Years ago, I read a finance book that opened with a question: how do you value an apple tree? The book walked through some unit economics—$100 to plant the tree, $20 in profits from the apples each year, $50 worth of firewood after ten years—and, after slapping a discount rate on the whole endeavor, came to a precise discounted cash flow valuation. Then, a few years later, while reading something unrelated, I got my first tutorial in Bayes’ Rule. The example walked through a classic example where the test for a rare disease had a low-but-nonzero false positive rate, and the striking conclusion that the lower the base rate, the more likely it is that scary test results are merely noise (this seems to be a universal way to illustrate Bayes’ Rule, for example here).

What both of these examples have in common is, first, that they provide a framework for answering otherwise unanswerable questions, and second, that they embed a deeper mystery in those answers. You can calculate what a known stream of future cash flows is worth if you know the appropriate discount rate, but you can only update your view on uncertain outcomes based on new information properly if you have some reasonable prior view.

And as it turns out, these questions are more deeply related: DCFs and Bayes' Rule are both useful tools for handling uncertainty, but each methodology has a key input that's arrived at mostly through experience. For Bayes, that's mostly domain-specific—trying to figure out what the base rate for some phenomenon is with historical data and probable selection effects. But for DCFs, it's partly a stylistic choice.

One of the key questions for any discounted cash flow analysis is whether to embed uncertainty into the discount rate or to do so when estimating the fundamentals. There are arguments for both styles:

1. An optimistic fundamental model coupled with a high discount rate basically says "This is a risky proposition, so I'm demanding a high return because it probably won't work out."

2. A more pessimistic model—say, one that splits the difference between "revenue will triple in the next three years" and "revenue goes to zero in three years because they're bankrupt" might just plug in 150% total revenue growth over that period. Since the fundamental model already includes the uncertainty, the required rate of return is lower.

But really, these are the same process expressed different ways: to get internally-consistent numbers, you might solve for a valuation in the second case at, say, a 9% discount rate (i.e. at current long-term rates plus the historical US equity risk premium of 5%), you'd then tweak your discount rate for the first valuation until it got to the same output. You want your valuation to be sensitive to meaningful inputs, like the probability that a business works out well and the cash flows it could get in that case. You don't want the valuation to be sensitive to how you phrase this process to yourself.

Where does that 9% rate come from? The idea of a discount rate is that it's some kind of fair-market value for the risk you're taking. You can calculate an ex ante long-term risk premium by looking at the relative performance of equities and bonds over long periods. You can also calculate it ex ante by looking at long-term growth rates in dividends per share, and at current dividend yields.¹

These thought experiments are really part of a broader category: they're implicitly the outputs of a monte carlo simulation. The basic idea there is to take a plausible range of fundamentals (say, "growth is somewhere between 5% and 15% annually, margins stabilize between 20% and 40%") and then run a bunch of simulations. The trouble with this is that it's hard to interpret: it's much easier to reason about specific estimates than about ranges, and it gets even harder if the ranges have feedback loops, like long-term margins depending on long-term scale.

Yet another way to run discounted cash flow analyses is to treat the discount rate, rather than the valuation, as the variable to solve for. In this case, you plug in fundamentals, your future estimates, and the current price, and solve for an expected return. This is very useful when you're looking at a set of heterogeneous opportunities, because it lets you establish a hurdle rate for new investments. If your portfolio of microcap value stocks has an implied forward rate of return of 15%, that's the number that other opportunities have to hit before they're worth considering, too. (But in this case, the trap that you can fall into is assuming that there's going to be some catalyst for realizing that value. For a company to be meaningfully cheaper than the fair market value of its assets, it needs to have had some reason to have systematically disregarded that cheapness. It does happen; some investors estimate the date of value realization actuarially by looking at how old the controlling shareholder is.)

So if choosing discount rates is so hard, how do normal investors handle it? The real secret is, they mostly don't. Short-term investors are really modeling changes in the behavior of long-term investors based on some catalyst, so they take the current price as a given and then ask how that price would change given some new piece of information. For longer-term investors, small gaps in DCF simply don't mean that much; if a stock is 10% too cheap and you're holding for five years, your excess return is 2% annualized if you get everything exactly right—and it's entirely possible for one bad quarter at the end of that five-year period to wipe out this excess return.

Instead, DCFs are a research tool. They're a way to ask what return is plausible given a set of fundamental assumptions, or, equivalently, what fundamental developments would be necessary to reach a particular return. The best ideas are obvious, to the point that arbitrary precision ("do we discount at 9% or 8.5%?") is pointless if it means the difference between saying something is trading at half of what it should be versus trading at a 53% discount to fair value instead. The value is in the process: the majority of the time something looks cheap, it looks that way for a reason, and building a model of when the cash flows arrive, how that happens, and what they're worth when they get there is the only way to test this.

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Read More in The Diff

Discounted cash flow analysis is at the heart of any attempt to value companies, either individually or in the aggregate. They've come up in The Diff many times, including:

• Why it can be worthwhile to structure a business so it's likely to go bankrupt from time to time ($).

• This piece goes into more detail on building DCF models as a tool ($).

• This piece raises the fun idea that sometimes, a business inflects in a way that simultaneously increases growth expectations and reduces discount rates ($).

• And a discussion of discounted cash flow would not be complete without some thoughts on reinvestment.

1. This sounds incredibly simplistic—we're omitting almost all fundamentals, and assuming zero change in valuation! But that's actually fair: valuation can't have an expected long-term drift in any direction, since that implies that multiples gradually approach either zero or infinity without companies changing how much stock they issue or buy back. And speaking of buybacks, they do factor into this model, but they tend to cancel out: if aggregate dividend yields drop by 1% because firms switch to buybacks, then expected future growth in dividends per share must move up by 1% to reflect the fact that the market is buying back ~1% of shares outstanding each year. Over long periods, it's remarkable how stable the dividend contribution has been, though in recent decades the exact switch mentioned above has happened. Meanwhile, valuations are an unpredictably big deal over the span of a decade, but will generally mean-revert over the course of a career. And of course if you knew which direction they'd move in over time, why bother with DCFs at all? You should be trading index futures with more leverage than single-name equities can provide, and monetizing your market-timing skill in a strategy whose capacity is effectively infinite!

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