Private Equity puzzles, solved step by step

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

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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
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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?
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?

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

r_nom	the rupee return, 7%
pi	inflation, 5%
r_real	the real return, growth in what the money buys

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

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 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