I. Module 1 Homework 2 [practice]: Lognormal Returns

  • Due No due date
  • Points 16
  • Questions 13
  • Time Limit None
  • Allowed Attempts 2

Instructions

This is a practice version of this assignment. You may attempt it two times, and it will grade your answers just as the actual assignment will. However, it is not graded for credit. 

For this set of problems, we'll explore the standard geometric Brownian motion model for stock returns,

\frac{dp_{t}}{p_{t}}=\mu dt+\sigma dz_{t}.

Recall the solution for the T period log and arithmetic return,

\begin{align*} R_{T} \equiv\frac{p_{T}}{p_{0}}&=e^{\left( \mu-\frac{1}{2}\sigma^{2}\right) T+\sigma\sqrt{T}\varepsilon_{t}} ~\varepsilon_{T} \sim N(0,1) \\ \log\left( R_{T}\right) &=\left( \mu-\frac{1}{2}\sigma^{2}\right) T+\sigma\sqrt{T}\varepsilon_{t} ~~\varepsilon_{T}\sim N(0,1)\end{align*}

As you can see, the log return works just like the discrete time log returns we explored in the last problem. In particular mean and variance scale with horizon:
\begin{align*} E\left[ \log\left( R_{T}\right) \right] &=\left( \mu-\frac{1}{2} \sigma^{2}\right) T\\ \sigma^{2}\left[ \log\left( R_{T}\right) \right] &=\sigma^{2}T \end{align*}
So the "Sharpe ratio" scales with the square root of the horizon:

\frac{E\left[ \log\left( R_{T}\right) \right] }{\sigma\left[ \log\left(R_{T}\right) \right] }=\frac{\left( \mu-\frac{1}{2}\sigma^{2}\right)}{\sigma}\sqrt{T}
The mean and variance of annualized returns are

\begin{align*} E\left[ \frac{1}{T}\log\left( R_{T}\right) \right] &=\left( \mu-\frac{1}{2}\sigma^{2}\right) \\ \sigma^{2}\left[ \frac{1}{T}\log\left( R_{T}\right) \right] &=\frac{\sigma^{2}}{T}.\end{align*}

(Why is \frac{1}{T}\log\left( R_{T}\right) the continuously compounded annualized return? Because e^{T\left( \frac{1}{T}\log\left( R_{T}\right) \right) }=R_{T}.) The latter result makes it look like returns are more stable in the long run, but that's a mistake.

In this set of problems, we'll look at the properties of the corresponding arithmetic return p_{T}/p_{0}.

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