To find the average annual gain using the geometric mean, we need to take into account the compounded nature of growth over multiple periods. Here are the steps:

1. Convert the percentages to growth factors by adding 1 to each percentage as a decimal:

   • 4% growth → $1 + 0.04 = 1.04$
   • 6% growth → $1 + 0.06 = 1.06$
   • 6% growth → $1 + 0.06 = 1.06$
   • 8% growth → $1 + 0.08 = 1.08$
   • 13% growth → $1 + 0.13 = 1.13$

2. Multiply all growth factors together to find the compounded growth factor over five years: $1.04 \times 1.06 \times 1.06 \times 1.08 \times 1.13$

3. Take the 5th root of the result to find the geometric mean, since we’re averaging over five years.

4. Subtract 1 and convert to a percentage to get the average annual gain.

Let's calculate:

$\text{Geometric mean} = \left(1.04 \times 1.06 \times 1.06 \times 1.08 \times 1.13\right)^{\frac{1}{5}} - 1$

Calculating, we get approximately:

$\text{Geometric mean} \approx 0.0731 = 7.31\%$

So, the average annual gain over the last five years is approximately 7.31%.

Would you like further clarification on any step?

---

Related Questions:

1. How does the geometric mean differ from the arithmetic mean in financial contexts?
2. What would happen to the geometric mean if one of the years had a negative return?
3. How does the geometric mean help in understanding compounded growth over time?
4. Can the geometric mean be used to calculate average losses, and how?
5. What is the impact of varying percentage changes on the overall average gain?

Tip:

The geometric mean is especially useful in finance for calculating average growth rates, as it accounts for compounding effects over time.