Anyone recall the old saying, “A bird in the hand is worth two in the bush”? Most of us do, but few of us know it has a mathematical component that can help investors.
Robert Hagsworth’s platform for explaining the many links between mathematics and investment is the subject of chapter eight of “Investing: The Last Liberal Art." It’s one of several subjects he marshals to create his own latticework of mental models, in a book inspired by Charlie Munger (Trades, Portfolio)’s multi-disciplinary approach to investing.
The story of a bird in the hand versus two in the bush is believed to originate with Aesop, who likely lived in Greece between 620 and 560 B.C.E. And what does that have to do with investing?
Warren Buffett (Trades, Portfolio), Munger’s partner and friend, explained it this way: “The formula we use for evaluating stocks and businesses is identical. Indeed, the formula for valuing all assets that are purchased for financial gain has been unchanged since it was first laid out by a very smart man in about 600 B.C.E.”
He went on to explain that in fleshing out this principle, you had to answer just three questions:
- How certain is it there really are birds in the bush?
- How many of them are there, and when will they emerge?
- What is the risk-free interest rate?
If you can answer those questions, then you will know the maximum value of the bush and how many birds that you hold now should be offered for the bush. Of course, he is really discussing how many dollars you should pay for a security or asset.
The calculations the Sage of Omaha uses to make decisions about whether to buy or not buy is almost as simple and can be done on the back of an envelope. Start by tabulating the cash; estimate growth probabilities of cash coming in and going out over the business’ lifetime; and then discount those cash flows to present value.
John Burr Williams, as noted in chapter six, gave investors the discounted cash flow analysis. When he first published his theory in 1937, economists believed asset prices were mainly based on investor expectations for capital gains. In other words, prices were based on opinions. John Maynard Keynes, the famous Cambridge economist, for example, called this the “beauty contest” approach.
On the other hand, Williams believed prices were ultimately determined by an asset’s value, not stock pickers’ opinions. In taking this direction, he moved the focus from technical analysis to measuring the underlying elements of asset value.
Rather than forecast stock prices based on investors’ opinions, he believed a quantitative value could be determined by looking at a corporation’s expected future earnings. From Williams’ work, we now think of a stock’s value as its intrinsic value, or the present value of all future net cash flows over the life of the investment.
Hagstrom then posed this question: “You may be asking yourself, if the discounted present value of future cash flows is the immutable law for determining value, why do investors rely on relative valuation factors, second-order models?” By “second-order models” he is referring to less comprehensive metrics such as price-earnings ratios.
The answer is that second-order models are much simpler and quicker than first-order models (such as DCF). Establishing DCF makes us examine the many forces that will influence future business valuations. However, forces such as the state of the economy, competition and innovation combine to create millions of variations. It is therefore difficult to reach a precise value from future cash flows.
We can, then, reach only approximations. But as Buffett said, “I would rather be approximately right than precisely wrong.” Or, can we get further than approximations?
With that, Hagstrom moved forward to discussions of risk, which arise out of the approximations of value. He began with a challenge put forward by a French nobleman and gambler, “How do you divide the stakes of an unfinished game of chance when one of the players is ahead?”
The 17th-century French mathematician Blaise Pascal took up that challenge. He partnered with Pierre de Fermat, the lawyer and mathematician who had invented analytical geometry. Out of their partnership came an early form of what we now know as “probability theory.”
Their work also set the course for what is now called “decision theory,” a process that allows us to make optimal decisions, even if the future is uncertain. According to financial historian and author Peter Bernstein, “Making that decision is the essential first step in any effort to manage risk.”
The idea that decision theory allows us to make optimal decisions under conditions of uncertainty glosses over something critical: The quality of the information used to make the probability estimate. Enter Jacob Bernoulli who made a distinction between the odds in a game of chance and the odds of getting a real-life decision right. He explained that nature’s patterns are only partially known, so real-life probabilities should be considered in degrees of certainty rather than absolute certainty.
Thomas Bayes, an 18th-century minister and amateur mathematician, made probability theory actionable. In a posthumous essay, he laid out what’s called the method of statistical inference. Hagstrom summarized the idea this way: “When we update our initial belief with new information, we get a new and improved belief,” and added, “Bayes’s theorem gives us a mathematical procedure for updating our original beliefs and thus changing the relevant odds.”
Bayesian analysis is also called “decision tree theory,” a diagram in which each branch of tree represents new information and as a result leads to new odds. Here’s an example:
You’ll recall that we just discussed a key problem with Williams’ DCF model: the uncertainty about future events. If we add Bayesian analysis (decision trees) to the DCF model, we can reduce the uncertainty or risk by adding branches that define the likelihood of future events and their financial implications.
We can also address the issue of linear extrapolation in a DCF case by expanding the decision tree to take in different time spans and growth rates.
That concludes Part 1 of chapter eight. In Part 2 there will be with more lessons, including Hagstrom’s thoughts on managing during sideways markets.
Read more here:Â
Learning to Think Like Charlie: Literature and Critical ThinkingÂ
Learning to Think Like Charlie: PhilosophyÂ
Learning to Think Like Charlie: The Psychology of MisjudgementÂ
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