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I have a series of monthly log-returns; let's assume the log-returns are normally distributed, but exhibit significant serial correlation.

In the case of normal, i.i.d. returns, I can annualize the the log-returns by multiplying by a factor 12, and annualise the volatility by a factor of sqrt(12).

Given the dependence in my returns, how do I correctly scale to annual results?

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  • $\begingroup$ I edited the question a bit. You do not annualise returns but you annualise their volatility. $\endgroup$ – Richard Dec 9 '14 at 8:30
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The correct answer has some intuition though it doesn't generalize to continuous time very easily:

Think about the paper below like this:

$Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)$

The generalization is slightly hard because the dynamics of $\mu$ and $\sigma^2$ could be dependent for arbitrary returns. You can use a GMM estimator to derive the asymptotic distribution for the required quantities and generalize:

The formula is given as:

$Q = \sqrt{T + 2\sum^{T-1}_{k=1} (T-k) \rho_k}$

where $Q$ is the annualization factor, and $\rho_k$ is the autocorrelation at the $\text{k}^{\text{th}}$ lag.

Andrew Lo analyzed this idea in the paper Statistics of Sharpe Ratios (2002).

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  • $\begingroup$ Thank you, that's great. Reading through Andrew Lo's paper, which is proving very useful. $\endgroup$ – Smackboyg Dec 8 '14 at 11:54
  • $\begingroup$ what is $T$ here? $\endgroup$ – Michael Aug 25 '17 at 21:05
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The answer is that it depends. In addition to the Lo paper above, there are a number of excellent references that go into depth about annualizing or time scaling non-i.i.d. returns, one of which is Roger Kauffman, "Long-Term Risk Management", 2005 which can be found at http://www.rogerkaufmann.ch/all-Budapest.pdf.

There are some well known cases where the variance of non-i.i.d. returns can still be time-scaled but here's one realistic example. Let's assume that we have a portfolio whose daily changes are given by a random variable $P_t$ that is not i.i.d. but are still identically distributed over time. It may still be possible to define the dependency relationship and use its properties to scale the volatility.

Assume that daily gain/losses on a portfolio $P_t$ follow a first-order autoregressive process with normal innovations, i.e. that

$ P_t \sim\phi_1 P_{t-1} + \epsilon_t \quad where \quad \epsilon_t \sim \mathcal{N}(0,\sigma_\epsilon^2) $

In this case, it can be shown that both the 1-day and the T-day P&L are normally distributed.

$ P_t \sim \mathcal{N}\left(0, \frac{\sigma_\epsilon^2}{1-\phi_1^2}\right) \qquad and\quad \sum\limits_{t=1}^T P_t \sim \mathcal{N}\left(0,\frac{\sigma_\epsilon^2}{(1-\phi_1)^2} \left(T - 2\phi_1 \frac{1-\phi_1^T}{1-\phi_1^2}\right)\right) $

You can then get an expression for the ratio of T-period volatility to 1-day volatility.

$ \frac{Vol(P_t)|time T}{Vol(P_t)|time 1}=\sqrt{ \frac{1+\phi_1}{1-\phi_1} \left(T-2\phi_1 \left(\frac{1-\phi_1^T}{1-\phi_1^2}\right)\right)} $

The right hand side is also the scaling factor you would apply to the 1-day volatility to get to the T period volatility. It is also clear that as $\phi_1 \to 0$ or in other words, no autocorrelation and so back to independently distributed, then as you would expect:

$ (Vol(P_t)|time=T) \to \sqrt{T}\times (Vol(P_t)|time=1) $

To the extent that the innovations are non-normal, the AR(1) solution may be biased. Also may go without saying, but even if there is no straightforward analytic means to scale daily returns, as long as the distribution is analytic and you can express returns at time t as a function of returns from prior periods $t-k$, then Monte Carlo simulation can be used to simulate an arbitrarily large number of paths $N$ over any arbitrary time period $T$. This yields a distribution of return outcomes $x_i$ at time T. The volatility (standard deviaiton really) could be computed in the conventional way, i.e.

$ s_N = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {\left( {x_i - \bar x} \right)^2 } } $

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