Venture Capital puzzles, solved step by step

20% if Rs 200 crore is the pre-money, 25% if it is the post-money. Pre-money is the value before the cheque, so post-money is 200 plus 50, Rs 250 crore, and 50 over 250 is 20%. If Rs 200 crore already includes the cheque, investors own 50 over 200, which is 25%, and the pre-money was only Rs 150 crore. One unstated word moves five points of the company.

What do pre-money and post-money actually mean?

Two friends own a food stall worth Rs 2,00,000. A third friend puts in Rs 50,000 for a new oven. The stall is now worth Rs 2,50,000 and the new friend owns a fifth of it, because the Rs 50,000 is inside the new total. Pre-money is the value before the new money arrives, post-money is pre-money plus the new money, and the investor's share is always the cheque divided by post-money. The only question is which of the two numbers the founder quoted.

Why does the post-money reading cost the founder more?

If Rs 200 crore is post-money, the pre-money is only Rs 150 crore: the founder sold Rs 50 crore of shares against a company valued Rs 50 crore lower. 25% against 20% is a quarter more of the company for the same cheque, and that extra slice is diluted alongside everything else in every later round. Check both readings the same way: 20% of 250 is 50, and 25% of 200 is 50. Both pass, which is exactly why the sentence alone cannot settle it.

Headlines tend to quote whichever number sounds bigger, and a term sheet settles it by stating the pre-money and the fully diluted share count. Some convertible instruments quote a post-money cap instead, so the habit of asking which one carries straight into later work.

Undiscounted, LTV is Rs 4 lakh and LTV to CAC is 3.3; discounted at 2% a month, LTV is Rs 2 lakh and the ratio is 1.7. Gross profit is Rs 8,000 a month and 2% churn means an average life of 50 months, so 8,000 x 50 is Rs 4 lakh. Discounting adds the 2% rate to the 2% churn, so LTV becomes 8,000 over 0.04, which is Rs 2 lakh.

Why is the average customer life one over the churn rate?

A gym that loses 2 of every 100 members each month, month after month, keeps the average member for 50 months: some leave in the first month, some stay ten years, and the steady 2% leak averages out to one over 0.02. With a constant monthly churn c, the expected customer life is 1 / c months, and lifetime value is the monthly gross profit times that life. Use gross profit, not revenue: Rs 10,000 at 80% margin is Rs 8,000 a month, and 8,000 x 50 is Rs 4 lakh, against a Rs 1.2 lakh acquisition cost, a ratio of 3.3.

What does discounting do to a stream that already decays?

A rupee of gross profit in month 40 is worth about 0.45 today at 2% a month. When churn and the discount rate are both monthly rates, discounted LTV is monthly gross profit divided by churn plus the discount rate, because each month the stream shrinks by survival and by time together. Here that is 8,000 / (0.02 + 0.02), exactly Rs 2 lakh if each month's profit arrives at the month end.

Is 2% a month a fair rate? It compounds to about 27% a year, a venture-style hurdle rather than a bank rate, so the halving here is at the harsh end. The point survives a gentler rate: a ratio that looks comfortable at 3.3 can sit much closer to 1 once time is priced, and the 15-month payback on gross profit is often the more honest single number to quote alongside it.

About Rs 24,000 crore a year on these assumptions, and supply, not demand, sets the ceiling. Take 60 crore phones in use on a 4-year cycle: 15 crore retire each year. If 40% are resold and half of those are refurbished, 3 crore units reach the refurbished channel at about Rs 8,000 each. Every input is an assumption to check against market data.

Why start from the phones people retire rather than the people who want one?

A second-hand bookshop can only sell the books people give up. However many students want cheap textbooks, the shop's shelves fill at the rate books are released. A refurbished market is a used-goods market, so its volume is capped by the flow of retired devices before anyone asks about demand. Size the flow first, then check demand only to see whether it is large enough to absorb that flow.

How do you build and defend each step?

Say every number is an assumption and give the reason for its size. Installed base over replacement cycle gives the retiring flow: 60 crore over 4 years is 15 crore phones a year. Of those, many stay in a drawer or pass to a parent, so assume 40% are sold or traded in, 6 crore. Some go straight to a buyer as they are; assume half pass through a grading, repair and warranty channel, 3 crore units. At an average Rs 8,000, that is Rs 24,000 crore.

How do you show that supply rather than demand caps it?

Run a quick demand check. If about 18 crore phones are bought each year, new and used, and a quarter of buyers would take a refurbished phone at a fair price, demand is about 4.5 crore units, more than the 3 crore supply. When demand exceeds supply, the market is the supply, so the replacement cycle is the lever to test. A 3-year cycle lifts the flow to 20 crore retired phones and the market to Rs 32,000 crore, a bigger swing than any plausible change in price. That also tells an investor where the business is won: in sourcing devices, not in finding buyers.

Without recycling the fund invests Rs 82.5 crore and needs 3.03x; with full recycling it invests Rs 97.5 crore and needs about 2.72x. Investors want Rs 250 crore back. Without recycling, Rs 82.5 crore must produce all of it. With recycling, Rs 97.5 crore must produce Rs 265 crore, because Rs 15 crore of proceeds was put back to work rather than paid out. Carry is ignored throughout.

Why can a fund not invest all of its commitments?

A family sets aside Rs 1,00,000 for a wedding and the planner's fee is Rs 17,500 of it: only Rs 82,500 buys food and flowers. Management fees are paid out of the same commitments that fund the investments, so a fund that never recycles invests less than its headline size. Rs 17.5 crore could be 2% a year for five years and 1.5% for five more; schedules vary, but the arithmetic is the same. To turn Rs 100 crore into Rs 250 crore for investors, Rs 82.5 crore has to earn 250 / 82.5 = 3.03x.

How does recycling change the multiple, and why is it not 250 over 97.5?

RecyclingReinvesting proceeds from early exits into new companies instead of distributing them, usually capped at about the amount of fees paid. lets the fund put Rs 15 crore of early exit money back into new companies, so invested capital reaches Rs 97.5 crore. But that Rs 15 crore was proceeds the portfolio had already earned and did not pay out, so the portfolio must produce Rs 250 crore for investors plus the Rs 15 crore it recycled. Gross proceeds of 265 on cost of 97.5 is 2.72x: still well below 3.03x, but not the 2.56x a quick division gives.

Say the limit too. Recycling delays money investors would have received early, so it can lift the multiple while lowering the IRR, and it works only if the fund finds good companies for the recycled rupees. It is a tool for getting fees back to work, not free return.

Faster than 33.3% a year; at exactly one third her stake's value stands still. Each round leaves her 75% of her previous ownership, so her stake's value changes by 0.75 times one plus the valuation growth. That equals 1 when growth is 1 over 0.75 minus 1, a third. At 20% growth her stake loses 10% of its value every year, even while the company grows.

Why is the break-even growth a third and not a quarter?

A shop sells an item at a 25% discount and wants its revenue back to where it was: it needs a third more units, not a quarter more, because 75 x 1.333 is 100. After giving up a quarter, she holds three quarters, and three quarters of anything must grow by a third to get back to the whole. This is the same asymmetry as recovering from a loss: the recovery is measured on a smaller base, so it needs a bigger percentage.

What happens at 20% and at 50% growth?

At 20%, each year multiplies her stake's value by 0.75 x 1.2 = 0.9. After five rounds the company is worth 2.5 times as much, and her stake is worth only 59% of what it was. At 50%, each year multiplies it by 1.125, and five rounds take it to 1.80 times. Same dilution, opposite outcomes, decided entirely by whether growth clears one third.

Why would an investor ask this?

Because it is the test every round must pass for existing holders. With a quarter sold, post-money growth of a third is the same as the new round's pre-money equalling the last round's post-money. A round only makes existing holders richer if the new pre-money beats the old post-money, which is what an up round means. Say that link and the puzzle becomes a point about pricing, not just arithmetic.

Let the first 7 founders go, then back the first one who is better than all of them; you back the single best about 38% of the time. The first 7 set the bar. You win when the best founder comes after them and nobody earlier than the best beats the bar first. Across all cut-offs, 7 gives the highest chance, 38.4%, against 5% for picking at random.

Why let good founders pass at all?

Renting a flat in a tight market works the same way: the first few viewings teach you what a good flat looks like, and you commit to the next one that beats them. The early meetings are the price of learning where the bar sits, and the rule trades a small chance that the best founder is among them for a much better read on everyone after. Commit too early and you have no bar; wait too long and the best has probably already gone.

How do you work out the chance for a given cut-off?

Say you let r founders pass. Suppose the best founder is at position i, after the first r. You back that founder only if nobody between r and i beats the bar first, which happens exactly when the best of the first i minus 1 founders sits inside the first r, a chance of r over i minus 1. Each position is equally likely to hold the best, one in 20, so add up across positions.

The curve is flat near the top: letting 6 pass gives 37.9% and 8 gives 38.2%. The general rule is to let about n/e go, 37% of the field, and win about 37% of the time; for 20 founders that is 7.4, which rounds to 7.

Where does the model stop describing real deal flow?

It assumes you only value the single best, can only rank founders against each other, see them in random order and never get a second chance. A real investor is happy with a top three founder, has an absolute sense of quality from past deals, and can sometimes return to a founder a week later. Each of those argues for committing earlier. Say that limit after the number; it shows you know what the model is for.

The cap bites at an exit of Rs 220 crore and converting takes over above Rs 300 crore; in between, the investor's payout is stuck at Rs 60 crore. Below Rs 20 crore the investor takes everything. Above it, the investor takes Rs 20 crore plus 20% of the rest until the total reaches 3x, Rs 60 crore. Only when 20% of the whole exit exceeds Rs 60 crore does converting pay more.

What does a capped participating preference mean in plain words?

Picture a relative who lends to your shop on the terms: my money back first, then a fifth of whatever is left, but I will never take more than three times what I put in; if a fifth of the whole shop is ever worth more, I will take that instead. The investor first takes back Rs 20 crore, then shares 20% of the remainder, until the total reaches the cap of 3x, Rs 60 crore. Above the cap, the investor can give up the preference and convert to plain common shares. Here the cap covers the total return, preference included, which is the usual reading; say so before you calculate.

Where exactly do the two kinks sit?

Set each piece equal to the cap. Participation reaches Rs 60 crore when 20 plus 20% of the exit less 20 equals 60, which is an exit of Rs 220 crore; converting beats Rs 60 crore when 20% of the exit exceeds it, above Rs 300 crore. At a Rs 100 crore exit the investor receives Rs 36 crore against Rs 20 crore as plain common; at Rs 400 crore it receives Rs 80 crore, exactly its 20%.

Why does the flat band matter at the negotiating table?

Inside the band the investor gains nothing from a better price, while every extra rupee goes to the common holders. Between Rs 220 crore and Rs 300 crore the investor is indifferent to the price, so a quick, certain sale at the bottom of the band can suit it more than a long push for the top. A founder who knows where the band sits knows when interests diverge, and can plan the board conversation before a buyer appears.

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.

Offer B leaves the founders 66.2% against 58.3% under Offer A, and its effective pre-money is Rs 69.5 crore against Rs 67.0 crore. A pool created inside the pre-money is paid for by the existing holders. Offer A's 20% pool is Rs 23 crore of a Rs 115 crore post-money, so the founders' shares are really valued at Rs 67 crore. Offer B's 10% pool costs only Rs 10.5 crore.

Why does a pool inside the pre-money lower the real price?

A flat sold for Rs 90 lakh, where the seller must also leave Rs 23 lakh of new furniture for the buyer, really sold for Rs 67 lakh. A pool created inside the pre-money is paid for entirely by the existing holders, so the effective pre-money is the headline pre-money minus the pool's value. Offer A: post-money is 90 + 25 = Rs 115 crore, the pool is 20% of that, Rs 23 crore, and the founders' shares are worth 90 - 23 = Rs 67 crore. Offer B: post-money Rs 105 crore, pool Rs 10.5 crore, effective pre-money Rs 69.5 crore.

What share do the founders keep under each offer?

Take the investor's share and the pool off the whole. The founders keep one minus the investor's share minus the pool: 58.3% under Offer A and 66.2% under Offer B. The investor gets a smaller share under A, 21.7% against 23.8%, which is why the headline looks better; the extra 10 points of pool more than cancel it. This assumes the founders own everything before the round and there is no existing pool.

What would you push back on?

The pool size, not the headline. A pool should be sized to the hires planned before the next round, not to a round number. If the founder can show that 10% covers the hiring plan, Offer A's pool falls to Rs 11.5 crore, its effective pre-money rises to Rs 78.5 crore and the founders keep 68.3%, better than either offer as written.

Split the envelopes into three groups of three and weigh two of the groups against each other. The heavier pan holds the cheque; if they balance, it is in the group left aside. Then weigh two envelopes from that group: the heavier one is it, and a balance means the third. Each weighing has three outcomes, so two weighings tell apart 3 x 3 = 9 envelopes.

Why is a balance scale worth three answers, not two?

A traffic signal carries three messages, not two, because amber counts. A balance scale is the same: left heavy, right heavy, or level. The balanced outcome is information too, so the best weighing splits the suspects into three equal groups, two on the scale and one off it. Most people instinctively halve, as they would with a yes or no question, and halving throws away the third answer.

How do you prove two weighings are enough and one is not?

Count end points. With k weighings and three outcomes each, the scale can produce at most 3 to the power k different results, and each envelope needs its own result. One weighing gives three results, too few for nine envelopes. Two give nine, exactly enough, and the tree in the figure shows a plan that uses all nine. Three weighings would handle 27 envelopes the same way.

What does the interviewer want to hear beyond the procedure?

Say the counting argument, not just the steps. It shows you can tell when a plan is optimal rather than merely working. The habit carries over to diligence: a good question is one whose every answer, including the dull one, changes what you do next. A reference call where only a glowing answer would move you is a weighing that wastes the balanced outcome. Then state the assumption the puzzle rests on: you know the odd envelope is heavier, not merely different.