Venture Capital puzzles, solved step by step

Puzzles: 100
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052You bought into a software company at 50x ARR, while mature peers trade at 10x. ARR grows 60% a year, and each year a new funding round dilutes your stake by 15%. Valuing the company at 10x ARR, how many years until your stake is worth what you paid?Growth and compoundingHardSaaS-focused VCGrowth equity
Try it first
Your first guess for the payback year?
Show the worked solution
About 5.2 years, not 3.4. Paying 50x when the exit multiple is 10x means ARR must grow 5x before you get your money back. At 60% a year that takes 3.4 years. But each round leaves you 85% of your previous stake, so your share of value grows by 1.6 x 0.85, or 36% a year. Solving 1.36 to the power n equal to 5 gives about 5.2 years, and that is only break-even.
What does paying 50x when peers trade at 10x actually commit you to?
Buying a mango sapling at the price of a grown tree is not a mistake if the sapling grows; it is a bet on how fast. When you pay five times the multiple a mature business earns, the company's revenue has to grow fivefold just for you to break even, because the multiple will fall back to the mature level by the time you sell. Growth first pays back the premium; only growth beyond that makes money. At 60% a year, 1.6 to the power n equals 5 gives 3.4 years.
Why does dilution stretch the answer by almost two years?
Think of a family shop that takes in a new partner every year, each time giving away 15% of everyone's share. The shop can double and your slice can still barely move. Your stake's value grows at the company's growth rate times what you keep after each round, so 60% growth with 15% dilution is 1.6 x 0.85, or 36% a year for you, not 60%. The bar the company must clear keeps rising: 5x becomes 5 divided by 0.85 after one year, then again after two, which is the red curve in the figure.
ARR growing 60% a year crosses the flat 5x break-even line at 3.4 years, but once each round dilutes the stake 15% the bar rises every year, and the ARR curve only catches it at 5.2 years.
The relationship
$\big[(1+g)(1-d)\big]^n = \frac{50}{10} \;\Rightarrow\; n = \frac{\ln 5}{\ln (1.6 \times 0.85)} = \frac{1.609}{0.308} = 5.2$
g ARR growth a year, 60%
d dilution from each year's round, 15%
50/10 entry multiple over the exit multiple, the growth needed
n years to break even
What it says in wordsYour stake compounds at growth times retention, and it has to compound until it covers the gap between the entry and exit multiples.
Say the limits before the interviewer does. The model holds 60% growth flat for five years, which few companies manage; growth that fades makes the wait longer. It treats each round's new cash as spent on the growth already assumed, rather than sitting on the balance sheet adding to value. And break-even after 5.2 years is a 0% return on money that a venture fund needs to multiply several times, so the honest conclusion is that the entry price has already spent most of the upside.
Where candidates lose it
The common answer is 3.4 years: candidates handle the multiple compression correctly and then forget that they own less of the company every year. The interviewer put dilution in the question precisely to see whether you apply it to the growth rate.
The second trap is subtracting, 60% minus 15% giving 45% a year. Growth and retention multiply: 1.6 x 0.85 is 1.36. At 45% you would answer 4.3 years and be wrong by nearly a year.
What the interviewer asks next
What ARR growth rate would get you back in three years with the same dilution?
How does a pro rata right change this calculation?
If growth falls by a fifth of itself each year, is break-even ever reached?
064A startup's revenue growth starts at 100% and loses a fifth of itself every year: 100%, 80%, 64%, 51.2%, 40.96%. What multiple of today's revenue does it reach after five years, compared with a steady 100% a year?Growth and compoundingHardSaaS-focused VCGrowth equity
Try it first
After five years of fading growth, revenue is about:
Show the worked solution
About 12.6x, against 32x at a steady 100%. Multiply the yearly factors one at a time: 2 x 1.8 is 3.6, x 1.64 is 5.9, x 1.512 is 8.9, x 1.41 is about 12.6. Steady doubling gives 2 to the fifth, or 32. A growth rate that fades by a fifth each year leaves you with less than 40% of the steady outcome, which is why the fade assumption moves a valuation more than the opening growth rate.
How do you compute this quickly without losing the thread?
Turn each year's growth into a factor and keep a running product, like a cricket scorer adding each over to the total rather than recomputing the innings. Revenue after five years is the product of the five yearly factors, so a fading rate must be multiplied year by year; there is no single rate you can raise to the fifth power. The running total goes 2, 3.6, 5.9, 8.9 and 12.6. Rounding each step to one decimal keeps it mental and still lands within a few per cent.
Steady 100% growth doubles revenue to 32x in five years, while growth that loses a fifth of itself each year reaches only 12.6x, and the gap between the two paths stays small for two years and then opens fast.
The relationship
$\frac{R_5}{R_0} = \prod_{t=0}^{4} \big(1 + g_0 k^{t}\big) = 2 \times 1.8 \times 1.64 \times 1.512 \times 1.4096 = 12.58$
g_0 first-year growth, 100%
k share of growth kept each year, 0.8
R_5 / R_0 revenue after five years as a multiple of today
What it says in wordsEach year's growth is the last year's times 0.8, and the revenue multiple is the product of all the yearly factors.
Which matters more, the starting growth rate or how fast it fades?
Run a few cases. A company that starts at 150% but keeps only 60% of its growth each year ends at about 11.6x, below the 100% company that keeps 80%, and well below a 100% company that keeps 90%, which reaches about 19.7x. The opening rate is the number in the pitch deck; the fade is the number that decides the outcome. Diligence time is better spent on why growth should persist, such as retention and new markets, than on last year's growth figure.
Starting growth Growth kept each year Revenue after 5 years
100% 90% 19.7x
100% 80% 12.6x
150% 60% 11.6x
80% 90% 12.3x
100% 100%, no fade 32.0x
Five-year revenue multiples under different starting growth rates and fade rates; the fade moves the answer more than the opening rate.
The limit: a constant fade is a convenient shape, not a law. Real growth can stall, then re-accelerate on a new product, and a smooth curve will miss both. Use it to show the sensitivity, then test the specific reasons this company's growth would or would not hold.
Where candidates lose it
The common error is answering near 32x, or averaging the five rates to about 67% and compounding that to about 13x. The second is close by luck; it is the wrong method, and on a different set of rates it will be far off.
The other loss is computing 12.6x and stopping. The interviewer wants the conclusion: the fade, not the opening rate, decides where revenue lands, so that is what diligence should test.
What the interviewer asks next
What steady annual growth rate gives the same five-year multiple?
What evidence would make you believe growth will fade by only 10% a year?
How would you put this fade into a valuation model?

g	ARR growth a year, 60%
d	dilution from each year's round, 15%
50/10	entry multiple over the exit multiple, the growth needed
n	years to break even

g_0	first-year growth, 100%
k	share of growth kept each year, 0.8
R_5 / R_0	revenue after five years as a multiple of today

Starting growth	Growth kept each year	Revenue after 5 years
100%	90%	19.7x
100%	80%	12.6x
150%	60%	11.6x
80%	90%	12.3x
100%	100%, no fade	32.0x

Try it first

What does paying 50x when peers trade at 10x actually commit you to?

Why does dilution stretch the answer by almost two years?

Where candidates lose it

What the interviewer asks next

Try it first

How do you compute this quickly without losing the thread?

Which matters more, the starting growth rate or how fast it fades?

Where candidates lose it

What the interviewer asks next