Explanation

The geometric mean return is calculated using the formula:

$\text{Geometric Mean} = \left[\prod_{i=1}^{n}(1 + R_i)\right]^{\frac{1}{n}} - 1$

Where:

• $R_i$ are the individual returns
• $n$ is the number of periods

Step-by-step calculation:

1. Convert percentages to decimal form:

   • Year 1: 12.2% = 0.122
   • Year 2: -8.5% = -0.085
   • Year 3: 6.7% = 0.067
   • Year 4: -3.3% = -0.033

2. Calculate (1 + R) for each year:

   • Year 1: 1 + 0.122 = 1.122
   • Year 2: 1 + (-0.085) = 0.915
   • Year 3: 1 + 0.067 = 1.067
   • Year 4: 1 + (-0.033) = 0.967

3. Multiply all (1 + R) values: $`$1.122 \times 0.915 \times 1.067 \times 0.967 = 1.0589$4`. Take the 4th root (since n=4): $$(1.0589)^{\frac{1}{4}} = 1.0144$5. Subtract 1 and convert to percentage: $`$1.01`44 - 1 = 0.0144 = 1.44%$$

Verification:

• Option A: 1.45% (closest to our calculated 1.44%)
• Option B: 1.78%
• Option C: 5.93% (this would be the arithmetic mean, not geometric mean)

Key Concept: The geometric mean accounts for compounding effects and is always less than or equal to the arithmetic mean when returns vary. The arithmetic mean would be: $\frac{12.2\% + (-8.5\%) + 6.7\% + (-3.3\%)}{4} = \frac{7.1\%}{4} = 1.775\%$ which is close to Option B, but this is incorrect for geometric mean calculation.