Continuously Compounded Average

本人数学不好，看了答案还是不懂啊。大家帮忙解释下为什么arithmetic average of the continuously compounded 为0啊？谢谢！

f a stock decreases in one period and then increases by an equal dollar amount in the next period, will the respective arithmetic average of the continuously compounded and holding period rates of return be positive, negative, or zero?

A)
Zero; zero.



B)
Positive; zero.



C)
Zero; positive.




The holding period return will have an upward bias that will give a positive average. For example, a fall from 100 to 90 is 10%, and the rise from 90 to 100 is an increase of 11.1%. The continuously compounded return will have an arithmetic average of zero. Since we can sum continuously compounded rates for multiple periods, the continuously compounded rate for the two periods (0%), means the rates for the two periods must sum to zero, and their average must therefore be zero.​