Hedge Funds puzzles, solved step by step

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  1. 097A fund makes 50% in year one on Rs 100 crore, then takes in Rs 400 crore of new money and loses 20% in year two. What are its time-weighted and money-weighted returns, and did its investors make money?Returns, compounding and feesHardFund of funds and allocatorsMulti-manager platforms

    Try it first

    The fund reports its two-year return. Which statement is right?

    Show the worked solution

    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.

    The 50% gain was earned on Rs 100 crore; the 20% loss hit Rs 550 crore100Start150End of year 1550After inflow440End of year 2+400 new+50%-20%Rs croreManager's recordtime-weighted1.5 x 0.8 = 1.20+20% over two yearsInvestors' resultmoney-weightedput in 500, hold 440lost Rs 60 croreabout -10.2% a year
    The fund earns 50% on Rs 100 crore, takes in Rs 400 crore, then loses 20% of Rs 550 crore, so the manager's time-weighted record is +20% while investors put in Rs 500 crore, hold Rs 440 crore, and earned about -10.2% a year.

    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.

    The relationship
    100(1+r)2+400(1+r)=440  ⇒  1+r=−4+33.62=0.898100(1+r)^2 + 400(1+r) = 440 \;\Rightarrow\; 1 + r = \frac{-4 + \sqrt{33.6}}{2} = 0.898
    100Rs crore invested at the start
    400Rs crore added after one year
    440Rs crore the investors hold at the end
    rthe money-weighted return, the internal rate of return on those flows
    What it says in wordsThe money-weighted return is the one rate that grows every rupee invested, from the day it arrived, into what the investors hold at the end.

    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.

    Where candidates lose it

    The common loss is reporting +20% and saying the investors made money. The chained return deliberately ignores that most rupees arrived just before the loss, and the interviewer built the numbers so the two measures point in opposite directions.

    The second loss is averaging the two yearly returns, +50% and -20%, to get +15% a year. That is neither measure: the time-weighted annual figure is the square root of 1.2 minus 1, about 9.5% a year.

    What the interviewer asks next

    • What is the time-weighted return expressed as an annual rate?
    • The Rs 400 crore had arrived at the start of year one instead. What are both returns now?
    • Why do fund fact sheets report time-weighted returns?
  2. 099You borrow in yen at 1% and invest in rupee deposits at 7% for a year. By how much can the rupee's value in yen fall over the year before the trade loses money?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds

    Try it first

    How big a fall in the rupee can the trade absorb?

    Show the worked solution

    About 5.6%. Borrow 1 of yen value, convert it, and the rupee deposit grows to 1.07 in yen terms at today's rate. You owe 1.01. If the rupee's yen value falls by d, the deposit is worth 1.07 x (1 - d), which equals 1.01 when d = 1 - 1.01/1.07, about 5.6%. The cushion is slightly below the 6-point gap because the fall hits the interest as well as the principal.

    Where does the carry trade's income come from?

    Think of borrowing from a relative at 1% to put the money in a deposit paying 7%, except that the relative wants to be repaid in a different currency. The 6 points look like easy income until the currency moves. A carry trade earns the interest gap only if the currency you hold does not fall by more than about that gap over the holding period. The rate gap is the cushion; the exchange rate is the risk.

    The 6-point rate gap cushions a rupee fall of only about 5.6%breakeven: 5.6% fall1.07 x (1 - d) = 1.01a 6% fall already loses Rs 0.42no fall: +Rs 610% fall: -Rs 4.700%2%4%6%8%10%Fall in the rupee's value in yen over the year-40+4Profit per Rs 100 borrowed
    Per Rs 100 of yen borrowed, the trade makes Rs 6 if the rupee holds its value, breaks even at a 5.6% fall, already loses Rs 0.42 at a 6% fall, and loses Rs 4.70 at a 10% fall.

    Why is the cushion 5.6% and not 6%?

    Work in yen per unit borrowed. At the end you hold 1.07 of rupee deposit, measured at today's exchange rate, and you owe 1.01 of yen; the rupee can lose value until 1.07 x (1 - d) = 1.01, so d = 5.6%. The fall applies to the whole 1.07, including the 0.07 of interest, which is why the cushion is a little smaller than 6. Borrow Rs 100 crore worth of yen and a 5.6% fall leaves you exactly even; a 10% fall loses about Rs 4.7 crore.

    The relationship
    1.07 (1−d)=1.01  ⇒  d=1−1.011.07=5.6%1.07\,(1 - d) = 1.01 \;\Rightarrow\; d = 1 - \frac{1.01}{1.07} = 5.6\%
    1.07what each unit becomes in the rupee deposit at 7%
    1.01what you owe on each unit of yen borrowed at 1%
    dthe fall in the rupee's value in yen over the year
    What it says in wordsThe trade breaks even when the grown rupee deposit, after the currency fall, buys back exactly the yen you owe.

    What does the forward market say about this?

    Covered interest parityThe rule that the forward exchange rate must offset the interest rate gap between two currencies, or else borrowing in one, lending in the other and locking in the forward would be an arbitrage. sets the one-year forward so that the rupee buys about 5.6% fewer yen forward than today, exactly the fall that wipes out the carry. So an unhedged carry trade is a bet that the rupee will fall by less than the forward already implies. The limitation to state: carry trades tend to earn small amounts steadily and then lose sharply when many holders unwind at once, so the spread of outcomes has a fat left tail that the breakeven alone does not show.

    Where candidates lose it

    The quick answer is 6%, the gap between the two rates. It misses that the currency loss applies to the interest earned as well as to the principal, so the true cushion is slightly smaller, and at exactly a 6% fall the trade is already losing.

    The second loss is treating the 6-point gap as a return and forgetting the currency altogether. Name the exchange rate as the risk before doing any arithmetic.

    What the interviewer asks next

    • The rupee weakens 10% against the yen over the year. What is the loss on Rs 100 crore of yen borrowed?
    • What one-year forward rate does covered interest parity imply, and what does it say about the trade?
    • Why do carry trades tend to lose money suddenly rather than gradually?