Mutual Fund Mastery puzzles, solved step by step

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.

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.

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.

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.

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 fair P/E rises from 20x to about 22.2x, about 11%, if growth holds. P/E equals 1 over (r minus g): 1/0.05 is 20x and 1/0.045 is 22.2x. But a cut often comes because the economy is weakening. If growth expectations fall to 6.5%, the gap is back to 5 points and the P/E is unchanged; at 6%, the P/E falls to 18.2x. The market may also have priced the cut in already.

Why does a small rate cut move the P/E so much?

Think of a ladder leaning against a wall: move its foot a few centimetres when it is nearly upright and the top barely moves; move it when the ladder is almost flat and the top drops a long way. The P/E depends on the gap between the required return and growth, and when that gap is small, a half-point change in it moves the P/E by a large fraction. Here the gap goes from 5 points to 4.5, a tenth smaller, and the fair P/E rises by about 11%.

So why might the market fall on a rate cut?

Three reasons, all visible in the formula. First, growth: central banks usually cut because they see the economy slowing, and if the market lowers its growth estimate by as much as the cut lowers r, the gap and the P/E do not change at all; lower it by more and the P/E falls, to 18.2x at 6% growth. Second, expectations: if a 50 basis point cut was fully expected, it was already in the price, and only the surprise moves the market; a cut smaller than hoped can disappoint. Third, earnings themselves: E in the ratio is next year's earnings, and a weakening economy can lower that number even as the multiple rises.

Say the simplification. Writing P/E as 1 over (r minus g) assumes all earnings are paid out and growth is steady forever; with partial payout, the numerator is the payout ratio instead of one. The direction of every effect survives, but the sizes are illustrative. The general point holds firmly: a rate cut raises fair value only if what it says about growth does not undo it.