# Discounted Cash Flow [Concepts Series]

Economics nerds always complain that you should never compare stocks to flows. It’s meaningless to say that one company’s cash on hand is bigger than a country’s GDP, for example, because GDP is quoted in dollars *per year* and cash on hand is a cumulative quantity. It’s like saying a plane is faster than the distance from New York to Boston. Does not compute.

There’s a giant exception, though: stocks, bonds, loans, and other financial products *explicitly exist to convert flows to ‘stocks’ in the economic sense*. The mechanics of this are worth understanding, because they underpin the value of so many financial assets.

The basic idea of discounted cash flow is to convert future estimate cash flows into present values, by “discounting” them at some interest rate. Suppose you can earn 1% annual interest on a certificate of deposit. The present value of $100 in a year is equal to the amount you’d have to put in a CD today in order to have $100 a year from now, or $99.01. The present value of $100/year for the next three years, at a 1% discount rate, is $100/1.01 + $100/(1.01 * 1.01) + $100/(1.01 * 1.01 * 1.01), or $294.01.

This can get pretty tedious as you extend it (here I’m using a higher discount rate):

As you can see, the amounts dwindle after a while. I’ve modeled it out for the first 100 years, and the value will keep accumulating after that point, albeit slowly.

There’s another twist to the model, though. What about growth? We can look at a series of cash flows that grow over time at a known pace:

But that’s not a realistic description of any business. No company grows at a steady pace *forever*. Usually, you expect growth to slow at some point. Here’s a multi-stage model, with fast growth for a while followed by a deceleration.

All of these models are *close to* correct, but they’re all below the theoretical value. There’s a nice trick for calculating an actual net present value given steady assumptions: cash flows / (discount rate — growth rate). So $1, growing at 1% per year in perpetuity, discounted at a 4% interest rate, is worth $33.33.

Normally, a two-stage model will calculate a “stub” value for the business after the growth stage is done, like so:

One interesting thing to look at is what’s called a sensitivity analysis: what happens to the net present value of a given cash flow when you adjust either the discount rate or the perpetual growth rate?

It looks like this:

You might notice that the chart is incomplete. There are blank spots. Let’s consider one. What is the present value of $1, growing at 5% per year, discounted at 2%? Year one’s earnings are $1.05, with a net present value of $1.03. Year two’s: $1.10, net present value of $1.06. Year three: $1.15 in future value, worth $1.09 in the present. Keep adding, and the net present value keeps compounding — it literally does not compute.

What this means in more practical terms is: as the long-term growth rate approaches the discount rate, the net present value rises at an exponential rate.

# What Discount Rate?

I’ve been using the term “discount rate” over and over again without defining it. Let’s get to it. In my first example, I said a discount rate for a guaranteed future cash payment is equal to whatever interest you could get on a guaranteed asset like a certificate of deposit. And then I switched, without warning, to a 4% discount rate.

Except for a risk-free asset, determining a discount rate is not an exact science. The discount rate for an uncertain asset is higher than the risk-free rate, but how much higher? However much you want — literally.

There are two ways to solve the net present value equation. One way is to say: for a given set of cash flows, what do you pay if you want a particular rate of return? And another way to look at them is: at a given price, what is the implied rate of return? Which one you choose is a matter of taste.

In general, if you’re aiming to buy an asset for capital appreciation, then sell it, you’d probably choose a discount rate that you think reflects the rate of return the average investor would require, and then buy the asset if its future growth implies that returns will exceed that discount rate.

If you’re buying an asset to hold it indefinitely — buying real estate, for example, or becoming a partner in a business — you’d probably think in terms of the expected return at a given price. Long-term investors like pension funds and endowments might also think in terms of the long-term return they expect from a given asset

# Discount Rates and Liabilities

For pension funds, the investment decision requires thinking about *two* discount rates. The one they expect to earn from a given investment, and the one they *have to* earn from all of their investments. Pension funds have a fund-wide discount rate — the actuarially assumed rate of return that measures the present value of their liabilities. If a pension fund assumes a 7% rate of return, it needs a portfolio whose average discount rate is at least 7%.

The discount rate pensions choose is set by both financial and political concerns. Finance tells you what rate of return you could expect from a blend of pension-appropriate assets: pension funds can afford to buy illiquid long-term assets (like real estate, timber, long-lived investment vehicles). They also need to buy safe assets like bonds, so they don’t permanently lose their capital. Take the weighted average expected return of an appropriate portfolio of assets, adjust your estimate down a bit to be on the safe side, and you know what discount rate to use. Politics has a different idea: the rate should be higher, please. A high discount rate means that the amount of funding a pension fund requires is lower: the liabilities are just one more stream of cash flows, discounted back to the present, and they respond to rates in the usual way.

Pension funds have been using high discount rates for a while now, as I wrote about here and more recently here (I wrote the second piece just after the crash and before most of the rally. Pensions are safe&dmdash;for now.)

As I noted in the first piece:

Right now, pension funds’ assumed rate of return is about 7.4%, compared to 8.0% in 2002, according to NASRA. Or, to benchmark that to a low-risk long-term investment: in 2002, they were expected to earn about 3% more than 10-year treasuries. Now, they’re expected to earn 4.8% more. However, nobody is explicitly calling for pension funds to take more risk, and nobody (as far as I know) believes that pension fund managers, in the aggregate, are adding 180 basis points more alpha per year than they used to.

That was then, this is now. 10-year treasuries yield 0.66%, so pension funds are planning to add about 7% annually from agile risk-taking and good asset selection. This is, to put it nicely, ambitious.

But it leads to an interesting form of wealth redistribution. Lower interest rates mean that long-term cash flows are worth more. Pensions and social security, for example, have a higher net present value than they used to. This is a fairly large wealth redistribution to the middle class, which as far as I know hasn’t been captured in traditional wealth inequality statistics. Of course, it’s not wealth creation in a real sense: pensions aren’t *actually* worth more because they promise high returns and haven’t made realistic adjustments. They’re just promising more. I continue to expect a pension bailout, and to the extent that it needs to be funded by taxes rather than deficit spending, it will have to be funded by taxes on relatively high-income taxpayers. While lower rates make the rich richer on paper, the rich will give some of that back when pensions with inadequate accounting finally choose to recognize reality.

*This is the first piece in an ongoing series explaining financial concepts that I cite on Medium and in my daily newsletter, *The Diff*. This morning’s subscribers-only post has some more detailed and current-events focused thoughts on discounted cash flows.*