WEBVTT 00:00.960 --> 00:06.330 This video is a short excursions to a more theoretical topic but it's a pretty important topic and I 00:06.330 --> 00:13.740 will try to make this as intuitive as possible with a very simple example and without complicated mathematical 00:13.740 --> 00:20.490 background or formulas so when calculating financial returns and mean returns we can either calculate 00:20.490 --> 00:26.580 simple returns and the arithmetic average of simple returns or we can calculate logarithmic returns 00:26.640 --> 00:31.050 or lucky returns and the arithmetic average of luck returns. 00:31.050 --> 00:32.130 That's another option. 00:32.130 --> 00:37.950 Calculating the geometric average of simple returns but I will not cover this here. 00:37.950 --> 00:45.570 So let's go to an easy example and let's have a look at the difference between those two concepts and 00:45.590 --> 00:45.980 uh. 00:46.010 --> 00:50.800 We import pandas and umpire and then we create a data frame with. 00:50.830 --> 00:57.970 And your stock prices the far end of the year 2016 to 17 and to 18. 00:57.990 --> 01:06.780 And uh the prices are 100 at the end of the sixteen fifty at the end of two seventeen and ninety five 01:06.810 --> 01:08.970 at the end of the year to 18. 01:08.970 --> 01:14.460 So let's do this here and we call the data frame D F so that's the variable here. 01:14.460 --> 01:15.520 So let's have a look. 01:15.700 --> 01:21.840 And now let's assume that we want to calculate returns and the first option would be to calculate a 01:21.840 --> 01:25.170 simple returns and we have already done this. 01:25.190 --> 01:32.620 And the last videos so we can calculate simple returns with the percentage change method. 01:32.700 --> 01:39.750 So we you see the percentage change method and then finally we also drop a rose with an A values. 01:39.750 --> 01:47.910 So that might be the very first row here into 16 and actually recreate here a return data frame simple 01:47.910 --> 01:48.710 returns. 01:48.720 --> 01:51.180 So let's do this. 01:51.360 --> 01:57.810 So here we have uh the negative return of minus 50 percent in 217. 01:57.960 --> 02:03.450 And the positive return of plus 90 percent in the year 2000 and 18. 02:03.600 --> 02:12.810 So minus 50 percent is actually the decrease from 100 to 50 and plus 90 percent is actually the increase 02:12.820 --> 02:16.670 of the stock price from 50 to 95. 02:16.710 --> 02:20.850 And next we can also calculate the mean or the average of simple returns. 02:20.850 --> 02:23.010 So this is actually nothing new. 02:23.400 --> 02:29.210 So we use uh the mean method and actually the mean returns. 02:29.210 --> 02:37.080 So that is the arithmetic average of four minus point five and plus 0 point 9 is 0 point 2. 02:37.080 --> 02:43.320 So we could say that the average or the mean annual your return for our stock is the 0 point to our 02:43.320 --> 02:50.640 20 percent and that's assume that uh we only know the stock price at the end of the year to 16. 02:50.910 --> 02:58.010 And the if you want to calculate today's stock price then that might be pretty intuitive to say okay 02:58.350 --> 03:05.730 the price and to 16 was one hundred and we had a mean return over two years of 20 percent. 03:05.820 --> 03:11.550 So today's stock price should be one hundred times the one point two times one point two. 03:11.580 --> 03:18.990 So we would actually expect to have at the end of the year 2000 and 18 a stock price of one hundred 03:18.990 --> 03:20.150 forty four. 03:20.190 --> 03:25.350 But this is actually pretty far away from our actual stock price 95. 03:25.500 --> 03:28.590 So actually something went wrong here. 03:28.860 --> 03:37.050 And uh what we can see here is actually a pitfall of simple returns and uh the mean are the ever of 03:37.170 --> 03:42.350 simple returns so having the average of simple returns of for example here. 03:42.360 --> 03:43.770 Oh point two percent. 03:43.770 --> 03:46.310 We cannot be 100 percent sure that. 03:46.340 --> 03:54.630 We um definitely realized a positive return over the last two years so that we increased our investment. 03:54.630 --> 04:01.530 In fact our investment dropped from one hundred to ninety five and having here only the average of simple 04:01.530 --> 04:07.740 returns does not give us the information that uh we really increased our investment. 04:07.860 --> 04:14.750 And with the mean or the average of simple returns we are not able to calculate starting from for example 04:14.750 --> 04:24.000 we added to a 16 price to calculate the ending price into 18 so there's no chance to do so and consequently 04:24.030 --> 04:27.480 the mean of simple returns can be pretty misleading. 04:27.480 --> 04:34.320 So here we have a positive mean return but actually we uh lost money here and we can actually overcome 04:34.320 --> 04:37.650 this problem by using logarithmic returns. 04:38.010 --> 04:40.310 So let's again have a look at our data frame 04:43.600 --> 04:51.460 and we can actually calculate logarithmic returns by dividing each price uh by the previous price and 04:51.460 --> 04:53.290 then we take the logarithm. 04:53.350 --> 05:01.510 So for example the a logarithmic return for the year 2017 is fifty divided by one hundred and then the 05:01.570 --> 05:10.000 logarithm and the logarithmic return for the year 2018 so ninety five divided by fifty and then taking 05:10.000 --> 05:18.250 the logarithm and coding wise we can actually do this by having our data frame divided by a shift of 05:18.250 --> 05:24.870 one of our data frame and then taking the logarithm with the NUM pi function and put out lock. 05:24.880 --> 05:25.980 So let's do this here. 05:28.090 --> 05:32.550 So here we have the logarithmic returns and no surprise into 16. 05:32.650 --> 05:36.220 We have a missing value and then four to 17. 05:36.280 --> 05:40.900 We have a negative return and into 18 a positive return. 05:40.900 --> 05:46.740 So now let's create a data frame with logarithmic returns and we also drop here. 05:46.750 --> 05:54.490 The first row with the drop and a method and we save the data frame and the variable lock returns and 05:54.490 --> 05:56.090 let's have a look here. 05:56.110 --> 06:05.340 So these are the two lock returns and then we can also calculate the mean logarithmic return so the 06:05.340 --> 06:07.250 mean logarithmic return. 06:07.440 --> 06:15.900 Minus 2.5 percent and if we can already see that we might lose some money over the two years period. 06:15.900 --> 06:19.240 So from one hundred to ninety five. 06:19.350 --> 06:22.950 So here we have a negative mean return. 06:22.950 --> 06:29.340 And in contrast to the mean are the average simple return here the mean or average logarithmic return 06:29.690 --> 06:30.610 gifts us. 06:30.780 --> 06:38.130 Actually the right to information or the impression that we have lost some money over the last two years. 06:38.220 --> 06:43.650 So this is one advantage of logarithmic returns so that the mean are the average logarithmic return 06:43.830 --> 06:50.510 gives us the correct information whether our investment increased or decreased during the whole period. 06:51.560 --> 06:56.370 And that's actually a second advantage of lock returns. 06:56.390 --> 07:03.770 So starting with the price and to 16 100 and having the mean or the average lock return over the last 07:03.770 --> 07:10.910 two years we can actually calculate today's stock price into 18 and this is actually impossible with 07:10.910 --> 07:12.440 simple returns. 07:12.440 --> 07:15.570 So with the logarithmic returns we take here. 07:15.590 --> 07:23.930 The stock price into 16 100 times e to the power off and then he apparently is this we have our mean 07:24.110 --> 07:26.260 lock returns times. 07:26.360 --> 07:29.260 Um the number of time periods. 07:29.390 --> 07:33.870 So our lock return is based on two time periods or on two years. 07:33.920 --> 07:39.970 So we have yet e to the power of uh two times minus 2.5 percent. 07:39.980 --> 07:45.060 And by multiplying this uh with 100 we get today's price. 07:45.110 --> 07:49.190 So into 18 and we end up here with 95 07:51.810 --> 07:54.540 so that are the two advantages of lock returns. 07:54.540 --> 07:58.080 And that's actually not completely wrong to work with simple returns. 07:58.560 --> 08:05.220 So there's actually nothing wrong in saying that the mean return was 0 point 2 percent over the last 08:05.220 --> 08:06.250 two years. 08:06.270 --> 08:13.200 The problem is that the interpretation of mean or the average simply returns can lead to wrong conclusions. 08:13.200 --> 08:19.110 So for example in our case that we increased our investment and also demeanor the average of simple 08:19.110 --> 08:23.210 returns can hardly be used to make a follow on calculations. 08:23.220 --> 08:30.660 So from a mathematical point of view we're lucky returns are the better and the more comfortable alternative. 08:30.740 --> 08:37.850 And also I must admit here that this was an extreme example with only two periods with quite different 08:37.850 --> 08:38.530 returns. 08:38.780 --> 08:43.130 So one highly positive return and one highly negative return. 08:43.130 --> 08:50.210 So in reality when we have dozens of monthly returns the difference between Lock returns and simple 08:50.210 --> 08:53.270 returns is actually fairly low. 08:53.270 --> 08:58.340 And again there's nothing wrong about using simple returns a lot of people in the finance and investment 08:58.340 --> 09:06.860 industry only work with simpler returns and in most cases that's a good approximation but with luck 09:06.870 --> 09:09.390 returns you are actually on the safe side. 09:09.440 --> 09:15.290 So in the next video we will see an example where it's definitely better to work with lock returns. 09:15.410 --> 09:17.270 So hope to see you there by.