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Types of returns

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Consider some asset ( is allowed to denote investor’s portfolio wealth as well)over the period  to . Time is measured in years. Let  and denote prices of  at time  and respectively.
Below are presented the definitions actively used in the current document. The corresponding terminology is not settled yet; therefore the divergences with the terms used in other sources might occur.

Types of Returns

Various definitions of return serve to measure the degree of price change over the given time period.
Definition. Quantity is called Simple return or Arithmetic return over the period to .
Definition. Quantity is called Log return or Geometric return over the period to .

Note. Applying Taylor series expansion up to fourth-order term, one can obtain the following approximation of  when  is close to

Rates of Return

Under Rates of return we will further denote returns that are normalized to annual basis. Rates of return may also be called Annualized returns or simply Annual returns. Another frequently used synonym is Growth Rate.
Definition. Quantity will be called Simple rate of return (other notations include Arithmetic rate of return or Rate of return without compounding) over the period  to .
Definition. Quantity will be called Logarithmic rate of return (Geometric rate of return or Continuously compounded rate of return respectively) over the period  to .

Important! For simplicity reasons everywhere hereinafter under the terms rate of return and growth rate a logarithmic rate of return will be implied. Then, when one requires using simple rates of return, the latter will be indicated explicitly.

Expected Rates of Return

The symbol below stands for Mathematical Expectation (averaging over all possible outcomes with weights equal to respective probabilities).

Definition. Quantity will be called Expected simple rate of return (Expected arithmetic rate of return or Expected rate of return without compounding) over the period to .

Note. Maximization of portfolio expected rate of return is equivalent to the maximization of expected portfolio wealth.