Private Equity puzzles, solved step by step

B has the higher IRR, about 26.0% against 17.0% for A. But if B's 2x can only earn 1.5x over the next four years, about 10.7% a year, it ends at 3x after seven years, exactly where A ends. A higher IRR is worth more only if the cash it returns early can be redeployed at a comparable rate.

How do you get the two IRRs in your head?

Use the rule of 72 and a couple of anchors. B doubles in three years, and 72 over 3 is 24, so B is in the mid twenties; exactly, 2 to the one-third is 1.26, so 26.0%. A triples in seven years. Tripling takes about 1.6 times as long as doubling, so a seven-year triple is like doubling in about 4.4 years, which the rule of 72 puts in the high teens. Exactly, 3 to the one-seventh is 1.170, so 17.0%.

Why does the higher IRR not make B the better deal?

Think of two fixed deposits: one pays 26% but matures in three years, after which the best rate on offer is 10%; the other locks in 17% for seven. Over seven years you end up in the same place. IRR assumes the cash a deal returns can be reinvested at the same IRR, and when it cannot, the IRR overstates what the investor actually ends up with. B ties A when its proceeds earn 10.7% a year after year 3; any less and A wins over seven years, any more and B does.

This is why fund investors look at both numbers. A fund with high IRRs from quick flips may return less money over a decade than one with lower IRRs and bigger multiples, if the investors cannot put the early cash back to work at a similar rate.

Expected loss is about 2% a year, 8 on 400, so the loan earns about 8% after losses. The chance of default is 5% and a default costs the 40% not recovered: 5% times 40% is 2%. That takes a fifth of the 10% coupon. Assuming a 5% base rate, the 5 point spread shrinks to about 3 points once expected losses are paid for.

Why multiply the default chance by the loss, not use it alone?

If you lend a friend Rs 1,000 and think there is a one in twenty chance they cannot repay, but you know you would get Rs 600 back from them anyway, your expected loss is not Rs 50 but one twentieth of Rs 400, Rs 20. Expected loss is the probability of default times the loss given defaultThe share of the loan a lender loses when a borrower defaults, after recoveries. A 60% recovery means a 40% loss given default., so recovery matters as much as the chance of default. Here: 5% times 40%, which is 2% a year.

What changes over the full three years?

Default can happen in any year, so the chance of at least one default over three years is 1 minus 0.95 cubed, about 14.3%. Expected loss in rupees falls slightly each year because a loan that has already defaulted cannot default again: 8, then 7.6, then 7.2, about 22.8 over the life. The simple 2% a year is a good approximation. It also ignores the coupon lost in the year of default, which a fuller model would include.

A credit interviewer then asks whether 8% is enough. Compare it with the fund's cost of capital and with what safer loans pay. If senior loans to stronger borrowers yield 9% with expected losses of 0.5%, this loan is the worse risk-adjusted deal despite its higher coupon.

The team needs 150 of new revenue at a 40% margin, and about 171 at 35%. A fixed cost is paid for out of contribution, and each rupee of revenue brings only 40 paise of it: 60 divided by 0.40 is 150. If the team wins sales by discounting and the margin falls to 35%, the hurdle rises to 60 divided by 0.35, about 171. Breakeven revenue moves from 750 to 900.

Why is the answer not simply 60?

A shop hires a helper for Rs 6,000 a month. If every Rs 100 of sales costs Rs 60 in stock, the helper must bring in Rs 15,000 of extra sales, not Rs 6,000, because only Rs 40 of each Rs 100 is left to pay wages. New fixed costs are paid out of contribution, so the revenue needed is the cost divided by the contribution margin. At 40%, 60 of cost needs 150 of revenue.

What would an operating partner ask before approving the hire?

Breakeven is the floor, not the case for the hire: the team should bring in well above 150 within a reasonable ramp, at a margin close to the existing 40%. Sales teams often win volume with discounts, which is exactly what drops the margin to 35% and lifts the hurdle to 171. The partner would also ask how long the ramp takes, because a team that needs a year to sell anything costs 60 before it earns a rupee.

About 393 of equity, and an entry enterprise value of about 786. Exit equity is 1,500 less 300 of net debt, 1,200. A 25% IRR for five years means multiplying the money by 1.25 to the fifth, about 3.05x, so the most the sponsor can invest is 1,200 divided by 3.05. If equity is half the price, the business is worth at most twice that.

Why work backwards from the exit?

A house buyer who wants to double their money by the time they sell, and expects to sell for Rs 1 crore after clearing the loan, knows they cannot put in more than Rs 50 lakh today. A target return plus an expected exit turns into a ceiling on the price, which is how sponsors set their bids. The exit equity is fixed by the assumptions; the only free number is what you pay today.

How do you get 1.25 to the fifth without a calculator?

Square twice and multiply once. 1.25 squared is 1.5625, about 1.56; squared again is about 2.44; times 1.25 is about 3.05. A 25% IRR over five years is roughly a 3x money multiple, worth knowing by heart because buyout targets are often quoted that way. Then check consistency: entry debt of about 393 falling to 300 at exit means about 93 repaid over five years, which you would test against the business's cash flow.

About 6 years. 1.2 to the sixth power is 2.99, so six years of 20% a year turns 1 into very nearly 3. Build it from anchors: 1.2 cubed is 1.728, and squaring that gives 2.99. The exact answer, the log of 3 over the log of 1.2, is 6.03 years.

How do you find the hold without logarithms?

Climb the powers of 1.2 until you pass 3. It is like counting how many 20% pay rises it takes to triple a salary: each rise is on the new salary, so they compound. 1.2 cubed is 1.728, and 1.728 squared is about 2.99, so six steps of 20% take you to 3. Seven would be 3.58, well past.

Why is this worth memorising for buyout interviews?

Interviewers ask for any one of the three numbers, multiple, IRR or hold, given the other two. Knowing a few powers of 1.2 and 1.25 answers most of them instantly: 2x at 20% is about 4 years, 3x is about 6, and 3x at 25% is about 5. The rule of 72 gives a cross-check: doubling at 20% takes about 3.6 years, and tripling takes about 1.6 times as long as doubling, near 5.8.

One caveat worth saying: this assumes all the money went in at the start and came out at the end. If proceeds came back in stages, the same 3x and 20% would imply a longer total hold, because early returns lift the IRR.

The blended cost is about 9.36%, and all-senior debt at 8% could reach 625, 6.25x, at 2.0x cover. Interest is 32 on the senior and 19.5 on the mezzanine, 51.5 on 550 of debt. A 2.0x floor on EBITDA of 100 allows 50 of interest, which buys 625 at 8%. The planned 5.5x mix costs 51.5, cover of 1.94x, just under the floor.

Why is the blended rate not the average of 8% and 13%?

If you borrow Rs 4 lakh from a bank at 8% and Rs 1.5 lakh from a relative at 13%, your average cost is pulled towards 8% because most of the money is cheap. A blended cost of debt is a size-weighted average: total interest divided by total debt. Here that is 51.5 divided by 550, about 9.36%, well below the simple average of 10.5%.

Why does cover limit the debt before the leverage multiple does?

A lender's cover test caps interest, not turns of debt. The cheaper the debt, the more of it fits under the same interest ceiling, so 8% senior debt supports 6.25x while debt at the blended 9.36% supports only about 5.34x. That is the surprise in this question: replacing the mezzanine with more senior debt would let the sponsor borrow more, not less, if a senior lender would go that far. In practice senior lenders also cap leverage directly, which is why mezzanine exists at all.

Close with the practical point. The planned structure fails a 2.0x floor by a whisker, 1.94x. A credit interviewer wants you to notice that and suggest a fix: trim the mezzanine to about 1.4x, or negotiate the cover test on a measure that adds back something, and say which you would try first.

The expected multiple is 2.2x, and the chance of at least one zero in three deals is 48.8%. Expected value is 20% x 0 plus 50% x 2 plus 30% x 4, which is 2.2. For the zero, take the complement: each deal avoids a zero 80% of the time, so all three do 0.8 cubed, 51.2% of the time. A good average hides a near coin-flip chance of a loss.

How do you get the expected multiple?

Weight each outcome by its chance and add. It is the same as asking what a hundred identical deals would return on average: twenty return nothing, fifty return 2x and thirty return 4x, which is 0 plus 100 plus 120, or 220 on 100 put in. The expected multiple is 2.2x, but no single deal ever returns 2.2x: each one returns 0, 2 or 4.

Why does at least one zero use the complement?

There are many ways to get at least one zero: the first deal, the second, the third, or any two, or all three. There is only one way to get no zero, so count that and subtract it from one. Adding 20% three times gives 60% because it counts the two-zero and three-zero cases more than once. The chance all three are zeros is 0.2 cubed, 0.8%.

The fund-level point is the one to close on: a manager with a 2.2x average still writes off a deal in about half of all three-deal stretches. That is why investors judge a manager over many deals, and why a single write-off early in a fund says little on its own.

About 352.5 is owed, about 32.5 more than a cash-pay note's 320. PIK interest is added to the balance each year: 224, 250.9, 281.0, 314.7, then 352.5. A cash-pay note returns 24 a year plus 200 at the end, 320 in all. The extra 32.5 is interest on interest, which is why a PIK claim grows faster than its coupon suggests.

What does paid in kind actually mean for the balance?

A PIKPayment in kind: interest that is not paid in cash but added to the amount owed, so the loan balance grows instead. note pays its interest with more debt. It is like a credit card where you pay nothing and the interest is added to the bill each month, so next month's interest is charged on the bigger bill. Because each year's interest joins the balance, the 12% is earned on a growing base, and the claim compounds. After one year the balance is 224, after two 250.9.

Why does a private credit lender or a sponsor care?

The borrower keeps its cash today, which helps a company that is growing or stretched. The price is a claim that grows every year, so at exit the sponsor's equity sits behind a bigger debt than the original 200. For the lender, PIK means more risk: nothing is received for five years, and if the company fails the larger balance may not be recovered. Lenders usually charge a higher rate for PIK than for the same loan paid in cash.

About 1.8 deals close, and about 111 must be screened to expect 5. Forty screened becomes 10 at investment committee, 3 signed and 1.8 closed. The stage rates multiply to 4.5%, so expecting 5 closes takes 5 divided by 0.045, about 111 deals screened. Conversion rates multiply, which is why sourcing volume matters so much.

Why do the stage rates multiply?

Each rate applies only to what survived the stage before, like a cricket trial where a quarter of players make the camp, a third of those make the squad and most of the squad play. A funnel's overall conversion is the product of its stage rates, so three reasonable-looking rates combine into a small number: 25% x 30% x 60% is 4.5%.

What would you do with this as a deal team?

You can lift closes by screening more deals or by improving any one stage, and improving a stage is often cheaper. Raising the committee rate from 25% to 35%, by screening sharper, takes closes from 1.8 to 2.5 without a single extra deal. One honest caveat: 1.8 is an expected number. In a year with 40 screens the fund could close zero or four, so a target of 5 closes should be planned with a margin above 111.

Interviewers in Indian mid-market funds often ask this to see whether you understand that origination is a numbers game. A candidate who says a team needs to see more than a hundred opportunities to close five has understood why associates spend so much time on screening.

About 36.9%, against 28.5% if the full 3.5x came back in year 5. Try 30%: the 150 and 200 are worth about 122 today, too much. Try 42%: about 87, too little. The rate that values them at exactly 100 is about 37%. The early 150 is discounted for three years, not five, which pulls the IRR up for the same money multiple.

How do you find an IRR by trial?

The IRR is the discount rate at which the money coming back is worth exactly what went in. Pick a rate, discount each inflow, and see whether the total is above or below 100. At 30%, 150 over 1.3 cubed is 68.3 and 200 over 1.3 to the fifth is 53.9, together 122.1, so 30% is too low; at 42% they come to 87.0, too high. Interpolating between the two lands near 37%, and the exact answer is 36.9%.

Why does the partial exit beat 3.5x in year 5?

Rs 150 received in year 3 is worth more than Rs 150 received in year 5, just as a bonus paid this year is worth more than the same bonus promised later. IRR measures speed, so pulling part of the money forward lifts it even though the total returned, 3.5x, is the same. A single year-5 exit of 350 is 3.5 to the one-fifth, 28.5%. That gap is why sponsors like early dividends and partial sales, and why investors check the money multiple alongside the IRR.