Mutual Fund Mastery puzzles, solved step by step

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.

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.

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.