Mutual Fund Mastery puzzles, solved step by step

A is the grocery retailer, B is the software firm and C is the cement maker. A thin 22% gross margin with stock turning every 45 days is a reseller living on volume. A 78% gross margin with no inventory is a product that costs almost nothing to copy. Capital employed of three times sales means huge fixed plant, which among these three only cement needs.

What does each number tell you about how the business makes money?

Picture three shops on one street. The kirana store buys packets for Rs 78 and sells them for Rs 100 within weeks. The tutor sells the same recorded lesson a thousand times. The brick kiln on the edge of town cannot sell a single brick until it has built a furnace. Gross margin tells you how much of each sale the company keeps after paying for what it sells; inventory and capital tell you what it must hold to make that sale. Company A keeps 22 paise of each rupee and holds 45 days of stock, which is about 9.6% of a year's sales sitting on shelves: a reseller. Company B keeps 78 paise and holds no stock, so its costs sit below gross profit, in salaries: software.

Why is the cement maker C and not A?

Cement is a commodity, so candidates often pin it to the lowest margin. But a cement maker does not resell anything; it turns limestone and fuel into a product, and its gross margin sits well above a pure reseller's. What gives it away is capital intensityHow much capital a business must tie up to produce a rupee of sales. Plant-heavy industries need several rupees; retailers and service firms need little.: capital employed of three times sales means Rs 300 of plant and working capital for every Rs 100 of sales. A grocer leases its stores and pays suppliers after it has sold the goods, so it might need only Rs 25 per Rs 100 of sales.

The table shows why the pattern matters to an analyst, not just to a quiz. The grocer earns a thin margin many times over; software earns a fat margin on little capital; the cement maker needs a fair margin just to cover the plant it had to build, which is why cement returns swing so hard with prices and plant utilisation. The limit: companies define gross margin differently. Some put power and freight inside cost of sales, some below it, so check the definition before reading a margin across two companies.

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.

About 8.00% for next year, and about 6.5% if the curve is inverted. Investing for two years at 7.5% must give the same as one year at 7.0% rolled into next year's rate, or someone could profit from the gap. So 1.075 squared equals 1.07 times (1 + f), and f is 8.00%. Flip the curve to 7.5% then 7.0% and the implied rate falls to 6.50%: the market is pricing lower short rates ahead.

Why must the two paths give the same answer?

Two routes to the same railway station, one direct and one with a change, must cost about the same, or everyone would take the cheaper one until the prices matched. Money for two years can be locked in at the two-year rate, or invested for one year and rolled; if one path paid more, investors would crowd into it, so the rate the curve implies for the second year is the one that makes them equal. That rate is the forward rateThe interest rate for a future period that is locked in today by the current yield curve, found by making a long investment equal to a chain of shorter ones.. Rs 100 at 7.5% for two years becomes Rs 115.56. Rs 100 at 7.0% becomes Rs 107 after one year, so the second year must turn Rs 107 into Rs 115.56.

What does an inverted curve tell you?

In your head, the forward is about twice the long rate minus the short rate: 2 x 7.5 - 7.0 = 8.0%. Turn the curve upside down, 7.5% for one year and 7.0% for two, and the same step gives 2 x 7.0 - 7.5 = 6.5%, exactly 6.50%. An inverted curve says the market expects short rates to fall, which usually happens when it expects the central bank to cut, often because it expects growth to slow. That is why inversions are watched as a recession signal.

State the limit, because it is the follow-up. Forward rates are not pure forecasts. Lenders usually want extra pay for tying money up longer, a term premium, so an upward curve partly reflects that premium rather than expected hikes. The forward rate is the break-even: if next year's actual one-year rate comes in below it, the investor who locked in two years did better.

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.

Alpha is 3.4%. The fund's exposures alone would have earned cash of 6%, plus 1.1 x 4% = 4.4% for its market beta, plus 0.3 x 4% = 1.2% for its tilt to small companies: 11.6% in all. It returned 15%, so 3.4% is what the factors do not explain. It beat the market's 10% by 5 points, but 1.6 of those were paid-for risk.

Why is beating the market by 5 points not 5 points of skill?

A delivery rider who earns more than others in the monsoon may simply be taking the rainy-day shifts that pay extra. A fund that takes more market risk, or leans towards small companies, is paid for those exposures in years when they do well, and that pay is not skill. A factor modelA way of explaining a fund return as cash plus a set of exposures, each times the return of a common driver such as the market or small companies, with what is left over called alpha. prices each exposure. This fund has a beta of 1.1, so it gets 1.1 times the market's 4% premium over cash: 4.4%. It has a size loadingHow strongly a fund return moves with the gap between small company and large company returns. A positive loading means a tilt towards small companies. of 0.3, so it gets 0.3 times the 4% that small companies beat large ones by: 1.2%.

What changes as you add each factor?

Each step strips out a reward that anyone could have bought cheaply. Against the market alone, the fund is 5.0 points ahead. After beta, the CAPMThe capital asset pricing model, which explains returns with one factor, the market, scaled by beta. alpha is 4.6%. After the size tilt as well, alpha is 3.4%, so about a third of the apparent outperformance was a small-company bet that a cheap index fund could have delivered. Add a value or momentum factor and the alpha could shrink further, or grow if the fund leaned against a factor that did well.

The limits are worth stating plainly. One year of data says almost nothing; loadings and alpha are estimated from many periods of returns, and a 3.4% alpha with typical noise needs years before it is distinguishable from luck. The answer also depends on which factors you include and how they are built, so two research teams can report different alphas for the same fund.

B has the higher IRR, 10.7% against 9.6%, but A has the higher NPV at 6%, Rs 0.49 lakh against Rs 0.19 lakh. IRR measures speed of growth; NPV measures rupees of value at your cost of money. B grows faster but stops after four years, while A keeps compounding above 6% for twelve. The two rankings swap at 9.05%: below that A wins on NPV, above it B does.

What does each measure reward?

Compare a short, well-paid contract with a long, steady job. The contract pays more per month, but it ends quickly and you must find something else. IRR ranks by speed: the rate at which the money grows. NPV ranks by size: how many rupees of value the option creates once everything is discounted at your own cost of money. A's IRR is 3 to the power 1/12, less one, 9.59%. B's is 1.5 to the power 1/4, less one, 10.67%. B is faster.

At a 6% discount rate, A's Rs 3 lakh in year 12 is worth 3 / 1.06 to the 12th, Rs 1.49 lakh today, an NPV of Rs 0.49 lakh. B's Rs 1.5 lakh in year 4 is worth Rs 1.19 lakh, an NPV of Rs 0.19 lakh. A creates more than twice the value, because it earns well above 6% for three times as long.

Where do the rankings swap, and what does that rate mean?

The NPV curves cross where both options are worth the same: 3 / (1 + r) to the 12th = 1.5 / (1 + r) to the 4th, so (1 + r) to the 8th = 2 and r = 9.05%. That crossover has a plain meaning: B returns Rs 1.5 lakh in year 4, and to match A it would need to double over the next eight years, which takes 9.05% a year. If you can reinvest B's proceeds above that, B is better; below it, A is. IRR silently assumes you can reinvest at B's own 10.7%, which is why it favours the short option.

For a fund desk this is the reinvestment question in plain clothes. A client choosing between a short product with a high stated yield and a long one with a lower yield is really asking what the short money will earn when it comes back. The limit: NPV needs a discount rate, and a different rate can flip the answer; at 10%, A's NPV turns negative, about Rs 4,411 below zero, while B's stays positive at about Rs 2,452, so B wins.

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.