Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
Topics: 12
Hard: 30

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092A private credit fund makes a 3-year loan of 400 at 10%. The borrower has a 5% chance of default each year and lenders would recover 60%. What is the expected annual loss rate, and what does the loan earn after expected losses?Credit and PIK mathsCorePrivate credit
Try it first
What is the expected loss each year, as a share of the loan?
Show the worked solution
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.
With a 5% chance of default and 60% recovered, a 400 loan loses 160 if the borrower defaults and nothing otherwise, an expected loss of 8, or 2% a year, a fifth of the 10% coupon, leaving about 8% after losses.
The relationship
$EL = PD \times LGD = 5\% \times 40\% = 2\% \qquad 10\% - 2\% = 8\%$
PD probability of default in a year, 5%
LGD loss given default, one minus the 60% recovery
EL expected loss each year, as a share of the loan
What it says in wordsExpected loss is how often default happens times how much it costs when it does; subtract it from the coupon to get the loss-adjusted yield.
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.
Where candidates lose it
The common slip is using the 5% default probability as the loss, which ignores recovery and overstates the expected loss by two and a half times.
The second is forgetting that expected loss is an average. A single loan either loses 160 or nothing; the 2% only describes a large, diversified book of loans like it.
What the interviewer asks next
What coupon would give the same 8% loss-adjusted yield if recovery fell to 30%?
Why do recoveries tend to fall exactly when defaults rise?
How would a PIK toggle change the expected loss on this loan?
093A portfolio company has fixed costs of 300 and a contribution margin of 40%. The sponsor adds a sales team costing 60 a year. How much new revenue must the team bring in to pay for itself, and how much if the new sales carry only a 35% contribution margin?Operating levers and margin mathsCorePortfolio operations team
Try it first
How much new revenue pays for a 60 sales team at a 40% contribution margin?
Show the worked solution
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.
Contribution at 40% of revenue covers fixed costs of 300 at revenue of 750; adding 60 of fixed cost moves breakeven to 900, so the sales team must bring in 150 of revenue, or about 171 if its sales carry a 35% margin.
The relationship
$\Delta R = \frac{\Delta F}{m} = \frac{60}{0.40} = 150 \qquad \frac{60}{0.35} \approx 171$
Delta F the new fixed cost, 60
m contribution margin on the new sales
Delta R new revenue needed to break even on the hire
What it says in wordsDivide the new fixed cost by the margin the new sales earn.
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.
Where candidates lose it
The fast wrong answer is 60: matching the cost with the same amount of revenue, as if revenue were profit. Variable costs take 60% of every rupee before anything is left to pay the team.
The second miss is assuming new sales carry the existing margin. A sales team that buys growth with discounts can raise revenue and still fail to pay for itself.
What the interviewer asks next
How many months of ramp can the sponsor afford if the team reaches 300 of annual sales by year end?
The team's sales carry a 50% margin because they sell a premium line. What is the hurdle now?
How does a commission-only sales structure change this calculation?
094A sponsor needs a 25% IRR over 5 years. It expects to exit at an enterprise value of 1,500 with net debt of 300. What is the most equity it can put in, and with half the entry price funded by debt, the most it can pay for the business?Valuation riddlesCoreLarge-cap buyout fundMid-market buyout fund
Try it first
Roughly what is the maximum equity cheque?
Show the worked solution
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.
Exit enterprise value of 1,500 less 300 of net debt leaves exit equity of 1,200, which divided by 3.05, 1.25 to the fifth, caps entry equity at about 393; with half the price in debt, the most the sponsor can pay is about 786.
The relationship
$E_0 = \frac{1{,}500 - 300}{1.25^5} = \frac{1{,}200}{3.052} \approx 393 \qquad EV_0 = \frac{393}{0.5} \approx 786$
E_0 maximum equity at entry
1.25^5 five years of compounding at 25%, about 3.05
0.5 equity's share of the entry price
What it says in wordsDivide exit equity by the required money multiple to get the maximum cheque, then gross up by equity's share of the price.
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.
Where candidates lose it
The usual slip is simple interest: 25% times 5 is 125%, so 2.25x, which gives a cap of about 533 and overpays. Returns compound.
The other is dividing the exit enterprise value, not the exit equity, by the multiple. The sponsor only owns what is left after the 300 of net debt.
What the interviewer asks next
The target IRR falls to 20%. How much more can the sponsor pay?
If exit EBITDA is 150, what entry multiple does the 786 imply against the exit multiple?
Why might a sponsor bid above this ceiling anyway?
095A deal returned 3x the money at a 20% IRR. Roughly how long was the hold?Returns mathsCoreMid-market buyout fundIndian mid-market PE
Try it first
Answer inside ten seconds.
Show the worked solution
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.
Raising 1.2 to the powers 1 to 7 gives 1.20, 1.44, 1.73, 2.07, 2.49, 2.99 and 3.58, so a 3x return at 20% a year needs about six years, 6.03 exactly.
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.
The relationship
$n = \frac{\ln 3}{\ln 1.2} = \frac{1.0986}{0.1823} \approx 6.03$
n years in the hold
ln 3 natural log of the money multiple
ln 1.2 natural log of one plus the IRR
What it says in wordsThe hold is the log of the multiple divided by the log of one plus the yearly rate.
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.
Where candidates lose it
The slip is simple interest: 3x means gaining 200%, and at 20% a year that looks like ten years. Compounding gets there in six.
The other loss is fumbling the powers under time pressure. Learn 1.44, 1.73, 2.07, 2.49 and 2.99 and the question takes five seconds.
What the interviewer asks next
A deal returns 2.5x in 4 years. Roughly what IRR?
Same 3x, but over 4 years. What IRR?
Why do sponsors often prefer a 2.5x in 4 years to a 3x in 6?
096A buyout is financed with senior debt of 4x EBITDA at 8% and mezzanine of 1.5x at 13%. What is the blended cost of debt? If the lender's floor is 2.0x interest cover, how much all-senior debt at 8% could EBITDA of 100 carry?Leverage and capital structureHardPrivate credit
Try it first
What is the blended cost of the 5.5x of debt?
Show the worked solution
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%.
Drawn with width equal to size and height equal to rate, the 400 of senior debt at 8% and 150 of mezzanine at 13% cost 51.5 of interest, a blended 9.36%; at a 2.0x cover floor EBITDA of 100 carries 625 of all-senior debt but only about 534 at the blended rate, so the 5.5x mix sits just over the limit.
The relationship
$r_{blend} = \frac{400 \times 8\% + 150 \times 13\%}{550} = \frac{51.5}{550} = 9.36\% \qquad D_{max} = \frac{100 / 2.0}{8\%} = 625$
400, 150 senior and mezzanine debt
51.5 total interest
100 / 2.0 the most interest a 2.0x cover floor allows
What it says in wordsBlend the rates by size; then divide the interest the cover floor allows by the rate to find how much debt it supports.
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.
Where candidates lose it
The common slip is averaging the two rates, 10.5%, which overweights the small expensive tranche. Weight by size every time.
The second is assuming more debt always means more interest cover pressure in the same proportion. Cover depends on the rate as well as the amount, so a cheaper tranche changes the capacity even at the same leverage.
What the interviewer asks next
How much mezzanine can stay in if senior is fixed at 4x and cover must be 2.0x?
Rates rise by 2 points on the senior debt only. What is cover now?
Why would a sponsor accept 13% mezzanine at all if senior debt is cheaper?
097Each deal in a fund has a 20% chance of returning 0x, a 50% chance of 2x and a 30% chance of 4x, independently. What is the expected money multiple per deal, and what is the chance that at least one of three deals is a zero?Probability and expected value in dealsCoreLarge-cap buyout fundSecondaries and fund of funds
Try it first
Chance that at least one of three deals returns zero?
Show the worked solution
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.
One deal returns 0x with 20% chance, 2x with 50% and 4x with 30%, an average of 2.2x; across three independent deals only the path with no zeros, 51.2%, avoids a loss, so the chance of at least one zero is 48.8%.
The relationship
$E[M] = 0.2(0) + 0.5(2) + 0.3(4) = 2.2 \qquad P(\geq 1 \text{ zero}) = 1 - 0.8^3 = 0.488$
E[M] expected money multiple of one deal
0.8 the chance one deal is not a zero
0.8^3 the chance none of three deals is a zero
What it says in wordsAverage the outcomes by their chances; for at least one, take one minus the chance of none.
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.
Where candidates lose it
The fast wrong answer to the second half is 60%, adding the probabilities. Probabilities of overlapping events cannot simply be added; the complement avoids the trap.
The other slip is treating the expected 2.2x as what each deal returns. Say plainly that the average is a property of many deals, not of one.
What the interviewer asks next
What is the chance that exactly one of the three deals is a zero?
With ten deals, what is the chance of at least one zero?
If outcomes are not independent, for example all deals in one sector, what changes?
099A fund screens 40 deals a year. 25% reach investment committee, 30% of those are signed and 60% of signed deals close. How many deals close? How many must be screened to expect 5 closes?Probability and expected value in dealsCoreMid-market buyout fundIndian mid-market PE
Try it first
How many deals close from 40 screened?
Show the worked solution
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%.
Forty screened deals narrow to 10 at investment committee, 3 signed and 1.8 closed, an overall conversion of 4.5%, so expecting 5 closes needs about 111 deals screened.
The relationship
$40 \times 0.25 \times 0.30 \times 0.60 = 1.8 \qquad \frac{5}{0.045} \approx 111$
0.25, 0.30, 0.60 the conversion rate at each stage
0.045 the overall share of screened deals that close
What it says in wordsMultiply the stage rates to get the overall rate; divide the target by it to size the top of the funnel.
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.
Where candidates lose it
The common slip is averaging or adding the rates, or applying each one to the original 40. Every rate applies only to the deals that survived the stage before.
The second is answering 111 and calling it certain. It is the screening volume for an expected 5 closes; to be confident of 5, you need more.
What the interviewer asks next
Which stage would you try to improve first, and why?
If closing takes six months, how many deals must be in the funnel at any time?
How would you estimate these conversion rates for a new fund with no history?
100A sponsor invests 100, sells half its stake for 150 in year 3 and the rest for 200 in year 5. The money multiple is 3.5x. Estimate the IRR by trial, and explain why it beats receiving the full 3.5x in year 5.Returns mathsHardLarge-cap buyout fundSecondaries and fund of funds
Try it first
Which is closest to the IRR?
Show the worked solution
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%.
The sponsor puts in 100 and gets 150 back in year 3 and 200 in year 5; the value of those flows falls as the discount rate rises, from +22.1 at 30% to -13.0 at 42%, crossing zero at an IRR of 36.9%, well above the 28.5% of a single year-5 exit.
The relationship
$-100 + \frac{150}{(1+r)^3} + \frac{200}{(1+r)^5} = 0 \quad\Rightarrow\quad r \approx 36.9\%$
r the IRR
150 proceeds from selling half the stake in year 3
200 proceeds from the rest in year 5
What it says in wordsThe IRR is the rate that makes the discounted inflows exactly equal to the 100 invested.
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.
Where candidates lose it
The common slip is treating 3.5x as if it all arrived in year 5 and answering 28.5%. The timing of the 150 is the whole question.
The other is guessing without a bracket. Name a rate that is too low and one that is too high, then interpolate out loud; an answer within a couple of points, reached that way, is what the interviewer is looking for.
What the interviewer asks next
If the 150 had come in year 2 instead of year 3, roughly what is the IRR?
Why might an investor prefer the single 3.5x exit despite the lower IRR?
How does a dividend recap in year 1 change the IRR and the money multiple?

PD	probability of default in a year, 5%
LGD	loss given default, one minus the 60% recovery
EL	expected loss each year, as a share of the loan

Delta F	the new fixed cost, 60
m	contribution margin on the new sales
Delta R	new revenue needed to break even on the hire

E_0	maximum equity at entry
1.25^5	five years of compounding at 25%, about 3.05
0.5	equity's share of the entry price

n	years in the hold
ln 3	natural log of the money multiple
ln 1.2	natural log of one plus the IRR

Try it first

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

What changes over the full three years?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not simply 60?

What would an operating partner ask before approving the hire?

Where candidates lose it

What the interviewer asks next

Try it first

Why work backwards from the exit?

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

Where candidates lose it

What the interviewer asks next

Try it first

How do you find the hold without logarithms?

Why is this worth memorising for buyout interviews?

Where candidates lose it

What the interviewer asks next

Try it first

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

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

Where candidates lose it

What the interviewer asks next

Try it first

How do you get the expected multiple?

Why does at least one zero use the complement?

Where candidates lose it

What the interviewer asks next

Try it first

Why do the stage rates multiply?

What would you do with this as a deal team?

Where candidates lose it

What the interviewer asks next

Try it first

How do you find an IRR by trial?

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

Where candidates lose it

What the interviewer asks next