Logarithmic rate of return

The logarithmic rate of return (also called continuous rate of return ) is a mathematical variable that plays a role in risk management when calculating volatilities (e.g. in the classic Black-Scholes model of option price valuation ).

Definition and characteristics

If there is a return (i.e. a ratio of the type ), then is the associated logarithmic return. ${\displaystyle r}$ ${\displaystyle {\frac {Wertzuwachs}{Ausgangskapital}}}$ ${\displaystyle \ln(1+r)}$

The logarithmized return is therefore the natural logarithm of the ratio of final capital to starting capital (or more generally also final value to starting value). The logarithmic return of successive periods is cumulated by addition.

With an expected logarithmic return (starting time t and time interval T ) for a given capital , the expected net present value in the following period is calculated as: ${\displaystyle r(t,T)}$ ${\displaystyle K(t)}$

${\displaystyle K(t+T)=K(t)\cdot e^{r(t,T)}}$

This calculation model does not only apply to returns, but to any rate of change or growth .

background

A main reason for using logarithmized returns is that they (in contrast to the actual returns) are defined on the entire set of real numbers, while "normal" (read: discrete) returns are defined on the left by the value −1 or a loss are limited by 100%. As a result, the empirical distribution of the returns can, for example, be better approximated by the normal distribution , but the empirical distribution of the returns usually deviates from the normal distribution.

See also

• Random Walk

Web links

• Calculation of volatility using logarithmized return
• Option valuation (PDF file; 430 kB)