Private Equity puzzles, solved step by step

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

Showing 1–7 of 7 · filtered from 100Clear filters

009Which pays more over a year: 12% compounded annually, or 11.5% compounded monthly?Compounding and time valueCoreCarlyle GroupNew York · 2015
Try it first
Your pick?
Show the worked solution
11.5% compounded monthly pays more: an effective 12.13% against 12.00%. Monthly compounding pays 0.958% each month, and each payment then earns interest for the rest of the year. Over twelve months that interest on interest adds about 0.63 points to the 11.5% headline. Compare effective annual rates, never headline rates with different compounding.
Why can a lower headline rate pay more?
Imagine two jobs paying the same annual salary, one paid monthly and one in a single lump in December. The monthly earner can put January's pay in a deposit and earn on it for eleven months. The more often interest is paid, the sooner it starts earning interest of its own, so the effective rate rises above the headline rate. 11.5% paid monthly is 0.958% a month, and each month's interest joins the balance that earns the next month's.
Paid annually, 12% is an effective 12.00%, while 11.5% paid monthly becomes 12.13% once the 0.63 points of interest on interest are counted, so the lower headline rate pays 0.13 points more.
How do you estimate the extra without a calculator?
Use the second term of the expansion. Compounding r over n periods adds roughly r squared times (n minus 1) over 2n on top of r, which for 11.5% monthly is about 0.0132 x 11/24, or 0.61 points. So 11.5% plus about 0.6 is roughly 12.1%, enough to beat 12%. The exact figure is 12.13%. A quicker sanity check: continuous compounding is the ceiling, e to the 0.115, which is 12.19%, and monthly sits just below it.
The relationship
$r_{\text{eff}} = \left(1 + \frac{0.115}{12}\right)^{12} - 1 \approx 12.13\%$
0.115/12 the monthly rate, 0.958%
12 the number of compounding periods in a year
r_eff the effective annual rate, comparable across compounding conventions
What it says in wordsCompound the periodic rate for a full year to get a rate you can compare with an annual one.
Where does this show up on a private equity desk?
In debt terms and in returns reporting. Loan margins, PIK interest and preferred returns are quoted with different compounding conventions, and comparing them on headline rates is the same mistake as answering 12% here. A PIK note that compounds quarterly costs more than its headline suggests. Fund returns quoted as an IRR are already annual effective rates, which is why they can be compared across funds with different cash flow timing.
Where candidates lose it
The trap is answering 12% because it is the bigger number. The interviewer has set the gap at half a point precisely so that monthly compounding is just enough to close it.
The second miss is saying monthly wins by a lot. The edge is 0.13 points; give the size, not just the winner, and show how you estimated it.
What the interviewer asks next
What monthly-compounded rate is exactly equal to 12% annually?
How much does 11.5% compounded daily give?
A PIK note compounds at 12% quarterly. What is its effective annual cost?
Asked at Carlyle Group, Generalist, New York, 2015 (Wall Street Oasis): Some math brainteasers and accounting questions ranging from compounding rates to how an inventory purchase would flow
017A fund compounds at 15% a year before fees, and fees take 2% a year, so the investor compounds at 13%. Over 20 years, what share of the gross wealth do the fees take?Compounding and time valueCoreSecondaries and fund of funds
Try it first
Roughly what share of the end wealth do fees take?
Show the worked solution
About 30% of the gross wealth. At 15%, 1 rupee grows to 16.37 in 20 years; at 13% it grows to 11.52. The investor keeps 11.52 over 16.37, about 70%, so fees take about 30%. Each year the fee removes 1 less 1.13/1.15, about 1.7%, of the wealth, and twenty years of that compounds to a loss of nearly a third.
Why is the fee's share so much bigger than 2%?
Think of a water tank with a small leak. A leak of a cupful an hour sounds trivial, but over a day it empties a good part of the tank, and every cup lost is water that would otherwise have been there. A fee taken every year removes not just that year's money but everything that money would have earned in all the years after, so the loss compounds just like the return. The fee is 2 points of a 15% return, but its effect on the end wealth is far larger.
Compounding at 15% turns 1 into 16.37 over 20 years while 13% turns it into 11.52, so the shaded gap widens every year and the 2% fee takes 16% of the gross wealth by year 10 and 30% by year 20.
How do you get to 30% without a calculator?
Work with the ratio, not the two big numbers. Each year the investor keeps 1.13 over 1.15 of what the gross fund keeps, about 0.983, so over twenty years the investor keeps 0.983 to the twentieth. Use the shortcut that (1 minus x) to the n is roughly e to the minus nx: 20 times 0.0174 is 0.35, and e to the minus 0.35 is about 0.70. So the investor keeps about 70% and the fees take about 30%. Checking against the full numbers, 11.52 over 16.37 is 0.704.
The relationship
$1 - \left(\frac{1.13}{1.15}\right)^{20} \approx 1 - e^{-20 \times 0.0174} \approx 0.30$
1.13/1.15 the share of each year's gross growth the investor keeps
20 years of compounding
0.0174 the yearly share of wealth lost to the fee
What it says in wordsThe share lost to fees is one less the yearly keep-ratio compounded over the holding period.
Why does an LP care about this arithmetic?
Because fees look small as annual rates and large as outcomes. The longer capital stays invested, the larger the share of wealth that a fixed annual fee takes, so long holding periods and multi-layer structures, such as a fund of funds charging on top of underlying funds, deserve the closest look. Say the limitation: real private equity fees are charged on committed or invested capital rather than as a clean drag on returns, and carry is a separate deduction, so this is the shape of the effect rather than an exact fund calculation.
Where candidates lose it
The common answer is 2%, or 2 over 15, about 13%. Both treat the fee as a one-off slice rather than a deduction that compounds every year.
The second loss is trying to compute 1.15 to the twentieth in your head and running out of time. Work with the ratio of the two growth factors; it is one small number raised to a power.
What the interviewer asks next
What share do fees take over 10 years?
If a fund of funds adds 1% on top, what share of the gross wealth is left after 20 years?
Why does the same 2% fee take a smaller share when returns are 5% instead of 15%?
047Fund A compounds at 18% a year for 10 years. Fund B earns 25% a year for the first 5 years, then 10% a year for the next 5. Which ends with more money?Compounding and time valueCoreSecondaries and fund of funds
Try it first
Which fund ends ahead after ten years?
Show the worked solution
Fund A ends ahead: 5.23x against 4.91x. 1.18 to the tenth is about 5.23. For B, 1.25 to the fifth is about 3.05 and 1.1 to the fifth about 1.61, which multiply to 4.91. B leads at year five, 3.05x against 2.29x, but its slow second half lets A catch up around year 9. B's ten-year compound rate is 17.3%, below A's 18%.
Why does the fast starter lose?
A batsman who scores fast for 20 overs and then crawls can end below a partner who kept a steady rate all innings. Over a long period what matters is the compound rate across the whole span, and two halves at 25% and 10% compound to 17.3% a year, below a steady 18%. The early lead is real, but the second half compounds on a big base at a low rate.
Fund B's 25% start puts it at 3.05x after five years against A's 2.29x, but at 10% thereafter it is overtaken around year 9.1 and ends at 4.91x against A's 5.23x.
The relationship
$1.18^{10} \approx 5.23 \qquad 1.25^5 \times 1.10^5 \approx 3.05 \times 1.61 \approx 4.91$
1.18^10 Fund A's ten-year multiple
1.25^5 x 1.10^5 Fund B's two halves multiplied together
What it says in wordsMultiply the growth factors of each period; never average the rates.
Why is the average rate misleading here?
The simple average of 25% and 10% is 17.5%, but the compound rate is the geometric mean of the factors: the square root of 1.25 x 1.10, less one, which is 17.3%. The geometric mean is always below the simple average when rates differ, so a fund with uneven returns compounds more slowly than its average suggests. That gap widens the more the rates vary.
This matters to a secondaries or fund-of-funds buyer reading a track record. A fund that shows a strong early IRR may owe it to a few quick exits; what the LP takes home depends on the whole life. Say the limitation: real funds call and return capital over time, so their IRRs are not simple compound rates on one cheque.
Where candidates lose it
The common loss is backing B because it starts faster and its rates average almost the same. Averaging rates is the error: compounding multiplies factors, and the geometric mean sits below the arithmetic one.
The second loss is computing 1.25 to the fifth correctly and then adding the second half's growth instead of multiplying. Say each half as a factor, then multiply.
What the interviewer asks next
What rate in B's second half would make the two funds end level?
Why is a fund's early IRR often higher than its final IRR?
What is the geometric average of +50% and -50%, and why does that matter for volatile returns?
049EBITDA grows from 100 to 250 over 6 years. What is the compound annual growth rate?Compounding and time valueWarm upMid-market buyout fundIndian mid-market PE
Try it first
Pick the CAGR.
Show the worked solution
About 16.5% a year. CAGR is the ratio of end to start, 2.5, raised to one over the number of years, less one. 1.15 to the sixth is about 2.31 and 1.17 to the sixth about 2.57, so the rate sits about three quarters of the way between them, near 16.5%. Dividing the 150% gain by 6 gives 25%, which would compound to 381.
Why is 25% wrong?
A savings account that pays interest on interest grows faster each year, so it needs a lower rate than you might think to reach a target. CAGR is the single rate that, compounded every year, turns the start into the end, so it is the sixth root of 2.5, not 150% divided by six. 25% compounded for six years would reach 381, far past 250.
Compounding at 16.5% a year takes EBITDA from 100 to 250 in six years, while the 25% from dividing 150% by six would compound to 381; bracketing between 1.15 and 1.17 to the sixth finds the rate.
The relationship
$CAGR = \left(\frac{250}{100}\right)^{1/6} - 1 \approx 16.5\%$
250 / 100 the ratio of end value to start value, 2.5
1/6 one over the number of years
What it says in wordsThe compound growth rate is the ratio of end to start, rooted by the number of years, less one.
How do you find a sixth root in your head?
Bracket it with rates whose sixth powers you can build. 1.15 squared is 1.3225, cubed that is about 2.31. 1.17 squared is 1.3689, cubed that is about 2.57. 2.5 sits about 0.74 of the way from 2.31 to 2.57, so the rate is about 15% plus 0.74 of 2 points, near 16.5%. A cross-check with the rule of 72: at 16.5% money doubles in about 4.4 years, and 2.5x in 6 years is a little more than one doubling, which fits.
Say what the number hides. A CAGR smooths the path: a business could have been flat for four years and then jumped, and the CAGR would be the same. A buyout investor asks for the yearly figures before trusting the rate, and checks whether the 250 includes acquisitions.
Where candidates lose it
The common loss is dividing the total growth by the years and saying 25%. That is the average simple growth, and it overstates the compound rate badly over six years.
The second loss is freezing on the sixth root. You do not need logarithms; bracket the rate between two you can compute and slide.
What the interviewer asks next
What CAGR turns 100 into 300 over 5 years?
If EBITDA grew 40% in year one and was flat after, what is the CAGR over six years?
How would you strip acquired EBITDA out of the growth rate?
068An investment returns 7% a year in rupees and inflation runs at 5%. What is the real return, and how much more can the money actually buy after 10 years?Compounding and time valueCoreIndian mid-market PE
Try it first
What is the real return per year?
Show the worked solution
The real return is about 1.9% a year, and after 10 years the money buys about 1.21x what it did. Real growth is the money growth divided by the price growth: 1.07 / 1.05 = 1.0190. Over ten years money grows 1.97x while prices grow 1.63x, so purchasing power grows 1.21x. Subtracting 5% from 7% gives 2% and 1.22x, close but slightly high.
Why divide instead of subtract?
Think in plates of biryani. If a plate costs Rs 100 and you have Rs 100, you can buy one. A year later you have Rs 107 and a plate costs Rs 105, so you can buy 107 / 105 plates, about 1.019. Real return measures how many more things your money buys, which is a ratio of two growth factors, not a difference of two rates. The subtraction shortcut ignores that inflation also eats into the 7% you earned, which is why it comes out slightly high.
Over 10 years money growing at 7% reaches 1.97x while prices growing at 5% reach 1.63x, so the money buys 1.21x as much, a real return of 1.90% a year; subtracting the rates gives a slightly high 1.22x.
The relationship
$1 + r_{real} = \frac{1 + r_{nom}}{1 + \pi} = \frac{1.07}{1.05} = 1.0190$
r_nom the rupee return, 7%
pi inflation, 5%
r_real the real return, growth in what the money buys
What it says in wordsOne plus the real return equals one plus the money return divided by one plus inflation.
When does the shortcut stop being good enough?
At low rates the gap is small: here 1.90% against 2.00%, and 1.21x against 1.22x after a decade. The error grows with the size of the rates, so with 20% returns and 15% inflation the shortcut says 5% when the truth is about 4.3%. A fund comparing returns across countries with different inflation, or across decades, should divide. In an interview, give the shortcut first, then the exact figure, and say which you would use when.
Where candidates lose it
Answering exactly 2% is the common slip, said fast because subtraction feels natural. It is a fair approximation, but the question is checking whether you know it is one.
The second miss is on the ten-year part: compounding 2% and calling it the answer, or subtracting the ten-year totals, 1.97 minus 1.63, and getting 0.34. Divide the totals: 1.97 over 1.63 is 1.21x.
What the interviewer asks next
A fund returns 18% in rupees with inflation at 6%. What is its real return?
Why might an Indian fund's real return still beat a lower-nominal foreign fund?
What real return do you need to double purchasing power in 10 years?
075A dollar-based fund earns a 15% IRR in rupees on an Indian investment. Over the holding period the rupee weakens steadily, with the rupee price of a dollar rising 3% a year. What is the fund's IRR in dollars?Compounding and time valueCoreIndian mid-market PELarge-cap buyout fund
Try it first
What is the dollar IRR?
Show the worked solution
About 11.65%. In rupees the investment grows by a factor of 1.15 a year. Converting back, each dollar now costs 1.03 times as many rupees, so the dollar value grows by 1.15 divided by 1.03, which is 1.1165. Over five years that compounds to 2.01x in rupees but only 1.74x in dollars. Subtracting 3 from 15 gives 12%, a close but slightly low shortcut.
Why divide by the currency move rather than subtract it?
Picture a relative abroad who sends you dollars to invest in a rupee deposit. The deposit grows, but when you convert the money back, each dollar costs more rupees than before. The dollar investor's growth is the rupee growth divided by the rise in the rupee price of a dollar, because both are growth factors applied to the same money. It is the same arithmetic as turning a nominal return into a real one: a ratio, with subtraction as the rough version.
A 15% rupee return becomes about 11.65% in dollars when the rupee price of a dollar rises 3% a year, because 1.15 divided by 1.03 is 1.1165; over five years 2.01x in rupees is 1.74x in dollars.
The relationship
$1 + r_{\$} = \frac{1 + r_{Rs}}{1 + d} = \frac{1.15}{1.03} = 1.1165$
r_Rs the IRR in rupees, 15%
d the yearly rise in the rupee price of a dollar, 3%
r_$ the IRR measured in dollars
What it says in wordsOne plus the dollar return is one plus the rupee return divided by one plus the currency move.
Does it matter how the depreciation is quoted?
Yes, and saying so earns credit. If instead the rupee loses 3% of its dollar value each year, the factor is 0.97 rather than 1 over 1.03, and the dollar IRR is 1.15 x 0.97 minus 1, 11.55%. The two conventions differ by only a tenth of a point here, but state which one you are using before you compute. The bigger point for a foreign fund is the size of the drag: 3 points a year off a 15% rupee return is a fifth of the return, every year, before any fees.
Where candidates lose it
The common slip is subtracting, 15% minus 3% gives 12%, and stopping. It is a fair first estimate but slightly low; give it, then correct it with the ratio.
The second is applying the currency move only once, at exit, as if three years of depreciation were a single 3%. The rupee weakens every year of the hold, so the drag compounds just like the return.
What the interviewer asks next
What rupee IRR does a dollar fund need to earn 15% in dollars?
The rupee strengthens 2% a year instead. What is the dollar IRR?
How could the fund hedge the currency, and what would the hedge cost it?
082Money compounds at 9% a year. Roughly how long does it take to double by the rule of 72, and how close is that to the exact answer?Compounding and time valueWarm upMid-market buyout fund
Try it first
Answer inside five seconds.
Show the worked solution
About 8 years by the rule of 72, and 8.04 years exactly. Divide 72 by the rate in per cent: 72 over 9 is 8. The exact answer is the log of 2 over the log of 1.09. The rule is near exact around 8% a year and drifts at very low or very high rates, where 69 or 70 works better.
Why does dividing 72 by the rate work?
Doubling needs the growth factor to reach 2, and the log of 2 is about 0.693. For small rates, the log of 1 plus r is close to r, so doubling time is about 69.3 divided by the rate in per cent. 72 is used instead of 69 because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because it corrects for the approximation at the rates people use most. It is a calculator in your head, like knowing that a dozen eggs at Rs 6 each is Rs 72.
At 9% a year money crosses twice its starting value at 8.04 years against the rule of 72's 8, and across rates the rule stays within about a tenth of a year from 8% to 12% but drifts to 0.33 years at 4% and 0.22 years at 24%.
The relationship
$t = \frac{\ln 2}{\ln(1.09)} = \frac{0.693}{0.0862} \approx 8.04 \qquad \text{rule: } \frac{72}{9} = 8$
t years to double
ln 2 natural log of 2, about 0.693
ln(1.09) natural log of the growth factor, about 0.0862
What it says in wordsExact doubling time is log 2 over log of one plus the rate; the rule of 72 approximates it with simple division.
Where does a buyout interviewer use this?
Everywhere returns are quoted. A deal that doubles the money in about four years has an IRR near 18%, and one that doubles in about three years is near 24%, because 72 over 4 is 18 and 72 over 3 is 24. That lets you check a quoted IRR against a quoted money multiple in seconds. At 24% the rule says 3 years and the truth is about 3.2, so for high-return deals you shade the rule slightly.
Where candidates lose it
The common slip is using simple interest and saying about 11 years, 100 divided by 9. Compounding means each year's interest itself earns interest, so doubling comes sooner.
The other is giving 8 and stopping when asked how exact it is. Know that the rule is closest around 8% and that the true figure here is just over 8 years.
What the interviewer asks next
How long does it take money to triple at 9%?
A deal returns 2x in 3 years. Roughly what IRR is that?
Why does the rule of 72 overstate doubling time at low rates and understate it at high rates?

0.115/12	the monthly rate, 0.958%
12	the number of compounding periods in a year
r_eff	the effective annual rate, comparable across compounding conventions

1.13/1.15	the share of each year's gross growth the investor keeps
20	years of compounding
0.0174	the yearly share of wealth lost to the fee

1.18^10	Fund A's ten-year multiple
1.25^5 x 1.10^5	Fund B's two halves multiplied together

250 / 100	the ratio of end value to start value, 2.5
1/6	one over the number of years

Try it first

Why can a lower headline rate pay more?

How do you estimate the extra without a calculator?

Where does this show up on a private equity desk?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the fee's share so much bigger than 2%?

How do you get to 30% without a calculator?

Why does an LP care about this arithmetic?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the fast starter lose?

Why is the average rate misleading here?

Where candidates lose it

What the interviewer asks next

Try it first

Why is 25% wrong?

How do you find a sixth root in your head?

Where candidates lose it

What the interviewer asks next

Try it first

Why divide instead of subtract?

When does the shortcut stop being good enough?

Where candidates lose it

What the interviewer asks next

Try it first

Why divide by the currency move rather than subtract it?

Does it matter how the depreciation is quoted?

Where candidates lose it

What the interviewer asks next

Try it first

Why does dividing 72 by the rate work?

Where does a buyout interviewer use this?

Where candidates lose it

What the interviewer asks next