The price at which you will receive a margin call on a long position in a stock is given by:
\[margin\ call\ price\ =\ P_0\left(\frac{1\ –\ initial\ margin}{1\ –\ maintenance\ margin}\right)\]
where:
- \(P_0\): initial price of the stock
The price at which you will receive a margin call on a short position in a stock is given by:
\[margin\ call\ price\ =\ P_0\left(\frac{1\ +\ initial\ margin}{1\ +\ maintenance\ margin}\right)\]
It might be useful to know how these formulae are derived, and, more importantly, why you do not need to know them.
How these Formulae are Derived
Let:
- \(P_0\): initial price of the stock
- \(P_m\): stock price that will generate a margin call
- \(i\): initial margin
- \(m\): maintenance margin
The initial margin that you have to post (in dollars) is:
\[P_0\ ×\ i\]
Long Position
When the price changes from \(P_0\) to \(P_m\), the price changes by an amount equal to \(P_m\ –\ P_0\). The value of the margin account for a long position changes by the same amount (e.g., if the price rises $5.00, the value of the margin account rises by $5.00), and becomes:
\[\left(P_0\ ×\ i\right)\ +\ \left(P_m\ –\ P_0\right)\]
At the margin call price, the value in the margin account will be the maintenance margin:
\[P_m\ ×\ m\]
Thus, at the margin call price,
\begin{align}\left(P_0\ ×\ i\right)\ +\ \left(P_m\ –\ P_0\right)\ &=\ P_m × m\\
\\
P_m\ –\ \left(P_m\ ×\ m\right)\ &=\ P_0\ –\ \left(P_0\ ×\ i\right)\\
\\
P_m\left(1\ –\ m\right)\ &=\ P_0\left(1\ –\ i\right)\\
\\
P_m\ &=\ P_0\left(\frac{1\ –\ i}{1\ –\ m}\right)
\end{align}
Short Position
When the price changes from \(P_0\) to \(P_m\), the value of the margin account for a short position becomes:
\[\left(P_0\ ×\ i\right)\ +\ \left(P_0\ –\ P_m\right)\]
and,
\begin{align}\left(P_0\ ×\ i\right)\ +\ \left(P_0\ –\ P_m\right)\ &=\ P_m × m\\
\\
P_m\ +\ \left(P_m\ ×\ m\right)\ &=\ P_0\ +\ \left(P_0\ ×\ i\right)\\
\\
P_m\left(1\ +\ m\right)\ &=\ P_0\left(1\ +\ i\right)\\
\\
P_m\ &=\ P_0\left(\frac{1\ +\ i}{1\ +\ m}\right)
\end{align}
Example
If the market price of a stock today is $25/share, the initial margin is 50% and the maintenance margin is 30%, then if you have a long position in the stock you will get a margin call if the price drops to:
\[$25\left(\frac{1\ –\ 0.5}{1\ –\ 0.3}\right)\ =\ $17.86\]
and if you have a short position in the stock you will get a margin call if the prices rises to:
\[$25\left(\frac{1\ +\ 0.5}{1\ +\ 0.3}\right)\ =\ $28.85\]
You can check these values to ensure that they make sense: your initial margin is $25.00 × 50% = $12.50. If the price drops to $17.86 and you have a long position, your margin account is now $12.50 – ($25.00 – $17.86) = $5.36, and,
\[\frac{$5.36}{$17.36}\ =\ 0.30\]
If the price rises to $28.85 and you have a short position, your margin account is now $12.50 – ($28.85 – $25.00) = $8.65, and,
\[\frac{$8.65}{$28.85}\ =\ 0.30\]
Why You Don’t Need to Know These Formulae
Note, however, that you do not need to be able to calculate the price at which you will get a margin call; all you need to be able to do is check whether a given price is correct. The reason is simple: on the exam, if you’re asked the price at which a margin call will occur, you know that one of the three answer choices given is correct; all you have to do is check one of them: the middle price (answer choice “b”). Calculate the value of the margin account, and divide that by the price; if that’s the maintenance margin percentage, then “b” is the correct answer; if it’s not, you can deduce the correct answer.
For example, suppose that for the stock above you are asked to compute the price at which the long position would get a margin call, and the answer choices are:
- $17.86
- $18.50
- $20.00
You check answer choice “b”: the margin account value is $12.50 – ($25.00 – $18.50) = $6.00, and,
\[\frac{$6.00}{$18.50}\ =\ 0.323\]
This value is too big, so you know that answer “b” is incorrect, and you also know that the correct answer is smaller than $18.50, because at $18.50 the margin percentage is above the maintenance margin; thus, the price hasn’t fallen enough to drop the margin account to 30% of the price, so the correct answer has to be “a”.
There are a lot of formulae in the Level I CFA curriculum that you need to memorize, but if you can check the value of the margin account as shown here, you don’t need to memorize the formulae for the margin call price; save some room in your brain for the others you do need to memorize.