Applied Math - Averaging Down in Stocks

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Introduction

A stock is a piece of ownership of a company. If we purchase shares of a company, we’re owning part of that company. Stock prices go up and down for a variety of reasons. One major driving factor behind the price of a stock is the fair market value of a company. If company A wanted to buy company B, they would have to pay something close to the current stock price for every share of the company.

Sometimes an organization or an individual buys a stock, then the price goes down. If the organization or individual decides that the company is still a good company, they might guess that the stock price will go up soon. They can buy more stock at the lower price. This is sometimes called "averaging down". If we bought a stock at $5 per share and then the price went down to $3 per share, how much stock would we have to buy in order to make our average price paid for the stocks to be $4.50?

It’s important to note that a stock price is never guaranteed to go back up, no matter how long we own it. Many companies go bankrupt, making the stock worthless.

Example:

1.

A person owns 686 shares of a company. They bought these shares at an average cost of $3.44 per share.

A year goes by. The stock price has currently fallen to $0.94 per share. How much money would they lose if they sell their shares right now?

You can calculate how much they have lost by calculating how much they have lost for one share, then multiply that loss by the 686 shares that they own.

Dollar lost per share: $3.44 - $0.94 = $2.50 per share.

Total loss: $2.50 loss per share \(\times\) 686 shares = $1,715 total loss



2.

Stocks can continue to decline in price. However, this person thinks that the company will improve and that the stock price will go up soon. They want to buy enough shares to bring their average price down to $2 per share. How many shares should they purchase at a price of $0.94 per share in order to bring their average price per share down to $2 per share?

We want to calculate

\[\frac{ \$ 2 }{1 \text{ share}}\]

We want dollar units in the numerator and shares in the denominator. Add the money we have in $3.44 shares to the money we will have at $0.94 per share, then divide by the total number of shares.

The one number we can control is the number of shares we purchase. We can call this \(x\).

\[\frac{\text{total money we have in shares}}{\text{total shares we own}} = \frac{ \$ 2 }{1 \text{ share}}\]

\[\frac{\text{cost for \$3.44 shares} + \text{ cost for new \$0.94 shares}}{686 \text{ shares } + x \text{ new shares}} = \frac{ \$ 2 }{1 \text{ share}}\], or

\[\frac{3.44 \cdot 686 + 0.94x}{686 + x} = 2\]

The cost for \(x\) shares at $0.94 per share is $0.94x.

Solve for \(x\) by getting \(x\) out of the denominator. Multiply both sides by \(686 + x\) to get \(x\) out of the denominator.

\[\frac{3.44 \cdot 686 + 0.94x}{686 + x} = 2\]

\[\frac{3.44 \cdot 686 + 0.94x}{686 + x} \cdot \frac{686 + x}{1} = 2 \cdot (686 + x)\]

\[3.44 \cdot 686 + 0.94x = 2(686 + x)\]

\[2,359.84 + 0.94x = 2(686 + x)\]

Distribute the 2.

\[2,359.84 + 0.94x = 2(686 + x)\]

\[2,359.84 + 0.94x = 2 \cdot 686 + 2 \cdot x\]

\[2,359.84 + 0.94x = 1,372 + 2 x\]

Subtract \(0.94x\) from both sides.

\[2,359.84 + 0.94x = 1,372 + 2x\]

\[2,359.84 + 0.94x - 0.94x = 1,372 + 2x - 0.94x\]

\[2,359.84 = 1,372 + 1.06x\]

Subtract 1,372 from both sides.

\[2,359.84 = 1,372 + 1.06x\]

\[2,359.84 - 1,372 = 1,372 - 1,372 + 1.06x\]

\[987.84 = 1.06x\]

Divide both sides by \(1.06\).

\[987.84 = 1.06x\]

\[\frac{987.84}{1.06} = \frac{1.06x}{1.06}\]

\[931.92 \approx x\]

Or, in other words, they should buy 932 shares to bring their average cost per share close to $2.

Practice Problems

1. A person owns 124 shares of a company. They bought these shares at an average cost of $30.92 per share.

A year goes by. The stock price has currently fallen to $15.26 per share. How much money would they lose if they sell their shares right now?



2. A person owns 124 shares of a company. They bought these shares at an average cost of $30.92 per share.

A year goes by. The stock price has currently fallen to $15.26 per share. How much stock should they buy the stock at the price of $15.26 per share in order to have an average share price of $20 per share?

Theory Questions

1. Consider the first equation in the example:

\[\frac{3.44 \cdot 686 + 0.94x}{686 + x} = 2\]

Say the stock price is now close to $0.

\[\frac{3.44 \cdot 686 + 0x}{686 + x} = 0\]

How many shares would an individual have to purchase in order to have an average price of $0?



2. What happens to a stock price when a company becomes bankrupt?



3. When should a person decide against averaging down on a stock?