Lesson 4 of 9
Annuities and perpetuities
A level stream and a forever stream each collapse to one line of arithmetic. The growing version is the terminal value in every DCF.
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
- An annuity is a level stream for a fixed number of years, and its whole present value is one multiplication.
- A perpetuity divides the annual amount by the rate. That is the entire formula.
- A growing perpetuity divides by the rate minus the growth rate, and that subtraction is the most dangerous denominator in valuation.
Adding a factor table works and is slow. When every year's cash is the same, or grows at a constant rate, the whole column collapses to one expression. Those expressions are the skeleton of every valuation model.
The same 1,000 a year, valued three ways at 8 percent
- Ten years, level$6,710
- Forever, level$12,500
- Forever, growing 3 percent$20,000
The annuity
A fixed amount for a fixed number of years. Its present value is the amount multiplied by the sum of the factors, and that sum has a closed form.
One thousand a year for ten years at 8 percent: 1 minus 1.08 to the power minus 10 is 0.53680, divided by 0.08 is 6.71008. Multiply by 1,000 and the answer is 6,710.08. A ten-year stream of 10,000 in total cash is worth two thirds of that today.
The perpetuity
Let the years run forever and the formula gets shorter, not longer. The term that counts the years disappears.
Perpetuity equals Annual cash divided by Discount rate.
Example1,000 / 0.08 = 12,500
Forever is worth 12,500 against ten years at 6,710. The extra 5,790 is every year from the eleventh onwards, discounted. That comparison is the useful one: an infinite tail is worth less than twice a ten-year stream at this rate.
The growing perpetuity
Now let the cash grow at a constant rate g every year. Subtract g from the discount rate and divide as before.
Growing perpetuity equals Next year's cash divided by Discount rate minus growth rate.
Example1,000 / (0.08 - 0.03) = 20,000
This is the terminal value in almost every discounted cash flow model that exists. It is also where models break, because the denominator is a difference between two numbers that are each uncertain.
The rule the formula enforces
Growth must be below the discount rate. At equal rates the denominator is zero and the formula returns nothing at all, which is the arithmetic refusing an impossible assumption rather than a bug. A company growing forever faster than the rate used to discount it would eventually be larger than the economy it sits in.
A perpetual growth rate above the long-run growth of the economy is the same claim in a quieter form. The honest handling is to state the growth assumption separately from the answer, and to show the answer at both ends of the range you would defend.
Check yourself
A perpetuity pays 500 a year and the discount rate is 10 percent. What is it worth?
- 500
- 5,000
- 50,000
Show the answer
Answer: B.
500 divided by 0.10 is 5,000. The number of years does not appear in a perpetuity formula, because there is no last year.In a growing perpetuity, what happens as the growth rate approaches the discount rate?
- The value falls towards zero
- The value rises without limit
- The value stops changing
Show the answer
Answer: B.
The denominator is the rate minus the growth rate. As the two converge, the denominator approaches zero and the quotient grows without bound, which is the formula refusing an assumption rather than describing a business.A ten-year annuity of 1,000 at 8 percent is worth 6,710. The matching perpetuity is worth 12,500. What is the 5,790 difference?
- The interest earned over ten years
- The present value of every year from the eleventh onwards
- An error in one of the two formulas
Show the answer
Answer: B.
A perpetuity is the annuity plus its own infinite tail. The tail is worth less than the first ten years because every year in it is discounted further.Sources
- Williams, The Theory of Investment Value, 1938: valuing a business as the present value of a perpetual dividend stream, which is where this formula enters investing.
- Damodaran, Investment Valuation: the constraint that a perpetual growth rate cannot exceed the growth rate of the economy.
What this leaves out
- These are closed forms for constant streams. A real forecast is rarely constant, and forcing one into this shape hides the years that are not.
- The growing perpetuity assumes the growth rate holds forever. No company has ever been observed over that horizon.
- Nothing here is advice about any security.