Statistical Methods for Finance

tags
Finance

Returns

Net Returns:

$$R_t = \frac{P_t}{P_{t-1}} - 1 = \frac{P_t - P_{t-1}}{P_{t-1}}$$

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

Gross Returns:

$$\frac{P_t}{P_{t-1}} = 1 + R_t$$

The gross return over the most recent $$k$$ periods is:

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

Log Returns:

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

$$r_t(k) = r_t + r_{t-1} + \dots + r_{t-k+1}$$

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

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

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