Venture Capital puzzles, solved step by step

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.

150 decks. Reduce everything to one unit first: 4 associates working 3 hours is 12 associate-hours, and 60 decks over 12 is 5 decks per associate-hour. Six associates for five hours is 30 associate-hours, so 30 x 5 = 150. As a check, scale the original: 60 x 6/4 x 5/3 is also 150. This assumes everyone works at the same steady rate.

Why reduce to a rate per associate-hour?

If 2 cooks make 40 rotis in an hour, one cook makes 20 an hour, and any kitchen size or shift length follows from that one number. Work problems become simple once you find the output of one worker in one unit of time, because the total is then that rate times workers times hours. Here 60 decks came from 4 x 3 = 12 associate-hours, so the rate is 5 decks per associate-hour. Everything else is multiplication.

How do you avoid the slip under time pressure?

Ask which way each change pushes the answer before you multiply. More associates and more hours both raise output, so both ratios must be greater than one: 6/4 and 5/3, never 4/6 or 3/5. The fast wrong answer, 90, applies the 6/4 and stops. On a timed test the danger is not hard arithmetic but a missed step, so a five-second check that the answer moved in the right direction by roughly the right amount is worth it: one and a half times the people for two thirds more time should give about two and a half times the output, and 60 x 2.5 is 150.

Where would this break down in a real deal team?

The model assumes every associate screens at the same steady pace and that output scales with hours. Real screening slows late in a long day, and a bigger team spends time coordinating and avoiding duplicate work. Saying so is unnecessary on the test itself, but it is the right instinct if the same question comes up in an interview: the arithmetic answer is the ceiling, and the real number is usually lower.