Mutual Fund Mastery puzzles, solved step by step

2%. Flip the P/E to get the earnings yield: a P/E of 20 means the company earns 1/20, or 5%, of its share price each year. It pays out 40% of those earnings, so the dividend is 40% of 5%, which is 2% of the price. A Rs 400 share earning Rs 20 pays Rs 8, and 8 / 400 is 2%.

How do you get from a P/E to a yield?

If a flat costs Rs 20 lakh and earns Rs 1 lakh a year in rent, it costs 20 years of rent, and the rent is 5% of the price. The same flip works for shares. A P/E of 20 means the share costs 20 years of earnings, so the earnings yieldEarnings per share divided by the share price: the P/E turned upside down. A P/E of 20 is an earnings yield of 5%. is 1/20 = 5%. Only part of those earnings reach the shareholder as cash. With a payout ratioThe share of profit a company pays out as dividends. The rest is retained and reinvested in the business. of 40%, the dividend is 0.4 x 5% = 2% of the price, and the other 3% stays in the company.

What does the other 3% do, and where does this identity help?

The retained 3% is not lost; it is reinvested. If the company earns 15% on what it reinvests, keeping 60% of profit lets it grow earnings by about 0.6 x 15% = 9% a year, and the dividend yield plus that growth, 11%, is a rough estimate of the shareholder's long-run return. That is the logic of a dividend discount model in one line. The identity also works backwards in an interview: a stock yielding 2% with a 40% payout must be on a P/E of 20.

The limits: the P/E uses one year's earnings, which may be unusually high or low, and the payout ratio can change from year to year. Buybacks return cash too, so a company paying low dividends but buying back shares can return more than its dividend yield suggests. And the growth estimate assumes the company keeps earning 15% on new money, which gets harder as it grows.

The expected payout is Rs 8.70 and the unit is worth about Rs 87. Weight each year by its chance: 0.40 x 12 = 4.80, 0.45 x 8 = 3.60, 0.15 x 2 = 0.30, which sum to Rs 8.70. A payment expected every year forever is worth that payment divided by the required return, so 8.70 / 0.10 = Rs 87. Valuing the likeliest year alone would give Rs 80.

How do you value income that changes every year?

A farmer whose crop is good four years in ten, ordinary in about half, and poor in the rest does not plan around a normal year. He plans around what the land produces on average over many years. When income is uncertain, you value the expected valueThe probability-weighted average of all possible outcomes: each outcome times its chance, added up. of the income, not the likeliest outcome and not the plain average of the outcomes. For this unit: 0.40 x Rs 12 = 4.80, 0.45 x Rs 8 = 3.60, 0.15 x Rs 2 = 0.30, a total of Rs 8.70 a year.

Where do the two common wrong answers come from?

The likeliest year pays Rs 8, so some candidates value the unit at Rs 80. That throws away the 40% chance of Rs 12, which is worth Rs 7 of value here. Others average 12, 8 and 2 to get Rs 7.33, as if each year were equally likely, which gives Rs 73 and punishes the unit for a bad year that happens only 15% of the time. The weighting is the whole answer.

Now the honest limits, because the follow-up is usually how you would assess such an asset in practice. The expected value hides the spread: the payout's standard deviation here is about Rs 3.36, large against Rs 8.70, and an investor needing steady income cares about that. Uncertainty usually shows up in a higher required return, not in a lower expected payout, so the 10% must be chosen with the spread in mind. A toll road's traffic also trends and its concession ends, so the forever assumption is a simplification to state openly.

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.

Weigh three coins against three. The pan that rises holds the light coin; if they balance, it is among the three left aside. Then weigh two of those three suspects against each other: the rising pan holds it, and a balance means it is the third. Each weighing has three possible outcomes, so two weighings can separate 3 x 3 = 9 coins.

Why split into three groups rather than two?

A shopkeeper asking a customer "yes, no, or not sure?" learns more from one answer than one asking only "yes or no?". A balance scale gives three answers, left pan rises, right pan rises, or the pans balance, so the best weighing splits the suspects into three equal groups and lets the scale tell you which group holds the light coin. Put coins 1 to 3 on the left and 4 to 6 on the right. If the left pan rises, the light coin is 1, 2 or 3. If the right pan rises, it is 4, 5 or 6. If they balance, it is 7, 8 or 9.

How do you know two weighings is the minimum, and how far does this go?

Count outcomes. One weighing has 3 outcomes, two have 3 x 3 = 9, and there are 9 possible light coins, so two weighings are just enough. The number of weighings needed is the smallest w with 3 to the power w at least the number of coins: 9 coins need 2, 27 coins need 3. One weighing can only handle 3 coins, so you cannot do 9 in one. Splitting into halves wastes the balance outcome and needs three weighings for nine coins.

The desk lesson is about information, not coins: a test with three outcomes is worth more than one with two, if you design it to use all three. The limit of the puzzle is that you are told the odd coin is lighter. If it could be heavier or lighter, each coin has two possible states, there are more cases to separate, and the strategy needs more care, which is the classic twelve-coin follow-up.

The yield to maturity is below 7%, and the current yield is 6.73%. Current yield is the coupon over the price: 7 / 104 = 6.73%. The yield to maturity is lower still, because you pay 104 and get back only 100, so the Rs 4 premium is a loss spread over the bond's life. For an assumed five years left, the yield to maturity is about 6.05%.

Why must the yield be below the coupon when the price is above 100?

Pay Rs 104 for a gift voucher worth Rs 100 that also pays Rs 7 of cashback a year. The cashback is generous, but you have overpaid for the voucher by Rs 4. A bond priced above par returns less than its coupon, because the buyer pays more than the Rs 100 that comes back at maturity, and that premium is lost along the way. The coupon of 7% is set on the face value of 100 and never changes. The current yieldThe yearly coupon divided by the bond price today. It ignores any gain or loss as the price moves to face value at maturity. divides the same Rs 7 by what you actually pay: 7 / 104 = 6.73%.

How far below 7% is the yield to maturity?

That depends on how long the bond has to run, which the question does not say, so name an assumption. With five years left, the Rs 4 premium is lost at roughly Rs 0.80 a year. A quick estimate takes the coupon less that yearly loss, Rs 6.20, over the average of the purchase and redemption prices, 102: about 6.08%. The exact yield to maturityThe single discount rate that makes all remaining coupons and the final repayment worth exactly the price paid today. is 6.05%. The longer the bond, the thinner the yearly slice of the premium and the closer the yield to maturity sits to the current yield.

For a debt fund this ordering is everyday arithmetic: when rates fall, older high-coupon bonds trade above par, and the fund's quoted portfolio yield sits below the coupons it receives. The limit of yield to maturity: it assumes the bond is held to maturity, never defaults and that coupons are reinvested at the same yield, none of which is guaranteed. A bond callable before maturity at par makes the premium an even bigger risk.

52%, the same as on any other day. If each day's move is independent, the five falls carry no information about tomorrow. A run of five falls was unlikely before it began, 0.48 to the fifth power or about 2.5%, but that probability belonged to days that are now finished. Believing a rise is due is the gambler's fallacy.

Why does the streak not make a rise more likely?

A captain who has lost five tosses in a row walks out for the sixth feeling owed a win. The coin has no record of the first five and no sense of fairness to restore; it is still 50:50. Independence means past outcomes do not enter the calculation for the next one, so tomorrow's chance of a rise is the 52% it always was. The feeling that things must even out is real, and it has a name: the gambler's fallacy. It is strongest exactly when a streak is long, which is when it does the most damage to decisions.

Where does the tiny probability come from, and why is it the wrong number?

It answers a different question. Standing on Monday morning, the chance of five falls in a row was 0.48 multiplied by itself five times, about 2.5%, and the chance of six in a row was about 1.2%, roughly one in 82. Once five falls have happened, the only uncertainty left is tomorrow, and for independent days the chance of a rise given the streak equals the chance of a rise on any day. Six falls in a row is rare only when viewed from the start; viewed from Friday evening, it needs just one more fall.

Is the independence assumption true for real markets?

Not exactly, and saying so earns credit. Day-to-day direction in a broad index is very hard to predict from the previous days, but the size of moves does cluster: a run of falls often comes with bigger swings, so tomorrow's move may be larger even if its direction is a near coin toss. For an adviser the danger is not the arithmetic but the conversation: a client who believes a bounce is due will add money for the wrong reason, or hold out for a rebound the odds do not promise. A streak on its own is not a reason to act; any case for investing has to rest on the client's plan and horizon, not on the last five days.

About 1,600 a year, on stated assumptions. Take a global narrow-body fleet of about 20,000 jets. If a jet flies for about 25 years, 800 retire each year and need replacing. If traffic grows about 4% a year and the fleet grows with it, another 800 are needed. Replacement plus growth gives roughly 1,600, with a sensible range of about 1,300 to 2,000.

Why split deliveries into replacement and growth?

Think about how many pairs of school shoes a town buys in a year. You do not count shoe shops; you count children. Shoes wear out after about a year, and each year the town has a few more children. Annual sales of anything durable are the flow that keeps a stock alive: what wears out plus what the stock grows by, and both are proportions of the stock. So the whole estimate hangs on one number, the fleet, and two rates you can reason about aloud: how long a jet lasts and how fast flying grows.

How do you get the fleet number if you do not know it?

Build it from passengers, and say that every input is an assumption. Suppose narrow-body jets carry about 4.5 billion passenger trips a year. A typical jet has 150 seats and flies 80% full, so 120 passengers a flight. If it flies five short sectors a day, every day, that is 1,825 flights and about 2.19 lakh passengers a year. Divide and you get roughly 20,500 jets. Two independent routes landing near 20,000 is the check an interviewer wants to hear, even when both routes are rough.

Where is this estimate weakest?

In three places. Service life is not fixed: when fuel is dear, airlines retire old jets early, and when new jets are scarce they keep old ones flying. Traffic growth is lumpy, and a shock year can turn it negative. And, most important for anyone valuing a maker, deliveries are capped by how many jets the factories can build, so demand can sit above actual deliveries for years, with the difference piling up as a backlog of orders. If the interviewer asks about one maker, split the 1,600 by an assumed market share and say the share is the assumption most worth checking.

About Rs 110.4 crore. Start from Rs 12 crore of gross potential rent and take off 8% vacancy to reach Rs 11.04 crore collected. Take off operating costs of 20% of that, Rs 2.208 crore, to reach net operating income of Rs 8.832 crore. Divide by the 8% cap rate: Rs 8.832 crore over 0.08 is Rs 110.4 crore.

Why does a cap rate turn one year's income into a price?

Imagine a flat that brings you Rs 30,000 a month after society charges and the odd empty month, Rs 3.6 lakh a year. If similar flats sell for Rs 72 lakh, buyers are accepting a 5% yield, and you can price any flat on the street by dividing its net rent by 5%. A cap rate is that street yield for commercial property: value equals net operating income divided by the rate buyers accept. It works like a perpetuity with expected rent growth folded into the rate, which is why a lower cap rate means a higher price.

Why does every cost line move the value so much?

Because each rupee of net income is multiplied by one over the cap rate, 12.5 times at 8%. A cost saving of Rs 10 lakh a year adds Rs 1.25 crore of value, and a rise in vacancy from 8% to 12% cuts value by about Rs 4.8 crore. That is why a buyer picks apart the rent roll and the service charge budget before arguing about the cap rate at all. The cap rate moves value too: the same income is worth Rs 117.8 crore at 7.5% and Rs 103.9 crore at 8.5%.

Say the limitation before the interviewer does. A single cap rate prices a stable, well-let building. A building with a large lease expiring next year, or one mid-refurbishment, needs its cash flows laid out year by year rather than one year's income capitalised. The cap rate method is a shortcut for the ordinary case, and the valuer's job is to spot when the case is not ordinary.

The true average was 10.2% a year, and a survivor-only database overstates it by 1.8 points. Seventy funds at 12% and thirty at 6% average to 0.7 x 12% plus 0.3 x 6%, which is 10.2%. A database that drops merged and closed funds shows 12%, because the funds that disappeared were mostly the weak ones, and the gap compounds every year.

Why do dead funds disappear from the numbers?

Think of a coaching centre that advertises the average marks of students who stayed to the final exam. The ones who struggled and left are not in the average, so the centre looks better than its teaching. A fund database that lists only live schemes does the same: the funds that did badly were merged or shut, their records left the table, and the category average rose without anyone earning it. Fund houses merge weak schemes into stronger ones as a matter of routine, so the losers vanish quietly rather than with a headline. The name for this is survivorship bias.

How much does the gap matter over ten years?

Compound it. Rs 1 lakh at 12% for ten years becomes about Rs 3.11 lakh; at 10.2% it becomes about Rs 2.64 lakh. A chart built on survivors shows roughly Rs 46,000 more per lakh than the category delivered to the average investor who chose a fund ten years ago. Compounding a category average is itself approximate, because each fund compounds on its own path, but the direction and the size of the gap are right.

The bias leaks into rankings too. A fund that looks top quartile among survivors may be only average once the vanished funds are put back, because the bottom of the table has been cut off. Before comparing a fund with its category, ask whether the category figure includes funds that no longer exist. Good research databases keep dead funds in; if yours does not, treat its averages as a ceiling rather than a middle.

Allocation cost 0.70% and selection added 0.75%, a net active return of about plus 0.05%. The IT overweight is 10 points in a sector that trailed the benchmark by 7 points: 0.10 x (5% minus 12%) is minus 0.7%. The bank stocks beat their index by 3 points on a 25% weight: 0.25 x 3% is plus 0.75%. Good stock picking almost exactly paid for a poor sector bet.

What is the difference between allocation and selection?

Picture a selector who picks four spinners for a pitch that suits pace, but whose four spinners bowl better than any other spinners in the country would have. Two separate decisions: how many of each kind, and which ones. Allocation measures the first decision, sector weights against the benchmark's weights; selection measures the second, how the stocks chosen inside a sector did against that sector. Splitting them tells a fund research team whether a manager's skill lies in calling sectors or in picking stocks, which matters more than the total when deciding what to trust next.

Why is allocation measured against the whole benchmark rather than against zero?

Because the extra 10% in IT had to come from somewhere, and the alternative was the benchmark itself. An overweight in a sector that makes money still costs you if that sector made less than everything else you could have held. Say the assumptions behind the split out loud: the bank weight matches the benchmark's, so banks carry no allocation effect; the IT stocks held matched the IT index, so IT carries no selection effect; and the 10 points taken from other sectors came from sectors that earned the benchmark's 12%.

One detail an interviewer may probe. Using the fund's 25% weight in the selection term folds in what the BrinsonThe Brinson method, named after the authors who set it out in the 1980s, splits a fund active return into allocation, selection and an interaction term. framework calls the interaction effect; the textbook version uses the benchmark's weight and reports interaction separately. With the bank weights equal here, both give the same 0.75%.