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Applying Expected Value Concepts to Option Investing

What is expected value?

Expected value is the average result that would occur if an experiment were to be repeated a very large number of times. For example, given a fair coin, if you gained $2 for each outcome of heads, and you lost $1 for each outcome of tails. If you flip the coin many many times, the average gain per toss (expected value) would be $0.50. This is computed by multiplying the probability of each outcome by its profit(loss) and summing these results.
(.5 x $2) + (.5 x $-1) = $0.50
If you flipped the coin one million times, you should expect to have gained about $500,000. Note that this result is not certain, you could lose $1,000,000 or gain $2,000,000, but the odds of either extreme occurring are miniscule. For any game of chance, if the expected value is in your favor and you can keep playing, you will accumulate winnings. If the "expected value" is against you, you will eventually go bankrupt.

Applying expected value to option investments

Expected value is computed by multiplying the profit(or loss) of each outcome by the probability of that outcome, then summing all of those results. Consider the below Bull Put Spread trade for ABC (now at $28.50)
Buy Put ABC ($24 strike expiring in 45 days) for $.20
Sell Put ABC ($26 strike expiring in 45 days) for $.45

Price Range Profit(Loss) Probability Profit x Probability
>$26 $.25 78% $0.195
$25.75 to $26 $.12 5% $0.006
$22 to $25.75 -$.85 5% -$0.043
<$24 -$1.75 >12% -$0.210
    Sum =  -$0.052

Break-even is at $25.75. In the price ranges of $24 to $25.75 and 25.75 to $26, the profit continuously changes. (When profit continuously changes, the analyzers cut each range into many very narrow slices to accurately calculate probability. To simplify this example an average profit for each range is used.) The expected value of -$.052 means that if you independently repeated this investment a large number of times, the average result of the investment would be a loss of $.052 per share ($5.20 per contract). The larger the expected value, the better your average profits will be. Avoid investments with negative expected values unless you know something that is in your favor that the analyzers have not considered. When the analyzers compute expected value, they include the effects of brokerage fees both in the reduction in your profit and in the likelihood that an option will be exercised. (The software offsets the exercise trigger point by $0.05 to account for the fees charged to the person exercising the option. Example: A Put option with a strike price of $26 will not be exercised if the stock finishes above $25.95 (strike price minus $.05), because such a trade would not be profitable to the person exercising the option once brokerage fees are considered.) Because these effects are included, the expected value displayed in reports is labeled "Net Expected Value".

Annualized Net Expected Return

The "Net Expected Value" of an investment is made more useful by taking into account the amount being placed at risk by the investment and the time period of the investment. Dividing "Net Expected Value" by the amount risked, yields the "Net Expected Return on Risk". Comparing investments requiring different time periods is facilitated by annualizing this return. The result is the "Annualized Net Expected Return".

"Annualized Net Expected Return" is intended to be used along with "maximum profit", "safety margin" and "probability of profit" as the primary means of ranking investments.

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