Distribution and Outliers
_ad51b7e8d85d4b80cf80097e092726b1_Lect02_5_Distribution-and-Outliers
Professor:
Now the question is, I was already suggesting that there is an issue of outliers. So, what do we mean by an outlier? Well, there’s something, I’m sure you’ve heard about the normal distribution or the bell shaped curve for random variables.The normal distribution is a typical distribution for random variables in natureand there are reasons to think that many random variables follow a distribution like this. The distribution has two parameters, its mean and its standard deviation. So in this case, I have plotted it for two different standard deviations but both a mean of zero. So, this is a theoretical probability distribution for let’s say a return on a stock and here the standard deviation is one on this pink curve, and it’s three on the blue curve. Many random variables in nature follow this distribution but not all of them and that’s important because in finance it tends not to follow this distribution,that we tend to have outliers or fat tails. The normal distribution, has two tails. This is the right tail which has high values of the random variable, and here’s the left tail which is low values of the random variable. And the height reflects the probability of getting that value. So, if this were the distribution of returns for Apple stock, and let’s say, a standard deviation of three percent, then the probability of getting a return of three percent is pretty good. Here’s three percent, and the probability of getting three standard deviations, that would be nine. You can see it’s just about negligible, and then I don’t even show it anymore. The probability of four standard deviations out is essentially zero. That’s what the normal distribution says. The normal distribution has been used to describe for example, human heights or human IQs or SAT scores or, lots of things seem to follow the random normal distribution. And you probably have become intuitive about this. If you see some random variable repeatedly, and it’s pretty much always been between say minus five and plus five, then intuitively you start to think it can’t happen, that it would be 15 because you’re intuitively trained by life’s experiences. But in fact, there are other distributions that are more characteristic of financial returns. So there’s another kind of distribution called the Cauchy distributionafter a famous mathematician. And what I’m showing here is, at 100 draws from the normal distribution in blue, and 100 draws from the Cauchy distribution in red. Now you see the difference between a Cauchy and the normal. The normal distribution has a kind of look to it, you see it’s going up and down about the same amount all the time, well, I’m not saying that’s exactly right. The probability that it will be 10 times the normal change, the usual change is negligible so you never see it deviating from… It has the kind of uniform look to it through time but with Cauchy, it also looks very much like normal. You can’t even tell them apart for long intervals of time and then bang! there’s some big positive value. In other words, the distribution under Cauchy is fat tailed so the Cauchy looks like a normal distribution except instead of just trailing off to zero, the distribution continues out above zero, way out. So, you can be deceived by a fat tailed distribution like the Cauchy into thinking that you’re living in a fairly stable world whose risk I understand, but the problem is, there are these big events that occur from time to time.The Central Limit Theoremin Statistics says that,averages of a large number of independent identically distributed shocks or random variables is approximately normally distributed, but that central limit theorem assumes that the underlying shocks do not have fat tails. So, if you’re taking the average of stock market returns which tend to be fat tailed, then your average is not a good indication of the real average over long intervals of time because you might well have gotten a sample where none of the fat tail outlier stocks.
My friend Nassim Talebhas written a book called “The Black Swan” which got a lot of attention. It referred to black swan events. So you’ve seen a lot of swans in your lifetime, and they’ve always been white, right? Have you ever seen a black swan? You might well conclude that black swans do not exist but in fact they do exist. There are black swans and so that’s a metaphor he uses for a fat tail.Here is a plot of the normal and the Cauchy distribution. The Cauchy distribution looks pretty much like the normal. It’s a bell-shaped curve and it trails off but there’s a subtle difference, that there are these rare very… they’re not quite as rare as as the normal would suggest. The real world puts fat tails in our lives. Here is a plot of the histogram of daily stock price changes since 1928. And what I have, what this thing is, how many days are there since 1928, but it’s tens of thousands of days. And so what we’re seeing here is that the stock market yielded a return of, this is for the S&P 500, or extended S&P 500 of between, I guess this is between… of one percent with some interval around that. It did that and suddenly like 9,500 days, it earned plus one percent one day. And then, on something like 2,500 days, it earned plus two percent. And then on Sunday like 800 days, it earned plus three percent and looks like it’s about, I don’t know, 400 days it was plus 4 percent. And then here, I can’t even figure that out, something at plus five. After that, they looked like you can’t even see them anymore. So you might conclude by looking at this histogram that stock market returns are always between say minus six percent and plus six percent. A matter of fact, on October 30, 1929 the stock market went up 12.53 percent in one day. And on October 19, 1987 the stock market fell 20.47 percent in one day. It was quite a shock. By the way, on this day I was lecturing giving my…this class I was teaching ECON 252, and one of the students was listening to a transistor radio. Do you know what a transistor radio is? It’s what they used to have before you had iPhones and things like that. So he raises his hand, I will never forget this, and he said, “Did you know the stock market is crashing right while you’re giving this lecture?”. So, instead of going back to my office, I just thought, “What is he saying?” I went downtown and I talked to my stockbroker, Merrill Lynch, right here in New Highland. I took the elevator up and just walked in on there just to see what was happening. And it was this turmoil and everyone, I did manage because I walked in. If you tried to call your broker, you couldn’t. He wouldn’t answer, he’s too many calls but I walked in, barged in on him and I said, “What’s happening?” And he said, “Don’t worry, don’t panic. There was this big event, horrible event.”. That was the biggest drop, one day drop, ever in the whole history of the U.S. stock market so I had the good fortune to be warned of it by my student with the transistor radio. The fact that my student, this is before laptops, but he did…we had problems with transistor radios back then. So that’s an outlier. The normal distribution with the same mean and standard deviation as this histogram says that the probability of a drop greater than 20 percent is equal to 3*10^-71. That’s awfully close to zero if you know. I think that the estimated number of atoms in the universe is bigger, that’s 10^80 but it’s getting on like that. So, it’s essentially zero but it’s wrong because it happened and I was there. I saw it happen and I saw the excitement that it generated.
1.中央極限定理：如果原有母體是非常態分佈，隨著樣本數n的增加（通常要求大於30），不論原來母體是否符合常態分佈，樣本平均值的抽樣分佈都將趨於常態分佈。
2.尾端/極端風險(Tail Risk)是指統計學上兩個極端值可能出現的風險，按照常態的鐘型分佈（Bell Shape），兩端的分佈機率是相當低的（Thin Tails）；但是兩個極端值的分佈亦有可能出現厚尾/肥尾（Fat Tails）風險，那就是距離中值（Mean），出現的機率提高。也就是原本不太可能出現的機率突然提高了，運用在金融市場上，那就是極端行情出現的可能性增加而且頻繁，這樣可能會造成市場行情的大幅震盪，造成的原因可能是市場上出現不尋常的事件，如2008年雷曼兄弟倒閉、2010年的南歐主權債信危機，皆產生肥尾效應。<柯西分佈(Cauchy distribution)>
3.黑天鵝作者Nassim Taleb就是用黑天鵝來形容這個情形，我們日常中幾乎沒看過黑天鵝，我們就會假設這世上沒有黑天鵝，但其實是有的，就如同我們在金融市場上認為一些暴跌不會發生，但實際上這些極端還是有可能發生。
recall 統計學：
Covariance 共變數
假設有兩家公司，兩間新創，都要追求風險投資，各有百分之五十的機率成功，成功會得到一百萬，失敗得到零元。兩間公司有一樣的機率成功或失敗，但問題在於，他們是完全互相獨立的嗎？如過我要投資他們兩間企業，他們是一樣的嗎？兩間企業的共變數是變異數的概念，但是是介於兩間企業之間。此時，各種情形包括兩者都成功、兩者都失敗、一者成功一者失敗，寫成共變數如下，Cov = 0.25(0.5*0.5) 兩者都成功+ 0.25(-0.5*-0.5) 兩者都失敗+ 0.5(-0.5*0.5)一者成功一者失敗，其和為零，是我們樂見的，不過這是兩個投資獨立的情形，但也有可能有如下兩者互相依賴的情形，Cov = 0.5(0.5*0.5) + 0.5(-0.5*-0.5) + 0(-0.5*0.5) = 0.25得到以上情形風險就是非常高的，也有可能是完全相反的情形，一者成功一者必失敗，Cov = 0.5(-0.5*0.5) + 0.5(0.5*-0.5) + 0(-0.5*0.5)會得到負的共變數，而共變數的概念，是很少大眾投資者會看到的，他們多注重於單一個表現，但共變數是很重要的概念，如果都投資一樣的企業，風險是很大的。
So you want to be careful. In other words, the basic says, that the market demands higher returns from higher beta stock. That means high covariance with the market stock. And they’re willing to take no returns if the beta is low, because that means it’s less contributing to risk in the portfolio.
市場風險是無法控制的，但這種風險是可以考慮的，考慮怎麼做投資組合的分配，和之前所學的Beta是一樣的概念，其實Beta = Cov(r, rmarket)/ Var(rmarket)。市場追求較高的報酬對於較高的Beta風險，而若beta風險較低，願意接受較低的報酬。
這個概念是進入金融市場的入門，幫助你減少犯錯。