthday problem are well known. The probability

then k birthdays can be in

elements of

for

Starting with n and a probability

Another classical approach is to look at the random variable X as the sum of all birthday-collisions of k people over n days, see for example [

a Bernoulli trial, so X is binomially distributed. Thus,

In the case of the

Two birthdays

The next definition is based on [

Definition 4 (±d Birthday Collisions) Consider n days in a year, spans of less than

In n days with a

there is a probability of at least 1/2 where at least two such random variables are fewer than d days from each other.

Definition 5 (Blocks of days) Let

A block of days contains a single birthday on one of its end-points. The birthday

The days between

Take k iid uniform random variables and consider

pings of the k random variables. Thus, to get the probability of at least one

Theorem 1 ( [

for

Using the bound

Note, Theorem 1 with

The falling factorial is

In these terms, Theorem 1 may be expressed as

The next classic result is important.

Lemma 1 (Classical) Let

The next definition is based on [

Definition 6 (Boy-Girl Birthdays) Consider n days in a year and two sets of dis- tinctly labeled iid uniform random variables all with range

For instance, in n days,

Stirling numbers of the second kind [

non-empty subsets is

Due to their nature, it is common to define Stirling numbers of the second kind

recursively [

The next classical equality counts the number of functions from

expressed as the number of non-empty i partitions of the

Theorem 2 ( [

The next Lemma is from [

Lemma 2 ( [

Consider a portfolio

Suppose portfolio

asset

Take ^{2} random variables

for

To apply a suitable version of Chernoff's bound ( [

So, for example, take

Select the probabilities that the number of buys and sales are the same, given

Let h be half the total trades

In fact,

Assuming the portfolio

Necessary conditions are given here for wash sales where a purchase and sale are within

Definition 7 (Boy-Girl

For example, starting with

and one boy have

The next result is based on [

Theorem 3. Consider n days in a year, a span of

Proof. This proof calculates the probability of not having no boy-girl

Given n days, a

The value

injective functions to

Now, consider placing the i and j partitions in separate locations among the

mappings of boys to i non-empty partitions and independently the number of injective

mappings of girls to j non-empty partitions.

This completes the proof.

Wash Sale Example 2: d = ±30 Calendar DaysStart with the same setup as the previous wash sale example from subsection 2.1.

Let h be half the total trades

Consider only a single asset type. The intuition behind these probabilities is straight- forward. For instance, consider

Capital gains or capital losses may be rounded to the nearest integer for US tax calculations. Provided all trades are rounded. Rounding drops the cents portion for gains whose cents portion is 50-cents or below. Rounding adds a dollar to the dollar portion of gains whose cents portion is greater than 50 cents while dropping the cents portion. Losses work the same way. Gains and losses must all be rounded or none must be rounded. So, from here on, let all gains or losses be integers.

Long term capital gains and losses are aggregated and at the same time short term capital gains and losses are aggregated. At the end of the tax year the long term and short term aggregates are added together to get the final capital gain or loss for taxation.

The focus here is capital gains or losses for capital assets that may have wash sales. Wash sales are losses, but losses may offset gains. The study of options and their associated premiums is classical [

In a portfolio, individual capital gain values and individual capital loss values are usually distinct. Though rare, identical capital gains and capital losses are possible. Identical capital gains or losses are possible for portfolios built using options. We are ignoring option premiums. That is, asset purchases may be done via the exercise of cash-covered American-style put options. Also asset sales may be done via the exercise of American-style covered-call options. In these cases with options that become in-the-money, a portfolio manager has no control of the asset sales or purchases or timing of such trades. See

Most often, put or call option strike prices are at discrete increments. For example, many put and call equity options have strike prices in $5 or $10 increments. Suppose a portfolio is built only using the exercise of American-style options. Many asset gains and losses may be for identical amounts. Of course, this depends on the size of the underlying positions or the number of options written. Options with the same expiry on identically sized underlying assets may have very different values [

In such option-based portfolios assume uniform, independent, and random capital gains and capital losses. This may be modeled by the Littlewood-Offord Problem.

Definition 8 is classical and extensive discussion may be found in the likes of [

Definition 8 (Littlewood-Offord Problem) The integer Littlewood and Offord’s problem is given an integer multi-set

Assuming equal probability of gains and losses and no drift [

Over a tax year, the total capital gain or loss is

In an optimal solution of this version of the Littlewood-Offord problem, [

the n-element multi-set

The next lemma’s proof follows immediately from the linearity of expectation given Rademacher random variables. See, for example, [

Lemma 3. Consider any integer multi-set

For any Rademacher random variable

Theorem 4. Consider any non-negative integer vector v and the random variable

Thus, the lowest variance,

Theorem 4 implies the next corollary.

Corollary 1. Assume

Corollary 1 highlights an exceptional case where all capital gains and capital losses are the same. Wash sales require the loss and gain to be from essentially the same security.

The generality of Theorem 4 asserts large variances too. Consider the set

follows since the sum is a geometric series.

Definition 9 (Distinct sums of a set or multi-set V) Consider a set or multi-set

are distinct iff there is some

Given any multi-set of positive integers

An important observation by [

In particular, take any distinct sums

Thus, the set

Theorem 5. Among all sets of distinct positive integers where no two distinct sums add to the same value, the set

Proof. Suppose, for the sake of a contradiction, that

Take the next enumeration of the 2^{n} distinct sums,

Let

The difference of any two distinct sums

giving

which must be even.

Starting from

Given a set of distinct positive integers V where

values, erasing a wash sale loss may have a very large impact. In particular, the multi-set

because all losses are of the form

Since

The following tail bound is given by [

Since by Theorem 4,

Suppose

Following

The term

The boy-girl

Given any number of boy-girl

This paper shows the probabilistic method may be used to model some tax implications for wash sales. Variations of the birthday problem and the Littlewood-Offord problem are applied to certain tax implications of wash sales.

Modeling and simulating taxes are important in both public policy settings as well as in practical tax planning. In public policy settings, conflicting fiscal and social policies make tax rules contentious. In tax planning, unexpected events may have serious consequences. Thus, reducing certain taxes to mathematical terms gives an unusual level of percision. Such percision can only benefit public policy and tax planning.

Thanks to Noga Alon and C.-F. Lee for insightful comments.

Bradford, P.G. (2016) Foundations for Wash Sales. Journal of Mathematical Finance, 6, 580-597. http://dx.doi.org/10.4236/jmf.2016.64044