Mutual Fund Mastery puzzles, solved step by step

The gap says a few star funds are pulling the average up; the typical fund made about 11%. The median is the middle fund, so half the category earned 11% or less. A mean of 14% needs a small group far out on the right: here six funds between 24% and 42%, and only 9 of the 40 beat the mean. A client choosing a fund at random should hear 11%, together with the spread around it.

Why can the average sit above most of the funds?

Ten people sit in a tea stall, each earning about Rs 30,000 a month. A business owner earning Rs 30 lakh a month walks in. The average income in the room jumps above Rs 2.9 lakh; the median, the middle person, still earns about Rs 30,000. A mean is pulled by every extreme value, while a median only cares about the middle, so when a few values are very large the mean overstates what a typical member got. Fund returns behave the same way: a few funds that caught one theme early can drag the category average up.

How do you check that the stars explain the gap?

Take them out and recompute. Without the six stars the other 34 funds average 10.9%, almost exactly the median. The six funds alone add about 3.1 percentage points to the category mean, which is the whole gap. A second check is to count: if the mean described a typical fund, about half the funds would sit above it. Here only 9 of 40 do.

Which figure should a client hear, and what else?

The median, because it is the honest answer to what a typical fund in the category did. Then the spread, because 5% to 42% is the real range of outcomes, and the chance of picking a star in advance is small. Two further cautions belong in the same breath. Category figures usually include only funds that survived the five years, and funds that closed or merged were often the weak ones, so even the median flatters. And past five-year returns, mean or median, are not a forecast.

No. Against a benchmark that matches its style, the manager lagged by 1 point. The fund's 4-point lead over the Nifty 50 came from holding mid caps in a year when mid caps beat large caps by 5 points. A plain mid cap index would have delivered that 5 points with no stock picking. The manager's own choices, after costs, cost the investor 1 point. Beating the wrong benchmark hid an underperformance.

Why is the Nifty 50 the wrong yardstick here?

A runner who trains at altitude and races at sea level will post faster times; the stopwatch is right but the comparison flatters him. A benchmark should hold what the fund holds, so that the gap measures the manager's choices and not the market segment the fund happens to sit in. A mid cap heavy fund compared with a large cap index is mostly measuring whether mid caps beat large caps that year, which no manager controls once the mandate is set.

How do you split the lead into style and skill?

Insert the style benchmark between the two numbers. Fund minus broad index splits into style benchmark minus broad index, which is the style effect, plus fund minus style benchmark, which is what the manager added. Here that is 5 plus (minus 1), giving the claimed 4. Illustrating with a Nifty 50 return of 12%, the style index made 17% and the fund 16%.

Two fairness points. The fund's return is after its expenses and an index's is not, so part of the minus 1 is cost; a passive mid cap fund would also have trailed its index by its own cost. And one year is a small sample: the right test of skill is the gap to the style benchmark over a full cycle, measured consistently. Regulators in India require schemes to show a benchmark that reflects their category; confirm the current rules before relying on any particular index choice.

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 76%. The easy route is the opposite event. Five positive years in a row needs a 75% chance to come up five times: 0.75 to the fifth is about 23.7%. Every other outcome includes at least one losing year, so the chance is 1 minus 0.237, about 76.3%. A small yearly chance of loss becomes close to a sure thing over an ordinary holding period.

Why work with the opposite event?

Ask a cricket fan the chance that a bowler concedes at least one boundary in an over, and the quick way is to ask the chance he concedes none. At least one is messy to count directly, because it covers one loss, two losses and every combination; none is a single clean path, so compute none and subtract from one. Here none means five positive years, each with a 75% chance, independent of the others: 0.75 times itself five times is 23.7%.

Why does an adviser care about this number?

Because it sets the client's expectation before the first bad year arrives. Over five years a losing year is the likely case, not the unlucky one, so a client who has not been told this reads the first negative year as something gone wrong. On the same assumptions the expected number of losing years in five is 1.25, the chance of exactly one is about 40%, and over ten years the chance of at least one rises to about 94%. A losing year is a calendar-year label; it says nothing about whether the five-year return was positive.

State the two assumptions. The 25% is an illustration, not a measured figure for any market, and real yearly returns are not fully independent: bad years cluster in some periods. Clustering lowers the chance of at least one losing year slightly while making the losing stretches longer. The method survives both caveats; the exact figure does not.

The maximum drawdown is 35%, from the peak of 140 to the trough of 91. Drawdown is always measured from the highest value reached before the fall, because that is the value an investor held and then lost. Measured from the start, 91 looks like a mild 9% dip, which hides how much was lost on the way. The fund later climbs to 120, still 14.3% below its peak.

Why measure from the peak and not from the start?

A house bought for Rs 1 crore, valued at Rs 1.4 crore at the top of a boom and then sold for Rs 91 lakh, did not lose 9%. Its owner watched Rs 49 lakh disappear. A drawdown measures the pain of falling from the best point reached, so it runs from the running peak, which keeps rising whenever the NAV sets a new high. Anyone who invested at 140, or held through it, lost 35%, and that is the risk the measure is built to show.

How do you compute it step by step?

Walk along the path and keep two numbers: the highest NAV so far and the current NAV. At each point the drawdown is current over peak, less one. The maximum drawdown is the worst of those readings, here 91 over 140 less one, minus 35%. The dip from 130 to 110 gives minus 15.4%, a smaller drawdown. The final 120 against a peak of 140 is a drawdown of minus 14.3% still open.

Add the recovery maths: from 91 the fund needs a 53.8% rise just to get back to 140. And the measure depends on how often you look. These are quarter-end NAVs; a daily series could show a deeper trough between the quarter ends, so a maximum drawdown should always be quoted with its data frequency and period.

About half: 49.5%. At 1% a month the corpus is about Rs 50.5 lakh after 15 years and Rs 99.9 lakh after 20, so the final five years add Rs 49.5 lakh. Only Rs 6 lakh of that is new instalments; about Rs 41.2 lakh is growth on the pot already built. An SIP's corpus is back-loaded, which is why stopping early costs far more than the missed instalments.

Why is the guess of a quarter so far off?

A quarter assumes the corpus grows by the same amount each year, like a piggy bank filled with the same coins every month. Compounding is not a piggy bank. Each year's growth is a percentage of a pot that is already larger than the year before, so the rupee gains get bigger every year and the corpus is heavily back-loaded. After 15 years of Rs 10,000 a month the pot holds about Rs 50.5 lakh, and five more years at 1% a month multiply that by 1.01 to the 60th, about 1.82, before a single new instalment is counted.

Where exactly does the last five years' money come from?

Split the Rs 49.5 lakh three ways. The 60 new instalments put in Rs 6 lakh. Those instalments earn about Rs 2.2 lakh of growth by year 20. The year-15 pot of Rs 50.5 lakh earns about Rs 41.2 lakh. The engine of the final years is the money already invested, not the money still to come. For contrast, the first ten years build only about Rs 23.2 lakh, 23% of the final corpus, though half the instalments are paid in them.

What does this mean for an investor thinking of stopping at year 15?

It depends what stopping means. Redeeming at year 15 gives up about Rs 49.5 lakh, half the eventual corpus, to save Rs 6 lakh of instalments. Pausing the instalments but staying invested is far cheaper: the year-15 pot alone grows to about Rs 91.7 lakh, so the cost is about Rs 8.2 lakh. The 12% is an assumption for the arithmetic; real returns vary year to year, and a bad last five years would cut the back-loaded gain sharply, which is the sequence risk of an SIP.

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.