Fight Finance

(a) Arithmetic average gross discrete return is $\text{AAGDR}_{0\rightarrow T} = \dfrac{\text{GDR}_{0 \rightarrow 1}+\text{GDR}_{1\rightarrow 2}+...+\text{GDR}_{T-1 \rightarrow T}}{T}$

(b) Geometric average gross discrete return is $\text{GAGDR}_{0\rightarrow T} = \dfrac{\text{GDR}_{0 \rightarrow 1} . \text{GDR}_{1\rightarrow 2} ... \text{GDR}_{T-1 \rightarrow T}}{T}$

(c) The geometric average gross discrete return can be quickly found using the first $(P_0)$ and last $(P_T)$ prices. $\text{GAGDR}_{0\rightarrow T} = \left( \dfrac{P_T}{P_0} \right)^{1/T}$

(d) The arithmetic average is always bigger than or equal to the geometric average, $\text{AAGDR} \ge \text{GAGDR}$.

(e) The arithmetic and geometric averages of returns will be equal if the variance of the stock's returns is zero.