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

Due No due date
Points 17
Questions 13
Time Limit None

Instructions

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 LaTeX: 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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