Compute gaussian VaR with a “1-month horizon”

I am trying to compute VaR for a long-only equity portfolio, under the assumption that all stock returns are normally distributed, using a "1-month horizon" methodology.

How do I do it? Do I compute my covariance using my daily 1-month returns? Or do I compute in the normal way using my daily returns, and then adjust for the different timespans?

This is the gaussian version of this question about 1-month horizons in VaR.

• Do I just convert my daily sigma to a monthly one by multiplying by root of 22 (being the number of business days in a month)? – lebelinoz Apr 3 '17 at 2:50

Most implementations of VaR are inherently parametric. If the data is indeed approximately normally distributed, more frequent (daily) sampling will result in greater sample size over shorter time intervals -- this affords greater statistical significance over shorter time periods. Greater significance is highly desirable whether your model employs frequentist and/or Bayesian calibration (i.e., "moment estimation") methods. If variance does indeed grow approximately linearly with respect to time, then basing VaR's parameters on daily returns is preferable.

That increasing sample size increases the statistical power of sampling moments from a parametric distribution is classically demonstrated through the Lindeberg–Lévy Central limit theorem (CLT) -- assuming returns are i.i.d. random variables from a stationary process.

$\sqrt{N}((\frac{1}{N}\sum_{n}^NX_n)-\mu)\to_{distribution} \varPhi(0,\sigma^2)$

where:

$N$ is sample size;

$X_i = \{X_1,X_2,...X_N\}$ is a sequence of i.i.d. variables with with an expected value equal to $\mu$ and expected variance equal $\sigma^2$; and,

$\varPhi(0,\sigma^2)$ is the cumulative distribution function of the standard, normal (i.e., Gaussian) distribution.

In order to test for the validity of implementing VaR on daily returns, I recommend a simple "unit-root" test on the variances at different sampling frequencies. The following might serve as the basis for a linearity test such that:

$$\sum_{i\times n}^N[(X_{n}-\mathbb{E}[X])^2] \propto i^a; \, a \approx 1 \ldots \forall i\in N$$

If returns exhibit varying amount of skewness and ex-kurtosis, or clustered variance, then observed variance will grow linearly with respect to time -- i.e., volatility sampled at varying frequencies and/or intervals will not behave according to the "square root of time" rule.

If, upon discovering this misbehavior, you remain committed to normative assumptions, the "correct" frequency really depends on investors' preferences to trade off between accuracy and significance. For example, sampling too frequently will imply unrealistic expectations. On the other, sampling too infrequently will surely result in spurious and insignificant estimates. Monthly frequency seems reasonable to me... it just depends...

If, however, you are willing to make increasingly specified parametric assumptions, one option is to incorporate higher level moments and/or a non-standard distributions. If you have reason to believe that the posterior distribution will be of a specific shape and/or will possess moments different from those implied by historical data, then more highly-specified parametric methods are more powerful than other approaches.

In my opinion, employing higher level moments to more precisely capture the posterior distribution without the expectation that these moments capture something fundamental is very unwise.

An alternative to parametric methods is employing non-parametric methods. If you are agnostic regarding the posterior distribution, non-parametric VaR estimators are more robust but may require greater sample sizes. Assuming statistically significant sample size, you can estimate the empirical x% return/loss interval in the absence of assumptions regarding the shape of the underlying distribution. Another class of (mildly) non-parametric estimators, based on kernel density estimation, can be robust at even smaller sample sizes.

The paper, "Estimating Value-at-Risk with a Precision Measure by Combining Kernel Estimation with Historical Simulation" appears to be on point with regard to non-parametric estimators.

Use monthly returns or, if you want only to scale, use daily data and multiply standard deviation by sqr root of 22 (or working days in the target market)

Notice that if you use a model based on monthly returns, you don't need to scale, but will have a model that takes more time to react to new information. This may be good if you are not a short term investor.