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Unformatted text preview: lity 1/2, or decreases by 50% with a probability 1/2. The second
stock either increases by 30% with a probability 2/3, or decreases by 50% with a probability
1/3. The ﬂuctuations in the two stocks are independent, so we have four scenarios: both stocks
go up (probability 2/6), stock 1 goes up and stock 2 goes down (probability 1/6), stock 1 goes
down and stock 2 goes up (probability 1/3), both stocks go down (probability 1/6). The
fractions of our capital we invest in stocks 1 and 2 are denoted by x1 and x2 , respectively.
The rest of our capital, x3 = 1 − x1 − x2 is not invested.
What is the expected growth rate of the log-optimal strategy x? Compare with the strategies
(x1 , x2 , x3 ) = (1, 0, 0), (x1 , x2 , x3 ) = (0, 1, 0) and (x1 , x2 , x3 ) = (1/2, 1/2, 0). (Obviously the
expected growth rate for (x1 , x2 , x3 ) = (0, 0, 1) is zero.)
Remark. The ﬁgure below shows a simulation that compares three investment strategies over
200 periods. The solid line shows the log-optimal...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
- Fall '13
- The Aeneid