The geometric mean return is calculated using the formula:

$GM = \left( \prod_{i=1}^{n} (1 + r_i) \right)^{\frac{1}{n}} - 1$

Where:

• $r_i$ is the return for each period,
• $n$ is the number of periods (in this case, 5),
• $\prod$ represents the product of terms.

Steps:

1. Convert percentages to decimals:

   • -16% = -0.16
   • 17% = 0.17
   • -18% = -0.18
   • 15.7% = 0.157
   • 16.1% = 0.161

2. Add 1 to each return:

   • 1 + (-0.16) = 0.84
   • 1 + 0.17 = 1.17
   • 1 + (-0.18) = 0.82
   • 1 + 0.157 = 1.157
   • 1 + 0.161 = 1.161

3. Multiply all the terms: $0.84 \times 1.17 \times 0.82 \times 1.157 \times 1.161 = 0.953$

4. Take the 5th root (since there are 5 periods): $GM = (0.953)^{\frac{1}{5}} - 1$ $GM = 0.988 - 1 = -0.012$

5. Convert to percentage: $GM = -1.2\%$

Conclusion:

The geometric mean return for the given data set is -1.2%.

Would you like a more detailed explanation or have any questions?

Here are some related questions for further practice:

1. How would the geometric mean change if there were all positive returns?
2. How does the geometric mean differ from the arithmetic mean?
3. What impact does a negative return in the data set have on the geometric mean?
4. How would you calculate the average rate of return for a series of periods?
5. What would happen if the data set included a zero return for one of the periods?

Tip: Geometric mean is especially useful when dealing with returns because it accounts for compounding, which the arithmetic mean doesn't.