Returns

Net Returns:

\begin{equation} R_t = \frac{P_t}{P_{t-1}} - 1 = \frac{P_t - P_{t-1}}{P_{t-1}} \end{equation}

\(R_t \ge -1\): a 100% loss occurs if the asset becomes totally worthless.

Gross Returns:

\begin{equation} \frac{P_t}{P_{t-1}} = 1 + R_t \end{equation}

The gross return over the most recent \(k\) periods is:

\begin{equation} 1 + R_t(k) = \frac{P_t}{P_{t-k}} = (1 + R_t)\dots (1 + R_{t-k+1}) \end{equation}

Log Returns:

\begin{equation} r_t = \log (1 + R_t) = \log \frac{P_t}{P_{t-1}} = p_t - p_{t-1} \end{equation}

\begin{equation} r_t(k) = r_t + r_{t-1} + \dots + r_{t-k+1} \end{equation}

log returns are similar for daily returns, less similar for yearly returns, and not necessarily similar for multi-year returns.

Random Walk Model

In the random walk model, single-period log returns are assumed to be independent:

\begin{align} \label{eq:rw} 1 + R_t(k) &= (1 + R_t)\dots(1 + R_{t-k+1}) \\
&= \textrm{exp}(r_t) \dots \textrm{exp}(r_{t-k+1}) \\
&= \textrm{exp}(r_t + \dots + r_{t-k+1}) \end{align}

It is also sometimes assumed that log returns are \(N(\mu,\sigma^2)\) for some constant mean and variance. Then \(k\) period log returns are \(N(k\mu, k\sigma^2)\).

Geometric Random Walks

From eq:rw, we have that

\begin{equation} \frac{P_t}{P_{t-k}} = 1 + R_t(k) = \textrm{exp}(r_t + \dots + r_{t-k+1}) \end{equation}

A process whose logarithm is a random walk is called a geometric random walk. If \(r_1\), \(r_2\), \(\dots\) are i.i.d \(N(\mu, \sigma^2)\), then \(P_t\) is lognormal for all \(t\) and the process is a lognormal geometric random walk. \(\mu\) is called the log-mean, and \(\sigma^2\) the log-standard deviation of thet lognormal distribution of \(exp(r_t)\).

Validity of the Random Walk Model

In the lognormal geometric random walk model we assume that:

  1. log returns are normally distributed
  2. log returns are mutually independent