$"11.$ Assume that our pal John $($see PS #$2)$ is still fixated on the stock of X Co$.$ John follows the financial news and notes that when X reports a positive earnings surprise, the X stock price goes up; when X reports a negative earnings surprise, the X stock price declines. In the past, positive surprises have been rare, occurring only twice in $10$ years. John estimates $($e$.$g$.,$ from studying past earnings announcement results$)$ that if there is good earnings news this year, $10$ shares of X will be worth $$4$k$.$ If X reports poor earnings this year, $10$ shares will be worth only $$500.$ X Co$.$ stock is currently $$100/$share$,$ and John is considering investing his hard$-$earned bucks in $10$ shares of X$.$

#b$.$ Assume the information in $($a$.)$ above applies. Now, John has been watching lots of T$.$V$.$; way too much, actually. He notes that when the Federal Reserve Open Market Committee meets each month, if the Chairman is smiling, the economy does well; if the Chairman is frowning, the economy does poorly. In the past, when X Co$.$ earnings surprise has been positive, the Chairman had smiled $60\%$ of the time. When X Co$.$ earnings surprise has been negative, the Chairman had frowned $70\%$ of the time. Guess what? X Co$.$ is soon to report annual earnings, and Chairman Bernanke just smiled. Is

the stock a good buy at $$100/$share$,$ according to John$$s model? How much would John be willing to pay?

$[$Hint: first, realize that this is a Bayes problem, about updating your beliefs due to new information. The probability that the Chairman smiles, given that earnings surprise is $$good$,$ is $\mathrm{.}6($e$.$g$.,$ p$($Cs$|$Sg$)=\mathrm{.}6).$ Also, p$($Cf$|$Sb$)=\mathrm{.}7.$ But this is backwards! We already know the Chairman is smiling; now we want to know the probability that surprise will be good, right? Or$,$ p$($Sg$|$Cs$)=?$ Once you calculate the probabilities of surprise being good$/$bad based on the Chairman smiling, then you can calculate expected utility from buying the stock. Second, you need to know how to work backwards from utility to price. Assume a payoff of $$36$ gives utility of $6($square root function$).$ A payoff of $$16$ gives utility of $4.$ So$,$ a $$fair$$ gamble $($p $=\mathrm{.}5)$ paying $$16$ or $$36$ has an expected utility of $5.$ What amount for certain gives the same utility? $$25,$ right? So when you realize you are about to spend $$1$k on a gamble that pays off either $$4$k or $$500,$ do the following: use Bayes$$ Theorem to calculate the appropriate probabilities; calculate the expected utility of the gamble; then calculate the dollar amount for certain that gives this same utility $($the $$certainty equivalent$).$ This is$,$ basically, how investors with their own beliefs about a company$$s prospects $($hopefully using valuation theory like Ohlson$$s models$)$ set stock prices, based on their information.$]"$