Final answer:
The statement that the geometric mean rate of return is always higher than the arithmetic mean rate of return is false, as the geometric mean generally accounts for the compounding of rates which can result in a lower value. It's also true that the Pythagorean theorem can be used to calculate the resultant of two vectors at right angles.
Step-by-step explanation:
The statement that the geometric mean rate of return is always higher than the arithmetic mean rate of return is false. When comparing geometric mean and arithmetic mean rate of returns, the former is typically lower because it accounts for the compounding of rates over time.
To explain this further, the geometric mean is calculated by multiplying all the numbers (1+rate of return), taking the nth root (where n is the number of rates) and subtracting one, which inherently takes into account the effect of volatility and sequence of returns. In contrast, the arithmetic mean is simply the sum of the rates of returns divided by the number of rates, which may overestimate the expected return as it does not consider the compounding effect.
For instance, if an investment has returns of +10%, -10%, and +10% over three years, the arithmetic mean would be (10-10+10)/3 = 3.33%, suggesting a profitable outcome. However, the geometric mean would be [(1.1*0.9*1.1)^(1/3)]-1, which equals approximately -0.13%, indicating a slight loss.
Additionally, regarding the question about the Pythagorean theorem, it is true that we can use it to calculate the length of the resultant vector when the vectors are at right angles to each other. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. When dealing with vectors in physics, this principle applies to situations where vector quantities like displacement, velocity, or force are perpendicular to each other, as in two-dimensional vector addition.