Private Equity puzzles, solved step by step

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

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002A deal returns 2.5x your money over five years. Without a calculator, what is the IRR, to the nearest percentage point?Returns mathsCoreMid-market buyout fundIndian mid-market PE
Try it first
Commit to a number first.
Show the worked solution
About 20%, and 20.1% exactly. Bracket it with round rates you can compound in your head. 1.2 to the fifth is 2.49 and 1.25 to the fifth is 3.05, so 2.5x sits right at the bottom of the bracket, a hair above 20%. Straight division, 150% over five years, gives 30% and overstates the return badly because it ignores compounding.
How do you compound 1.2 five times in your head?
Do it by squaring, the way you would fold a sheet of paper. 1.2 squared is 1.44. Square again to get the fourth power: 1.44 times 1.44 is about 2.07. One more 1.2 gives about 2.49. Chaining small multiplications you trust is safer than reaching for a formula you half remember, and it is exactly what the interviewer wants to watch. The same chain at 1.25 runs 1.56, then 2.44, then 3.05.
Compounding for five years turns 15% a year into 2.01x, 20% into 2.49x and 25% into 3.05x, so a 2.5x deal sits just past the 20% rung and its IRR is 20.1%.
Why is the answer not 30%?
Think of a savings account that pays interest on interest. If it pays 20% a year, the gain in year five is far larger than the gain in year one, because it is earned on a bigger balance. An IRR is a compound rate, so the five years of gains are not equal slices of the 150% total; the later years carry more of it. Dividing 150 by five assumes every year earns 30 on the original 100, which is simple interest. The compound rate needed is lower: 20.1%.
The relationship
$\text{IRR} = \text{MOIC}^{1/n} - 1 = 2.5^{1/5} - 1 \approx 20.1\%$
MOIC multiple on invested capital, money back divided by money in
n years the money is invested, here 5
What it says in wordsWith one cash flow in and one out, the IRR is the yearly growth rate that turns the money in into the money out.
What table is worth memorising before a buyout interview?
A small grid of multiples by holding period. Once you know that 2x in five years is about 15%, 2.5x is about 20% and 3x is about 25%, most return questions become a lookup with a small adjustment. If you need more precision, interpolate between rungs: 2.5 is 0.01 of the way along a 0.56 gap from 2.49 to 3.05, which adds about a tenth of a point to 20%, giving 20.1%. State that this assumes a single cash flow in and a single cash flow out; interim dividends would change it.
Where candidates lose it
The classic slip is 30%: dividing the total gain by the years. It sounds reasonable when said quickly and is wrong by ten points, which an interviewer in a returns-driven job notices immediately.
The second miss is guessing 25% because 2.5x feels like a strong result. Bracket it with two rates you can compound, then say which end of the bracket the deal sits at.
What the interviewer asks next
What multiple does 20% a year give over three years?
What IRR does 3x over seven years imply?
If half the proceeds come back in year three, is the IRR higher or lower than 20.1%, and why?
016A deal is on track to return 2x in three years. The exit slips to year six, but by then the equity value has doubled again, to 4x. Has the IRR improved?Returns mathsCoreMid-market buyout fund
Try it first
Compared with 2x in three years, the IRR on 4x in six years is:
Show the worked solution
No, the IRR is unchanged at about 26%. 2x in three years means the money doubles every three years. 4x in six years is two doublings in six years, the same pace. The cube root of 2 is 1.26, and the sixth root of 4 is the same number, so both IRRs are 26.0%. The multiple doubled; the annual rate did not move.
Why does a bigger multiple not mean a better IRR?
Think of two runners: one covers 2 km in 10 minutes, the other 4 km in 20 minutes. The second went further, but neither ran faster. IRR measures pace, and the multiple measures distance, so doubling the multiple over double the time leaves the pace exactly where it was. Three years to double, then another three years to double again, is the same doubling time throughout.
Compounding at 26.0% a year reaches 2x at year 3 and 4x at year 6, so both outcomes sit on the same curve with the same IRR, while 3x at year 6 would sit below it at 20.1%.
How do you show it in two lines?
Write both as roots. The first IRR is 2 to the power of one third, less one. The second is 4 to the power of one sixth, less one. Since 4 is 2 squared, 4 to the one sixth is 2 to the two sixths, which is 2 to the one third: the same number, 1.26. So both are 26.0%. If the slip had produced only 3x by year six, the IRR would fall to 20.1%, a real cost of the delay.
The relationship
$4^{1/6} = (2^2)^{1/6} = 2^{1/3} \approx 1.260$
2^{1/3} the yearly growth factor for doubling in three years
4^{1/6} the yearly growth factor for quadrupling in six years
What it says in wordsQuadrupling in six years is the same yearly pace as doubling in three.
So is the slipped exit just as good?
Not necessarily, and the judgement is the point of the question. Whether 4x in six years is as good depends on what the money could have done in years four to six: if the fund could have redeployed it at 26%, the two are equal; if not, the larger multiple is better. For the GP, 4x means more carry in absolute terms. For the LPs, the delay holds capital in an older fund for longer. And three extra years of exposure carry three more years of risk that the second doubling does not happen.
Where candidates lose it
The trap is answering that the IRR improved because the multiple doubled. Candidates who think in multiples alone fall into it every time.
The second loss is answering correctly and stopping. The follow-on judgement, about redeployment, carry and risk, is what separates a calculator from an investor.
What the interviewer asks next
What multiple would the deal need at year six to beat 26%?
Why do LPs track both net IRR and net multiple?
If you could redeploy at 15%, which outcome would you prefer?
028A sponsor invests 100. In year 2 it takes a dividend recap of 60, and in year 5 it exits for 180. What is the money multiple, and is the IRR higher or lower than a deal that simply turns 100 into 240 at year 5?Returns mathsCoreMid-market buyout fund
Try it first
Both deals return 2.4x. Which has the higher IRR, and by roughly how much?
Show the worked solution
The multiple is 2.4x in both cases, but the recap deal's IRR is about 24.1% against 19.1%. Money back is 60 plus 180, which is 240 on 100. The single exit's IRR is 2.4 to the power one fifth, less one. The recap returns part of the money in year 2, and cash that comes back sooner lifts the IRR even though the total is unchanged.
Why does the same multiple give a different IRR?
Lend a friend Rs 100 and get Rs 240 back. If Rs 60 of it comes back after two years, you have that money in hand for three years while waiting for the rest. The multiple counts how much money comes back; the IRR counts how fast it comes back, so pulling cash forward raises the IRR without touching the multiple. The two deals below differ only in timing.
Both deals turn 100 into 240 over five years, but the deal that returns 60 in year 2 earns an IRR of 24.1% while the single exit at year 5 earns 19.1%, because early cash shortens the average time the money is out.
How do you get the recap IRR without a spreadsheet?
The single exit is a one-line calculation: 2.4 to the power 0.2 is about 1.191, so 19.1%. The recap needs a guess and a check, because there are two inflows. Try 24%: 60 divided by 1.24 squared is about 39.0, and 180 divided by 1.24 to the fifth is about 61.4. Together that is 100.4, just above the 100 invested, so the true rate is a touch higher. Guess, discount each cash flow, and nudge the rate until the present values add back to the cheque. The answer is 24.1%.
The relationship
$100 = \frac{60}{(1+r)^2} + \frac{180}{(1+r)^5} \quad\Rightarrow\quad r \approx 24.1\%$
r the IRR, the rate that makes the discounted inflows equal the investment
60 the recap dividend in year 2
180 the exit proceeds in year 5
What it says in wordsThe IRR is the single rate at which the discounted cash coming back exactly repays the cash put in.
Say what the recap costs, because an interviewer will push. The dividend was paid with new debt, which is why the exit cheque is 180 and not 240. The company carried more leverage for three years, so the higher IRR came with higher risk of a covenant problem. That is the honest trade: an LP sees a better IRR and early cash back, and the business sees a thinner cushion.
Where candidates lose it
The fast wrong answer is that the IRRs are equal because the multiples are equal. Candidates who think of IRR as a multiple spread over years miss that IRR weights early cash more heavily.
The second loss is getting stuck on a two-cash-flow IRR. You do not need the exact figure in your head; bracket it with one trial rate, say it is a little above 24%, and explain why.
What the interviewer asks next
Why might a GP favour a dividend recap late in a fund's life?
If the recap had been 100 in year 2 and exit 140 in year 5, what happens to the IRR and to the multiple?
What does an LP look at besides IRR to judge whether the recap added value?
031On a 5-year hold, how much more exit equity do you need to lift the IRR from 20% to 25%?Returns mathsCoreLarge-cap buyout fund
Try it first
Pick the closest answer.
Show the worked solution
About 22.6% more exit equity: from 2.49x the cheque to 3.05x. 1.2 to the fifth is 2.49 and 1.25 to the fifth is 3.05. Their ratio is (1.25 / 1.2) to the fifth, about 1.226. Five points of IRR compound every year, so over five years they add up to almost a quarter more money at exit, and more again over longer holds.
Why is the answer not five per cent?
Two runners leave together, one about 4% faster. After one lap the gap is small. After five laps it is five small gaps stacked, each measured on a longer distance. An extra five points of IRR is an extra growth factor of 1.25 / 1.20, about 1.042, applied every year, so over five years the exit has to be 1.042 to the fifth, about 1.226 times larger. That is a 22.6% increase in exit equity.
The relationship
$\frac{(1.25)^5}{(1.20)^5} = \left(\frac{1.25}{1.20}\right)^5 = 1.0417^5 \approx 1.226$
1.25^5 the exit multiple at 25% a year for five years, 3.05x
1.20^5 the exit multiple at 20% a year, 2.49x
What it says in wordsThe extra exit value needed is the ratio of the two yearly growth factors, raised to the number of years.
Lifting the IRR from 20% to 25% over five years needs exit equity of 3.05x the cheque instead of 2.49x, 22.6% more, and the same five points need 33.1% more over a seven-year hold.
How much more enterprise value does that take?
Less than 22.6%, if the deal carries debt at exit. Say exit EV is 1,000 with 400 of debt, so equity of 600 earns 20%. Equity needs to rise to about 736, and with debt unchanged that needs EV of about 1136. Leverage works in your favour here: a 13.6% rise in enterprise value delivers the 22.6% rise in equity. That is also why sponsors stress-test exit multiples so carefully, since small EV moves swing equity hard either way.
Say the limitation: the sums assume a single cheque in and a single exit. With interim dividends or staged investments, the required change in exit value differs, and the IRR needs a full cash flow schedule.
Where candidates lose it
The common loss is thinking linearly: five more points sounds like a small tweak, so candidates say exit value needs to rise by 5% or so. Compounding turns five points a year into about 23% over five years.
The second loss is dividing 25 by 20 and answering 25%. That compares the rates, not the money multiples. Raise the ratio of the growth factors, 1.25 over 1.20, to the number of years.
What the interviewer asks next
What exit multiple of the cheque does 30% over five years need?
Over a three-year hold, how much more exit equity does the same move need?
If exit equity can only rise 10%, how much shorter must the hold be to reach 25%?
042Which makes more money on the same cheque: a 25% IRR for 3 years, or a 20% IRR for 5 years?Returns mathsWarm upMid-market buyout fund
Try it first
Which ends with more money?
Show the worked solution
The 20% for 5 years makes more money: 2.49x against 1.95x. 1.25 cubed is about 1.95 and 1.2 to the fifth is about 2.49. On a cheque of 100 that is a profit of 149 against 95. The 25% deal is faster, but the money is back after three years, and it only catches up if it can be reinvested at about 13% for the remaining two.
Why does the higher IRR make less money?
A car doing 100 km an hour for three hours covers 300 km; one doing 80 km an hour for five hours covers 400. Speed and distance are different questions. IRR measures how fast money grows, the multiple measures how much money you end with, and a longer hold at a lower speed can end further ahead. Here 1.25 cubed is 1.95 and 1.2 to the fifth is 2.49.
Compounding at 25% for three years ends at 1.95x, while 20% for five years ends at 2.49x, so the higher IRR makes less money unless its proceeds can be reinvested at about 12.9% for the two years it is not running.
So which would an LP prefer?
It depends on what happens to the money after year three. If the LP can redeploy A's proceeds at more than about 12.9% a year, A ends ahead; below that, B wins. Reinvested at 20%, A reaches 2.81x by year five; parked at 8%, only 2.28x. That is why LPs read IRR and the money multiple together, and why GPs who sell early to protect a high IRR are sometimes accused of leaving money on the table.
The relationship
$1.25^3 = 1.95 \qquad 1.20^5 = 2.49 \qquad (1+r)^2 = \frac{2.49}{1.95} \Rightarrow r \approx 12.9\%$
1.25^3 A's money multiple after three years
1.20^5 B's money multiple after five years
r the reinvestment rate at which A catches B by year five
What it says in wordsCompare the two multiples, then ask what rate A's money must earn in the gap years to catch up.
Say the limitation. Both deals here are a single cheque in and out. A fund's IRR also depends on when capital is called and returned, and a high IRR on a small, quick deal can flatter a fund that made little money overall.
Where candidates lose it
The common loss is picking the higher IRR on reflex. The interviewer is checking whether you know that IRR is a rate and says nothing on its own about how much money comes back.
The second loss is giving the right answer without the reinvestment point. The full answer is that B makes more money, and A wins only if its proceeds can be redeployed at about 13% or better.
What the interviewer asks next
What IRR over 3 years would match 2.49x?
Why might a GP sell a winner early even though holding would make more money?
How does the timing of capital calls affect a fund's IRR but not its multiple?
058A deal is worth 2.0x the equity after four years, an IRR of about 18.9%. Holding one more year would add 10% to the exit equity, taking it to 2.2x. Does the IRR rise or fall, and what growth in the extra year would keep the IRR flat?Returns mathsHardLarge-cap buyout fundMid-market buyout fund
Try it first
Hold a fifth year for 10% more equity. What happens to the IRR?
Show the worked solution
The IRR falls, from 18.9% to about 17.1%, even though the money multiple rises from 2.0x to 2.2x. IRR is an average yearly growth rate, and a year that adds 10% is below the 18.9% the deal has been earning. To keep the IRR flat, the fifth year must grow equity by 18.9% itself, to about 2.38x. The hurdle for holding on is the IRR already achieved.
Why can a bigger multiple mean a lower IRR?
Think of a batsman averaging 19 runs a match over four matches. If he scores 10 in the fifth, his total rises but his average falls. IRR is the average yearly growth of the equity, so any year that grows it by less than the current IRR lowers the average, even while the total keeps rising. The money multiple is the total; the IRR is the average. Here the deal has averaged 18.9% a year and the fifth year offers 10%.
At 2.0x after four years the IRR is 18.9%; a fifth year that adds 10% lifts the multiple to 2.2x but drops the IRR to 17.1%, and only 2.38x at year 5 would hold the IRR flat.
The relationship
$2.0^{1/4} = 1.189 \qquad 2.2^{1/5} = 1.171 \qquad 2.0 \times 1.189 = 2.378$
2.0^(1/4) one plus the IRR of 2.0x over four years
2.2^(1/5) one plus the IRR of 2.2x over five years
2.378 the multiple at year 5 that keeps the IRR at 18.9%
What it says in wordsTake the root of the multiple by the number of years to get one plus the IRR; to hold it, the extra year must grow by the same factor.
How would a fund actually weigh the choice?
Both numbers matter to an investor, and they point opposite ways. The extra year adds 0.2x of money but costs about 1.8 points of IRR, so the right call depends on what the cash would earn if it were returned now. If the investors could put it to work at more than 10% elsewhere, selling at year 4 is better; if not, the extra 0.2x is real money. Say this trade-off out loud, then say the limit: the 10% is a forecast, while the 2.0x offer may be on the table today.
Where candidates lose it
Candidates hear that equity grows and say the IRR rises. They are describing the money multiple. IRR is a rate per year, and a below-average year pulls a rate down.
The second miss is on the flat-IRR part: answering 10%, or the old IRR divided over five years. The extra year has to match the rate already earned, about 18.9%, which takes the multiple to roughly 2.38x.
What the interviewer asks next
What if the extra year adds 25%? What IRR then?
Why do investors track both IRR and the money multiple rather than one of them?
How does a dividend recap at year 4 change this decision?
067You make 8x your money in 6 years. Estimate the IRR using the rule of 72, then check it exactly with nothing more than a basic calculator.Returns mathsCoreWarburg PincusNew York · 2012
Try it first
What does the rule of 72 give, and is the true IRR higher or lower?
Show the worked solution
The rule of 72 gives 36%; the exact IRR is about 41.4%. Eight times is three doublings, so the money doubles every two years, and 72 divided by 2 is 36. The exact rate is 8 to the power one sixth, which is the square root of 2, so the IRR is 1.414 minus 1, 41.4%. The rule runs low because it is tuned for rates near 8%.
How does the rule of 72 apply to a multiple bigger than 2?
Think of folding a sheet of paper: each fold doubles the thickness, so eight layers is three folds. Any money multiple that is a power of 2 is a count of doublings, and once you know the years per doubling, the rule of 72 gives the rate. Eight is 2 x 2 x 2, so six years hold three doublings of two years each. The rule says a doubling every two years needs 72 over 2, which is 36% a year.
Eight times in six years is three doublings of two years each, which the rule of 72 turns into 36% a year; the exact rate is 41.4%, and 36% compounded for six years only reaches 6.33x.
How do you get the exact answer with a basic calculator?
You need the sixth root of 8. Split it: the sixth root is the square root of the cube root, and the cube root of 8 is 2, so the answer is the square root of 2. One press of the square root key gives 1.414, an IRR of 41.4%. Without a square root key, use trial and error: 1.4 squared is 1.96, and 1.96 cubed is about 7.53, a little short of 8, while 1.42 to the sixth is about 8.20, a little over. The answer sits between 40% and 42%.
The relationship
$(1+r)^6 = 8 \;\Rightarrow\; 1 + r = 8^{1/6} = \sqrt[3]{8}^{\,1/2} = \sqrt{2} = 1.414$
r the IRR
8 the money multiple
6 years held
What it says in wordsThe IRR is the sixth root of the multiple, less one; here that is the square root of two, less one.
Then say why the rule ran low. The rule of 72 is built around rates near 8%; the higher the rate, the bigger the number you should divide into. For a two-year doubling the exact rate implies a numerator of about 83, not 72. Checking the shortfall out loud, 36% for six years gives only 6.33x, shows the interviewer you know the rule's limit.
Where candidates lose it
The most common slip is dividing 72 by the six years and answering 12%, as if 8x were a single doubling. Count the doublings first: three.
The second loss is giving 36% as the final answer. The interviewer who says use the rule of 72 is often waiting to hear that the rule understates at high rates, and that the exact answer is the square root of 2 minus 1.
What the interviewer asks next
What IRR does 3x in 5 years give, by rule of thumb and exactly?
4x in 6 years: rule of 72 and exact?
Why does the rule of 72 understate at high rates and overstate at very low ones?
Asked at Warburg Pincus, Private Equity, New York, 2012 (Wall Street Oasis): If I make 8 times my money in 6 years, what's my IRR? You have to use the rule of 72 to figure this out.
091Deal A returns 3x in 7 years. Deal B returns 2x in 3 years. Which has the higher IRR? Then show that if B's proceeds can only be reinvested to earn 1.5x over the remaining 4 years, both end at 3x after 7 years.Returns mathsHardLarge-cap buyout fundSecondaries and fund of funds
Try it first
Which deal has the higher IRR?
Show the worked solution
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%.
Deal A compounds steadily at 17.0% to 3x in year 7, while deal B reaches 2x in year 3 at 26.0% but, reinvested at only 10.7% a year, also ends at 3x in year 7, so its higher IRR adds no wealth.
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.
The relationship
$2 \times 1.5 = 3.0\times \qquad 1.5^{1/4} - 1 = 10.7\% \qquad 3^{1/7} - 1 = 17.0\%$
2 B's money multiple at year 3
1.5 what the proceeds earn over the next four years
3^(1/7) the seven-year rate both deals deliver
What it says in wordsB's two stages multiply to the same 3x over seven years as A, so the seven-year rate is identical.
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.
Where candidates lose it
Candidates pick A because 3x sounds better than 2x, confusing size with speed. Or they pick B on IRR alone and never ask what happens to the cash after year 3.
The point of the second half is reinvestment. Say the condition: B is better only if its proceeds can be redeployed above about 10.7% a year for the remaining four years.
What the interviewer asks next
What reinvestment multiple makes B end at 3.5x after seven years?
Why do secondaries buyers often care more about the money multiple than the IRR?
How can a fund raise its reported IRR without raising its money multiple?
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?
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?

MOIC	multiple on invested capital, money back divided by money in
n	years the money is invested, here 5

2^{1/3}	the yearly growth factor for doubling in three years
4^{1/6}	the yearly growth factor for quadrupling in six years

r	the IRR, the rate that makes the discounted inflows equal the investment
60	the recap dividend in year 2
180	the exit proceeds in year 5

1.25^5	the exit multiple at 25% a year for five years, 3.05x
1.20^5	the exit multiple at 20% a year, 2.49x

Try it first

How do you compound 1.2 five times in your head?

Why is the answer not 30%?

What table is worth memorising before a buyout interview?

Where candidates lose it

What the interviewer asks next

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Why does a bigger multiple not mean a better IRR?

How do you show it in two lines?

So is the slipped exit just as good?

Where candidates lose it

What the interviewer asks next

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Why does the same multiple give a different IRR?

How do you get the recap IRR without a spreadsheet?

Where candidates lose it

What the interviewer asks next

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Why is the answer not five per cent?

How much more enterprise value does that take?

Where candidates lose it

What the interviewer asks next

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Why does the higher IRR make less money?

So which would an LP prefer?

Where candidates lose it

What the interviewer asks next

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Why can a bigger multiple mean a lower IRR?

How would a fund actually weigh the choice?

Where candidates lose it

What the interviewer asks next

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How does the rule of 72 apply to a multiple bigger than 2?

How do you get the exact answer with a basic calculator?

Where candidates lose it

What the interviewer asks next

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How do you get the two IRRs in your head?

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

Where candidates lose it

What the interviewer asks next

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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

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