Investor has a diversified portfolio of stock?

Question

The portfolio contains shares of 1,000,000,000
companies each initially worth $1.

Each day every share either gains or loses 1% of its
value with 50-50 probability, wrt previous day.
What is expected value of the portfolio after 1000 days?

Answer 1

Let's say P(v) is the probability that a particular stock has value v on day t, then the expectation value on day t+1 is:

=
= Sum over v: P(v) * (1.01*v + .99*v)/2
= Sum over v: P(v) * v
=

That is, the expectation value doesn't change. The expectation value of a sum of random variables is the sum of the expectation values so that doesn't change either.

--> Ans = 1,000,000,000

Answer 2

Two answers:

(1) if the stock value is rounded off to a penny each night, it's $0.49 per share. The stock can go down one penny from $0.50, but then it would get "stuck" at $0.49 (because 1% up rounds off to no change). Over a long trial starting at $1.00, the stock would be very likely to hit all the values around $1, so I'd go with $490,000,000.00.

(2) Otherwise, if the stock value is carried out to infinite precision, there's a slight bias toward lower numbers -- the geometric mean of 1.01 (1% up) and 0.99 (1% down) is not 1.0 but instead 0.9999499987.... In that case the answer is (0.99994...^1000) which is 0.951227046272. In that case the value is $951,227,046.27

Answer 3

Expected value=x*p(c)

= SUM (1 to 1000) ( .5*1.01)+(.5*.99)

=_____

I don't have a calculator handy