Question

You observe the following annual rates of return over the previous 6 years:

Year	Rate of return
1	20%
2	30%
3	10%
4	-40%
5	-20%
6	30%

What is the arithmetic annual average return? Please state your answer in percentage form with two digits after the decimal, e.g., 10.00 represents 10%.

What is the geometric annual average return? ? Please state your answer in percentage form with two digits after the decimal, e.g., 10.00 represents 10%.   

If these past 6 returns are representative of typical years, and what we expect next year, which average is the expected return for the upcoming year? (Arithmetic, or Geometric)

Answer 1

Answer a.

Arithmetic Average Return = [r1 + r2 + r3 + r4 + r5 + r6] / 6
Arithmetic Average Return = [0.20 + 0.30 + 0.10 + (-0.40) + (-0.20) + 0.30] / 6
Arithmetic Average Return = 0.30 / 6
Arithmetic Average Return = 0.05 or 5.00%

Answer b.

Geometric Average Return = [(1 + r1) * (1 + r2) * (1 + r3) * (1 + r4) * (1 + r5) * (1 + r6)]^(1/6) - 1
Geometric Average Return = [(1 + 0.20) * (1 + 0.30) * (1 + 0.10) * (1 - 0.40) * (1 - 0.20) * (1 + 0.30)]^(1/6) - 1
Geometric Average Return = 1.070784^(1/6) - 1
Geometric Average Return = 1.0115 - 1
Geometric Average Return = 0.0115 or 1.15%

Answer c.

If these past 6 returns are representative of typical years, then you should expect to earn geometric return of 1.15% for the upcoming year.