Show that (left(frac{1}{2}, frac{1}{2} ight)) is the optimal policy for Example 18.2 . Example 18.2 (Volatility pumping)

Show that \(\left(\frac{1}{2}, \frac{1}{2}\right)\) is the optimal policy for Example 18.2 .

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Example 18.2 (Volatility pumping) Suppose there are two assets available for investment. One is a stock that in each period either doubles or reduces by one-half, each with a probability of 50%. The other just retains value-like putting money under the mattress. Neither of these investments is very exciting. An investment left in the stock will have a value that fluctuates a lot but has no overall growth rate. The other clearly has no growth rate. Nevertheless, by using these two investments in combination, growth can be achieved. To see how, suppose that we invest one-half of our capital in each asset each period. Thus we rebalance at the beginning of each period by being sure that one-half of our capital is in each asset. Under a favorable performance, our capital will grow by the factor +x2 = +1. Under an unfavorable performance, the factor will be+x+. Hence the expected growth rate of this strategy is = == m= In(+1)+In(+)~.059. Therefore em = 1.0607, and the gain on the portfolio is about 6% per period.