Annualizing Discrete-Return Volatility with Log-Return Assumptions

# Annualization of discrete returns


# Annualization of discrete returns












There is a well known approach to annualize volatility of log-returns for a given frequency. Let $P(t)$ a price process and define a log return $r_l(t)$ as $$r_l(t) = \ln \left( \frac{P(t)}{P(t-1)} \right).$$

An aggregate return over $n$ periods is $$ \begin{equation} \begin{aligned} r_l^A(t) &= \ln \left( \frac{P(t)}{P(t-n)} \right) \\ &= \sum_{t-n+1}^t r_l(i). \end{aligned} \end{equation} $$

Let $\sigma_l^2$ denote variance of log-returns. Then, variance of an aggregate return is $$ \begin{equation} \begin{aligned} Var[r_l^A] &= Var \left[ \sum_{i=t-n+1}^t r_l(i) \right] \\ &= \left( n + 2\sum^{t-1}_{i=1} (n-i) \rho(i) \right) \times \sigma^2, \end{aligned} \end{equation} $$ where $\rho(i)$ is an autocorrelation of order $i$.

If observed series has autocorrelations zero for all lags than this simplifies to $$Var[r_l^A] = n \times \sigma^2.$$

The formula is, however, not valid for discrete returns. One could argue that it is a good approximation for higher frequencies since log-returns are close to discrete ones, but how would annualize volatility of discrete returns for, say, weekly or monthly data?

### Normal log-returns case

For a case when log-returns are normally distributed, we can derive a closed form solution for a variance of aggregated dicrete returns. In the below I assume that all autocorrelations are zero to simplify formulas and that $E[r_l(t)] = \mu$.

(Aggregated) Discrete return is related to a log-return via $$r_d^A(t) = \exp (r_l^A(t)) - 1,$$ which means that $r_d^A(t)$ is log-normally distributed with variance $$Var[r_d^A(t)] = \exp \left( n Var[r_l^A(t)] - 1 \right) \times \exp \left( 2n\mu + nVar[r_l^A(t)] \right).$$

This is a monotonically increasing function in a mean of log-returns, $\mu$, as well as in their variance, $Var[r_l^A(t)]$. Below are two figures that illustrate that the divergence between the two volatilities (took square root of the above variances) increase as $\mu$ and $Var[r_l^A(t)]$ increase (number of aggregation periods is $12$ for monthly data).

## Answer by Nipper (score 0)

https://quant.stackexchange.com/a/68028

If I have understood your question correctly then you might be interested in the following paper: Weber, A. E. Annual risk measures and related statistics. Applied paper Ortec Finance Research Center (8 2017).

You should be able to download a copy here.