Lesson 1 of 7
A multiple is a compressed DCF
Derive a price to earnings ratio from a discounting identity in four steps, and the three assumptions hidden inside every multiple become visible.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- A price to earnings ratio is a growing perpetuity rearranged: payout divided by the discount rate less the growth rate.
- That means every multiple already contains three assumptions, and a comparison between two multiples is a comparison between two sets of them.
- Two companies can carry the same multiple for opposite reasons, which is why a multiple is a question rather than an answer.
Nobody derives a multiple before using one, which is why multiples get compared across companies that share nothing. The derivation takes four steps and it changes how the number reads.
From a discounting identity to a P/E
- Start with the value of a growing stream. A cash flow that grows forever at rate g and is discounted at rate r is worth that cash flow divided by r less g. This is the same perpetuity every terminal value uses.
- Say which cash flow. Shareholders receive dividends and buybacks, not earnings, so the cash flow is earnings times the share paid out.
- Divide both sides by earnings. The price over earnings is the payout share divided by r less g.
- Read it. A 60 percent payout, a 9 percent required return and 3 percent growth give 0.60 divided by 0.06, which is 10 times earnings.
The same identity, one input moved at a time
- Payout 60%, return 9%, growth 3%10.0x
- Growth 4% instead of 3%12.0x
- Return 8% instead of 9%12.0x
- Payout 45% instead of 60%7.5x
What the identity exposes
Three inputs, and none of them is visible when someone says "it trades on 10 times".
The payout share. Two companies earning the same amount are not worth the same if one can distribute 60 percent and the other must reinvest 80 percent to stand still. The multiple is a claim about how much of the profit ever reaches an owner.
The required return. A point of discount rate moved the multiple from 10 to 12 above. That point is a judgement about risk, and it is the term with the least evidence behind it.
The growth rate. Note where it sits: subtracted from the return, so it acts on a small divisor and moves the answer hard. It is also the term most often justified by nothing.
Why two identical multiples are not the same fact
The multiple has compressed two different arguments into one number, and it cannot be decompressed without going back to the three inputs. That is the whole case for reading a multiple as a question: what payout, what return and what growth would make this number reasonable, and does the company look like that.
What this does not fix
The identity assumes a single growth rate forever and earnings that translate into distributable cash. Neither holds for a company whose growth is falling, or whose earnings are consumed by capital spending. In those cases the compressed form breaks before the model does, which is why the chapters after this one go back to the filings.
Check yourself
In the identity derived above, price over next year's earnings equals what?
- One divided by the growth rate
- The payout share divided by the required return less the growth rate
- Earnings divided by the discount rate
Show the answer
Answer: B.
It is a growing perpetuity divided through by earnings. The payout share is on top because shareholders receive distributions, not reported profit.Why does one point of growth move a multiple so much?
- Because growth compounds over many years in the formula
- Because it is subtracted from the required return, so it acts on a small divisor
- Because it raises the payout share
Show the answer
Answer: B.
The divisor is the gap between two rates. Shrinking a 6 point gap to 5 raises the multiple by a fifth.Two companies trade on 15 times earnings. What does that tell you?
- They are equally risky
- Very little on its own: the same multiple can come from a high payout and low growth or the reverse
- They will produce the same return
Show the answer
Answer: B.
One number is holding three assumptions. Two companies reaching it by different routes have nothing in common except the arithmetic.Sources
- Gordon's growing perpetuity, the identity this derivation rearranges.
- Williams, The Theory of Investment Value, 1938: value as the present value of what a security distributes, which is why the payout share sits on top.
- Damodaran, Investment Valuation: multiples as compressed discounted cash flow models and the fundamentals that drive each one.
What this leaves out
- The identity is built on next year's earnings and a single perpetual growth rate. A published P/E is usually on the last filed year, so the two are not the same number.
- All figures here are round invented inputs, chosen so the arithmetic can be checked by hand.
- The derivation assumes the payout share can be sustained. A company distributing more than it earns cannot hold one.
- Nothing here is advice about any security.