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# What is this stock's expected return?

J. Harper Inc.'s stock has a 50% chance of producing a 35% return, a 30% chance of producing a 10% return, and a 20% chance of producing a -28% return. What is Harper's expected return?

### 3 Answers

- John WLv 710 years agoFavorite Answer
Sounds like a textbook question asking for an arithmetic expectation which would be:

0.5 * 35% + 0.3 * 10% + 0.2 * -28% = 12.4%

But using an arithmetic expectation to evaluate an investment prospect puts you at risk of the St. Petersburg Paradox. Basically, if you subscribed to such a methodology you would be sinking everything you have into PowerBall tickets every time the Jackpot reached 200 million but we know that would be foolish. The solution proposed in the 18th century by Bernoulli is to adopt the log utility of wealth and take into consideration your net worth and the amount or your net worth you are to invest in the opportunity. Hence you would invest such that the following geometric mean of the outcomes is at a maximum (if W is your net worth and X is the amount you are to invest):

e^( 0.5 * ln( W + X * ( 1.35 - 1 ) ) + 0.3 * ln( W + X * ( 1.10 - 1 ) ) + 0.2 * ln( W + X * ( 0.72 - 1 ) ) )

Note that the scenario provided is very favourable and fundamentally unrealistic as it does not include the probability of total loss hence the above equation is at a maximum when X is 200.6% of the total net worth W which means you should use leverage on this investment.

In reality, the average lifespan of a Fortune 500 company is between 40 to 50 years so if you assume that there is a reasonable chance that a company would go out of business in 40 to 50 years (reasonable being 1-1/e or 63%) then there is a 2% chance that it would go out of business in any given year so you should always include that as one of the possible outcomes and the calculations would then say that you should invest no more than 78.1% of your net worth in this investment.

With geometric analysis, you wind up with reasonable limits to your investments and a minimum value for the amount of diversification needed. With the standard arithmetic reward to risk ratio analysis, you periodically get into situations where you are at risk of losing everything.