Explanation

Let's calculate each type of return to determine which one equals -5.34%.

1. Arithmetic Mean Return: $\text{Arithmetic Mean} = \frac{6\% + (-37\%) + 27\%}{3} = \frac{-4\%}{3} = -1.33\%$

2. Geometric Mean Return: $\text{Geometric Mean} = \left[(1 + 0.06) \times (1 - 0.37) \times (1 + 0.27)\right]^{1/3} - 1$ $= \left[1.06 \times 0.63 \times 1.27\right]^{1/3} - 1$ $= \left[0.848\right]^{1/3} - 1$ $\approx 0.945 - 1 = -5.5\%$

3. Money-Weighted Return (IRR): This requires cash flow information which is not provided. The money-weighted return depends on the timing and size of cash flows into and out of the portfolio.

4. Time-Weighted Return: The time-weighted return is calculated as the geometric mean of the period returns: $\text{TWR} = \left[(1 + 0.06) \times (1 - 0.37) \times (1 + 0.27)\right] - 1$ $= 1.06 \times 0.63 \times 1.27 - 1$ $= 0.848 - 1 = -15.2\%$

Comparison:

• Arithmetic mean: -1.33%
• Geometric mean: -5.5% (approximately -5.34% with rounding)
• Time-weighted return: -15.2%

The manager's reported return of -5.34% is closest to the geometric mean return of approximately -5.5% (the slight difference is due to rounding).

Why not the others?

• Arithmetic mean (-1.33%) is too high
• Money-weighted return cannot be calculated without cash flow data
• Time-weighted return (-15.2%) is too low

Therefore, the manager has reported the geometric mean return.