My Math Forum Shares break-even point question

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 December 28th, 2018, 08:08 AM #1 Newbie   Joined: Dec 2018 From: UK Posts: 3 Thanks: 0 Shares break-even point question I’ve posted this here as I’m sure it’s an algebra problem and it's driving me nuts. I want to calculate the break-even point of 2 positions via a formula. For example if I have a long position of (+ 5) @ 1261 and a short position of (- 3) @ 1251 What is the easiest way of calculating break-even point if the market keeps trending up? I know the answer is 1276 ,as I’ve plotted it manually, but just can figure it out via calculation. It has to do with the ratio of the size which is 1.66 Any insight would be much appreciated.
December 28th, 2018, 02:15 PM   #2
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 Originally Posted by Anton Chigurh I’ve posted this here as I’m sure it’s an algebra problem and it's driving me nuts. I want to calculate the break-even point of 2 positions via a formula. For example if I have a long position of (+ 5) @ 1261 and a short position of (- 3) @ 1251 What is the easiest way of calculating break-even point if the market keeps trending up? I know the answer is 1276 ,as I’ve plotted it manually, but just can figure it out via calculation. It has to do with the ratio of the size which is 1.66 Any insight would be much appreciated.
$p = \text {break even price.}.$

$5(p - 1261) + 3(1251 - p) = 0 \implies 5p - 6305 + 3753 - 3p = 0 \implies$

$2p = 6305 - 3753 = 2552 \implies p = 1276.$

December 29th, 2018, 02:04 AM   #3
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 Originally Posted by JeffM1 $p = \text {break even price.}.$ $5(p - 1261) + 3(1251 - p) = 0 \implies 5p - 6305 + 3753 - 3p = 0 \implies$ $2p = 6305 - 3753 = 2552 \implies p = 1276.$
Thank you so much for your time JeffM1

This is great and does solve that problem, but there seems to be a variable missing.

For example, if I have +8@1261 and -5@1251 using this calculation does not work out:

2p = 10088 - 6225 = p = 1931.5

December 29th, 2018, 05:39 AM   #4
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 Originally Posted by Anton Chigurh Thank you so much for your time JeffM1 This is great and does solve that problem, but there seems to be a variable missing. For example, if I have +8@1261 and -5@1251 using this calculation does not work out: 2p = 10088 - 6225 = p = 1931.5
That is because, in your second case, the difference in shares long and shares short is 3, not 2. That is why I worked out your first example: understanding the logic behind a formula is far more important than memorizing a formula. Formulas that are not understood are dangerous because they frequently are misused.

$8(p - 1261) = \text { realized gain (or loss) on } 8 \text { shares long.}$

$5(1251 - p) = \text { realized gain (or loss) on } 5 \text { shares short.}$

$\therefore 8(p - 1261) + 5(1251 - p) = 0 \implies 8p - 10088 + 6255 - 5p = 0 \implies$

$8p - 5p = 10088 - 6255 \implies 3p = 3833 \implies p =1277 \text { and } \dfrac{2}{3}.$

Does that work out?

$8 \left ( 1277 + \dfrac{2}{3} - 1261 \right ) = 8 * \left ( 16 + \dfrac{2}{3} \right ) = 128 + \dfrac{16}{3} = 133 + \dfrac{1}{3}.$

$5 \left ( 1251 - \left \{ 1277 + \dfrac{2}{3} \right \} \right ) = -\ 5 * \left ( 26 + \dfrac{2}{3} \right ) = -\ \left ( 130 + \dfrac{10}{3} \right ) = -\ \left ( 133 + \dfrac{1}{3} \right ).$

Yes, that works exactly. In practice, of course, it may not be possible to achieve the exact breakeven price due to fractions.

The general formula is as follows:

$q = \text { purchase price on long transaction;}$

$r = \text { sales price on short transaction;}$

$m = \text { number of shares long;}$

$n = \text { number of shares short; and}$

$p = \text { breakeven price.}$

$m = n \text { and } q \ne r \implies p \text { does not exist.}$

$m = n \text { and } q = r \implies p \text { is anything.}$

$m \ne n \implies p = \dfrac{mq - nr}{m - n}.$

To derive the last:

$m(p - q) + n(r - p) = 0 \implies mp - mq + nr - np = 0 \implies mp - np = mq - nr \implies$

$p =\dfrac{mq - nr}{m - n}.$

 December 30th, 2018, 08:48 AM #5 Newbie   Joined: Dec 2018 From: UK Posts: 3 Thanks: 0 Ah of course. I feel so stupid. Thank you so much for this. Sent you a PM. Last edited by skipjack; February 4th, 2019 at 12:02 AM.

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