 # Logarithmic returns Question

Ok, I’m not really a quantitative oriented mind, so can someone explain to me why anyone would calculate risk (VaR) using logarithmic scaled returns not actual scaled returns? Basically, I’m reading some Lehman research where they lay out Relative Value analysis in FX. Their goal is to find a Cross that has 1) positive carry (carry = interest rate differential btwn countries) and 2) has lower VaR than the long currency and the base currency which is USD in their case. For example, they want to see if BRL/MXN has a lower VaR than BRL/USD. Long BRL/MXN has positive carry; therefore they need to see if it also has lower VaR. The steps they take: 1) Calculate the correlation of daily total returns over a 1-yr sampling period of BRL/USD and MXN/USD. It needs to be high and positive because (for students of portfolio theory) a high correlation between assets is sufficient to reduce the risk of a portfolio. The portfolio in this case is long BRL and short MXN. Weight(BRL) = 1, Weight(MXN) = -1. (I understand this first step) (Note: I’m not going to delve into how total return is computed because it’s not important for the purpose of this post) 2) They calculate 1-month VaR from logarithmic total returns over a 1-yr sampling period. They do this by evaluating the left side of a normal distribution w/ a mean representing the monthly total return and with the standard deviation of daily total returns over the past yr at the 5% confidence level. Next, they compute and compare the VaR of BRL/USD and BRL/MXN using the portfolio standard deviation formula. (I don’t understand why in this step they are using logarithmic total returns, what’s the point of using log returns? Also, why use log returns here, but not in the calculation of correlation? Maybe they do use log returns when calcing correlation, but they don’t state that they are in this paper. Also, what do they mean by “w/ a mean representing the monthly total return”? Do they mean they’re using the average monthly return? I guess the wording of their statement is a little odd to me.) Feedback is greatly appreciated. Thank you!

I suppose they think FX returns are log-normal not normal (hmm…)

I suppose to make the data more linear (compact) and that way you’ll get better results i.e t or f stats would be lower.

Isn’t it low correlation that is supposed to reduce portfolio risk? The purpose of the log returns is to address a problem that you get from the regular normal distribution. The problem is that for most investments, there is a theoretical maximum amount that you can lose (for unlevered stuff, up to 100% of your investment), but a theoretically unlimited amount that you can gain (n times your investment, and there is no limit to n on the upside). If you say that your daily returns are drawn from a normal distribution (i.e. if you plot out a histogram of your daily returns over a long time, it will take the form of a normal distribution), you have a problem, which is that you will never see returns smaller than -100%. So all the possible normal returns from negative infinity up to -100% will be missing from the distribution, no matter how long you sample. So the normal distribution doesn’t really fit the data. The solution that many analysts use is to say that your returns come from a log-normal distribution. This says that the returns don’t come from a normal distribution, but the logarithm of the returns does come from a normal distribution. This has the advantage that the logarithm of a very very large negative number will still map to a number between 0 and 1 (and log(x) gets closer and closer to 0 as x gets closer to negative infinity). If you do this, then you get returns that are more accurately mapped. Log returns also have the advantage that for small changes (similar to daily changes in prices), log(x)+1 is approximately x. (i.e. log(1.01) is approximately 0.01) So you can look at log returns and know approximately what the daily percentage returns are, provided that it’s not an extreme day.

i thought you use lognormal returns b/c of compounding. it assumes continous compounding vs having to compound hourly/daily/monthly etc…

bchadwick Wrote: ------------------------------------------------------- > > This has the advantage that the logarithm of a > very very large negative number will still map to > a number between 0 and 1 (and log(x) gets closer > and closer to 0 as x gets closer to negative > infinity). Sounds completely wrong. The log(x) function has a domain of x>0, so negative x is out of the question.

you’re right that I got my axes reversed - but the basic purpose of using logarithms is as I stated. If the logarithms are normally distributed, then the antilogarithm of returns - exp(X) - gets closer to 0 the closer X gets to negative infinity. Therefore it is possible to map a normal distribution to the range 0->infinity by using the lognormal distribution. As for geometric compounding, logarithms also have that nice property that log(x*y) = [log(x) + log(y)], so that by adding up all the log(returns), and taking the antilog, you get the cumulative return. You also get the geometric average of all the returns by taking the average of the log(returns) and taking the antilog, IIRC.

There is a difference between log-normal(x) and log(x)

ok I’m done.

The posts got crossed, so I didn’t read your reply. You have to communicate better dude, your post is like a huge barf… intended to impress us AF folks. You probably know your stuff, but got to put it across better.