The signal is worth double if you can short

Question

An asset is worth \0 or \100, each equally likely, fair price \50. A signal is 80\% accurate (says "high" 80\% of the time when worth \100, "low" 80\% when worth \0). This time you may buy at \50 or short-sell at \50 (a short profits if the asset ends worth less than \50). What is the most you should pay for the signal now?

Accepted Answer

Without the signal. Both buying and shorting at the fair price of \50 have expected value 0, so profit is \0. With the signal. By symmetry each reading has probability 	frac12, with posteriors 0.8 after "high" and 0.2 after "low." After "high" (posterior 0.8): buying is worth 0.8(100) - 50 = +\30, so you buy. After "low" (posterior 0.2): now you short at \50. A short gains \50 if the asset is worthless and loses \50 if it is worth \100: 0.8(+50) + 0.2(-50) = 40 - 10 = +\30. So the "low" reading, which was worth nothing when you could only buy, is now worth +\30. Expected profit with the sign…

The signal is worth double if you can short

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