Question

3.5 Assume the returns of each asset are independent from each other, are the mean returns statistically different from each other?

Answer 1

Yes, it is true that mean value differ from each other, means there is some level of standard deviation attached to the mean values. Expected returns measures the mean values of profitability of investment returns. The expected return of a portfolio is the anticipated amount of returns that a portfolio may generate, whereas the standard deviation of a portfolio measures the amount that the returns deviate from its mean.

Expected Return can be calculated by multiplying the weight of its value by expected return and adding all the values up. For example there are three investments A,B,C. A takes 40% of the investment , B takes 35% and C takes 25%. A has an expected return of 5%, B has an expected return of 8% and C has an expected return of 2%. Thus,

Expected Return = [40% * 5% + 35% * 8% + 25% * 2% ] *100 = 5.3%

Standard deviation says how much the investment values differs from the mean values. The standard deviation of a two investment can be calculated in some steps:

1. 1. Squaring the weight of the first investment and multiplying it by the variance of the first investment and add it to the square of the weight of the second investment multiplied by the variance of the second investment.
2. Add this value to 2 multiply by the weight of the first investment and second investment multiplied by the covariance of the returns between the first and second asset.
3. Take the square root of that value, and the portfolio standard deviation is calculated.

Consider a two-asset investment with equal weights, variances of 6% and 5%, respectively, and a covariance of 40%. The standard deviation can be found by taking the square root of the variance. Therefore, the portfolio standard deviation is 16.6% (√(0.5²*0.06 + 0.5²*0.05 + 2*0.5*0.5*0.4*0.0224*0.0245))