# Discrete returns versus log returns of assets

There have been similar posts here already but nevertheless I find the question worth posting: why do some people claim that log returns of assets are more suitable for statistics than discrete returns.

E.g. in the ESMA CESR guidliens about SSRI log returns are used. I personally think that discrete returns are as good for means of risk management as continuous returns. Furthermore in portfolio context I can calculate the portfolio return by weighting the discrete returns of the assets which does not work with log returns. The time-aggregation of log returns is easier that's true. But people rather think in discrete returns. If my NAV drops from $100$ to $92$ then I have lost $8\%$ and that's it.

Is there any study on this - any good reference? Anything that I can tell my regulator why I stick to discrete returns.

• I'm not sure there needs to be a "study." You seem well aware of the reasoning. Arithmetic returns allow for easier cross-sectional aggregation and log returns allow for easier time-aggregation. The reason people use log returns is that (for equities) they are approximately invariant and are easier to work with in estimating distributions. However, proper procedure is to convert the log returns to arithmetic returns for the purposes of portfolio optimization and risk management. – John Sep 20 '12 at 16:17
• @ John What do you mean by 'approximately invariant'? And how/why can you estimate distributions more easily? Can't we fit distributions for both kinds of returns? – Richard Sep 20 '12 at 19:00
• If you take normally distributed log returns and convert them to arithmetic, then they will become log normal. That's what I mean by estimating distributions easier. Also, it is easier to project log returns to the appropriate horizon due to time aggregation. As for invariance, see: symmys.com/node/85 – John Sep 20 '12 at 20:23
• Agree with John here, an almost exact identical post as yours was answered by me in the same fashion : quant.stackexchange.com/questions/3979/… – Matt Sep 21 '12 at 6:00
• @John thank you for the comment. I have not realized these issues although dealing with this for years now. If you make it an answer then I will accept it. Thanks again. – Richard Sep 23 '12 at 19:12

## 1 Answer

Arithmetic returns allow for easier cross-sectional aggregation and log returns allow for easier time-aggregation.

The reason people use log returns (for equities) is that they are approximately invariant and hence easier to work with in estimating distributions. Meucci does better justice in describing invariance here. The basic idea (again, for equities) is that the distribution of security prices is log-normal, so the arithmetic returns will also be. However, making a log transformation results in approximately normal returns, which are easier to work with. Also, if you do assume them to be normally distributed, then there are convenient results for the convolution of multivariate normal series. This is what allows for easier time-aggregation.

However, you shouldn't take log returns and use them to obtain the arithmetic portfolio return. This is because while you can link them through time, the math doesn't work out, particularly at long horizons, cross-sectionally. Hence, after estimating the distribution of the log returns, proper procedure is to convert the them to arithmetic returns for the purposes of portfolio optimization and risk management.