Hedge Funds puzzles, solved step by step

Investor A pays the fee on 10 per unit, a fee of 2; investor B pays it on 30, a fee of 6. Each high-water mark is the highest value that investor's own units have reached: 100 for A, who lived through the fall, and 80 for B, who bought at the bottom. The fund made 37.5% in year two for both of them, yet B pays three times A's fee, because every point B made is new profit for B.

Why does the same year produce two different fees?

Two shopkeepers pay a helper a bonus only when monthly sales beat their own best month so far. One had a record month last year and a slump since; the other opened last month. The same good month earns the helper a bonus from the new shop and little or nothing from the old one. A high-water mark is personal: it is the peak value of that investor's own units, so it depends on when they came in. A bought at 100 and watched it fall to 80, so the rise back to 100 only repairs A's loss. B bought at 80, so the whole rise to 110 is new money for B.

How do funds keep the two investors apart?

If the fund kept one NAV and one mark for everyone, either A would be charged on a recovery or B would ride free on 20 points of profit. Funds solve this with per-investor accounting: a separate series of units for each subscription date, or equalisation adjustments, so each investor pays on their own gain and nobody else's. Series are the easier version to explain in the room: B's units are a new series that starts life with a mark of 80. The limitation is worth a line: not every fund does this, so an allocator reads the offering document before assuming it.

What does this do to the manager's incentives?

A manager whose older investors sit below their marks earns no performance fee on them until the loss is repaid. For an allocator, a high-water mark is a fee holiday on the recovery, and it belongs only to the investors who stayed through the loss. For a manager deep under water it can mean years of work for no incentive fee, which is why some funds in that position close and relaunch rather than climb back to their marks, and why allocators ask about it.

About 8% a year, two points below the 10% average. Compound growth is roughly the average return minus half the variance: 10% minus 0.5 x 0.20 squared, which is 10% minus 2%. Check with two years of +30% and -10%: the average is 10% and the volatility 20%, but 1.30 x 0.90 is 1.17, which compounds at 8.17% a year. The drag grows with the square of volatility.

Why does the average return overstate what you end up with?

A shopkeeper whose sales rise 50% one month and fall 50% the next has not broken even: 100 becomes 150 and then 75. A loss is applied to a bigger base after a gain, and a gain to a smaller base after a loss, so swings always drag compound growth below the simple average. The bigger the swings, the bigger the drag. The simple average of yearly returns is called the {term('arithmetic mean', 'The plain average of the yearly returns, adding them up and dividing by the number of years.')}; the rate your money actually grows at is the geometric mean, and it is always the lower of the two when returns vary.

How big is the drag, and where does half the variance come from?

For returns that are not too large, the geometric mean is close to the arithmetic mean minus half the variance. With 20% volatility the variance is 0.04, half of that is 0.02, so a 10% average compounds at about 8%. The two-year example makes it concrete: +30% and -10% average 10% with a standard deviation of 20%, and 1.30 x 0.90 = 1.17, a compound rate of 8.17% a year. The rule of thumb says 8.00%, close enough to trust in an interview.

Add why an allocator asks this. Two funds with the same average return and different volatility do not leave investors with the same money. Cutting volatility from 20% to 10% raises compound growth by 1.5 points with no change in the average, which is part of why lower-volatility strategies can be worth more than their averages suggest. The limitation: the half-variance rule is an approximation that weakens for very volatile or fat-tailed returns.

The index ends down 1% and the 2x fund down 4%, not 2%. The index goes 100, 110, 99. The fund goes 100, 120, 96, because it doubles each day's move: up 20%, then down 20% of a bigger number. Two days of index returns of plus and minus r cost the index r squared, 1%, but cost the 2x fund 4r squared, 4%. The drag comes from resetting leverage daily, and it grows with volatility.

Why is the index down at all after up 10% and down 10%?

A salary cut of 10% after a 10% raise leaves you below where you started, because the cut is taken on the bigger salary. An up move of r followed by a down move of r multiplies to (1 + r)(1 - r) = 1 - r squared, so the index loses r squared, here 1%. This is the same volatility drag that makes compound returns lower than average returns. It is small for the index because r squared is small.

Why does the 2x fund lose four times as much?

Because it resets its leverage each day. The fund multiplies to (1 + 2r)(1 - 2r) = 1 - 4r squared, so its drag is four times the index's, not two times. With r = 10% that is 4%. A buyer expecting twice the index return over the period expected 2 x (-1%) = -2% and got -4%. The extra 2 points is the price of re-levering after the gain, which buys more exposure at the top, and de-levering after the loss.

Say where the drag does not apply. In a steady trend, daily resetting helps: two days of +10% take the index to 121 and the 2x fund to 144, more than the naive 142. Daily-reset leveraged funds win in trends and lose in chop, so holding one for months is a bet on the path, not just the direction. That is why allocators treat them as trading tools rather than long-term holdings.

1.5% a month compounds to about 19.6% a year, and 40% over three years is about 11.9% a year. Compounding multiplies growth factors rather than adding rates: 1.015 to the twelfth is 1.196. Going the other way, take the cube root of 1.40, which is 1.119. Simple arithmetic gives 18% and 13.3%, understating the first answer and overstating the second.

Why is twelve times 1.5% not the annual return?

A savings account that credits interest every month pays interest on last month's interest. Each month's 1.5% is earned on a base that already includes every earlier month's gain, so the growth factors multiply: 1.015 times itself twelve times. That is 1.196, a 19.6% year. The extra 1.6 points over 18% are interest on interest, tiny in any one month and not tiny over a year.

How do you go back from a total to an annual rate?

Take the root, not the division. A 40% total over three years means the yearly growth factor cubed is 1.40, so the factor is the cube root of 1.40, about 1.119, an annual rate of 11.9%. Dividing 40 by 3 gives 13.3%, and 1.133 cubed is 1.456, a 45.6% total: division overstates the yearly rate because later years grow on earlier gains. Allocators compare managers on the compound annual growth rateThe single yearly rate that, compounded over the period, turns the starting value into the ending value., so dividing can misrank two funds.

How do you do it in your head?

Add the square-term correction. Compounding adds roughly n(n - 1)/2 times r squared to n times r: for 12 months at 1.5% that is 66 x 0.000225, about 1.5 points, taking 18% to about 19.5%. That is close enough to show you know the direction and the size. For the cube root, guess and check: 1.12 cubed is about 1.405, a shade over 1.40, so the answer sits just under 12%. Checking by cubing is faster and safer than estimating a root directly.

About 31% of the ending wealth. Rs 1 lakh growing at 10% a year becomes Rs 6.73 lakh in 20 years; at 8% after the fee it becomes Rs 4.66 lakh. The fee takes Rs 2.07 lakh, about 31% of what the investor would otherwise have had, although each year it looks like only a fifth of the return.

Why does a fifth of each year's return become almost a third of the wealth?

Think of a mango tree whose fruit you replant. If a neighbour takes one mango in every five each year, you lose those mangoes and every tree they would have grown. A fee taken every year removes money that would itself have compounded for the remaining years, so its cost grows faster than the fee rate suggests. The share lost is 1 minus (1.08/1.10) to the 20th: each year the investor keeps 98.2% of what the gross path would have held, and 0.982 to the 20th is about 0.69.

How do you estimate it without a calculator?

Use the rule of 72. At 10% money doubles about every 7.2 years, so 20 years is about 2.8 doublings, near 7 times; at 8% it doubles every 9 years, about 2.2 doublings, near 4.6 times. The exact numbers are 6.73 and 4.66, so the estimate lands within a few per cent, and 4.6 over 6.9 already tells you roughly a third is gone. Say the estimate first, then the exact figure.

What does an allocator take from this?

That fees should be judged against the return they leave, over the holding period, not as a percentage in a single year. A manager charging 2% needs to beat a cheaper alternative by about 2 points a year, every year, just to leave the investor level; over 20 years the difference is 31% of the pot. Hedge fund fees often add a share of profits on top, which widens the gap further. The illustration assumes a steady 10%; real returns vary, but the compounding of the fee does not depend on that.

About 27.3%. At 2 and 20 the manager earns Rs 20 crore of management fee and 20% of the remaining Rs 100 crore gain, Rs 20 crore: Rs 40 crore in all. At 1%, the management fee is Rs 10 crore and the gain after it is Rs 110 crore. To keep Rs 40 crore the performance fee must bring in Rs 30 crore, which is 30/110, or 27.3%. The two deals match only at a 12% gross return.

What does the manager earn today?

Think of a tailor who charges a fixed stitching fee plus a share of whatever the finished suit sells for above cost. Cut the fixed fee and the share must rise to keep the same income, but the share now applies to a slightly larger base. At 2 and 20 on Rs 1,000 crore earning 12%, the manager takes Rs 20 crore of management fee plus 20% of the Rs 100 crore gain left after it, Rs 40 crore in total. Investors keep Rs 80 crore, a net return of 8%.

How do you find the new performance fee?

Keep the total at Rs 40 crore. The management fee falls to Rs 10 crore, so the performance fee must bring in Rs 30 crore, and it is charged on a gain of Rs 110 crore, not Rs 100 crore. 30 divided by 110 is 27.3%. The base grows because less has been taken off the top before the performance fee is worked out. State the assumptions as you go: no hurdle rate, and no earlier losses to recover below a high-water markThe highest value an investor has paid a performance fee on; no new performance fee is charged until the fund climbs back above it..

Is the new deal really the same for investors?

Only at a 12% gross return. The new deal pays the manager less in poor years and more in good ones, so it moves risk from the investors to the manager. At a 4% gross return the old deal pays Rs 24 crore and the new one Rs 18.2 crore; at 20% the old pays Rs 56 crore and the new one Rs 61.8 crore. That is why allocators push for a lower fixed fee even at the price of a higher share: they would rather pay for performance than for size.

The time-weighted return is +20%; the money-weighted return is about -10.2% a year, and investors as a group lost Rs 60 crore. Rs 100 crore grows to Rs 150 crore, then Rs 400 crore arrives and the Rs 550 crore falls 20% to Rs 440 crore. Chaining 1.5 x 0.8 gives +20%, the manager's record. But Rs 500 crore went in and Rs 440 crore remains, because most of the money arrived just before the loss.

What does each measure answer?

Think of a restaurant that gets a glowing review after a good year, triples its tables, and then has a poor year. Its record over the two years looks fine, but most of the diners ate in the poor year. The time-weighted return measures the manager, chaining each period's return so the timing of money coming in or out does not count; the money-weighted return measures the investors, weighting each period by the money actually exposed to it.

How do you compute the money-weighted return?

Treat the investors' money as a set of cash flows and find the single annual rate that links them: Rs 100 crore in at the start, Rs 400 crore in after a year, Rs 440 crore out at the end. Solve 100(1 + r) squared + 400(1 + r) = 440; the root is 1 + r = 0.8983, so the money-weighted return is about -10.2% a year. It is negative because most of the money sat through the losing year: Rs 550 crore lost 20%, Rs 110 crore, while the good year earned only Rs 50 crore on Rs 100 crore.

Which number should an allocator look at?

Both, for different questions. To judge the manager's skill, use the time-weighted +20%, because the manager did not choose when investors arrived. To judge whether the fund was good for the people in it, or whether money chased performance, use the money-weighted figure. A wide gap between the two, as here, is a warning about hot money: investors piling in after a strong year and bearing the next year's loss on a much bigger base. Had all Rs 500 crore arrived at the start, both measures would agree and the investors would hold Rs 600 crore.