Return (finance)
In finance, the money generated by an investment is called the return.
Percent Return
For initial investment value <math>V_0<math>, final value <math>V_T<math> at time <math>T<math>, and dividends <math>Div<math> paid out in time period <math>T<math>, the percent return is given by:
<math>R_P = {{V_T + Div - V_0}\over V_0} = {{V_T + Div}\over V_0} - 1<math>.
This return has some characteristics such as <math>R_P>0<math> when the investment increases and <math>R_P<0<math> when the investment decreases, <math>R_P=+100%<math> when the initial investment is doubled and <math>R_P=-100%<math> when all is lost.
Log Return
The above definition is problematic in that a +10% return and a –10% return do not add up to 0%. For example, starting with $100, a +10% return would result in $110. A subsequent –10% return would result in $99.
To correct this, academics use a natural log return defined as:
<math>R_L = \ln({{V_T+Div}\over V_0})<math>.
This return has similar characteristics for when <math>R_L>0<math> or <math>R_L<0<math>; a doubling occurs when <math>R_L=\ln2=69.3%<math> and loss of everything occurs when <math>R_L\to-\infty<math>.
