Venture Capital puzzles, solved step by step

About Rs 69.3 crore of ARR, a burn multiple of 1.5 and about 18 months of runway. 42 x 1.65 is 42 plus 21 plus 6.3. A quarter's burn is 9.3, and 9.3 over 6.2 is 1.5 because both are multiples of 3.1. Cash of 55 over 3.1 a month is a shade under 55 over 3, so about 17.7 months.

How do you multiply by 1.65 without a calculator?

Break the awkward multiplier into pieces you already know. A shopkeeper adding 65% to a Rs 42 cost does not multiply by 0.65; he adds half, then a bit more. 65% is 50% plus 15%, so 42 x 1.65 is 42 plus 21 plus 6.3, which is Rs 69.3 crore. Fifteen percent is itself 10% plus half of that again: 4.2 plus 2.1. Every step is a halving or a shift of the decimal point, which is fast and hard to get wrong out loud.

Why does the burn multiple need care before any arithmetic?

Because the two inputs come in different periods. Burn is quoted per month and net new ARR per quarter. A burn multiple divides the cash burned by the new annual recurring revenue added over the same stretch of time, so three months of burn, Rs 9.3 crore, goes over the quarter's Rs 6.2 crore. Then notice that 9.3 is 3 x 3.1 and 6.2 is 2 x 3.1, so the ratio is exactly 3 over 2, or 1.5. Interviewers choose numbers like these on purpose; spotting the common factor is part of the test.

What do you say once the three numbers are out?

Check them against each other, because that is what a growth investor does next. Growing ARR by Rs 27.3 crore next year needs about Rs 6.8 crore of net new ARR a quarter, a little above the Rs 6.2 crore just achieved, so the 65% plan is plausible but not banked. At a 1.5 burn multiple that growth costs about Rs 41 crore of burn in the year, and with Rs 55 crore of cash and about 18 months of runway, the company will want to raise again within roughly a year, before the runway gets short. That one sentence turns arithmetic into a view on the company.

About 1.97 and about 0.48. For the first, the rule of 72 says 7% doubles money in about 10.3 years, so ten years gives just under 2. For the second, ln 0.93 is about minus 0.0725, ten years of it is minus 0.725, and e to that is about 0.48. They are not reciprocals because 0.93 is not 1 over 1.07: undoing a 7% rise takes only a 6.5% fall.

How do you get each number without a calculator?

Start with the one you know. The rule of 72A shortcut for compounding: money growing at r per cent a year doubles in roughly 72 divided by r years. says money growing at r% doubles in about 72 over r years, so at 7% it doubles in about 10.3 years and ten years leaves you just short of 2. For the fall, work in natural logs, which turn compounding into adding. ln(1 minus x) is about minus x minus half of x squared, so ln 0.93 is about minus 0.07 minus 0.00245, which is minus 0.0725. Ten years gives minus 0.725. e to minus 0.693 is exactly one half, and minus 0.725 is a little further down, so the answer is a shade under a half: about 0.48.

Why are the two answers not mirror images?

Everyday version first: a shirt marked up 7% and then marked down 7% ends below its starting price, because the markdown is taken on the higher price. The x squared term in the log has the same sign whichever way the rate goes, so it drags both paths down: the up path gains a little less than 7% a year in log terms and the down path loses a little more. The true mirror of a 7% rise is a 6.5% fall. Put the two paths together and 1.07 x 0.93 is 0.9951 a year, so ten years of each leaves 0.952: a round trip that still loses about 5%.

Then say why a growth investor cares. Swings cost compound growth even when the average yearly change is zero. A company whose revenue alternates between up 7% and down 7% averages zero change but ends smaller, losing roughly half the square of the swing every year. The limit is that this is a small effect at 7%; it becomes large at the 30% and 50% swings early-stage revenue can show.

Up 50% in all, or about 14.5% a year. Turn each round into a multiplier: up 150% is x 2.5, down 60% is x 0.4, up 50% is x 1.5. The shortcut is that 2.5 x 0.4 is exactly 1, so the Series C fall wiped out the Series B gain, and the net is just Series D's x 1.5. The annual rate is the cube root of 1.5, which sits between 14% and 15%.

Why can the three percentages not simply be added?

A shopkeeper who marks a shirt up 150% and then runs a 60% off sale is not ahead by 90%. The markup took Rs 100 to Rs 250, and the sale took 60% of Rs 250, landing back at Rs 100. Each percentage is measured against whatever the value was just before it, so moves in a chain multiply rather than add. Adding gives +140%, a number that describes nothing in this company's history. Multiplying gives 2.5 x 0.4 x 1.5, which is 1.5: a valuation of 100 went to 250, back to 100, then to 150.

How do you find the annual rate without a calculator?

You need the number that, cubed, gives 1.5. Bracket it. 1.14 cubed is about 1.48 and 1.15 cubed is about 1.52, so the rate sits just below 14.5%, and saying 'about 14.5% a year' is the right precision for the room. The exact figure is 14.47%. Check it the other way: half of 50% is 25%, far too high, which is the arithmetic average and the error the question is fishing for.

One sentence of judgement helps. A company that fell 60% in one round and still ended up 50% ahead over three years has had a volatile path, and the steady 14.5% hides that volatility completely. Investors who entered at the Series B price are down 40% at Series D, which is why the entry round matters as much as the company's overall path.

About 26%; the exact figure is 25.7%. Two anchors bracket it: 2x in 3 years is 26.0% and 3x in 5 years is 24.6%. For a sharper figure, take ln 2.5, about 0.92, divide by 4 to get 0.23, and add half its square: about 25.6%. Check by squaring twice: 1.26 squared is 1.59, and 1.59 squared is about 2.5.

Which anchors should you carry into the room?

You judge a distance on a road by the milestones, not by pacing it out. Carry a few multiple and years pairs and interpolate between them rather than calculating from scratch. 2x in 3 years is 26.0%, 3x in 5 years is 24.6%, 2x in 4 years is 18.9% and 3x in 4 years is 31.6%. The target, 2.5x in 4 years, lies between the last two and is bracketed by the first two.

How do you land on 26% out loud?

Route one: interpolate on a log scale. 2.5 sits about 55% of the way from 2 to 3 in log terms, so the IRR sits about 55% of the way from 18.9% to 31.6%, near 26%. Route two uses the log directly. The continuous growth rate is the log of the multiple divided by the years, and adding half its square converts it to an annual rate: 0.23 plus 0.026 is about 25.6%.

Then check. Squaring twice is the fastest test of a four year rate: 1.26 squared is 1.59, and 1.59 squared is 2.52, close enough to 2.5. The check matters more than the method, because it catches a slip in either route in five seconds.