Given the following historical returns (HPR):

   Year                                              Return

   2017                                                18%

   2018                                                  -6%

   2019                                               30%

   2020                                              18%

  

1. Calculate the arithmetic mean.
2. Calculate the geometric mean.
3. Is the geometric mean lower or higher than the arithmetic mean?
4. Under what circumstances would the geometric mean equal the arithmetic mean?
5. Calculate the historical standard deviation (σ).

Given the following expected returns with the indicated probabilities:

  Probability                               Return

     30%                                       5%

     50%                                     10%

     20%                                     15%

  

1. Calculate the expected return.
2. Calculate the expected standard deviation (σ).