Someone plans to use x dollars buying some stock share, the stock price is $a$ dollars per share now. One year later, the stock price will possibly increase to $ra$ or decrease to $a/r$ (r>1) which the probability is $p$, or remain the price now which probability is $1-2p$. If he want to own more shares, should he buy the stock share or one year later?
Solution I:
The expected value of the share price one year later is $a(rp+\frac{p}{r}+1-2p)$.
So one year later he can buy $\frac{x}{a(rp+\frac{p}{r}+1-2p)}<\frac{x}{a}$.
So buy the stock share now.
Solution II:
One year later, if price is $ra$, he can buy $\frac{x}{ra}$ shares; if price is $\frac{a}{r}$, he can buy $\frac{xr}{a}$ shares.
Then the expected value of the shares he can buy one year later is $\frac{x}{ra}*p+\frac{xr}{a}*p+\frac{x}{a}*(1-2p)>\frac{x}{a}$.
So buy the stock one year later.
What's wrong with Solution II.