Item P2: 0.2 0.1 Ope 1.54 0.15 0.05 0 -0.05 M drift u 0.1 -0.1 Hexcess available portfolios -0.15 0.05 Hriskless 0.1 0.2 0.3 0,4 0.5 volatility o Figure 1. The efficient frontier (green straight line) intersects the riskless asset R (here /riskless = 0.05 per time unit) and is tangent to the space of available portfolios (green oval), touching at the point defined as the market portfolio, M (here UM =0.1 per time unit, resulting in an excess expected return of M compared to R Of Mexcess =0.05 per time unit; the volatility of the market portfolio OM =0.18 per square-root of time unit), corresponding to leverage /=1, left arrow. The color coding shows the expected, more specifically time-average, growth rate. g = u -02/2; the portfolio of optimal leverage (see section 3, here /opt ~1.54) along the efficient frontier is indicated by the right arrow. Both for fixed volatility o and fixed expected return, which we call the ensemble-average growth rate, u, there are no obtainable portfolios (those below the efficient frontier) whose time-average growth rates exceed that at the efficient frontier. Zero time-average growth rate is indicated white.All estimators are of the general form er no . J is the maximum likelihood estimator of the mean log-return from a sample of length 7. The investment horizon is H years. The values for & are as follows: Arithmetic estimator (A): k = 1 Geometric estimator (G): k -0 Unbiased estimator (U): ku- 1 - HIT Minimum MSE estimator (M): ky=1 - 3 HIT Condition Ordering Bias 0 T> >H ky= ky = 1 M = U = A G T > 3H 0 T > H ky T KM