Statistical Methods for Finance

tags: Finance

Returns

Net Returns:

$R_{t} = \frac{P_{t}}{P_{t - 1}} - 1 = \frac{P_{t} - P_{t - 1}}{P_{t - 1}}$

$R_{t} \geq - 1$ : 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:

$\begin{aligned} 1 + R_{t} (k) & = (1 + R_{t}) \dots (1 + R_{t - k + 1}) \\ = exp (r_{t}) \dots exp (r_{t - k + 1}) \\ = exp (r_{t} + \dots + r_{t - k + 1}) \end{aligned}$

It is also sometimes assumed that log returns are $N (μ, σ^{2})$ for some constant mean and variance. Then $k$ period log returns are $N (k μ, k σ^{2})$ .

Geometric Random Walks

From eq:rw, we have that

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

A process whose logarithm is a random walk is called a geometric random walk. If $r_{1}$ , $r_{2}$ , $\dots$ are i.i.d $N (μ, σ^{2})$ , then $P_{t}$ is lognormal for all $t$ and the process is a lognormal geometric random walk. $μ$ is called the log-mean, and $σ^{2}$ the log-standard deviation of thet lognormal distribution of $e x p (r_{t})$ .

Validity of the Random Walk Model

In the lognormal geometric random walk model we assume that:

log returns are normally distributed
log returns are mutually independent