Mutual Fund Mastery puzzles, solved step by step

Modified duration is about 3.99. Bracket first: a coupon bond's Macaulay duration sits below its 5-year maturity but not far, because the principal dominates, so somewhere in the low fours; dividing by 1.08 takes it just under 4. Exactly, the Macaulay duration is 4.312 years, and 4.312 divided by 1.08 is 3.993: a 1 point rise in yield cuts the price by roughly 4%.

How do you bracket it before doing any arithmetic?

Picture a see-saw with one heavy child at the far end and four small children spread along the plank. The balance point sits close to the heavy child but is pulled in a little by the others. A bond's Macaulay duration is the balance point of its discounted cash flows, so it can never exceed maturity and sits close to it when the final payment dominates. A zero coupon 5-year bond sits exactly at 5; this one pays 8 a year along the way, so it lands a little inside.

The final payment of 108 is worth 73.5 today, about 74% of the price of 100. The four coupons carry the rest at years 1 to 4. A balance point roughly three quarters of the way at 5 and a quarter spread between 1 and 4 lands a bit above 4, and dividing by 1 plus the yield lands just under 4.

Is there a shortcut for the exact figure?

For a bond priced at par there is a closed form. At par, Macaulay duration equals (1 + y) over y, times one minus the discount factor at maturity: 13.5 times (1 minus 0.6806), which is 4.312 years. Divide by 1.08 for modified duration, 3.993. Saying you know the par shortcut, then checking it against the bracket, is a strong answer in the room.

Say what the number is for. A modified duration of 3.99 means a 1 percentage point rise in yield costs roughly 3.99% of price, and a 0.25 point rise roughly 1%. The estimate is linear, so it drifts for large moves; convexity handles that.

The manager must beat the index by 1.6 points a year before costs just to tie, and with zero skill the investor ends about Rs 10.4 lakh behind. Both funds earn the index's 12% before costs. Net, that is 11.8% against 10.2%. Rs 10 lakh compounds to Rs 53.3 lakh in the index fund and Rs 42.9 lakh in the active fund, a gap of about 19% of the final wealth.

Why is the break-even the whole cost gap and not the active fee?

Two taxis to the same station, one charging Rs 18 per km and one Rs 2. The expensive one only wins if it is a much shorter route. The investor's alternative is not zero cost, it is the index fund, so the active manager has to earn back the difference in costs, 1.6 points a year, before adding anything. Beating the index by 1% before costs sounds like skill and still leaves the investor 0.6 points a year behind the cheaper fund.

Why does 1.6 points a year become a fifth of the money?

Because the fee is charged on the balance every year, and the rupees it takes would themselves have compounded. A cost gap compounds exactly like a return gap: the ratio of final wealth is (1.102 over 1.118) to the 15th power, about 0.806, so the active investor keeps about 81% of what the index investor has. The longer the horizon, the larger that share becomes; over 30 years it would be over a third.

The limitation: the 12% index return is an assumption for the arithmetic, and some active managers do beat their index after costs. The question is not whether active management can win. It is how large the hurdle is, and 1.6 points a year, every year, is a high bar to clear consistently.

About 2.5% a year of earnings growth just to tie, and about 5.5% if you demand an assumed 3 point premium for equity risk. Equity's expected return is roughly its earnings yield plus growth. At 22 times earnings the yield is 4.5%, 2.5 points below the bond. On starting yield the bond is cheaper; equity is cheaper only if you expect earnings growth comfortably above 5.5% a year.

How do you put a stock index and a bond on the same scale?

Think of two shops for sale. One pays its owner a fixed Rs 7 for every Rs 100 of price, forever. The other pays Rs 4.5 today, but its takings rise every year. Turn the P/E upside down to get an earnings yield, then add the growth the earnings will carry, and you have a number you can hold against the bond's yield. One over 22 is 4.55%, so the index starts 2.5 points behind the bond.

So which is cheaper?

Answer with a condition, not a verdict. On starting yield the bond is cheaper, and equity is cheaper only if earnings can grow faster than about 5.5% a year for a long time. The 3 point premium is an assumption for this arithmetic; you should say you would set it from your own view of equity risk. Then compare the required growth with a sensible estimate of nominal earnings growth in that economy, and state which side of the line you think it falls.

Name the limitation that marks you out. The bond yield is nominal, while earnings rise with inflation, so comparing 4.5% directly with 7% mixes a real yield with a nominal one. This comparison, often called the Fed model, is a quick screen, not a valuation. It also ignores that some earnings are reinvested, and the payout and the return on that reinvestment both shape the growth you can expect.

About Rs 150 crore a month, with a range of roughly Rs 100 to 200 crore. One crore people at four per household is 25 lakh households. Assume 15% have at least one SIP, 3.75 lakh households, with 1.6 SIPs each, 6 lakh SIPs. At an average Rs 2,500 per SIP that is Rs 150 crore a month, about Rs 1,800 crore a year. Penetration is the assumption to test first.

Where do you start, people or money?

Think of estimating how much a housing society spends on milk. You would count flats, then flats that buy from the dairy, then litres per flat, then price. A sizing answer is judged on the chain: each step one assumption, stated with a number and a reason, so the interviewer can challenge any link without the whole answer falling over. Start with people because they are the one number given, and move to households because SIP decisions are made at home.

How do you defend each assumption?

Give a reason in one breath for each. Four per household reflects a mix of nuclear and joint families in a large city. Fifteen per cent penetration is the weakest link: it assumes a minority of salaried and business households invest monthly through funds, and you should say you would check it against published folio or SIP account data for the city. Families with an SIP often run more than one, one per goal or per earner, hence 1.6. A Rs 2,500 average blends small starter SIPs with larger ones.

Then check the result a second way. Rs 150 crore a month is Rs 150 per person, or Rs 4,000 per investing household. If an investing household in this city earns around Rs 50,000 a month, that is about 8% of income going into SIPs, which is plausible. A figure that implied half of income would tell you an assumption has gone wrong.

One-day 95% VaR is about Rs 9.9 crore, and ten-day VaR about Rs 31 crore. On 19 days out of 20 the fund should lose less than 1.645 times 1.2%, which is 1.97% or Rs 9.87 crore. Scaling by the square root of 10 gives Rs 31.2 crore. What VaR does not say is how much is lost on the one bad day in 20.

What does a 95% one-day VaR actually promise?

Think of a flood marker on a riverbank that says the water passes this line in one year out of twenty. It tells you where the bad years start. It says nothing about whether the water stops a foot above the line or ten feet above it. VaR is a threshold: on 95% of days the loss should stay below it, and on the other 5% it is exceeded by an amount VaR does not measure. For a normal distributionthe bell-shaped curve where most outcomes sit near the average and extreme ones are rare in a fixed, symmetric way, the 5% left tail starts 1.645 standard deviations below the average.

The working is three multiplications. Daily standard deviation 1.2%, times 1.645, is 1.974%. On Rs 500 crore that is Rs 9.87 crore. The quiet assumption is that the average daily return is about zero, which is reasonable over one day for an equity fund.

Why does ten days scale by the square root of ten and not by ten?

Because daily moves partly cancel. If each day is independent, variances add, so ten days carry ten times the variance and the square root of ten, about 3.16, times the standard deviation. Ten-day VaR is Rs 9.87 crore times 3.162, about Rs 31.2 crore, not Rs 99 crore, because a bad day is often followed by a better one. Multiplying by ten would assume every one of the ten days is a 5% tail day, which almost never happens.

Now the limitation, which is what the interviewer is waiting for. VaR is silent on the size of the tail. On a normal curve the average loss on those worst 5% of days, the expected shortfallthe average loss on the days that are worse than the VaR line, is about Rs 12.4 crore, but real daily returns have fatter tails than the normal, so the true figure can be larger. The square-root rule also assumes independent days and an unchanged portfolio; in a falling market, losses cluster and correlations rise, and both assumptions break when VaR is needed most.

The growth stock falls about 33% and the value stock about 9%. With a constant-growth model, price is dividend over required return minus growth. The value stock goes from 5 over 0.05, 100, to 5 over 0.055, 90.9. The growth stock goes from 1 over 0.01, 100, to 1 over 0.015, 66.7. Their implied durations are 20 and 100 years.

Why can a stock have a duration at all?

Think of two people selling their future income for cash today. One will earn a steady amount starting now; the other will earn little for years and a lot later. If the lender's interest rate rises, the second one's offer falls much more, because every rupee of it has to be discounted over a longer wait. A stock is a stream of future cash, and the further out the cash sits, the more its price moves when the discount rate moves: that sensitivity is its duration. In the constant-growth modela valuation that prices a share as the dividend expected next year divided by the required return minus a steady growth rate, price is D over (r minus g), and duration works out to 1 over (r minus g).

How do you work the two falls quickly?

Watch the gap, r minus g, not r. For the value stock the gap goes from 5 points to 5.5 points, ten per cent wider, so the price falls by 1 minus 5 over 5.5, which is 9.1%. For the growth stock the gap goes from 1 point to 1.5 points, fifty per cent wider, so the price falls by a third, 33.3%. The same half-point move is a small change to a wide gap and a large change to a narrow one.

The duration numbers also show their own limit. Twenty years times half a point predicts a 10% fall, close to the true 9.1%. A hundred years times half a point predicts 50%, well off the true 33.3%, because a straight-line estimate fails when the move is large next to the gap. And the model assumes growth runs forever at 9% against a 10% required return, which is fragile; real growth stocks are valued with a high-growth phase that fades, but the direction of the answer holds.

A shared driver that moves slowly lifts or sinks both funds together across years, while short-term noise pushes them in opposite directions day to day. Within a month the slow driver barely changes, so daily correlation reflects only the opposing noise. Across years it dominates. With daily noise of 0.8% at -0.5 correlation and a shared yearly drift of 20% standard deviation, yearly correlation comes out at about +0.57.

What kind of situation produces this?

Think of two ice cream stalls on the same beach. On any given day, a customer who buys from one does not buy from the other, so their daily sales move against each other. Across years, both do well in hot summers and badly in wet ones. Correlation is not one fixed number between two things; it depends on which driver dominates at the horizon you measure, and different drivers dominate at different horizons. For funds, the slow driver might be the economy's earnings cycle, which both portfolios share; the fast one might be money rotating between their two styles day to day.

Can you show it with numbers?

Build each fund's yearly return from two parts. A shared drift, the same for both within a year but different from year to year, with a standard deviation of 20%. Plus daily noise of 0.8% a day for each fund, correlated at -0.5 between them, over 250 trading days. Within a month the drift is a constant, so it drops out of any correlation measured around the month's average, and the daily figure is the noise's -0.5. Across years, the drift contributes 0.2 squared, 0.040, to covariance, and the noise contributes -0.5 times 250 times 0.008 squared, -0.008, leaving +0.032.

Give the condition and a second mechanism. The sign flips only if the shared drift's variance is larger than the summed noise covariance; with a drift of 10% instead of 20%, covariance would be 0.010 minus 0.008 and correlation barely positive. A second route is timing: if one fund's holdings are priced with a lag, its daily moves look unrelated or even opposite to the other's, while over a year the lags wash out. Either way, the lesson for a fund analyst is that a diversification benefit measured on daily data may not exist at the horizon a client actually holds.

Roughly Rs 560 to 784 crore, on Rs 56 crore of operating profit at assumed multiples of 10 to 14 times. Fees are 8,000 times Rs 2 lakh, Rs 160 crore; a 35% margin leaves Rs 56 crore. The case for buying is durability: a brand, accreditation and a campus that keep the seats full for decades. The multiple you ask for depends on how sure the buyer can be of that.

What is the interviewer really testing?

Think of selling the tea stall outside a busy railway station. The buyer is not paying for the kettle; he is paying for the queue that shows up every morning and the confidence it will keep showing up. Any institution can be valued as a stream of cash plus a judgement about how long and how reliably that stream lasts, and the question has two halves because those are the two halves of a valuation. "Why should I buy" asks for the durability story; "how much" asks for the numbers.

How do you get from fees to a value?

Revenue is students times fees: 8,000 times Rs 2 lakh is Rs 160 crore a year. A 35% operating margin leaves Rs 56 crore, and the multiple you put on that profit is where the durability argument turns into a number. At 10 times it is Rs 560 crore, at 14 times Rs 784 crore. These multiples are assumptions for the exercise; say you would set them against what comparable education businesses have changed hands for, and against a cash flow valuation.

Then sell the durability and test it. The selling points are a waiting list larger than the intake, accreditation that a new entrant would take years to earn, land owned rather than leased, and alumni who send their children. The tests are the risks: if enrolment falls 10% while costs stay fixed, operating profit drops from Rs 56 crore to about Rs 40 crore, because every rupee of lost fees falls straight to profit. A per-seat check helps too: Rs 560 crore over 8,000 seats is Rs 7 lakh a seat, three and a half years of fees.

Name one structural limit. Many colleges are run by trusts or societies that cannot distribute profit, so in practice a buyer may be acquiring a management contract, the land or a related company rather than the college itself; the cash a buyer can actually take out may be smaller than the operating profit. Asking who can receive the cash shows the interviewer you think like an owner.

200 miles. The car covers 100 miles at 50 miles an hour, so the journey lasts two hours. The bird flies the whole time at 100 miles an hour, so it covers 200 miles however many times it turns. The long way agrees: the bird's round trips are 133.3, then 44.4, then 14.8 miles, each a third of the last, summing to 133.3 times 1.5, 200.

What is the question really asking?

Think of a dog running back and forth between you and your front door while you walk home. You could trace every turn, or you could notice the dog runs the whole time you are walking. When something moves at a constant speed for a known length of time, distance is speed times time, and the path it takes in between does not matter. The bird's speed is given; the only thing you need is how long it flies, and that is the car's travel time.

Can you prove it the long way, too?

Yes, and doing so briefly earns credit. The car and bird close a 100 mile gap at 150 miles an hour, meeting after two thirds of an hour, 66.7 miles from B. The bird flies 66.7 miles back to B, arriving at 1 hour 20 minutes, when the car is 66.7 miles from A. Each new chase starts with a gap a third of the last, so each round trip is a third as long: 133.3, 44.4, 14.8 and so on. A geometric series with ratio one third sums to the first term times 1.5, which is 200.

Why would a fund interviewer ask this? It tests whether you look for the quantity that is fixed before diving into detail, the same habit that makes you ask what an investor actually earned before reconciling every trade. The limitation is that it is a pure puzzle; turning instantly at each meeting is an idealisation that makes the infinite number of turns harmless.

Rs 39 a share. Enterprise value is the house: Rs 1,000 crore for the business itself. The debt is the mortgage, Rs 300 crore owed to the bank, which leaves the owner Rs 700 crore of the house. The cash is money in the drawer that the owner keeps on top, Rs 80 crore. Together that is Rs 780 crore for the shareholders, which over 20 crore shares is Rs 39 each.

What does the house stand for, and what does the mortgage stand for?

A family owns a house worth Rs 1 crore with a Rs 30 lakh home loan, and keeps Rs 8 lakh in a drawer. If they sold everything and settled up, they would walk away with the house's value, less the loan, plus the drawer: Rs 78 lakh. Enterprise value is the value of the operating business, the house; lenders are paid first out of it, and whatever cash the company already holds belongs to the shareholders on top. Scale the same family up a thousand times and you have this company.

Why is cash added back rather than subtracted?

Because enterprise value was built to leave it out. A buyer of the whole company would pay for the business and inherit the cash, so the price of the business alone nets the cash away. Going from enterprise value back to equity, you reverse that step: take off what the lenders are owed and add back the cash the shareholders already own. Subtracting cash as well, the common slip, gives 1,000 minus 300 minus 80, Rs 620 crore, or Rs 31 a share, which hands the drawer to the bank.

Add the refinements an interviewer will probe. Anything else with a claim ahead of the ordinary shareholders comes off too, such as preference shares or a minority partner's share of a subsidiary, the way a second loan on the house would. And the share count should include options and convertibles that are likely to become shares. The limitation of the analogy is that a house is easy to price on its own; a business's enterprise value is itself an estimate, and every rupee of error in it lands on the equity.