# How is VaR calculated with mixed return-periods

For example, if you have a dataset of returns that are not daily or yearly, but span 24 days, 1 day, 5 days, 7 days, etc., how do you calculate or interpret the VaR of that? I've tried linearly scaling each return to be daily. E.g., (24 day return / 24) or even doing (24 day return / round(24 * (252/365), 0) to account for trading days. Then I can calculate daily VaR, 10 day, 15 day, and 30 day. But I'm not sure what's correct. Is there a better way of interpreting or calculating the VaR of the original dataset?

• Remember that garbage in = garbage out. What you are attempting to do will result in garbage, unfortunately. Try to get the price data over at least several months (3 at a minimum). Calculate the log returns. From here, you can generate the correlation matrix and volatilities for your VaR simulation. I'm afraid your initial dataset won't get you to your intended solution. Most VaR simulations use EWMA (0.06 decay) scaling so less returns are usually needed. I've written VaR engines for 3 companies, just giving you the basics. Get better data. Hope that helps.
– Matt
Commented Nov 24, 2021 at 17:57

There are several aspects:

The holding period that you want to measure: Usually, you want to calculate VaR for a specified holding period. For USCITS funds it is e.g. 20 days for bank pillar 2 regulations it is an annual holding period. What is your holding period?

Returns over periods If you have daily returns for certain assets, then you can aggregate them to get e.g. 5 days (i.e. weekly) or monthly returns. You could transform to log-returns and then add as many returns as you need. If your daily return is $$r_t = \frac{P_t-P_{t-1}}{P_{t-1}}$$ then the log-return $$R_t$$ is $$R_t = \ln(1+r_t)$$ Doing some maths you can see that the return over $$n$$ days is just the sum of $$n$$ log-returns. You can transform the final result back to the percentage return scale by $$r_t = \exp(R_t)-1.$$

Portfolio returns You could choose the lowest frequency (e.g. monthly) and calculate returns for all your assets using this frequency and a common period (e.g. monthly returns from 2011-2021). Then you can compare VaR estimates (VaR of the monthly return) for your assets or form a portfolio and measure its VaR.

Scaling VaR Finally, it is quite usual to scale VaR from one holding period to another. There are many assumptions that come with this, but it is done - keeping the limitations of this approach in mind. E.g. under the assumption that returns follow a normal distribution, the VaR scales with the square-root of time. This means that $$VaR(\text{annual return}) = \sqrt(VaR( \text{monthly return})).$$