Simple Returns vs. Log Returns

Alexander Hagmann
A free video tutorial from Alexander Hagmann
Data Scientist | Finance Professional | Entrepreneur
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Analyze Stocks with Pandas, Numpy, Seaborn & Plotly. Create, analyze & optimize Index & Portfolios (CAPM, Alpha, Beta)

26:46:46 of on-demand video • Updated November 2021

  • Step into the Financial Analyst role and give advice on a client´s financial Portfolio (Final Project)
  • Import large Financial Datasets / historical Prices from Web Sources and analyze, aggregate and visualize them
  • Calculate Return, Risk, Correlation and Rolling Statistics for Stocks, Indexes and Portfolios
  • Create, analyze and optimize financial Portfolios and understand the use of the Sharpe Ratio
  • Intuitively understand Modern Portfolio Theory (CAPM, Beta, Alpha, CML, SML, Risk Diversification) with Real Data examples
  • Create Interactive Price Charts with Technical Indicators (Volume, OHLC, Candlestick, SMA etc.)
  • Create Financial Indexes (price-, equal- and value- weighted) and understand the difference between Price Return and Total Return
  • Easily switch between daily, weekly, monthly and annual returns and understand the benefits of log returns
  • Start from Zero and learn all the Basics of the powerful Pandas Library
English [Auto] This video is a shot, excrescence, to a more theoretical topic, but it's a pretty important topic and I will try to make this as intuitive as possible with a very simple example and without complicated mathematical background or formulas. So when calculating financial returns and mean returns, we can either calculate simple returns and the arithmetic average of simple returns, or we can calculate logarithmic returns or Lucke returns and the arithmetic average of returns. That's another option. Calculating the geometric average of simpler returns. But I will not cover this here. So let's go to an easy example and let's have a look at the difference between those two concepts. And, uh, we import Pendas, an umpire, and then we create a data frame with annual stock prices for end of the year 2016 to a 17 and to 18. And, uh, the price is one hundred at the end of to sixteen fifty at the end of to seventeen and ninety five at the end of the year to eighteen. So let's do this year and we call the data frame D.F.. So that's the variable here. So let's have a look and let's assume that we want to calculate returns. And the first option would be to calculate a simple returns. And we have already done this and the last videos so we can calculate simple returns with the percentage change method. So we use the percentage change method. And then finally we also drop rows with an A values. So that might be the very first row here into 16. And actually we create here a return state frame, simple returns. So let's do this. So here we have the negative return of minus 50 percent and to a 17 and the positive return of plus 90 percent and the year 2018, so minus 50 percent is actually the decrease from 100 to 150 and plus 90 percent is actually the increase of the stock price from fifty to ninety five. And next, we can also calculate the mean or the average of Semba returns. So this is actually nothing new. So we use the mean method and actually the mean returns. So that is the arithmetic average of four minus four point five and plus four point nine point two. So we could say that the average are the mean annual return for our stock is 0.01 20 percent. And that's assumed that we only know the stock price at the end of the year to 16. And if you want to calculate today's stock price, then that might be pretty intuitive to say, OK, the price then to 16 was one hundred and we had a mean return over two years of 20 percent. So today's stock price should be one hundred times one point two times one point two. So we would actually expect to have at the end of the year 2018, a stock price of 144. But this is actually pretty far away from our actual stock price. Ninety five. So actually something went wrong here. And what we can see here is actually a pitfall of similar returns and the mean or the Everetts of simple returns. So having the average of temperature and so, for example, zero point two percent, you cannot be 100 percent sure that we definitely are realised a positive return over the last two years so that we increase our investment. In fact, our investment dropped from 100 to 95. And having here only the average of similar returns does not give us the information that we really increase our investment. And with the mean or the average of simple returns, we are not able to calculate, starting from, for example, Heidi, to a in price to calculate the ending price and to 18. So there's no chance to do so. And consequently, the meaning of simple returns can be pretty misleading. So here we have a positive return, but actually we lost money here and we can actually overcome this problem by using logarithmic returns. So let's again have a look at our data frame. And we can actually calculate logarithmic returns by dividing each price by the previous price, and then we take the logarithm. So, for example, the logarithmic return for the year 2017 is fifty, divided by one hundred. And then the logarithm and the logarithmic return for the year 2018 is the ninety five divided by 50 and then taking the logarithm and coding wise, we can actually do this by having our data frame divided by a shift of one of our data frame and then taking the logarithm with the non-pay function and Portlock. So let's do this here. So here we have the logarithmic returns and no surprise, and to 16, we have a missing value and then four to 17 we have a negative return and into 18, a positive return. So now let's create a data frame with logarithmic returns. And we also drop the first row with the drop and a method and we save the data frame and the variable returns. And let's have a look here. So these are the two lock returns and then we can also calculate the mean logarithmic return. So the mean logarithmic return is minus 2.5 percent. And if we can already see that we might lose some money over the two year period, so from one hundred to ninety five, so here we have a negative mean return. And in contrast to the mean, the average simper return here, the ever it's logarithmic return gives us the right information, not the impression that we have lost some money over the last two years. So this is one advantage of logarithmic return. So that dimino, the average logarithmic return, gives us the correct information whether our investment increased or decreased during the whole period. And that's actually a second advantage of lock returns, so starting with the price into 16, 100 and having the means, the average return over the last two years, you can actually calculate today's stock price into 18. And this is actually impossible with simple returns. So with logarithmic returns, we take the stock price into 16, 100 times a year to the power of and then he and parenting's us. We have our mean lock returns times the number of time periods. So our lock return is based on two time periods or on two years. So we have to eat with the power of two times minus 2.5 percent. And by multiplying this with 100, we get today's price so into 18 and we end up here with 95. So that are the two advantages of lucky returns, and it's actually not completely wrong to work with simple returns, so that's actually a nothing wrong and saying that the mean return was 0.01 percent of the last two years. The problem is that that the interpretation of mine or the ever as simple returns can lead to a wrong conclusions. So, for example, in our case, so that we increase our investment and also, dimino, the Everetts of similar returns can hardly be used to make a follow on calculations. So from a mathematical point of view, a lot of returns, the better and the more comfortable alternative. And also, I must admit here that this was an extreme example with only two periods, with quite different returns, so one highly positive return and one highly negative return. So in reality, when we have dozens of monthly returns, the difference between locker returns and the simple returns is actually fairly low. And again, there's nothing wrong about using simple returns. A lot of people and the finance and investment industry only work with simple returns. And in most cases it's a good approximation. But with luck returns, you are actually on the safe side. So in the next video, we will see an example where it's definitely better to work with lucky returns. So I hope to see you there by.