# Measuring returns

always come across the issue of which return to use. There a three types that I know about. The simple return, the log return and the geometric return.

Now I wonder whether it depends on the subject which return to use. For example to calculate the average daily return of a portfolio and then annulize the average daily return, it shouldn't make a difference since for small returns the log return is appxrimately equal to the simple return.

However, on some days you have higher returns like 5% or more, which is a log return of 4.88% which is a significant difference. So if the series has daily returns of 5%, is it still legitimate to use the log return? The problem here is that the log return underestimates positives returns, but corrrectly estimates average returns. For example. Assume that the price of a stock trades from 100 to 105 and back to 100. The return over the entire period is 0%. However the the averafe simple return is not 0%. Geometric and log returns yield 0%, which is correct in my opinion.

So when annualizing the daily return, I always get higher returns then expected.

Then when reporting the annualized return, should the average return be retransformed to the simple return through the exp() function? When I do this, can I also transform back the standard deviation with exp()?

Then I haven't had any good solution for the following: I want to create a portfolio and measure the returns on a daily basis. My approach was to take the log return of, say 2 stocks, and weight each with 50% and sum them up each day. Then I calculate the mean and standard deviation over a certain period of time. Intuitively, it feels wrong to add up transformed returns in a portfolio. I did this, because I have know idea how create the proper returns of a portfolio.

Maybe someone can help me out.

## 2 Answers

You do not have to think too much about return formulas and get confused, just go to the basics. Return is simply:

$$Return = Ending Value / StartingValue - 1$$

Log returns are used in places where it provides model simplicity in defining returns in a logarithmic format. Also when assumptions are made on log-normality rather than normality. Log returns were popularized for models used in relation to derivative modelling.

$$log(1+r) \approx r$$ holds true when $$r$$ is small, which is usually the case, but consider the case of monthly returns, recently in December 2018, S&P fell by 9%. Now $$r$$ is no longer small.

If you calculate the Log returns for S&P TRI from 2000 to today, it will understate the cummulative returns by more than 5%!

As for standard deviation, it ideally should not matter that much unless you are measure the standard deviation over a very small period.

As for portfolio returns, the best practice is to compute the daily portfolio value as a weighted average of stock values and then compute the return on the portfolio series itself.

Stick to the basics.

• Yes the log return underestimates the true return, but the simple return overestimates it in a much worse way. I'll give you an example: The S&P500 total return yields annualized simple return of 12.5% and log returns of 10.2% from 1988 to 2019. This is 22% more for simple. On daily average basis it is still 15% more for simple returns and the german DAX is even worse at 30% more! Now lets get real numbers: Using the average daily simple return of 0.0456% and aggregating them for 30 years (starting in 1988 with 256 points) gives a value of 9531 points instead of 5931 on 20th of July 2019. – Chris H. Jun 26 '19 at 6:11
• Hey Chris, the return aggregation formula is an approximation, the actual returns from 2019 to 1988 would just be total return index at 2019 divided by total return index at 1988 minus one, which would sure come out to be a cagr of 12.5% – Dhruv Mahajan Jun 26 '19 at 6:15
• Thats true, but I am working with other statistics and measures as well. Like the standard deviation or VaR, Sharpe... in order to get that from daily average to annual value, you need to annualize it. I can't see the advantage of your approach when further statstics are needed. Don't get me wrong; I appreciate your help and maybe this discussion illuminates some hidden thoughts. – Chris H. Jun 26 '19 at 6:27
• @ChrisH., the advantage is that one's correct and one isn't. Log returns are a simplifying approximation used in a lot of ways and places because they're easier to work with. Any time you need to know your actual return over some period, you should be using arithmetic returns. That doesn't change with needing to annualize for Sharpes or what have you. I'm also not sure what's more complicated about annualizing a Sharpe ratio based on arithmetic return versus one using daily log returns (which, again, is technically incorrect). – Chris Jun 26 '19 at 6:38
• Ok, then, now lets assume we follow your approach. Taking the daily simple return is plainly wrong, since it overestimates the true return by a lot. I showed that. If I am mistaken, please correct me. Now, to get annual returns, the geometric mean is a proper and correct way. How could I get the corresponding standard deviation? It just seems wrong to annualize the standard deviation of simple returns and use it for a risk-return comparison along with geometric mean. Is there something like a 'geometric standard deviation'. – Chris H. Jun 26 '19 at 8:22

Since the discussion above is not coming to an end, I am going to show a summary of my assumptions. If you don not agree, let me know where exactly you do not agree.

1) The return

i) the simple annual return of a 1-year period is: $$r_{1989}=value_{1989}/value_{1988}-1$$ for $$value_{1988}=100, value_{1989}=110, value_{1990}=100$$ we get $$r_{1989}=10\%, r_{1990}=-9,1\%$$

ii) the simple annual return of a 2-year period is $$r^2_{1990}=value_{1990}/value_{1988}-1=0\%$$

iii) the mean of the sample of returns is $$r_{mean}=(r_{1989}+r_{1990})/2 = 0.45\%$$

iv) to calculate the end value we use the mean and take it to the power of the number of periods (2 in this case) $$endvalue = (r_{mean}+1)^2*startvalue=100.09$$

But this is not our endvalue. We could easily employ this method to daily returns, calculate the mean of daily returns and annualize them. We could take the daily returns the number if periods required to to get the end value of that 2-year period. However to get a real annualized value we simply take the mean of daily returns and take it to the power of one year (annualizing). However, this procedure will still overestimate the true end value. As I showed above, it will overestimate the endvalue by a lot.

v) the log return is calculated for a 1-year-period as: $$logr_{1989}ln(value_{1989}/value_{1988})$$

for a two year period

$$logr_{1990}=ln(value_{1990}/value_{1988}) = ln(value_{1989}/value_{1988})+ln(value_{1990}/value_{1989})$$

and the mean

$$logr_{mean} = logr_{1990}/2$$

in this example you get

$$logr_{mean}=(0.95-0.95)=0\%$$

which is the true return of that period.

To get from an annual average to the end value you multply by the number of periods (in this case 2)

$$endvalue=(1+0\%*2)*startvalue=100$$

Now we see the log method leads to the correct endvalue. However for a very specific one-year period, the simple return is the most correct return to get. For a period that is greater than 1, to get an annual return, you cannot annualize the simple daily returns, since the degree of overestimation is way too worse. If you can proof me wrong, I am glad to know. However the annualized mean of the daily logreturns underestimate the true return by a bit with greater sampling size

vi) To get an average return annual return from a two-period case you do this:

$$r_{geometric\ mean}=(value_{1990}/value_{1988})^{(1/2)}-1=(100/100)^{1/2}-1=0\%$$

2) the standard deviation (of one year)

The standard deviation is the average deviation around the mean. Mean is the keyword: Around which mean? Typically around the arithmetic, not the geometric mean. But you can show me a formula of a geometric standard deviation if you have one.

Now, I have shown that the mean of the simple return leads to very incorrect annual return, when taking the daily return and annualizing it. Again, if you use a very specific one year period, the simple return is the most correct choice. If you have more than one period and still want the average yearly standard deviation your compared mean return will be the geometric mean from vi) or the logreturn from v). Since the standard deviation is the average deviation from the arithmetic mean (of a sample (=here: of daily returns)), and since the simple daily return is wrong, I see only one return that is left: that is the log return. What sence would is make to say in the same sentence: "the geometric mean is 10% and the standard deviation of the daily log returns is 9% (while the standard deviation is the average deviation around its mean (the mean of log returns and not of the geometric mean))" It gets even worse when we report measures like the sharpe ratio or the VaR that a calculated using different approaches. Now, you can for sure use the standard deviation of the simple daily returns but then you should also report the annualized mean of the daily returns. And if you do not believe me, please try it out yourself: Using daily simple returns and trying to get to the endvalue will dramaticall overstimate the true return of an entire period.

3) reporting

This is what my initial post intented to ask: Might it be not the return itself that matters so much but rather on what purpose we are using it for? For statistcal purposes like regressions, t-statistics, annualization, key metrics like the sharpe ratio, the log return seems to me favourable. For Sales, I definitively would use the simple return.

final remarks

As a private investor that tries to optimize his performance, I am trying to apply carefull methods that rather underestimate return and rather overestimate risk. Because, if something goes wrong, I will still have a cusion. And if nothing goes wrong I can happily get more return than I expected. I have proven that the simple return is not only wrong, but even more wrong than the logreturn, especially if you consider risk of false decisions caused by too high expectations or wrong risk-return portfolios. If you do not think that the simple return is more wrong than the logreturn, I am fully open to hear other opinions but I'd like to have some proof of that.