How two-stage DCF works

Discounted cash flow is the time value of money pointed at a going concern. Each future free cash flow is divided by $(1 + r)$ once for every year of waiting, and those present values are added. A firm is not a five-year project, so the model splits the life in two: a stretch of years written out by hand, and everything after them, capitalised as a growing perpetuity. The sum of the two present values is enterprise value, the value of the operations.

The rate $r$ is not invented on this page. It is the return the capital funding those cash flows requires, which is the subject of cost of capital. That page is the hurdle. This one is the valuation that uses the hurdle. A careful rate next to a sloppy forecast, or a careful forecast next to a rate built for a different risk, produces a number that looks finished and is not.

$$V = \sum_{t=1}^{n} \frac{F_t}{(1+r)^t} + \frac{F_n(1+g)}{(r-g)(1+r)^n}$$

$F_t$ is free cash flow in year $t$, written as a cash amount at the end of that year. $r$ is the discount rate as a decimal, usually WACC when the cash flow is to the whole firm. $g$ is the perpetual growth rate after year $n$, and $n$ is how many years the explicit forecast runs. The first term is the explicit stage. The second is Gordon growth on the year-after-forecast flow, then discounted $n$ years so it sits in today's money with the rest. $g$ has to stay strictly below $r$, or the second term is not a number.

The DCF calculator above is that formula. Its default is a flat $100,000 a year for 5 years, discounted at 10 percent, with 3 percent terminal growth. The explicit stage is $379,078.68. The terminal stage is $913,641.38 in today's money. Enterprise value is $1,292,720.05. About 71 percent of the answer is the terminal stage. That split is normal for a stable growing perpetuity, and it is the reason a change in $g$ moves the result more than a change in year 1.

A project that ends when the last cash flow is paid does not need the second term. Discount what is written down and stop: that is net present value. Two-stage DCF is the same discounting with a life after the forecast still inside the model. Enterprise value is that present value of operations. It is not the net present value of buying the firm, which would subtract the price paid for those operations.

The explicit stage is the part you can argue about year by year. It is also, on the default, the smaller part of the value. Those two facts are not in conflict. The years you type are there to set the last cash flow the perpetuity is allowed to grow from, and to charge the right number of years of waiting on that perpetuity. They are not there to be most of the enterprise value.

Free cash flow is cash from operations minus the capital spending needed to keep going, after tax and after reinvestment in working capital. Variants exist. Free cash flow to the firm is before interest, and it is the series that belongs against WACC, because WACC already prices the debt. Free cash flow to equity is after interest and after net borrowing, and it belongs against the cost of equity. Discount the equity series at WACC, or the firm series at the cost of equity, and the capital structure has been charged twice or not at all. The calculator above takes a series and a WACC, so the series it wants is cash to the firm.

Timing is a convention, and this page uses one: every year's cash flow arrives at the end of that year. Year 1 is divided by $(1+r)$ once, year 5 by $(1+r)^5$. A mid-year convention would treat the cash as arriving in the middle of each year and would raise every present value a little. The formula does not change. The exponent does.

On a flat $100,000 for 5 years at 10 percent, each year is the same cash divided by a larger number:

Year 1's $90,909.09 is the same figure the one-year worked example isolates on its own. The five rows add to $379,078.68, which is $100,000 times the five-year annuity factor $(1 - 1.1^{-5}) / 0.10 = 3.790787$.

Nothing in those five years grows. That is a teaching choice, not a claim about firms. A firm whose cash flow is truly flat for five years, then grows at 3 percent forever from that flat level, is a different firm from one whose cash flow starts growing at 3 percent from year 2. The explicit path is the launchpad. Change the last year and you change the perpetuity, which is why the forecast is doing real work even when it is not most of the value.

The usual source for $r$ is the WACC calculator. Match the rate to the risk of these cash flows, not to the firm's average risk if this business is a different one, which is the warning the cost of capital page spends its last section on.

The terminal stage exists because writing cash flows out to infinity by hand is not a model, it is a spreadsheet with no end. Gordon growth replaces that tail with one assumption: after year $n$, free cash flow grows at a constant rate $g$ forever. A level growing perpetuity that pays $F_{n+1}$ next year, growing at $g$ and discounted at $r$, is worth $F_{n+1} / (r - g)$ on the date just before that first perpetual payment.

That date is the end of year $n$. So:

$$TV_n = \frac{F_n(1+g)}{r-g} = \frac{F_{n+1}}{r-g}$$

On the default, year 5's cash flow is $100,000, so the year-6 flow is $100000 \times 1.03 = 103000$, which is $103,000. Divide by $0.10 - 0.03$:

$$TV_5 = \frac{103000}{0.07} = 1471428.57$$

That $1,471,428.57 is a value at the end of year 5, not a value today. Discount it five years at 10 percent: $1471428.57 / 1.1^5 = 913641.38$, which is $913,641.38 in today's money. Add the explicit stage and enterprise value is $1,292,720.05. The terminal stage is about 71 percent of it. Rounding each present value to the cent first and then adding would land a cent higher, which is why the addition is done on the unrounded figures and rounded once.

The same Gordon figure, $1,471,428.57, appears in the one-year worked example, because that case also takes a last explicit flow of $100,000 and grows it by 3 percent. What changes is how far the terminal value has to be discounted. After one year it is worth $1,337,662.34 today. After five years it is worth $913,641.38 today. Same future terminal value, more waiting.

The one-year case has a cleaner identity behind it. If year 1 pays $100,000 and growth at 3 percent starts immediately afterwards, the whole firm is a growing perpetuity whose first payment is a year away:

$$V_0 = \frac{F_1}{r-g} = \frac{100000}{0.07} = 1428571.43$$

Enterprise value is $1,428,571.43, of which $90,909.09 is the explicit year and $1,337,662.34 is the terminal stage in today's money. About 94 percent is terminal, because you only wrote one year by hand.

That identity is also why the five-year flat case is worth less ($1,292,720.05) than the one-year growing case ($1,428,571.43). They are not the same firm with a longer forecast. The flat path holds cash flow at $100,000 for four extra years instead of letting it grow at 3 percent from year 2, so the launchpad at year 5 is still $100,000 rather than a grown figure. The explicit path is the input. The terminal stage multiplies whatever it is given.

A single-stage Gordon model with no explicit forecast at all is the dividend discount calculator, aimed at dividends rather than free cash flow, and discounted at the cost of equity rather than at WACC. The algebra of $F / (r - g)$ is the same object.

Gordon growth divides by $r - g$. If $g$ equals $r$, the denominator is zero and the perpetuity has no finite value. If $g$ exceeds $r$, the denominator is negative and the formula prints a negative enterprise value for a growing firm, which is a contradiction wearing a minus sign. In words: a firm cannot grow faster than the return required on the capital that funds it, forever, and still be worth a finite amount of money. The calculator reports n/a in that case rather than a huge or negative figure.

A usable $g$ is a growth rate the whole economy could bear for a very long time, not a growth rate one firm posted in a good stretch of years. Nominal economic growth, inflation plus real output, is the usual ceiling people have in mind. A firm can beat that ceiling for a while, which is what the explicit stage is for. It cannot beat it forever without eventually becoming the economy. Three percent on the default is in that long-run band. Eight percent is a forecast-stage rate. Putting 8 percent in as $g$ is the next section's mistake.

Raising $r$ shrinks both stages, and it shrinks the terminal stage more. The explicit years are nearby. The terminal value is a claim on cash from year $n+1$ to infinity, so it behaves like a long-dated asset: a small change in the rate moves it a long way. That is why a WACC that is half a point too low does not just lift the answer a little. It lifts the piece that was already most of the answer. On the default, about 71 percent of $1,292,720.05 is that long-dated piece.

The rate and the cash flows have to be in the same units of inflation. A WACC built from nominal bond yields and a nominal cost of equity is a nominal rate. It belongs against cash flows that already include future price rises. Discount a forecast written in today's prices at a nominal WACC and inflation is charged without ever being added, which understates the firm, more so the longer the tail. The other mix, a real rate against nominal cash flows, overstates it. The real return calculator converts a nominal rate to a real rate. Pick a pair and keep it. WACC already contains the cost of debt, so it also belongs against free cash flow to the firm, not against a series that has already deducted interest.

Two-stage DCF exists so the growth rate you believe for the next few years does not have to be the growth rate you believe forever. The explicit stage can grow fast. The terminal stage cannot.

Start at $80,000 and grow 8 percent a year for 5 years:

$$80000,\quad 80000 \times 1.08 = 86400,\quad 86400 \times 1.08 = 93312$$

$$93312 \times 1.08 = 100776.96,\quad 100776.96 \times 1.08 = 108839.1168$$

The series used on this page rounds that last flow to the cent: $80,000, $86,400, $93,312, $100,776.96, $108,839.12. Discount those five flows at a 9 percent WACC and the explicit stage is $360,300.53.

Year 6 is the last explicit flow grown at the terminal rate, not at 8 percent: $108839.12 \times 1.025 = 111560.098$, which is $111,560.10 to the cent. Gordon growth at 2.50 percent then capitalises that flow at the end of year 5:

$$TV_5 = \frac{108839.12 \times 1.025}{0.09 - 0.025} = 1716309.20$$

That terminal value is $1,716,309.20. Discounted five years at 9 percent it is $1,115,483.22 today. Add the explicit stage: enterprise value is $1,475,783.75. About 76 percent of it is the terminal stage.

Read the two rates again. Eight percent is the explicit growth. Two and a half percent is $g$. The whole point of two stages is that those are allowed to differ. Feeding 8 percent into Gordon growth, at a 9 percent WACC, puts $r - g$ at one percentage point and parks almost all of the value in a perpetuity one tick from being undefined. That is the usual way a DCF explodes, and it is not a finding about the firm. It is a finding about a forbidden $g$.

Growth is not free. Free cash flow is already after reinvestment, so a high growth rate in the explicit years is a claim that the firm is putting capital back in and earning a return on it. A firm cannot grow at 8 percent forever on zero reinvestment, and it cannot grow at 8 percent forever in an economy that grows at 3. The arithmetic on this page will discount whatever series it is given. The job of the forecast is to give it a series that a firm could actually produce.

The last explicit year is the lever that looks small and is not. On this growing case it is $108,839.12. Grow that at 2.50 percent and capitalise it, and you have the $1,716,309.20. Miss the last year and the terminal stage, about 76 percent of enterprise value here, misses with it.

The number this page produces is enterprise value: the operations, in today's money, before any claim other than the operations themselves has been taken off. It is not the equity value, and it is not a share price.

Equity value starts from enterprise value, subtracts net debt (interest-bearing debt minus surplus cash), subtracts other claims that rank ahead of ordinary shareholders, and adds non-operating assets the cash-flow forecast did not contain. None of those balance-sheet items is an input to the formula above, so the formula stops where its inputs stop. Treating $1,292,720.05 as what the shares are worth is the same error as treating a firm's operations as if they had no debt and no spare cash.

Two other substitutions get made because the algebra looks familiar.

• The explicit years alone. That is $379,078.68 on the default, and it is the value of a five-year project that then vanishes. A going concern is not that project. Dropping the terminal stage because it is only an assumption does not make the model conservative. It makes it a different model, of a firm that is wound up at year 5. The NPV calculator is the right tool when that is actually the question.
• Gordon growth on dividends. That is a cost-of-equity model of the residual claim, and the dividend discount calculator is the tool for it. WACC against dividends, or the cost of equity against free cash flow to the firm, prices the wrong claim.

What a reader can do with the number is compare it with another enterprise value: the one implied by a traded price, which is the market value of the equity plus net debt, minus non-operating assets. The gap is only as good as the forecast, the WACC and $g$. Move any of the three and the gap moves, which is the method working rather than failing.

The figures on this page are educational material, not financial advice. A DCF is a translation of a forecast and a rate into a present value of operations. The cost of capital page is where the rate itself is built.