General: anualization in performance / risk / inflation / ...

Owned by Wolfram Stacklies
Jan 07, 2025
2 min read

Background: values such as performance or inflation rate can be measured on different intervals. E.g. you might measure the inflation rate for the month of May, or for the year 2024. The same is true for any other performance | risk | yield statistic, which always has an observation interval.

To make values for different intervals COMPARABLE, we typically use a technique called anualisation. This is, from an interval which is unequal to a year, we try to approximate how the value would have been, if the interval WOULD have been one year.

How anualisation is done for performance related values

Typically, annualisation of performance related data (stock index, inflation rate, …) is done using the formula Math.pow(rate, (365.0G / days)) where days are the observation interval. In practical terms that would yield for a rate of 1.5 the formula would read

  • 1.512 for an interval of one month

  • 1.51 for an interval of one year

  • 1.51/3 for an interval of three years

Calculating monthly values from a yearly value or an annualized value

To calculate a monthly rate from an annualized rate, you must take the square root to the power of 12. This is because your yearly value is the product of each monthly rate. So, let’s say our yearly (or annualized) rate is 2 %. Then the monthly rate is 21/12 which is approx. 1,0595 %, as

1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 * 1,0595 ≈ 2

Calculating yearly values from a set of monthly rates

Assume we have a set of monthly ROI values. E.g. we did a montly ROI of 1 % in the first 6 months and a monthly ROI of 2 % in the last six months of the year. Then the yearly rate is calculated as

1,01 * 1,01 * 1,01 * 1,01 * 1,01 * 1,01 * 1,02 * 1,02 * 1,02 * 1,02 * 1,02 * 1,02 ≈ 1,195 or 19.5 %

We use this recursive formula, because we need to include the ROI of the previous period in the next one.

How anualisation is done for standard-deviation based values (risk)

Annualisation for values based on variance is done significantly different. Here, the assumption is that the variance is smaller the smaller the measurement interval. This makes sense as the standard deviation for day-to-day changes will usually be significantly smaller than the standard deviation for year-on-year changes. Simple, as changes on the stock market typically happen slowly, and the day-to-day change is expected to be usually very small.

Here, the formula for annualization is

standardDeviation * Math.sqrt(AnnualizationHelper.annualizationFactor(interval))

where the annualization factor is again based on the measurement interval, typical values are

  • day: 365

  • week 52

  • month: 12

  • year: 1 (no annalization needed)