Venture Capital puzzles, solved step by step

Puzzles: 100
Traced to a firm: 7
Topics: 12
Hard: 30

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019A portfolio company says its revenue doubles every 9 months. What is that as an annual growth rate, and as a quarterly one?Growth and compoundingCoreSeed and early-stage VCConsumer internet VC
Try it first
What annual growth rate does doubling every 9 months imply?
Show the worked solution
About 152% a year and 26% a quarter. A year holds 12/9, or 4/3, doubling periods, so revenue grows by 2 to the power 4/3, which is 2.52x. A quarter is a third of a doubling period, so it grows by the cube root of 2, 1.26x. Scaling the 100% in a straight line, 133% a year, understates the compounding.
Why is the answer not 100% times 12/9?
Picture a savings account that doubles every 9 months. In the three months after the first doubling, it grows on twice the original balance. Doubling compounds, so a fraction of a doubling period multiplies the amount by 2 raised to that fraction, not by 1 plus that fraction of 100%. A year is 4/3 of a doubling period, so revenue grows by 2 to the power 4/3. Since 2 to the power 1/3 is about 1.26, that is 2 x 1.26, about 2.52, or 152% growth.
The relationship
$g_{year} = 2^{12/9} - 1 \approx 152\% \qquad g_{quarter} = 2^{3/9} - 1 \approx 26\%$
12/9 doubling periods in a year
3/9 doubling periods in a quarter
2 the growth factor per doubling period
What it says in wordsRaise 2 to the number of doubling periods in the time window, then subtract one.
Doubling every 9 months reaches 2x at month 9 and 2.52x at month 12, so each year multiplies revenue by 2.52, 152% growth, and three years compound to 16x.
How do you get the cube root of 2 in your head?
Remember one anchor: 1.26 cubed is very close to 2, because 1.26 squared is about 1.59 and 1.59 x 1.26 is about 2.0. So a quarter, one third of a doubling period, grows by about 26%, and four quarters give 1.26 to the power 4, about 2.52, which checks the annual figure. The same anchor gives the monthly rate: 2 to the power 1/9 is about 1.080, roughly 8% a month.
What do you say about the founder's claim?
Translate it into something you can test. Doubling every 9 months means revenue 16 times larger in three years; ask what has to be true about the market and the team for that, and how long it has lasted. Founders often quote the doubling period from their best stretch. Ask for the doubling time over the last 24 months and over the last 6; if the recent one is slower, the growth is decaying, which is normal but changes the valuation.
Where candidates lose it
The trap is 133%, scaling 100% by 12/9 as if growth were a straight line. The interviewer is checking whether you compound fractional periods.
The quieter loss is getting the annual number right and then dividing it by four for the quarter, 38%. A quarter is a third of a doubling period, so it is the cube root of 2, about 26%.
What the interviewer asks next
What doubling time corresponds to 100% annual growth?
If growth slows so that the doubling time lengthens by 3 months every year, what is revenue after three years?
How would you check a founder's doubling claim from monthly revenue data?
031A SaaS company at Rs 20 crore of annual recurring revenue follows the triple, triple, double, double, double path. What is its ARR after five years, and what compound annual growth rate is that?Growth and compoundingCoreSaaS-focused VCSeries A to C VC
Try it first
What is the ARR at the end of year five?
Show the worked solution
Rs 1,440 crore after five years, a compound rate of about 135% a year. The yearly multipliers chain: 3 x 3 x 2 x 2 x 2 is 72, and Rs 20 crore times 72 is Rs 1,440 crore. The annual rate is the fifth root of 72, about 2.35, so revenue grows about 135% a year on average, even though no single year actually grew at that rate.
Why do the multipliers multiply rather than add?
Think of a sapling that triples its height in its first year and triples again in its second. A one metre sapling is three metres after a year and nine after two, not six, because the second tripling acts on the three metres it has already grown. Each year's multiplier applies to the revenue the company already has, so five multipliers combine by multiplication into a single 72x. Adding them gives 12, which is the wrong operation and the most common slip.
Walk the path aloud so the interviewer can follow: Rs 20 crore, then 60, 180, 360, 720 and 1,440. Saying each year also gives you the intermediate figures that a follow-up question usually asks for, such as when the company crosses Rs 500 crore, which here happens during year four.
Triple, triple, double, double, double takes Rs 20 crore of ARR to Rs 1,440 crore, a 72x gain that equals a steady 135% a year; the real path runs above the steady line in the middle years because the big multipliers come first.
How do you find the annual rate in your head?
Bracket the fifth root first. 2 to the fifth is 32 and 3 to the fifth is 243, so the answer is between 2 and 3, and much closer to 2. Try 2.5: 2.5 squared is 6.25, to the fourth about 39, to the fifth about 98, too high. The fifth root of 72 is about 2.35, so the compound rate is about 135% a year. If you know your logs, ln 72 is ln 8 plus ln 9, about 4.28; a fifth of that is 0.855, and e to 0.855 is about 2.35.
The relationship
$\text{ARR}_5 = 20 \times 3 \times 3 \times 2 \times 2 \times 2 = 1{,}440 \qquad \text{CAGR} = 72^{1/5} - 1 \approx 135\%$
ARR annual recurring revenue, the yearly value of subscriptions in force
72 the combined five-year multiplier
CAGR compound annual growth rate, the single yearly rate that gives the same end point
What it says in wordsMultiply the yearly multipliers to get the five-year multiple, then take its fifth root to get the equivalent steady yearly rate.
What does the average rate hide?
The 135% is a summary, not a description. The path is front-loaded: after one year the company is at Rs 60 crore against Rs 47 crore on the steady path, and the gap persists until the two meet at year five. That matters to an investor because the early years carry most of the growth, and a company that misses its first tripling cannot recover the 72x by doubling later. Say the limitation too: this path is a widely quoted benchmark for the best software companies after they find their market, not a forecast, and very few companies sustain it.
Where candidates lose it
The fast wrong answer is Rs 240 crore, from adding 3 + 3 + 2 + 2 + 2 to get 12. The interviewer is checking whether you know that yearly growth multipliers compound.
The second loss is stopping at Rs 1,440 crore and dodging the rate, or dividing 72 by five and calling it 1,400% a year. The rate is the fifth root, and bracketing between 2 and 3 gets you there in under a minute.
What the interviewer asks next
If the company only doubles every year for five years, what ARR does it reach and what is the rate?
Which matters more for the five-year multiple, a slip in year one or a slip in year five?
At what point on this path does the company first cross Rs 500 crore of ARR?
043One investment returns 3x in 3 years. Another returns 5x in 7 years. Which has the higher IRR, and by how much?Growth and compoundingCoreGrowth equityFund of funds and LPs
Try it first
Which investment has the higher IRR?
Show the worked solution
The 3x in 3 years, at about 44.2% a year against 25.8%, a gap of about 18 points. IRR for a single cash in and out is the multiple's root over the years held. The cube root of 3 is about 1.44 because 1.44 cubed is 2.99. The seventh root of 5 is about 1.26 because 1.26 to the seventh is just over 5. Time sits under the multiple, so the smaller, quicker multiple earns the higher rate.
Why can a smaller multiple have the higher IRR?
Two runners: one covers 3 kilometres in 3 minutes, the other 5 kilometres in 7 minutes. The second went further, but the first was faster. A multiple says how far the money went; the IRR says how fast, so time sits in the denominator and a long hold drags the rate down. For a single cheque in and a single exit, the IRR is the yearly growth rate that turns 1 into the multiple over the holding period: the multiple's root over the years.
The relationship
$\text{IRR} = M^{1/t} - 1 \qquad 3^{1/3} - 1 \approx 44.2\% \qquad 5^{1/7} - 1 \approx 25.8\%$
M the money multiple, cash out divided by cash in
t the holding period in years
IRR the annual rate that grows 1 into M over t years
What it says in wordsThe annual rate is the multiple's root over the years held, less one.
The 3x deal climbs at 44.2% a year and the 5x deal at 25.8%, so the smaller multiple earns the higher rate; the 3x proceeds only need to grow at 13.6% a year afterwards to match 5x by year seven.
How do you take those roots without a calculator?
Guess and cube. 1.4 cubed is 2.744 and 1.5 cubed is 3.375, so the cube root of 3 is a little above 1.4; 1.44 cubed is 2.986, close enough, so 44%. For the seventh root of 5, use doublings: 5x is about 2.3 doublings, since 2 to the 2.3 is about 5, and 2.3 doublings in 7 years is one every 3 years, which the rule of 72 puts at about 24%. The rule of 72 runs a little low at rates this high, so nudge up: 1.26 to the seventh is just over 5, giving about 26%.
Does the higher IRR make the 3x the better investment?
Not on its own. The 3x deal wins on rate, but it hands money back after three years, and whether that is better depends on what the money can earn next. To match 5x over the same seven years, the 3x proceeds need to grow 5/3 times over the remaining four years, about 13.6% a year. If the fund or its LPs can redeploy above that, the quick 3x is better; if not, the 5x leaves more money in total. This is why LPs look at both IRR and multiple: a high IRR on a short hold can return fewer rupees.
Where candidates lose it
The common loss is picking the 5x because 5 is bigger than 3, or dividing multiples by years, 1x a year against 0.71x a year, which gets the order right but the rates badly wrong.
The second loss is giving the IRRs and stopping. The interviewer often wants the follow-through: the higher IRR is not automatically the better deal, because the quicker exit has to be reinvested, and the break-even redeployment rate of about 13.6% settles it.
What the interviewer asks next
What multiple over 7 years matches a 44.2% IRR?
If the 3x deal had a two-year delay between investment and the first rupee of value, how would its IRR change?
Why might a fund that reports a high IRR still have a low multiple, and which number should an LP trust more?
093A software company's ARR is Rs 10 crore and grows 8% every month. If that growth holds, how many months until ARR reaches Rs 100 crore?Growth and compoundingCoreSaaS-focused VCSeed and early-stage VC
Try it first
Pick the closest before you calculate.
Show the worked solution
About 30 months. By the rule of 72, 8% a month doubles ARR roughly every 9 months: Rs 20 crore at month 9, 40 at month 18, 80 at month 27. Ten times is about 3.3 doublings, so about 30 months. The exact figure is ln 10 over ln 1.08, which is 29.9, so ARR first passes Rs 100 crore in month 30. Few companies hold 8% a month that long.
How do you get to 30 in your head?
Money in a deposit that doubles every nine years has gone up eight times after twenty seven. Turn the growth rate into a doubling time with the rule of 72, then count doublings: ten times is a little over three doublings, because 2 to the power 3.3 is about 10. 72 divided by 8 is 9 months per doubling, and 3.3 x 9 is about 30 months. The exact doubling time at 8% is 9.01 months, so the rule is very close here.
Growing 8% a month from Rs 10 crore, ARR doubles to about Rs 20 crore at month 9, Rs 40 crore at month 18 and Rs 80 crore at month 27, and crosses Rs 100 crore at month 29.9, so a tenfold rise takes about 30 months.
The relationship
$10(1.08)^n = 100 \;\Rightarrow\; n = \frac{\ln 10}{\ln 1.08} = \frac{2.303}{0.0770} \approx 29.9$
n months of growth
1.08 one month's growth factor at 8%
ln 10 natural log of the tenfold rise, about 2.303
What it says in wordsDivide the log of the total rise by the log of one month's growth.
Check the answer: at month 29 ARR is Rs 93.2 crore and at month 30 it is Rs 100.6 crore. The crossing falls just before the end of month 30, so say about 30 months, or month 30 if the interviewer wants a whole number.
Is 8% a month believable for two and a half years?
8% a month compounds to about 152% a year, two and a half times ARR every twelve months. Young companies from a small base sometimes do that for a while; holding it from Rs 10 crore to Rs 100 crore is rare. Monthly growth rates usually decay as the base grows, so a plan built on a constant 8% overstates how soon the company reaches scale. Saying that limit turns a compounding exercise into a judgement about the plan in front of you.
Where candidates lose it
The linear slip is the expensive one: 8% of Rs 10 crore is Rs 0.8 crore a month, so Rs 90 crore of growth takes about 112 months. That ignores compounding entirely and is off by a factor of almost four.
The second loss is going silent to grind logs. Say the rule of 72 estimate first, then refine. The interviewer wants a fast, defensible number, and 30 months with a reason beats 29.92 after a long pause.
What the interviewer asks next
How many months to reach Rs 100 crore at 5% a month?
If growth decays from 8% a month by a tenth of a point each month, does it ever reach Rs 100 crore within three years?
Why do investors often prefer to see growth quoted per year rather than per month?

12/9	doubling periods in a year
3/9	doubling periods in a quarter
2	the growth factor per doubling period

ARR	annual recurring revenue, the yearly value of subscriptions in force
72	the combined five-year multiplier
CAGR	compound annual growth rate, the single yearly rate that gives the same end point

M	the money multiple, cash out divided by cash in
t	the holding period in years
IRR	the annual rate that grows 1 into M over t years

n	months of growth
1.08	one month's growth factor at 8%
ln 10	natural log of the tenfold rise, about 2.303

Try it first

Why is the answer not 100% times 12/9?

How do you get the cube root of 2 in your head?

What do you say about the founder's claim?

Where candidates lose it

What the interviewer asks next

Try it first

Why do the multipliers multiply rather than add?

How do you find the annual rate in your head?

What does the average rate hide?

Where candidates lose it

What the interviewer asks next

Try it first

Why can a smaller multiple have the higher IRR?

How do you take those roots without a calculator?

Does the higher IRR make the 3x the better investment?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get to 30 in your head?

Is 8% a month believable for two and a half years?

Where candidates lose it

What the interviewer asks next