Calculating Geometric Mean Returns
The monthly returns on a portfolio during the past year are gathered, and the simple average of these returns was 1.18%. The portfolio rose in value from JPY 1.20 million to JPY 1.37 million over the course of the year. Which of the following statement is _least accurate_ about the sample mean monthly return?
Correct!
This statement is not true. The geometric mean is always less than or equal to the arithmetic mean. The arithmetic mean is given as the simple average of 1.18%, and it is also possible to calculate 1.18% by taking the holding period return and dividing by 12.
However, the geometric mean monthly return is calculated as:
$$\displaystyle \left[ \frac{AUD 1.37 \mbox{ million}}{AUD 1.20 \mbox{ million}} \right]^{1/12} - 1 = 1.11 \% $$.
Without this calculation, it's possible that the geometric mean would be equal to the arithmetic mean if each monthly return was the same. But the starting and ending portfolio values show that there was some variance to the monthly returns throughout the year.
Incorrect.
This statement is true. The simple average of monthly returns is 1.18%, which is then the arithmetic mean monthly return. The arithmetic mean return is the best unbiased estimate of future returns.
Incorrect.
This statement is true. The arithmetic mean is the sum of all returns divided by 12, the number of observations, which is a simple average.
However, the question asks for the "least accurate" statement.
ORIGINAL TABLE
| Month | Return (%) |
|-----------|------------|
| January | –1.9 |
| February | 3.3 |
| March | 3.6 |
| April | –3.2 |
| May | 45.7 |
| June | 9.2 |
| July | 5.6 |
| August | –0.6 |
| September | 11.8 |
| October | –3.1 |
| November | 6.8 |
| December | 10.6 |
The arithmetic mean return is 1.18%
The geometric mean return is 1.18%
The best estimate of next month's return is 1.18%
