Graham's Mr. Market is a fellow who turns up every day at the stock holder's door offering to buy or sell his shares at a different price. Often, the price quoted by Mr. Market seems plausible, but often it is ridiculous. The investor is free to either agree with his quoted price and trade with him, or to ignore him completely. Mr. Market doesn't mind this, and will be back the following day to quote another price.
Graham's point is that the investor should not regard the whims of Mr. Market as determining the value of the shares that the investor owns. He should profit from market folly rather than participate in it. The investor is best off concentrating on the real life performance of his companies and their dividends, rather than being too concerned with Mr. Market's often irrational behavior.
Here's an excerpt from The Intelligent Investor by Benjamin Graham, Revised Edition 2005, page 204-205:
"Imagine that in some private business you own a small share that cost you $1,000. One of your partners, named Mr. Market, is very obliging indeed. Every day he tells you what he thinks your interest is worth and furthermore offers either to buy you out or to sell you an additional interest on that basis. Sometimes his idea of value appears plausible and justified by business developments and prospects as you know them. Often, on the other hand, Mr. Market lets his enthusiasm or his fears run away with him, and the value he proposes seems to you a little short of silly.
"If you are a prudent investor or a sensible businessman, will you let Mr. Market's daily communication determine your view of the value of a $1,000 interest in the enterprise? Only in case you agree with him, or in case you want to trade with him. You may be happy to sell out to him when he quotes you a ridiculously high price, and equally happy to buy from him when his price is low. But the rest of the time you will be wiser to form your own ideas of the value of your holdings, based on full reports from the company about its operations and financial position.
"The true investor is in that very position when he owns a listed common stock. He can take advantage of the daily market price or leave it alone, as dictated by his own judgment and inclination. He must take cognizance of important price movements, for otherwise his judgment will have nothing to work on. Conceivably they may give him a warning signal which he will do well to heed—this in plain English means that he is to sell his shares because the price has gone down, foreboding worse things to come. In our view such signals are misleading at least as often as they are helpful.
"Basically, price fluctuations have only one significant meaning for the true investor. They provide him with an opportunity to buy wisely when prices fall sharply and to sell wisely when they advance a great deal. At other times he will do better if he forgets about the stock market and pays attention to his dividend returns and to the operating results of his companies."
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Benjamin Graham, the "Father of Value Investing" and Warren Buffett's teacher developed a low-risk system for finding good value investments.