# Comparison between Absolute return and Relative return

In typical VaR calculation, we need to calculate the time series of historical realised return. There are mainly 2 ways to calculate such returns viz. Absolute return (i.e. just the difference between levels of consecutive time series data points) and Relative return (i.e. Percentage return)

Let we calculate these two return for the time series of interest rate (assuming all are non-negative). In an article, I found one statement stating that: In a falling interest rate environment, Absolute returns are more conservative.

I could not understand this statement, what could Author mean to say as Conservative?

Any pointer will be very appreciated.

It's easy to see why absolute returns are "more conservative" as the authors put it. Assume the yield of a bond at $$t$$ is 10bp and at $$t+1$$ it's 20bp. The absolute return is 10bp but the percentage return is 100%.

Conversely, if you take the same bond in a high interest rate environment and assume it moved from 5% to 5.1%. We get the same absolute return as in the falling interest rate regime but a 2% percentage return.

Now try calculating the volatility of absolute returns vs percentage returns in the two regimes. You'll get two wildly different results. In the falling rates environment your volatility estimate (simple standard deviation of historical returns) might explode as rates approach zero. For example using a set of daily yields: You can see the two absolute vols are the same but percentage wise you get a massive difference.

• Your answer would address the OP's question if we replaced falling with low in the latter. It does not do that without that change. Jul 12 at 5:38

The important point is there is nothing inherently wrong with either estimation, provided that the relevant historical series is stationary (i.e. the points are tied in a meaningful way with each other and hopefully with the future).

Let's say you are in a low rates environment and suddenly move to a high rates environment. Then the percentage return series will not be stationary (in particular vol decreases with time) and thus there is no point working with percentage returns.

If rates are falling then it means you're likely to obtain a series whose percentage returns are non stationary so I wouldn't want to use percentage returns in that case. So work with absolute returns in this case - this is my best guess of what they mean by "conservative".