Statistics for Business: Decision Making and Analysis, 3rd Edition

10.5 IID Random Variables

Along with picking a portfolio, day traders need to decide how long to invest. The day trader we’ve been helping gives up the same amount of interest if she invests $200 for one day or $100 for two days. Is one plan better than the other?

If she invests $200 on Monday in IBM, then the random variable that describes the change in the value of her investment is 2X. Both shares change the same way; both rise, fall, or stay the same. The mean is E(2X) = 2E(X) = 0.20, and the variance is Var(2X) = 4Var(X) = 4 × 4.99. The Sharpe ratio is thus

$$S\left(2X\right)=\frac{2{\mu }_X-2r_f}{\sqrt{\text{Var}\left(2X\right)}}=\frac{0.20-0.03}{\sqrt{4\times 4.99}}\approx 0.038$$

That’s the same as S(X) because doubling her money has no effect on the Sharpe ratio.

Now let’s figure out what happens if she invests $100 ...