I should have posted this last week when we covered this in class, but better late than never!

In sections 6-7 of the Hull’s “Wiener Processes and Ito’s Lemma” chapter and my teaching note entitled “Applying Ito’s Lemma to determine the parameters of the probability distribution for the continuously compounded rate of return“, it is shown (via the application of Ito’s Lemma) that *T-*period log returns are normally distributed with mean (*μ-σ*^{2}/2)*T* and variance *σ*^{2}*T*. In the geometric Brownian motion equation (equation (6) in Hull’s chapter),

*dS/S *= *μdt + σdz,*

*μ *corresponds to the expected return in a very “short” time, *dt**,* expressed with a compounding frequency of *dt*; in other words, it corresponds to the *arithmetic* mean return). *μ-σ*^{2}/2 on the other hand corresponds to the expected return in a “long” period of time, *T-t,* expressed with continuous compounding; i.e., it corresponds to the *geometric* mean return.

To see the difference between the *arithmetic *and *geometric* mean return, consider the following numerical example. Suppose that returns in successive years are *r*(1) = 15%, *r*(2) = 20%, *r*(3) = 30%, *r*(4) =−20% and *r*(5) = 25%. If you add these returns up and divide by 5, this gives you the *arithmetic *mean value of 14%. The arithmetic mean of 14% is analogous to *μ **. *However, the annualized return that would actually be earned over the course of a five-year holding period is only 12.4%. This is the *geometric *mean return which is analogous to *μ-σ*^{2}/2. It is calculated with the following equation:

[1(1.15)(1.20)(1.30)(.80)(1.25)]^{(1/5)} – 1 = .124.

The “problem” with volatility is that the higher the volatility, the more it lowers the 5-year holding period return. We can create a mean preserving spread of the (*r*(1) = 15%, *r*(2) = 20%, *r*(3) = 30%, *r*(4) =−20%, *r*(5) = 25%) return series by resetting *r*(1) to 0% and *r*(5) to 40% ; both return series have arithmetic means of 14% but the (*r*(1) = 0%, *r*(2) = 20%, *r*(3) = 30%, *r*(4) =−20%, *r*(5) = 40%) return series has a higher variance (.058 versus .039 for the original return series). This increase in variance results in a lower geometric mean:

[1(1)(1.20)(1.30)(.80)(1.40)]^{(1/5)} – 1 = .118.

On the other hand, if we lower volatility, then this increases the geometric mean return. To see this, instead of resetting r(5) in the original return series from 25% to 40%, let’s leave r(5) at 25% and instead reset r(4) in the original return series from -20% to -5%. This change generates the following return series: (*r*(1) = 0%, *r*(2) = 20%, *r*(3) = 30%, *r*(4) =−5%, *r*(5) = 25%), which has a 14% arithmetic mean and variance of .024. With lower variance, the new return series has a higher geometric mean:

[1(1)(1.20)(1.30)(.95)(1.25)]^{(1/5)} – 1 = .131.