let say exchange rate as ER. the take natural log, as ln(ER)for the time period t. then return is defined as rt= ln(Er)t-ln(ER)t-1.

Exchange rate returns are well-known to be unconditionally symmetric but highly

leptokurtic. Standardized daily or weekly returns from ARCH and related stochastic volatility

models also appear symmetric but leptokurtic; that is, the distributions are not only

unconditionally, but also conditionally leptokurtic, although less so than unconditionally. A sizable literature explicitly attempts to model the fat-tailed conditional distributions

let say exchange rate as ER. the take natural log, as ln(ER)for the time period t. then return is defined as rt= ln(Er)t-ln(ER)t-1.

Sir

When you get rt like this, (difference in ln of ER at time t and t-1), the result is in log. Should we not find antilog to get the return? Dr Durairajan

hi it is rt=ln(ERt/ERt-1)= this can be proved as compound growth rate . no need to take antilog. if you search goole you can find it. or read some financial econometrics book.

With respect to the above given answers, it should be noted that log returns assume and hence require normally distributed (Gaussian) returns. Normally distributed returns are, of course, very hard to find in financial markets. Mandelbrot's "(Mis)Behaviour of Markets" deals with this topic. I just did the math on DAX returns from 2003 to 2013 (daily data) and the difference of log returns vis-á-vis actual returns is quite significant. in the 10yr period, the difference amounts to c. 60ppts. Whereas log returns give a return of 92%, the actual return was 152%. Given that the DAX is an aggregate, the difference is remarkable and therefore using log returns strikes me as being questionable, although I know that in academia, the use of log returns is common. However, in my opinion, the usage of log returns is more than a trivial offence, and I am confident that people who actually had to base a trading strategy on it, would never use log returns.

Bottom line, altough some of the mathematical properties of using log returns may seem tempting, I wouldn't use it.

I understand the problem of log returns. But when return values are very small, log approximation is valid. Taylor series proof help to understand this.

Yes, but the initial question states that monthly data will be used.

Arithmetic return,or geometric return formula all depends on the data type. We have to check and apply. When you ask how to calculate return of exchange rate then we usually say log difference. But when we do research we have to consider all

siva

Agreed, I just wanted to mention the pitfalls of using logs.

As already suggested, I would go with log returns. And this is not only a question of providing good approximation to returns when the latter are small (which is true) but it is more fundamental. Saying that log-return follow a GARCH model, is different from saying that returns follow a GARCH model. In particular, if the first statement is true, log-prices are an integrated process of order one, while if the latter is true, it is difficult to obtain such a nice characterization. When time series analysis is concerned, you often play with differencing (subtracting) and integrating (summing) observations and the interpretation of these operations is straightforward on the series of log-prices or log-return, while it is not on the series of return and prices. More in general, if you want to think to the price dynamics has a stochastic process, using log-return allow for a easier application of martingale theory, stationary increments, and so on... but this is leading us a little bit too far from the original question.

Hi Thomas can u refer some research articles / papers in which they used exchange return with out log.

Data (pre) treartement analysis often suggest to transform to log (argue with Box Cox) and first difference (argue with Unit Root test ). This may be a statistical fundation to the log story. the mathematical one is Taylor expension that allow to interpret this log-first difference as a geometric return.

Now I just want to point that log(r_t) - log(r_t-1) may be a little confusing, to have a return, you have to consider the log-difference with the prices since (in my sense) r_t = log(P_t) - log(Pt-1)... Then to clarify interpretations, make sure that ER is to be seen like a price (for a money to another).

Good Luck

Soufiane

Er is not return. Log difference gives return like compound growth rate. Return is a new variable. Not log (er). Pl refer some financial econometric books

ss

Hello Debadatta,

Sorry for only answering now. I can't think of any articles from the top of my head. As has been pointed out, articles in top-tier academic journals usually use log returns. There is quite a network effect in academic writing, it seems to me. However, if you look at more practice-orientated articles (for example, applied research like that of the CFA or MTA) you'll find actual returns used. Think of it like that: If you are a hedge fund or in the asset management industry, you will likely use returns instead of log returns because you will typically be more concerned with making money and therefore use data and methodology that reflect the real world than use a methodology that is seriously flawed just because it has some neat mathematical and statistical properties which may or may not will make your article being published. However, I reiterate, using log returns is purely academic and will make your research inapplicable to the real world because log normality doesn't hold there - in no market.

Agreed with this. Why we transform a simple thing to more complex situation if the problem can be solved without making it complicated. If the necessary assumptions are satisfied under the model without transforming to log, then we can proceed with actual returns (or percentage returns). For the purpose of modeling sometimes we may have to use log returns to if model with actual returns does not comply with the usual model assumptions. However, when interpreting the model we can transform it back to actual return which is more realistic..

I want to calculate natural log returns for stock prices so kindly help me with its formula and also tell me whetheri  need to consider it in  % or not .