alpha vs mu(t) ASM Problem 7.2 7th edition Question

[undergrad01]

In problem 7.2 you are asked to find the probability that the rate of return will be less than the expected rate of return.

I initially interpreted "expected rate of return" as mu(t), but obviously this is wrong and alpha is defined as the "expected rate of return".

My question is: how would the problem ask you to find the probability that the rate of return is less than mu(t)? Mu(t) is the E[ln(St/So)] ...so wouldn't you also call this the expected rate of return?

On page 138, ASM (7th ed 2nd printing) calls E[ln(S/So)] "the expected value of the annual rate of return". Isn't this definition the same thing as "expected rate of return"?

[xiaofan4]

The probability that rate of return less than mu(t) is N((mu(t)-mu(t))/sigma(t))=0.50.

I recommend reading section 7.2.1 again. The paragraph that starts with "Therefore, when we refer to the continuously compounded annual rate of turn..." answers all your questions.

E[ln(S/So)] is indeed "the expected value of the annual rate of return". It is not the same thing as "expected rate of return." Expected rate of return is alpha = ln(E[(S/So)]). Expected rate of return is generally a constant, while ln(S/S0) "rate of return" is a normally distributed random variable.