Joe and Peter each got a New Year's gift of one share of Google stock from their grandmother. The price of this share at the end of the year is given by the random variable X. Joe simply waits until the end of the year and sells his share then, receiving $X at the end of the year. Peter actively buys and sells fractions of his share (yes, you can do that!), and the number of shares he has at the end of the year is denoted by the random variable Y. At the end of the year Peter sells all his shares, receiving $XY. Both Joe and Peter measure the performance of their strategies by the mean-variance criterion: i.e., they compute he metrics EX-Var(X) and E(XY) – Var(XY), respectively, and see which one is larger (this determines the winner of the game).
Joe and Peter each got a New Year's gift of one share of Google stock from their grandmother. The price of this share at the end of the year is given by the random variable X. Joe simply waits until the end of the year and sells his share then, receiving $X at the end of the year. Peter actively buys and sells fractions of his share (yes, you can do that!), and the number of shares he has at the end of the year is denoted by the random variable Y. At the end of the year Peter sells all his shares, receiving $XY. Both Joe and Peter measure the performance of their strategies by the mean-variance criterion: i.e., they compute he metrics EX-Var(X) and E(XY) – Var(XY), respectively, and see which one is larger (this determines the winner of the game).
Chapter8: Sequences, Series,and Probability
Section: Chapter Questions
Problem 41CT: On a game show, a contestant is given the digits 3, 4, and 5 to arrange in the proper order to form...
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![Joe and Peter each got a New Year's
gift of one share of Google stock from their grandmother. The price of this share at
the end of the year is given by the random variable X. Joe simply waits until the
end of the year and sells his share then, receiving $X at the end of the year. Peter
actively buys and sells fractions of his share (yes, you can do that!), and the number
of shares he has at the end of the year is denoted by the random variable Y. At the
end of the year Peter sells all his shares, receiving $XY. Both Joe and Peter measure
the performance of their strategies by the mean-variance criterion: i.e., they compute
the metrics
EX - Var (X) and
E(XY) – Var(XY),
=
respectively, and see which one is larger (this determines the winner of the game).
Assuming that E[Y|X] 1, show that one of the players cannot possibly win (de-
termine which one). You are expected to provide a rigorous argument in support of
your conclusion.
(Hint: you need to compare the expectations and variances of X and XY. To do
this, you need to use the tower property of conditional expectation and the Jensen's
inequality)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3298c6a-8b99-4df5-b07f-e611d4d793cb%2Fc8c3ccee-fcd6-4692-b6b5-497d115fab71%2F5y1zz3f_processed.png&w=3840&q=75)
Transcribed Image Text:Joe and Peter each got a New Year's
gift of one share of Google stock from their grandmother. The price of this share at
the end of the year is given by the random variable X. Joe simply waits until the
end of the year and sells his share then, receiving $X at the end of the year. Peter
actively buys and sells fractions of his share (yes, you can do that!), and the number
of shares he has at the end of the year is denoted by the random variable Y. At the
end of the year Peter sells all his shares, receiving $XY. Both Joe and Peter measure
the performance of their strategies by the mean-variance criterion: i.e., they compute
the metrics
EX - Var (X) and
E(XY) – Var(XY),
=
respectively, and see which one is larger (this determines the winner of the game).
Assuming that E[Y|X] 1, show that one of the players cannot possibly win (de-
termine which one). You are expected to provide a rigorous argument in support of
your conclusion.
(Hint: you need to compare the expectations and variances of X and XY. To do
this, you need to use the tower property of conditional expectation and the Jensen's
inequality)
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