# linear algebra – How can I solve for Max Risk / Max Reward variables in both scenarios without having to flip the equations used?

My question is related to finance, but really it’s (I think) a fairly straightforward math problem.

There are two opposing (inverse) investment vehicles with the same equations that solve their Max Risk / Max Reward values, except the solutions to the two variables (Max Risk / Max Reward) are flipped depending on the scenario. To compute them, you have to identify which is which to determine which equation should be used. But it seems to me (intuitively) that since they’re just inversed it should be possible to write a single equation that would calculate the values regardless of which one is is a credit spread or a debit spread. See the following:

``````DISH 3/17 28c               DISH 3/17 30c
current value: 254.99       current value: 163.75
original cost: 273.75       original cost: 186.24
strike: 28                  strike: 30
``````

When placing a debit spread, the risk amount is the debit price plus any transaction costs. The potential reward equals the spread width minus the debit price, less transaction costs. For example, let’s look at a spread in DISH consisting of the purchase of the 28-strike call for \$273 and the sale of the 30-strike call for \$186. Resulting in a trade credit of -\$87. (debit of \$87)
In this case, the risk amount would be \$87 per contract. The potential reward would be the difference between the strikes (\$2) x 100 (\$200) plus the negative credit (or minus the debit) amount (-\$87) = \$112 per contract (less transaction costs).

Bought DISH 28c, Sold DISH 30c, Equations:

• Credit: sold.cost(\$186) – bought.cost(\$273) = -\$87
• Max Risk (Cost): Credit(-\$87)
• Max Reward: sold.strike(\$30) – bought.strike(\$28) = \$2 * 100 = \$200 + Credit(-\$87) = \$112
• Current Reward: Credit(-\$87) – sold.value(\$163) + bought.value(\$254) = \$4
• Percent Return: (Current Reward(\$4) / Max Reward(\$112)) * 100 = 3%