Mutual Fund Mastery puzzles, solved step by step

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.

Expected value says take the coin toss on the gain and the sure loss, the opposite of what most people do. The toss is worth Rs 60,000 against a sure Rs 50,000; on losses it costs Rs 60,000 against Rs 50,000. Combined, the popular pair gives +Rs 50,000 or -Rs 70,000, which is worse in every outcome than +Rs 70,000 or -Rs 50,000. In a portfolio the same pattern sells winners early and holds losers.

What does expected value say for each choice?

Weight each outcome by its probability and add. In choice 1 the coin toss is worth 0.5 x Rs 1,20,000 = Rs 60,000, Rs 10,000 more than the sure thing. In choice 2 the toss costs 0.5 x Rs 1,20,000 = Rs 60,000, Rs 10,000 worse than the sure loss. So a person who maximises expected value gambles on the gain and accepts the loss, and most people do the reverse. Taking a sure Rs 50,000 is defensible on its own if the person cannot afford to walk away with nothing; the oddity is the switch to gambling the moment the frame becomes a loss.

Why is the popular pair worse in every outcome, not just on average?

Put the two choices together, because in real life they arrive together. The popular pair, sure +50,000 plus the loss gamble, ends at +50,000 if the toss goes well and -70,000 if it goes badly. The expected value pair, the gain gamble plus the sure loss, ends at +70,000 or -50,000. The second pair beats the first by Rs 20,000 whichever way the coin falls, so the popular choice is not a matter of taste; it is a mistake caused by judging each choice in its own frame. This is the pattern Kahneman and Tversky described in prospect theoryA description of how people actually choose under risk: they judge outcomes as gains or losses from a reference point and feel losses more sharply than equal gains..

Now the portfolio. A stock up 30% is a gain, so the investor wants the sure thing and sells. A stock down 30% is a loss, so the investor prefers to gamble on a recovery and holds. This is the disposition effectThe habit of selling investments that have risen too early and holding investments that have fallen too long., and it leaves a portfolio full of its weakest holdings while the strongest have been cashed out. The same pull shows up in fund redemptions: investors exit a fund that has done well to lock in the gain and stay in one that has done badly to avoid booking a loss. The limit of the lesson: expected value ignores how much a person can bear to lose, and for a large loss relative to wealth a sure thing can be the right choice.

About 20 full-time distributors. Five lakh people in households of four is 1,25,000 households. If 12% invest Rs 3 lakh each, the town holds Rs 450 crore. Suppose half comes through distributors: Rs 225 crore, which at an assumed 0.8% trail pays Rs 180 lakh a year. A distributor needing Rs 9 lakh a year in gross trail means about 20 can live on it.

What actually limits the number of distributors?

Ask how many tailors a colony can support and you do not count the people; you count the stitching work and divide by what keeps one tailor going. A distributor is paid from trail commissionA yearly commission paid to a distributor as a percentage of the client assets they hold in a fund, for as long as the money stays invested., so the town's capacity is the trail pool divided by one livelihood, not the population divided by some ratio of advisers to people. That tells you the order of the chain before you guess a single number: people, then households, then investors, then assets, then the share that pays a distributor, then the trail.

How do you defend each guess, and which one moves the answer most?

Say each assumption with a reason. Four people to a household is a fair Indian average. Twelve percent of households holding funds is a cautious guess for a district town. Rs 3 lakh is a modest balance once systematic plans have run a few years. Half through distributors leaves the rest to direct plans, apps and bank branches. Trail rates vary by scheme and over time, so 0.8% is an assumption to confirm. The answer is a product of guesses, so a change in any one moves it in proportion: halve the trail and the town supports half as many.

Close with the sanity check and the limit. Twenty distributors for 15,000 investing households is about 750 families each, which is a full book for one person with a small office. In reality many towns have far more registered distributors than this, because most are part-time, also sell insurance or deposits, or work for a bank, so their fund trail is only part of their income. The estimate sizes the full-time capacity, not the head count on a register.

The deposit, by about Rs 1,418 on Rs 10 lakh. Taxed yearly, the deposit compounds at 7.2% x 0.7 = 5.04% and reaches Rs 11,58,949. The fund grows to Rs 12,25,043 and keeps 70% of its gain: Rs 11,57,530. Deferral alone is worth about Rs 3,399, roughly the cost of a 0.14% yield gap, so a 0.2% lower yield more than uses it up. Tax rates here are assumptions; confirm the current rules.

Where does the value of deferral come from?

Think of a shop that lets you pay your bill at the end of the year instead of every month. You keep the money a little longer and can earn something on it, but over a short time that is a small favour. Tax deferral is the same favour: the tax you would have paid each year stays invested and earns until redemption, so the gain comes from interest on money that would otherwise have gone to the tax office. Over three years at 7% that extra earning is modest. Over twenty years it is large, because it compounds.

Work both. The deposit pays 7.2%, of which 30% goes in tax each year, so it compounds at 5.04%: Rs 10 lakh becomes Rs 11,58,949. The fund compounds untaxed at 7.0% to Rs 12,25,043; on redemption 30% of the Rs 2,25,043 gain goes in tax, leaving Rs 11,57,530. The deposit is Rs 1,418 ahead.

How do you separate the deferral from the yield gap?

Give the fund the same 7.2% and the only difference left is the timing of tax. It would end at Rs 11,62,348, Rs 3,399 above the deposit: that is the value of deferral. Then the fund's 0.2% lower yield takes away Rs 4,818, slightly more than the deferral gave, which is why the deposit edges ahead. The fund would break even at a yield of about 7.06%, so the deferral is worth roughly 0.14% a year over three years.

The limits matter more than the arithmetic here. The tax treatment of debt funds has changed in India in recent years, and slab rates, surcharge and cess all move the answer, so confirm the current rules before using any rate. A deposit has no price risk, while a debt fund's NAV moves with interest rates and credit events. And deferral grows with time: over fifteen years the same set-up would give the fund a clear lead.

About Rs 1.68 lakh from the liquid fund, against about Rs 1.58 lakh from the overnight fund. Short-term fund income accrues daily, so it is amount x yield x days / 365: Rs 10 crore x 6.8% x 9 / 365 = Rs 1,67,671. At 6.4% it is Rs 1,57,808. The gap of about Rs 9,863 is what the treasurer is paid for accepting the liquid fund's small extra rate and credit risk.

How do you turn an annual yield into nine days of income?

A shopkeeper who rents a room for Rs 36,500 a year earns Rs 100 a day, and nobody would charge a guest for a nine-day stay by quoting the year. Yields are quoted per year, but liquid and overnight funds accrue income every calendar day, so the money earned is the yearly yield scaled by days over 365. Rs 10 crore at 6.8% earns Rs 18,630 a day; nine days is Rs 1,67,671. The overnight fund, at 6.4%, earns Rs 17,534 a day and Rs 1,57,808 over the same nine days.

Is the extra Rs 9,863 worth taking?

An overnight fundA debt fund that lends only for one business day at a time, mostly through overnight repo-style borrowing, so it carries almost no rate or credit risk. lends for one day at a time; a liquid fundA debt fund holding money market paper and short bonds that mature within about three months, so its NAV moves only slightly with rates and credit events. holds paper running out to about three months. The 0.4% yield gap is pay for that extra term and credit exposure, and over nine days it comes to Rs 9,863 on Rs 10 crore, about 0.01% of the money. A treasurer weighs that against the small chance of a mark-down in the liquid fund during the same nine days. For a company that needs every rupee back on day nine, the overnight fund's certainty can be worth more than Rs 9,863.

Two practical limits. Liquid funds in India can charge an exit load on very short holdings, and the cut-off times for same-day NAV decide which day's NAV applies, so check the scheme's current terms before working out a short stay. And the quoted yield is the portfolio's yield; the realised return depends on what happens to that paper over the nine days, so the figures above are estimates, not quotes.

The equity fund's fee takes 15% of its expected return; the liquid fund's takes about 3.8%. A fee is charged on the money invested, but it comes out of the return, so it should be measured against the return. 1.5 / 10 is 15%; 0.25 / 6.5 is 3.85%. Measured against the 3.5% extra that equity is expected to earn over cash, the 1.5% fee takes 43%.

Why does 1.5% sound smaller than it is?

An agent who charges you 1.5% of your house price to sell it sounds cheap until you remember you only made Rs 10 lakh of profit on a Rs 1 crore house: the Rs 1.5 lakh fee takes 15% of the profit. Fund fees are quoted on the money invested, but they are paid out of the return, so the honest size of a fee is the fee divided by the return it takes from. For the equity fund, 1.5 / 10 = 15%: almost one rupee in every seven the fund is expected to earn goes to costs. For the liquid fund, 0.25 / 6.5 = 3.85%.

What is the sharper way to measure the equity fee?

An investor holds equity instead of cash to earn the extra return, the equity risk premiumThe return equities are expected to earn above cash or short government bills, as pay for their extra risk.. If cash pays 6.5% and equity is expected to make 10%, the extra is 3.5%. A 1.5% fee takes 1.5 / 3.5 = 43% of that extra, so close to half of the reason for owning equity goes in costs. That is why low-cost index funds draw so much money: the fee matters most where the expected extra return is thin.

Two limits keep this honest. Expected returns are assumptions, not promises, and in a year when the fund falls the fee takes more than all of the return. And a higher fee can still be worth paying if the manager adds more than the fee in return after costs, which is rare enough that the evidence for it should be asked for, not assumed.

Because the extra ten years come first, and the earliest rupees compound the longest. At 1% a month, the early starter reaches about Rs 6.50 crore and the late starter about Rs 1.90 crore, 3.4 times less, though the gap in money paid in is only Rs 12 lakh. That Rs 12 lakh, saved from 25 to 35, grows to about Rs 4.60 crore by 60, 71% of the early starter's corpus.

Why is the early starter so far ahead for so little extra?

Plant a mango sapling at 25 and another at 35, and at 60 the first tree is not slightly bigger; it has had ten more seasons of growth on a trunk that kept getting larger. Money saved early does not just add ten more years of deposits; it gives every rupee in those years twenty-five more years of compounding after them. The late starter pays in Rs 30 lakh over 25 years and ends with Rs 1.90 crore. The early starter pays Rs 42 lakh over 35 years and ends with Rs 6.50 crore.

How much does the first decade contribute on its own?

Split the early starter's plan in two. From 25 to 35 she saves Rs 12 lakh, which is about Rs 23.2 lakh by 35. From then she saves exactly what the late starter saves, so that part ends at Rs 1.90 crore. The Rs 23.2 lakh from the first decade compounds for 25 more years at 1% a month and becomes about Rs 4.60 crore, 71% of her final corpus. Her first ten years are worth more at 60 than the late starter's entire 25 years of saving.

State the assumptions, because the size of the gap depends on them. A steady 12% every year is a simplification; real equity returns swing, and the order of good and bad years matters for a monthly saver. Inflation shrinks both corpora in today's rupees, though not the ratio. And the gap narrows if the late starter saves more each month; to match the early starter at 60 she would have to invest about 3.4 times as much, around Rs 34,000 a month.

The fund ends up about 4.0%, while the index is flat. It falls 80% of the index's 20%, so 16%, to 84. It then gains 95% of the index's 25%, so 23.75%, which takes 84 to 103.95. Falling less matters more than it looks, because a smaller hole needs a smaller climb: the fund only needed 19% to get back to 100 and got 23.75%.

Why does missing part of the rally still leave the fund ahead?

If you fall into a 2 metre ditch you need to climb 2 metres out; fall into a 1.6 metre ditch and the same climbing gets you above the rim. Losses and gains compound on different bases: after a 20% fall, the index needs 25% just to get back, but a fund that fell only 16% needs just 19%. Downside captureThe fund return in falling markets as a share of the index return in the same periods. 80% means the fund fell 8% when the index fell 10%. of 80% means the fund lost 16% when the index lost 20%. Upside captureThe fund return in rising markets as a share of the index return in the same periods. 95% means the fund rose 9.5% when the index rose 10%. of 95% means it gained 23.75% when the index gained 25%.

What does the comparison fund show, and what are the limits?

Take a fund that captures 110% in both directions, a bolder version of the index. It falls 22% to 78, then rises 27.5% to 99.45, ending below the index. A fund that amplifies both moves loses ground on a round trip, because the bigger fall needs an even bigger recovery; a fund that softens the falls more than the rises gains ground. That is why many research teams read the two capture ratios together and look for downside capture well below upside capture.

The limits matter. Capture ratios are measured over past periods and change with the manager's positioning, so a defensive fund in one cycle can be caught out in the next. Over a long bull market with few falls, the 95% upside capture costs more than the 80% downside capture saves. And the answer here depends on the index ending flat; the ratio between the two captures decides the outcome only for that kind of round trip.