A stock had returns of 5.10%, 31.60%, and -21.20% in each of the past three years. Over the past four years, the arithmetic average annual return for the stock was 4.75%. What was the geometric annual return for the stock over the past four years?a. 2.91% (plus or minus 0.02 percentage points)b. 3.06% (plus or minus 0.02 percentage points)c. 3.37% (plus or minus 0.02 percentage points)d. 5.07% (plus or minus 0.02 percentage points)e. None of the above is within 0.02 percentage points of the correct answer

The correct option is c. 3.37% (plus or minus 0.02 percentage points).

Explanation:

Note that since we are asked to find the geometric annual return for the stock over the past four years but we are only given the stock returns for  each of the past three years, this implies that the stock return to be used for the first year of the past four years will be the arithmetic average annual return for the stock over the past four years.

Therefore, the geometric annual return for the stock over the past four years can now be computed using the following formula:

Geometric average return = {[(1 + r1) * (1 + r2) * (1 + r3) * (1 + r4)]^(1/4)} - 1 (1)

Where;

r1 = Stock return in year 1 = arithmetic average annual return = 4.75%, or 0.0475

r2 = Stock return in year 2 = 5.10%, or 0.0510

r3 = Stock return in year 3 = 31.60%, or 0.3160

r4 = Stock return in year 4 = -21.20%, or -0.2120

Substituting the values into equation (1), we have:

Geometric average return = {[(1 + 0.0475) * (1 + 0.0510) * (1 + 0.3160) * (1 - 0.2120)]^(1/4)} - 1

Geometric average return = 1.14166543988^(1/4) - 1

Geometric average return = 1.03367666813442 - 1

Geometric average return = 0.03367666813442, or 3.367666813442%

Rounding to 2 decimal places, we have:

Geometric average return = 3.37%

Therefore, the correct option is c. 3.37% (plus or minus 0.02 percentage points).

a manufacturer of printed circuit boards uses exponential smoothing with trend to forecast monthly demand of its product. at the end of december, the company wishes to forecast sales for january. the estimate of trend through november has been 200 additional boards sold per month. average sales have been around 1000 units per month. the demand for december was 1100 units. the company uses α = 0.20 and β = 0.10. make a forecast including trend for the month of january.