VaR Estimation from Returns with Mixed Holding Periods
Summary
The document considers how to estimate value at risk when a dataset mixes returns measured over different spans, such as one day, several days, and multiple weeks. Its central guidance is to first define the holding period the VaR is meant to represent, since estimates only make sense relative to a specified horizon.
For combining observations, the answer describes converting simple returns to log returns, adding log returns across consecutive periods, and converting the aggregate back to percentage returns. It also suggests choosing a common observation frequency when comparing assets or constructing a portfolio. Scaling VaR between horizons is another option, but depends on assumptions: square root of time scaling follows under normally distributed returns. The response cautions that scaling has limitations and does not establish that linear scaling of mixed-period observations is generally valid. It provides conceptual methods rather than a worked comparison or empirical validation.
Key ideas
- VaR should be tied to a clearly specified holding period.
- Consecutive simple returns can be combined by converting them to log returns, summing them, and converting back.
- Using a common return frequency supports more consistent asset and portfolio comparisons.
- Scaling VaR across horizons relies on assumptions, including normality for square root of time scaling.
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# How is VaR calculated with mixed return-periods
# 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?
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/68894
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 $$ VaR(\text{annual return}) = \sqrt(VaR( \text{monthly return})). $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.