Mutual Fund Mastery puzzles, solved step by step

She holds 10,000 main units at Rs 23.75, worth Rs 2,37,500, plus 10,000 segregated units valued at zero. The fund has 32 crore units; Rs 760 crore of healthy bonds over those units gives Rs 23.75. If 60% of the bond comes back, Rs 24 crore is spread over 32 crore segregated units, Rs 0.75 each, so she receives Rs 7,500. Her total is Rs 2,45,000 against Rs 2,50,000 before.

Why split the fund instead of simply marking the bond to zero?

Picture a housing society that lends money to a builder who then stops paying. If the society writes the loan off and lets members leave with their share of what is left, the members who leave first give up their slice of any money later recovered, and the ones who stay collect it all. A plain write-down hands the value of a future recovery to whoever is still in the fund when the cash arrives, not to whoever owned the fund when the loss happened. A segregated portfolioA separate pool, created on a credit event, that holds only the troubled bond. Everyone who held the fund that day gets matching units in it. fixes that by giving every holder on the day a separate, frozen claim on the bad bond.

Here the fund has Rs 800 crore at a NAV of Rs 25, so it has 32 crore units. The bond of Rs 40 crore moves out; Rs 760 crore of healthy bonds stays behind. Divide by the same 32 crore units and the main NAV is Rs 23.75. Her 10,000 units there are worth Rs 2,37,500, and she also receives 10,000 segregated units, valued at nothing today.

What does the recovery pay, and who would have got it without segregation?

A 60% recovery brings back Rs 24 crore. Spread across 32 crore segregated units it is Rs 0.75 a unit, so she receives Rs 7,500 whether or not she has since redeemed her main units. Without segregation, the same Rs 24 crore would land on whoever held the single fund on recovery day. Suppose half the units redeem at Rs 23.75 after the write-down: the recovery then lands on only 16 crore units, Rs 1.50 each. The holders who stayed collect Rs 15,000 per 10,000 units, double their fair share, and the holders who left get nothing.

Say the limit too. Segregation does not reduce the loss; she is still Rs 5,000 down on Rs 2,50,000. It only makes sure the loss and any recovery fall on the same people. Indian rules allow it only after a defined credit event such as a rating downgrade, with conditions set in SEBI circulars that should be checked before quoting them.

About 18.2x. Enterprise value is 10 x 200, Rs 2,000 crore. Take off Rs 500 crore of net debt and shareholders own Rs 1,500 crore. Earnings for shareholders are EBITDA of 200 less depreciation of 40 and interest of 50, which is 110 before tax and 82.5 after 25% tax. Rs 1,500 crore over Rs 82.5 crore is 18.2x.

Why can you not read the P/E straight off the EV multiple?

Think of a house worth Rs 1 crore with a Rs 40 lakh home loan. The owner's stake is Rs 60 lakh, and the rent left for the owner is the rent after the loan interest is paid. An EV multiple compares the whole business with profit before lenders and the tax office are paid; a P/E compares the shareholders' slice with the profit left for them. So you must take both the value and the earnings down to the shareholder line before you divide. Converting one side and not the other is the usual slip.

Why does the P/E come out higher than the EV multiple here?

Both sides shrink on the way down, but not by the same share. Value falls from 2,000 to 1,500, a quarter lost to lenders. Earnings fall from 200 to 82.5, well over half lost to depreciation, interest and tax, so the denominator shrinks faster and the multiple rises. Heavy depreciation, heavy interest or a high tax rate all push the P/E above the EV/EBITDA; a debt-free, low capex business has the two much closer together.

A quick check catches the common half-conversion. If you remember the tax but forget the interest, earnings are 160 x 0.75, which is 120; divide the full EV of 2,000 by that and you get 16.7x, a number that mixes the whole business with a shareholders' profit line. State the assumption that net debt is the only claim between EV and equity: minorities or preference capital would sit there too.

The best rule shows about 3.7% of alpha in the backtest, and you should expect about zero from it afterwards. The largest of 20 normal draws sits about 1.87 standard deviations above the mean, and 2% noise times 1.87 is 3.7%. Picking the winner selects the luckiest draw, and luck does not repeat, so its expected alpha out of sample is the true alpha of every rule: zero.

Why does the best rule look good when none of them is?

Ask twenty people to toss a coin ten times and one of them will probably get eight heads. Nobody concludes that person has a talent for heads. When you choose the best of many attempts, you are choosing the luckiest draw, and the more attempts you make, the luckier the winner looks. One rule on its own shows more than 2% alpha only about one time in six. Among 20 rules, the best one shows more than 3% about 75% of the time.

What should you expect out of sample, and how would you check a real rule?

Out of sample the noise is drawn again, fresh, and the rule has no skill, so its expected alpha is zero. The gap between 3.7% and zero is the price of searching, sometimes called selection biasThe distortion that comes from reporting the best of many tries as if it were the only try. The winner looks better than its true quality.. The honest checks follow from that. Hold back data the rules never saw and test only the winner on it. Raise the bar with the number of rules tried: a one-rule test might accept 2 standard deviations, but after 20 tries the best result is expected to reach 1.87 on luck alone. Ask whether the rule has an economic reason to work.

This is also why fund ranking tables are a weak guide on their own. A category with 20 funds and no skill still produces a fund that beat its peers by about 3.7% a year over the backtest window, and the marketing for that fund writes itself. The limit of the arithmetic: real rules are correlated, which shrinks the effective number of tries and the size of the winner's luck.

The 10-year zero, with a duration of 10.0 years. A zero pays everything at maturity, so its duration equals its maturity. The 15-year 12% bond has a duration of about 8.6 years, because its large coupons bring value forward, and the 10-year 9% bond is about 7.1. The zero moves most when rates move, even though it is not the longest bond.

Why is maturity the wrong ruler?

Think of two loans to a friend. One friend repays everything in one go after ten years. The other repays a large slice every year for fifteen years. Most of the second loan is back in your hands long before the first friend pays a rupee. Duration measures when, on average, a bond's value comes back to you, weighted by the present value of each payment, and rate risk follows that average, not the final date. A zero returns all of its value at one date, so its duration is exactly its maturity.

How do you rank them without a calculator?

Use two rules and one check. A zero's duration is its maturity. A coupon bond's duration is always below its maturity, and the higher the coupon, the further below. So the only real contest is between the zero at 10 and the 15-year bond, and a 12% coupon pulls hard: about 60% of that bond's present value arrives in the first ten years. The 10-year 9% bond cannot beat the 10-year zero, because it has the same final date and pays some value earlier.

Add the sizing, because desks price risk in rupees. Modified durationMacaulay duration divided by one plus the yield. It gives the approximate percentage price change for a one point move in yield. is Macaulay duration over 1.08, so the zero loses about 9.3% of its price for a one point rise in yield and the 15-year bond about 7.9%. The limit: this is a small-move estimate, and it ignores the curve changing shape. Even a 30-year 12% bond only reaches about 11.5 years, because a coupon bond's duration can never exceed that of a perpetuity, 13.5 years at 8%.

The equity multiple is 1.45x and the IRR is about 8.7%. You get back five payments of Rs 21 and Rs 330 on sale, Rs 435 in all, on Rs 300 in. The IRR is the rate at which those flows are worth exactly Rs 300 today. The multiple counts rupees and ignores time, so the same Rs 435 received in one lump at year 10 is still 1.45x but only about 3.8% a year.

What does each measure actually count?

Lend a friend Rs 300 and get Rs 435 back. Whether it came back in five years or fifteen, you can say you made 1.45 times your money, but you would not call the two loans equally good. The equity multipleTotal cash received divided by cash invested. It counts rupees and ignores when they arrive. counts how many rupees come back; the IRR counts how fast they come back. Here total cash is 5 x 21 plus 330, Rs 435, and 435 over 300 is 1.45x.

How do you get the IRR quickly in the room?

Split it into income and growth. The payment of Rs 21 on Rs 300 is a 7% yield. The price rises from 300 to 330, 10% in five years, which compounds to just under 2% a year. Add them and you are near 9%; the exact answer is a little lower, 8.7%, because the price gain arrives only at the end. Then offer the check: at 8.7% the five payments and the sale discount back to Rs 300.

Now the comparison the interviewer wants. The same Rs 435 in a single payment at year 5 is 7.7% a year, lower than 8.7% only because the distributions no longer arrive early. At year 10 it is 3.8%. The limit of IRR: it assumes the early cash can be reinvested at the same rate, and it says nothing about size, so a 20% IRR on Rs 1 lakh for one month is not better than 9% on Rs 1 crore for five years.

About 0.48% a year, roughly half a percent. Turnover of 120% means the fund sells and rebuys 1.2 times its portfolio in a year. Each rupee replaced costs one sale and one purchase, 0.4% together, so the drag is 1.2 x 0.4%. That cost shows up as a lower NAV, never as a line in the expense ratio, so the investor pays it without seeing it.

Why does a cost this size not appear on the factsheet?

Think of a shopkeeper who keeps rearranging the stock. The rent is printed on the lease, but every time he returns goods and reorders, the supplier keeps a small handling margin, and that never appears on any bill he shows you. The expense ratioThe annual charge for running a fund: management fee, administration, distribution and similar costs, shown as a percentage of assets. is the rent. Trading costs are the handling margin: brokerage, taxes on trades and the impact costThe amount a price moves against a large buyer or seller while the order is being filled. It is paid through a worse price, never through a bill. of moving prices, all paid through the prices the fund gets, so they land in the NAV and not in the expense ratio. Which explicit trading charges may be loaded inside the expense ratio is set by regulation and should be checked for the market you are in; impact cost is never there.

How do you turn turnover into a cost without double counting?

Portfolio turnoverThe share of a portfolio replaced in a year, usually measured as the smaller of purchases or sales divided by average assets. counts how much of the book is replaced. At 120%, every rupee of assets is sold and rebought 1.2 times in the year. Because the 0.4% already covers both legs of a replacement, the cost is simply turnover times the round-trip cost: 1.2 x 0.4% = 0.48%. The common slip is to say a replacement has a buy and a sell and double it to 0.96%, counting each leg twice. A fund turning over 20% of its book pays only 0.08%.

Put rupees on it, because half a percent sounds small. Rs 10 lakh compounding for 20 years at 11.3% grows to about Rs 85.1 lakh; at 10.82% it grows to about Rs 78.0 lakh. The gap is about Rs 7.0 lakh, paid quietly. The limit of the estimate: impact cost depends on how liquid the stocks are and how large the fund is, so the same turnover costs a small-cap fund of Rs 20,000 crore far more than a large-cap fund of Rs 2,000 crore.

It is a 0.3 percentage point cut, which is 30 basis points, and a 20% relative cut in the fee; "a 0.3% cut" is the sloppy one. Basis points remove the ambiguity. The 20% is true but invites a client to imagine returns rising 20%; on Rs 10 lakh the saving is Rs 3,000 a year, and a net return of 8.5% becomes 8.8%.

Why are there two different answers that are both right?

A shop raises the price of milk from Rs 50 to Rs 60. It went up by Rs 10, and it went up by 20%. Nobody confuses those, because rupees and percent look different. With rates, the change and the base are both percentages, so "percent" can mean the gap between two rates or the change relative to the old rate, and you must say which. The gap between 1.5% and 1.2% is 0.3 percentage pointsThe plain difference between two percentages. From 1.5% to 1.2% is a fall of 0.3 percentage points.. Relative to the old fee, 0.3 is a fifth of 1.5, a 20% cut.

Which description will a client get wrong, and what should you say instead?

"A 0.3% cut" fails one way: a careful reader can take it as 0.3% of the fee, which would leave 1.5% x 0.997 = 1.4955%, a cut a hundred times too small. "A 20% cut" fails the other way: it is true of the fee, but a client hears 20% and imagines the investment doing 20% better. The fee is a slice of a slice; a 20% smaller slice lifts a net return of 8.5% to 8.8%, only 3.5% better in relative terms. The clean form for a desk is basis pointsHundredths of a percentage point. 30 basis points equal 0.30 percentage points.; the clean form for a client is rupees: Rs 15,000 a year on Rs 10 lakh becomes Rs 12,000.

The same confusion runs through everything a fund desk reports: a yield moving from 7.0% to 7.5% is 50 basis points, not 0.5%, and a fund outperforming by 2% might mean two points or a fiftieth more. The limit worth stating: neither unit tells the client the money, so pair any rate change with the rupee effect on their actual holding.

About 25% in any one year, and about 2% over ten years. One year: zero sits 12/18 = 0.67 standard deviations below the mean, which leaves 25.2% of outcomes below it. The ten-year average keeps the 12% mean but its spread falls to 18 divided by the square root of 10, 5.7%, so zero is 2.11 standard deviations away and the chance is 1.8%.

How likely is a losing year?

Picture the daily commute. Any one day might be twenty minutes late because of rain or a breakdown; your average over a month is almost never more than a few minutes off. Single outcomes are noisy; averages of many independent outcomes are much less noisy, because the bad days and the good days partly cancel. For one year, the question is how far zero sits below a 12% mean when the spread is 18%. That distance is 12/18 = 0.67 standard deviations, and the normal table puts 25.2% of the curve below it. One year in four is a loss, which matches what investors in equity funds actually live through.

Why does the ten-year chance collapse to about two percent?

Averaging ten independent years keeps the centre at 12% but divides the spread by the square root of ten. The standard errorThe spread of an average. For independent draws it equals the spread of one draw divided by the square root of the number of draws. of the ten-year average is 18 / 3.16 = 5.7%, so zero is now 2.11 standard deviations below the mean instead of 0.67. The tail beyond 2.1 standard deviations is 1.8%. Five years sits in between: a spread of 8.0% and a chance of about 7%.

Now the limits, because a sharp interviewer will push. A negative arithmetic average is not quite the same as losing money: compounding knocks roughly half the variance off growth, so the fund compounds nearer 10.4% than 12%, and the chance of ending ten years below the starting amount is a little higher than 1.8%. Real returns also have fatter tails than a normal curve and are not fully independent, since bad years cluster. The direction survives all of that: time narrows the spread of the average, not the risk of a bad single year.

You are indifferent at about 7.18% a year. Rs 50 lakh at 50 matches Rs 1 crore at 60 only if it doubles in ten years, so (1 + r) to the 10th = 2, which gives 7.18%. The rule of 72 says money doubles in 72 / rate years, so doubling in 10 years needs about 72 / 10 = 7.2%. At 8% the early money grows to Rs 107.9 lakh; at 7% only Rs 98.4 lakh.

What is the question really asking?

An uncle offers you his old car now or a new one in ten years. You cannot answer until you know what you would do with the car meanwhile. Money received earlier can be invested, so the fair comparison is what the Rs 50 lakh grows to by 60, set against the Rs 1 crore paid then. At 8%, Rs 50 lakh x 1.08 to the 10th is Rs 107.9 lakh, more than a crore, so the early money wins. At 7%, it is Rs 98.4 lakh, just short, so the crore wins. The answer flips somewhere between.

Why is the crossover the rule of 72?

The crore is exactly twice the 50 lakh, and the wait is ten years. So the choice reduces to one question: can you double your money in ten years? The rate that does that is 7.18%, and the rule of 72A shortcut for compounding: money doubles in roughly 72 divided by the yearly rate in years, so 7.2% doubles in about ten years. gives 72 / 10 = 7.2% in your head. The rule works because the doubling time is ln 2 / ln(1 + r), and ln 2 is 0.693; using 72 instead of 69.3 corrects for the rates people usually quote being near 8%.

The arithmetic is not the whole decision, and the interviewer will want you to say so. The rate that matters is the after-tax rate you can actually earn, not a headline rate. Rs 1 crore promised at 60 carries the risk that the promiser does not pay; Rs 50 lakh in hand does not. And a person who needs the money at 50, for a child's education say, values it more than the arithmetic does.

About 0.89% each. Uncorrelated risks add in squares. Twenty positions of active risk s give a total variance of 20 s squared, which must equal 4 squared, 16. So s squared is 0.8 and s is 0.89%. The straight split of 4% over 20, 0.20%, would use under a quarter of the budget, because it assumes every position fails at the same time.

Why can each position carry more than a twentieth of the budget?

Think of twenty friends each guessing the weight of a cake. Each guess is off by about 100 grams, but in random directions, so the errors partly cancel and the total of their errors is nowhere near 2 kilograms. Independent errors grow with the square root of how many there are, not in proportion, because some push up while others push down. Tracking errorThe standard deviation of the gap between a fund return and its benchmark return, usually quoted per year. works the same way. Twenty uncorrelated bets of equal size s give a total of s x sqrt(20), about 4.47 s.

How do you solve it, and what happens if the bets are not independent?

Set the total equal to the budget and solve. 4 = s x sqrt(20), so s = 4 / 4.47 = 0.89%: each position may carry almost four and a half times the naive answer. The catch is the word uncorrelated. If every pair of bets has a correlation of 0.2, the variance picks up 20 x 19 cross terms, each worth 0.2 s squared. The total variance becomes 96 s squared and each position can carry only 0.41%. Correlated bets behave more like one large bet.

This is why risk teams care about the true number of independent bets more than the number of line items. Twenty stocks that are all quietly a bet on falling interest rates are close to one position, and they would breach the budget at a fraction of the size the arithmetic above allows. The limit: tracking error is a one-number summary of normal-times behaviour, and correlations between bets tend to rise in a sell-off, exactly when the budget matters.