Suppose we have two options for investments, a bond that always returns 6% increase each year, and a stock that increases your investment by 50% with 0.8 probability or decreases your investment by 100% with probability 0.2 each year. We can invest a percentage p of our investment in the stock and 1-p of our investment in the bond.If we define a random variable In to be 1 when the stock goes up and 0 when the stock goes down for the year n, then find functions f(p) and g(p) such that:

Xn=f(p)Xn1+g(p)Xn1In . (This means that if stocks do well, the second term will evaluate to g(p)Xn1 and if the stocks do bad it will evaluate to 0.)