PEG ratio - why is 1 the magic number?

I don't think it has any theoretical basis. I think it's more of a rule of thumb derived from Peter Lynch's quote in his 1989 book, "The P/E ratio of any company that's fairly priced will equal its growth rate."

More fully ( LINK):
"The p/e ratio of any company that's fairly priced will equal its growth rate ... If the p/e of Coca-Cola is 15, you'd expect the company to be growing at about 15 percent a year, etc. But if the p/e ratio is less than the growth rate, you may have found yourself a bargain. A company, say, with a growth rate of 12 percent a year ... and a p/e ratio of 6 is a very attractive prospect. On the other hand, a company with a growth rate of 6 percent a year and a p/e ratio of 12 is an unattractive prospect and headed for a comedown."

I suppose this rule of thumb made sense for the particular interest rate and inflation conditions of 1989, but I'm not sure how well it holds today. From 1989 to 2016, S&P 500 earnings ( LINK) increased from 43.84 to 87.17, for a real growth rate of less than 2.6%. You can throw in inflation, but that was less than 2.5% per year for that time period, so you still only get a little over 5% nominal growth rate. Does that mean the S&P 500 stocks should have P/E ratios of 5?

What about cash cow stocks that might have slightly negative growth rates? Why include an arbitrary multiplication by 100 to get a percent figure? I suppose there might be something I'm missing, but it looks to me like a merely coincidental rule of thumb that worked well in the 1980s.