Mutual Fund Mastery puzzles, solved step by step

An NPV of about Rs 8.1 crore at 12%, and the NPV reaches zero at about 15.2%, the IRR. Five payments of Rs 30 crore discounted at 12% are worth 30 times 3.605, about Rs 108.1 crore, against a cost of Rs 100 crore. Raising the rate shrinks the payments' present value, and at 15.2% they are worth exactly Rs 100 crore.

How do you value the five payments quickly?

A rupee promised next year is worth less than a rupee in your hand, the same way a friend's promise to repay in five years is worth less than cash today. Discount each payment back to today, and because the payments are equal you can use one annuity factor instead of five divisions. At 12% the five-year annuity factor is 3.605, so Rs 30 crore a year is worth about Rs 108.1 crore today. Subtract the Rs 100 crore cost: the NPV is about Rs 8.1 crore. Positive NPV means the project earns more than 12%.

How do you find the rate where NPV is zero without a calculator?

Bracket it. You need an annuity factor of 100 over 30, about 3.333. At 12% the factor is 3.605, too high, so the rate is above 12%. At 15% the factor is about 3.352, still a touch above 3.333; at 16% it is about 3.274, below. The IRRInternal rate of return: the discount rate at which the present value of a project inflows equals its cost, so the NPV is zero. is simply the discount rate at which the NPV curve crosses zero, here about 15.2%. In an interview, saying between 15% and 16%, closer to 15%, and showing the bracket, is a full answer.

State the limits of IRR alongside it. It assumes the payments can be reinvested at the IRR itself, it can mislead when comparing projects of very different size, and a project with cash flows that change sign more than once can have more than one IRR. NPV at the right cost of capital is the cleaner decision rule; IRR is the useful headline.

After about 1.1 years, roughly 13 months. The ETF's trading costs are paid once: 0.03% brokerage on the way in and out, 0.06%, plus the 0.10% spread, 0.16% in all. Its fee is 0.15% a year below the index fund's. Divide the one-off cost by the yearly saving, 0.16 over 0.15, and the ETF catches up after 1.07 years. Hold longer and it wins; trade in and out and it loses.

How do you compare a one-off cost with a yearly one?

Think of buying a monthly rail pass against paying per ride. The pass costs more up front and less per trip, so it only pays if you ride enough. The ETF is the pass: it costs 0.16% to get in and out, then saves 0.15% a year against the index fund, so the break-even holding period is the one-off cost divided by the yearly saving. Count both sides of the trade: brokerage is paid when you buy and again when you sell, and crossing a 0.10% spread means buying a little above the middle price and selling a little below it.

What does the gap look like in rupees?

On Rs 10 lakh held for five years, the ETF costs about Rs 4,100 and the index fund about Rs 10,000, ignoring the small effect of compounding. Held for six months, the order flips: the ETF costs 0.185% against the index fund's 0.10%. ETFs win on long holds and lose on short ones, so the answer depends on the investor, not on the product.

Name what the simple sum leaves out. Spreads on a thinly traded ETF can be much wider than 0.10%, and the price can sit at a premium or discount to NAV. Some investors pay yearly demat charges that matter on small balances. An index fund may carry an exit load in the early months, and both products have tracking differences that can outweigh a few basis points of fee. Each of these moves the break-even, so the useful answer is the method, with the inputs confirmed for the actual products.

About 7.2% a year: the portfolio's yield to maturity less the expense ratio. Holding to the fund's maturity removes most of the price risk, so the bonds earn roughly their 7.4% yield and the fund keeps 0.2% a year. It will not be exactly 7.2% because coupons are reinvested at future yields nobody knows, the fund tracks its index imperfectly, and the yield you lock in is the one on the day you invest. On these numbers the drift is about plus or minus 0.13% a year.

Why is yield to maturity a fair starting point?

A fixed deposit tells you its rate because the bank promises to hold it to maturity for you. A bond does the same if you hold it to maturity: price swings along the way wash out, and what remains is the yield to maturityThe single discount rate that makes the present value of a bond coupons and principal equal its price today; the return earned if the bond is held to maturity and coupons are reinvested at that same rate.. A target maturity fund holds bonds that mature near one date and then pays out, so an investor who stays to that date earns close to the portfolio's yield to maturity, less the fund's costs. 7.4% less 0.2% is about 7.2% a year.

Why will the realised return not be exactly 7.2%?

Yield to maturity quietly assumes every coupon is reinvested at the same yield, and the future reinvestment rate is unknown. Take a 5-year bond bought at par with a 7.4% coupon. If its coupons can only be reinvested at 6.4%, the realised return is about 7.27% a year; at 8.4% it is about 7.53%. After the 0.2% fee that is 7.07% to 7.33%. The fund also holds bonds that do not mature on exactly one date, keeps some cash, and replicates its index imperfectly, each worth a few basis points either way.

What else should the investor be told?

Three things. The estimate holds only to maturity: an investor who exits after two years takes the market price then, which can be well above or below the path to 7.2%. The yield locked in is the one on the day of investment, not the one quoted at launch. And the figure is before tax; how a debt fund's gains are taxed has changed in recent years, so confirm the current treatment before turning 7.2% into an after-tax number. The estimate is the honest best guess, not a promise.

His average cost is Rs 75 a unit and he needs a 25% rise from 60 to break even, but the fund is no better than before. Rs 1 lakh bought 1,000 units at 100 and 1,667 units at 60, so Rs 2 lakh buys 2,667 units, Rs 75 each. Before the top-up he needed 66.7%. Averaging down moves the break-even point, not the fund's prospects, and it doubles the money exposed to the next move.

Why is the average 75 and not 80?

Spend Rs 100 on mangoes at Rs 100 a kilo and another Rs 100 at Rs 60 a kilo. You have 1 kilo plus 1.67 kilos, 2.67 kilos for Rs 200, which is Rs 75 a kilo. Equal rupees buy more units when the price is low, so the average cost leans towards the lower price; it is the harmonic mean of the two NAVs, not the simple average. That is the same arithmetic that makes rupee-cost averaging in an SIP work, applied here to two purchases.

Does a lower break-even mean the decision was good?

No, and this is the behavioural point. The break-even number is about his purchase history; the fund's next move does not know or care what he paid. His position today is 2,667 units worth Rs 1.6 lakh, with an unrealised loss of Rs 40,000, the same loss he had before the top-up. A further 10% fall now costs Rs 16,000 instead of Rs 6,000. The question that decides whether to add is whether he would buy this fund today at 60 with fresh money if he had never owned it, given his goal and his allocation.

There are good reasons to add after a fall: a disciplined rebalance back to a target equity weight, or an SIP that keeps running through the dip. There are bad ones: wanting to feel closer to breaking even, or refusing to accept that the first purchase was a mistake. The arithmetic is the same in both cases; only the reason differs, and the adviser's job is to ask which one it is.

About 10 crore 5G phones a year, on stated assumptions. Take 70 crore smartphone users replacing every 4 years: 17.5 crore replacement phones a year. Add about 1.5 crore first-time buyers for 19.0 crore phones. If 55% of new handsets are 5G, that is about 10.5 crore. Each input is an assumption to state and defend; reasonable changes give a range of about 8 to 14 crore.

Why start from replacements and not from users?

A town with ten thousand households does not buy ten thousand refrigerators a year. It buys the ones that wear out, plus a few for new homes. The user base is a stock and yearly sales are a flow, and the replacement cycle is what converts one into the other. Stating that structure first is most of the marks, because it shows the interviewer how you will build the number before you pick any inputs.

How do you justify each input?

Say where each comes from and how you would check it. Users: a large share of a population of roughly 140 crore, here taken as 70 crore, half the population, to be confirmed against current telecom data. Replacement cycle: budget phones tend to be replaced sooner and premium phones later; 4 years is a middle assumption. First-time buyers: people moving from basic phones, assumed at 1.5 crore a year. 5G share: the share of new models sold with 5G, assumed at 55%. None of these are facts; they are reasoned guesses, and the interviewer is grading the reasoning and the sense check, not the decimal.

How do you sense check the answer?

Two quick checks. First, 19 crore phones a year for a country of roughly 140 crore people is about one phone per seven people each year, which is plausible for a market where most adults already own one. Second, the value: at an assumed average price of Rs 18,000, 10.5 crore phones is about Rs 1.9 lakh crore of sales. If either check looks absurd, revisit the inputs. Then give the range: a 3.5-year cycle and a 65% share give about 14.0 crore; a 4.5-year cycle and a 45% share give about 7.7 crore. The replacement cycle and the 5G share move the answer most, so those are the two to research first.

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.

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.

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.