# Sequencing Stock Sales

A decision that individual investors commonly have to make when selling stocks is choosing which shares to sell given the different cost bases of different lot of shares. For investors that have multiple lots of the same stock, it’s good to go through a simple exercise to provide a quantitative decision calculus.

Suppose an investor Alice owns two different lots of shares in the stock of fictitious company BigCo, and both lots have been held for more than a year as to receive the same long-term capital gains tax treatment:

• Lot A: Alice has 100 shares of BigCo purchased at $10 • Lot B: Alice has 75 shares of BigCo, purchased at$15.

Given the current price of BigCo at, say, $22.50, and some belief about the future prospects of BigCo, should Alice sell lot A today followed by lot B at a later date, or sell lot B today followed by lot A at a later date? Let’s first establish a baseline. If Alice has no opinion and expects the future price of BigCo to be unchanged, then selling A then B would result in the same gains as selling B then A. At a price of$22.50, both actions would result in a gain of $1,812.50. Suppose now that Alice expects a bright and rosy future for BigCo, and that the price will rise by 20% some time in the future. If Alice sells lot A today at$22.50 followed by lot B at $27 in the future after the 20% increase, she stands to gain Selling lot B today followed by lot A at a later date, she stands to gain which is an improvement of$112.50! Since we’re already dealing with what might be obvious to many seasoned investors, let’s make the calculations explicit and see what the key variable is in deciding between selling lots A or B.

For the current price of $p$ and a future increase of $r$ to a price of $(1+r)p$, the breakeven point for selling $s_A$ shares at price $k_A$ in lot A then selling $s_B$ shares at price $k_B$ in lot B, versus selling lot B then lot A is

which reduces simply to $s_A = s_B$. That is, if the number of shares in lot A is equivalent to the number of shares in lot B, it doesn’t matter the order in which the lots are sold. Likewise, if the price goes up in the future, selling the lot with more shares in the future will be the more lucrative course of action. Intuitively, the more shares you have in a lot, the more you stand to gain (or lose) should the price go up (or down) in the future. That’s why Alice stands to gain more by selling her larger lot A at a later date, when the price has gone up.

The final piece of the equation for Alice is to factor in her beliefs about the future prospects of BigCo. If Alice expects a dismal future for BigCo, and thinks that the price will fall by 20% some time in the future, her losses will be symmetric to her gains if the price rises by 20%. Selling lot A then B after the price falls, Alice stands to walk away with $(22.5-10) \times 100 + (18-15) \times 75 = 1475$. Selling lot B then A, the proceeds come out to be $(22.5-15) \times 75 + (18-10) \times 100 = 1362.50$, which is \$112.50 less than the proceeds from the other course of action.

Ultimately, the best way for Alice to model these outcomes then is to assign probabilities to the events of the price going up or down at a future date, and then compute the expected value of each action. If Alice thinks that there’s a 50-50 chance that BigCo stock will be either 20% up or 20% down in the future, then her expected proceeds of selling lot A then B are

and proceeds from selling lot B then A are

In other words, regardless of the order in which Alice sells lot A and B, the expected value of her proceeds is the same. The important takeaway here is that the key factor in deciding which lot of stocks to sell depends on your beliefs about the future price of the stock. If you have no strong beliefs and simply assume that the price will follow a random walk, it doesn’t matter which lot you sell first. In that case, you should strive to optimize for diversification, utility, or some other personal metric.