Week 5: Leverage

The one failure mode this course refuses to let you buy with real money, and the arithmetic that explains the refusal.

22 March 2027rationedhanded to you, never assessedOpen the slides

Every other week on this syllabus you are allowed to go and do the thing. This week you are not, and the reason is the subject of the lecture.

Under the bounded-loss policy, real-money positions are long-only and fully paid: you own the thing and the most you can lose is what you paid. That rule exists because of one number, which we are going to derive.

Borrowing against a position#

Take the 610 units of B at $18.00 from last week — $10,980 of your own money. Now borrow the same again and buy twice as much:

1,220 units at $18.00  =  $21,960
  your money           =  $10,980
  borrowed             =  $10,980

Your equity is what you would have left after repaying the loan:

equity  =  1,220P − $10,980

where P is the price. At $18.00 that is $10,980, as it should be.

The number the broker is watching#

Whoever lent you the money wants the position to stay worth comfortably more than the loan. The usual form is a maintenance margin: keep your equity above some fraction of the position’s value, or the position is closed for you. Take 25%:

(1,220P − 10,980) / 1,220P  =  0.25
 1,220P − 10,980            =  305P
                      915P  =  10,980
                         P  =  $12.00

$12.00. Not a price you chose, not a price you were warned about, and not a price at which you have any say. It falls out of three numbers: what you borrowed, how many units you bought, and a fraction in a document you did not read.

What that costs when the price comes back#

Suppose B falls to $11.50 and then recovers to $14.00.

Unleveraged. You hold 610 units throughout. You are down, and you are still holding:

610 × $14.00  =  $8,540

Leveraged. You were closed out at $12.00 on the way down:

equity  =  1,220 × $12.00 − $10,980
        =  $3,660

and the move from $12.00 to $14.00 happened to somebody else.

Now compare that to what you would have guessed. B went from $18.00 to $14.00, a fall of 22.2%. Twice the exposure, twice the loss, so you would expect

$10,980 × (1 − 2 × 0.222)  =  $6,101

You have $3,660. The missing $2,441 is not the size of the fall. It is the shape of it. Leverage did not multiply your loss by two; it removed your ability to still be holding when the price came back, and it did so at a price computed by somebody else.

This is what “the path matters” means, and it is the sentence to take out of this week.

The same effect without any borrowing#

Leverage makes this worse but does not create it. An instrument that falls 50% and then rises 50% leaves you at

0.50 × 1.50  =  0.75

down 25% after two moves that sound like they cancel. To get back from 0.50 to 1.00 the price has to rise 100%, not 50%. Every percentage fall demands a larger percentage rise to undo it, and leverage enlarges the falls.

Why the policy says no#

Go back to the borrowed position and ask what happens if B keeps falling. The position is worth the loan when

1,220P  =  $10,980  →  P  =  $9.00

At $9.00 your equity is zero. Below $9.00 you owe money you never deposited. At $8.00 the position is worth $9,760 against a $10,980 loan and you are $1,220 in debt, on top of having lost everything you put in.

Unleveraged, B at $9.00 means you are holding 610 units worth $5,490 — a bad week, and yours.

That is the whole argument. A $20 cap is not a cap if the instrument can send you a bill, and instruments that can send you a bill are sold to retail customers with the word “amplify” in the advertising. So this course teaches you to derive the $12.00 and forbids you from paying to experience it.

This week’s work#

On the paper track, run the overlay: take your own position history since week 1, apply 2:1, and compute the price at which you would have been closed out. Use your real numbers, not the example’s. Put the result in the log.

Most of you will find the number sits inside the range your position has already travelled this semester.