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.

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.

A duration of 4.05, rising to 7.25 after the switch. Weight each duration by its share of the Rs 100 crore: 0.40 x 1 + 0.35 x 4 + 0.25 x 9 = 4.05. Move the Rs 40 crore of 1-year duration into the 9-year bonds and it becomes 0.35 x 4 + 0.65 x 9 = 7.25. A 1% rise in yields now costs about Rs 7.25 crore instead of Rs 4.05 crore: the rate risk is about 79% higher.

Why is portfolio duration a weighted average?

A household's average commute is not the average of the three commutes, it depends on who travels most. If the person with the one-kilometre walk makes most of the trips, the household's average is short. A portfolio's duration is the money-weighted average of its bonds' durations, because each bond's price change counts in proportion to the rupees held in it. Here the biggest holding is the shortest, so the portfolio sits at 4.05, below the simple average of 4.67.

What changes in the portfolio when its duration rises?

The portfolio becomes more sensitive to interest rates in both directions: a 1% fall in yields gains about 7.25% instead of 4.05%, and a 1% rise loses as much. On Rs 100 crore, that is a swing of about Rs 7.25 crore for each 1% move instead of Rs 4.05 crore. On an upward sloping curve the portfolio's yield usually rises too, because longer bonds pay more, so the switch buys extra income with extra rate risk. Convexity also rises, which slightly cushions large moves, and the portfolio loses the cash-like buffer the short bonds gave it for meeting redemptions.

One caveat worth saying: duration is a linear estimate. For a 1% move it is close; for a 3% move, convexity makes the true loss smaller and the true gain larger than duration alone suggests. The durations here are given, and they drift as bonds age and yields move, so a portfolio's duration has to be re-measured, not set once.

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 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.