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An investor has $100,000 to make an investment for one year. The investor is considering two options: placing the money in a bank, which guarantees a fixed annual gain of 15%, or an investment plan whose annual profit can be considered as a random variable whose values depend on prevailing economic conditions. Based on the past history of the second plan, an analyst has determined the possible values of the gain and calculated its probabilities, as shown in the table. Taking into account the expected gain of this second option, which of the two plans should be selected?

Let $X$ be the possible percentage of gain:

  • $P(X=30)=0.20$
  • $P(X=25)=0.20$
  • $P(X=20)=0.30$
  • $P(X=15)=0.15$
  • $P(X=10)=0.10$
  • $P(X=05)=0.05$

I chose the second option, because the expected value gave me $20.5$, was it okay?

The fact that the expected value is $20.5$ means that in the long run the investor is expected to earn 20%?

The expected value, if it does not belong to the range, is at least this between the minimum and the maximum of the range (to be finite)?

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    Well, if the investors utility function is simply expected return, then yes. Your computation is correct. Of course, some investors may be extremely risk averse and therefore prefer the stable return. And, yes, the expected value lies between the minimum and maximal values. – 2017-02-09
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    So, in the long run the investor is expected to earn 20%? – 2017-02-09
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    Well, you have to take compound and such into effect. All the expected return calculation shows is that the expected return in any one year is $20.5\%$. You said the investor has a one year time horizon so this is all that matters. – 2017-02-09
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    Just to stress the point: suppose the investment yields double or nothing with equal probability. Then the expected return is $0\%$, but if you keep making this bet you are sure to lose all your money eventually. – 2017-02-09
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    On the other hand, the data here indicates that at least you will never take a loss. The gain is predicted to be strictly positive, and the only randomness is by magnitude. There is a $15\%$ probability for not making as great a gain as the stable plan, but a $70\%$ probability of gaining over that level. It seems a good bet if you can manage your budget in case of a downturn (plan to fail before you plan to win). – 2017-02-09

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