ECON 251: Financial Theory
Spring 2012
Professor: John Geanakoplos
Yale University
TA: Bin Li
Problem Set #6: Answer Key
1.
A fund’s quarterly returns are independent.
Its long run annual volatility is 64%. In the long run,
what is its quarterly volatility?
Solution
:
Let random variable
be the quarterly return and Y be the long run annual return, then
( )
(
)
(
)
Since
are independent for each i. hence,
2.
Oklahoma and Miami play a series of games to determine who NBA champion is. Miami has a
70% chance of winning each game. Use the spread sheet to figure out what the odds are of
Miami winning a 5, 7, 9, or 11 game series.
Solution
:
We use backward induction to calculate the 5 game series.
1.
Define V(O,M) is the probability of Miami wins finally, where O is the number of Orlando
wins, and M is the number of Miami wins.
2.
Thus, V(0,3) = V(1,3) = V(2,3) = 1 and V(3,2) = V(3,1) = V(3,0) = 0
3.
By backward induction, when we know the value of game in (O+1, M) and (O, M+1),
V(O,M) = 0.7*V(O,M+1)+0.3*V(O+1,M)
4.
Hence,
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
From the Excel spread sheet, we could see, the probability of Miami winning a 5, 7, 9, and 11
game series are: 83.69%, 87.40%, 90.12%, and 92.18%, respectively.
3.
(Monte Hall problem): You
don’t
.t know which of three closed doors has a prize behind it. You
pick a door. Then Monte Hall (who does know which door has the prize) opens a door without a