CHAPTER 05: SPECIAL TOPICS: The Lurking Dangers of Extrapolation
In this segment, we're going to talk about the dangers of extrapolation. Extrapolation is not a scientific principle, as such, although we use extrapolation all the time to extrapolate what the population of the city would be, what the stock market index would be, and so on and so forth. So let's go ahead and look at an example to show that how extrapolation can be a dangerous idea. This is the example of the NASDAQ index, which is a measure of about maybe 3000 companies now to come up with an index of the stock market, how the stock market is doing. NASDAQ started in 1971 with a base index of 100, and now as you can see that in the years of 1994 to 1999, it went from anywhere from 752 to 4000. It went as high as about 5000 in February of 2000 . . . February of 2000. Nowadays, the value of NASDAQ is about 1500 or so, we're talking about April, 2009. So you can very well see that this index has seen quite an amount of turbulence. So what happened was that, this was during the Internet boom era, the value of NADSAQ, which is mainly made of many of the technological companies, it increased quite a bit as time went by, and as you can see that right between 98 and 99, it almost doubled, as time . . . just in a single year. So there were a lot of who were dreaming of retiring at the age of 40, retiring at the age of 45, because they thought that it would double every year, so if they put in some money in there, in a few years they will have so much money that they don't need to work anymore, but as we know that those things don't last very long. In fact, even the housing bubble, which took place in 2008, which burst in 2008, and we are seeing the repercussions of that kind of bubble in today's climate. So let's go ahead and see that why extrapolation's a bad idea. Here I'm plotting those values which are given to you in this table. So you are given values in this table for six years, and those are the six values which are shown, and as you can see there is some kind of upward trend going on, and people thought that this upward trend would continue. I've also, what I have done is that these are the end of the year NASDAQs, and I numbered them 1, 2, 3, 4, 5, 6, just to not to have too much round-off error, so that's why you find out that when I used extrapolation, I used the numbers 1, 2, 3, 4, 5, 6 as my x values, just for avoiding any kind of large round-off errors. Now, this is what I would get by extrapolation. So what I have done is that I took the . . . I took one, two, three, four, five, and six points which are given to me, and I used a fifth-order polynomial, so I used a fifth-order polynomial by in Excel or in MATLAB, this is from MATLAB, actually, and I took a fifth-order polynomial, and said, hey, let's go ahead and . . . let me go ahead and interpolate it to a fifth-order polynomial, and then I said, hey, let me see what happens in year 7 and year 8, which is corresponding to year . . . end of year 2000, and end of year 2001, and this is what I got as the extrapolated value, I got, hey, that at the end of year 7, my NASDAQ value, which would be 9128, so if I had invested in a stock or a mutual fund which mimicked the stock index, it would have doubled at the end of 2000, and at the end of 2001, it would have further doubled, and become 20000 or so, and as we know that that's not true, because the maximum value of NASDAQ took place in February, 2000 to be about 5000 or so. So see what happens now here. So this gives you the comparison of what you extrapolated, so this is what you extrapolated at the end of year 2000, and this is what you got as was the actual value of NASDAQ at the year . . . in the year of 2000. In fact, if you would have . . . if you would have used this to . . . so keep in mind that this extrapolation is only just one point ahead, so it's not that we're extrapolating way ahead, we're extrapolating just to one year ahead. So we had data from year 1 through 6, and we're trying to find out what the value in year 7 will be, and this turned out to be the extrapolated value and this turned out to be the actual value, where the difference is about 270 percent between the actual value and the extrapolated value. Now, if I had used the same fifth-order polynomial to predict what happened at the end of last year, which is 2008, I would have got that the NASDAQ would have been over a million, that the value of the stock index would have been over a million, when actually it was only 1570. In fact, it's hovering around about 1600 at this point of time in April, 2009. So those are . . . these are one of the many, many cases where you can show that extrapolation is a dangerous idea. So please, whenever you are doing extrapolation, you do need to think about whether you are using some good basis for extrapolation. Again, it is inexact, because it's something which is extrapolated for the time ahead, and nobody knows what happens in time ahead in many of these situations, such as population growth, or stock market index, and so on and so forth. And that's the end of this segment.