M21:12
嗯，你指出了一个很有意思的数学关系，因为如果你用的是现值贴现公式，而你又像你刚才假设的那样，代入一个高于贴现率的增长率，那答案当然就会是无穷大。
Well, you put your finger on an interesting mathematical relationship, because if you're using a present value discount formula and you put in a growth rate that is higher than the discount rate, as you have postulated, the answer of course will be infinity.
而且有很多管理层都很喜欢觉得他们公司的股票值无穷大，不过我们到现在还没发现哪一家真是这样。
And there are a lot of managements around who like to think their stocks are worth infinity, but we haven't found one yet.
这个非常具体的话题，大概三十年前，有个叫 Durand 的人写过一篇论文，叫 The St. Petersburg Paradox。
That precise subject was covered in a paper called The St. Petersburg Paradox by a fellow named Durand, probably 30 years ago.
我们办公室里大概也还有一份。
And somewhere we probably have a copy at our office.
我猜如果你去 Google，输入 Durand 这个名字，再输入 St. Petersburg，你也许能把那篇文章搜出来，虽然他们在老文章这方面不一定特别强。
My guess is if you go to Google and you put in the name Durand and you put in St. Petersburg, you may be able to call up that article, although they aren't necessarily terrific on old articles.
所以如果你想看的话，我们——我也想看看。
So if you'd like it, we'd— I'd like to see it.
如果你跟我们办公室的人说一声，我们会找一找，看看能不能把它找出来。
If you'll let somebody know at our office, we'll look around a little and see if we can find that.
去预测长期的高增长率是很危险的，因为你就会掉进这种悖论里。
It gets very dangerous to project out high growth rates because you get into this paradox.
如果你说一家公司的增长率从现在到 Judgment Day 都会是百分之九，而你用的是百分之七的贴现率，那这个结果就会跑飞，你知道，就会变成无穷大。
If you say the growth rate of a company is going to be 9% between now and Judgment Day, and you use a 7% discount rate, it goes off, you know, you get into infinity.
很多人就是在这里栽了大跟头。
That's where people get in a lot of trouble.
那种把极高增长率往很长时间去外推的想法，已经让投资者损失了非常非常大笔的钱。
The idea of projecting out extremely high growth rates for very long periods of time has caused investors to lose, you know, very, very large sums of money.
没多少公司能做到——你就看看 Fortune 500，往前倒五十年，他们现在还在纪念这个。
There aren't many companies— just take a look at the Fortune 500, go back 50 years, it's— they're commemorating that.
再看看当时在榜上的那些公司，有多少家真的能长期维持明显高于百分之十的增长率。
And look at the companies that were there and how many have really maintained rates much above 10%.
这不是一个容易迈过去的门槛。
It's not an easy hurdle.
等你到了百分之十五，你知道，那某种程度上已经是在高空里了，是一种很稀薄的空气。
And when you get up to 15%, you know, you're in the atmosphere, in a way, a rarefied atmosphere.
所以说，去外推高增长率，确实有很大的风险。
So that's— there's a real danger in projecting out high growth rates.
Charlie 和我很少会——嗯，几乎从来不会——把增长率估到很高的个位数以上。
And Charlie and I will very seldom— well, virtually never— get up into high digits.
这么做，你可能会亏很多钱。
You can lose a lot of money doing that.
有时候你也许会因此错过某个机会，但我还没见过谁靠这么做能长期持续地成功。
You may miss an opportunity sometime, but I haven't seen people who have been consistently successful doing that.
而且你确实会碰到你刚才提到的这个悖论。
And you do run into this paradox you mentioned.
嗯，当你得到一个数学结果是无穷大的时候，你显然应该往后退一步，意识到那是不可能发生的。
Well, you're obviously right when you get a mathematical result that is infinity to back off and realize that can't happen.
当然，人们通常的做法是，假设增长率会逐步下降，而且最终会停下来。
And of course, what people do is they project that the growth rate will reduce and indeed eventually stop.
这样一来，你得到的数字就会更现实一些。
And then you get more realistic numbers.
不然谁还能怎么做呢？
What else could anyone do?