Example
Setup:
Suppose you were going to make an investment into only one of three investment vehicles: stock, mutual fund, or certificate of deposit (CD). Further suppose, that the market has a 50% chance of increasing, a 30% chance of staying even, and a 20% chance of decreasing. If the market increases the stock investment will earn $1500 and the mutual fund will earn $900. If the market stays even the stock investment will earn $300 and the mutual fund will earn $600. If the market decreases the stock investment will lose $800 and the mutual fund will lose $200. The certificate of deposit will earn $500 independent of the market's fluctuation.
Question:
What is the expected value of perfect information?
Solution:
Expectation for each vehicle:
The maximum of these expectations is the stock vehicle. Not knowing which direction the market will go (only knowing the probability of the directions), we expect to make the most money with the stock vehicle.
Thus,
On the other hand, consider if we did know ahead of time which way the market would turn. Given the knowledge of the direction of the market we would (potentially) make a different investment vehicle decision.
Expectation for maximizing profit given the state of the market:
That is, given each market direction, we choose the investment vehicle that maximizes the profit.
Hence,
Conclusion:
Knowing the direction the market will go (i.e. having perfect information) is worth $350.
Discussion:
If someone was selling information that guaranteed the accurate prediction of the future market direction, we would want to purchase this in only if the price was less than $350. If the price was greater than $350 we would not purchase the information, if the price was less than $350 we would purchase the information. If the price was exactly $350, then our decision is futile.
Suppose the price for the information was $349.99 and we purchased it. Then we would expect to make 1030 - 349.99 = 680.01 > 680. Therefore, by purchasing the information we were able to make $0.01 more than if we didn't purchase the information.
Suppose the price for the information was $350.01 and we purchased it. Then we would expect to make 1030 - 350.01 = 679.99 < 680. Therefore, by purchasing the information we lost $0.01 when compared to not having purchased the information.
Suppose the price for the information was $350.00 and we purchased it. Then we would expect to make 1030 - 350.00 = 680.00 = 680. Therefore, by purchasing the information we did not gain nor lose any money by deciding to purchase this information when compared to not purchasing the information.
Note: As a practical example, there is a cost to using money to purchase items (time value of money), which must be considered as well.
Read more about this topic: Expected Value Of Perfect Information
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