Mutual Fund Mastery puzzles, solved step by step

A P/E of 20x, rising to 25x if ROE falls to 12% while the P/B stays at 3. Earnings are ROE times book value, so P/E equals P/B divided by ROE: 3 over 0.15 is 20. At 12% ROE the same P/B gives 3 over 0.12, or 25. The shares look no dearer on book, but each rupee of earnings now costs 25% more.

How do P/B and ROE give you the P/E?

Suppose a shop is sold for three times the money the owner has put into it, and the shop earns 15% a year on that money. A buyer paying 300 for every 100 put in gets 15 of profit a year for her 300, which is 20 years of profit. ROE is the bridge between book value and earnings, so P/E is simply P/B divided by ROE. Price sits on top of both ratios, and it cancels.

What does an unchanged P/B after a fall in ROE tell you?

If returns fall and the price-to-book does not, the shares have become more expensive per rupee of profit, even though nothing on the book-value screen moved. For a lender, book value is the natural anchor, and many investors screen on P/B alone. This is the case where that screen misleads: the same 3.0x now buys 12% returns instead of 15%. To hold a 20x P/E at 12% ROE, the P/B would have to fall to 2.4x.

There are fair reasons the market might hold the P/B: it may expect ROE to recover, or the fall may be a one-off provision. Say that, then say the test: if ROE stays at 12%, a P/B of 3.0 is paying for returns that are no longer there. The ratios here are illustrations; the relationship holds for any company whose book value is meaningful.

For this client the active fund is far riskier, though the two volatilities look almost identical. Her pain is falling behind the index, and that gap is measured by tracking error: 6% a year against 0.3%. With no edge, the active fund trails the index by 5 points or more in about 20% of years; the index fund almost never does. Volatility of 16% and 17% describes how much each fund swings, which is nearly the same.

Why does the same volatility not mean the same risk?

Two students both score between 60 and 90 across a year of tests. One sits every test next to her twin and scores within a mark of him every time; the other drifts ten marks above or below him. If what upsets the family is doing worse than the twin, only the second student is a worry. Risk depends on what the investor measures herself against: total volatility answers how much the fund swings, and tracking error answers how far it strays from the index.

How do you put a number on the regret?

Treat the yearly gap to the index as roughly normal, centred on zero if the manager has no edge, with a standard deviation equal to the tracking errorThe standard deviation of the difference between a fund return and its benchmark return.. A shortfall of 5 points is 5 over 6, or 0.83 standard deviations below the centre, which the normal table puts at about 20%. For the index fund, 5 points is about 17 standard deviations away: effectively never.

When is volatility the right measure?

Volatility is the right measure when the client cares about losing money outright, and tracking error when she cares about falling behind a benchmark. A retiree drawing income cares about the first; an investor who reads the index level in the newspaper every morning cares about the second. Most people care about both, which is why a suitability conversation asks which one hurts more. The normal assumption is a simplification and real gaps have fatter tails, so treat 20% as an order of magnitude.

About Rs 95,000 before costs, and the trade itself pushes the premium down. The participant buys the underlying basket at Rs 245.60 a unit, swaps it for 50,000 new ETF units, and sells them at Rs 247.50, keeping Rs 1.90 a unit. That is a 0.77% premium on Rs 1.23 crore. Selling the new units adds supply, so the price falls towards NAV until the premium no longer covers costs.

Where does the profit come from?

Picture a sweet shop that sells a box of twelve laddoos for more than the twelve laddoos cost loose. Someone will buy loose laddoos, box them, and sell the boxes until the gap closes. An ETF unit is a box of shares, and when the box trades above the value of what is in it, the participant who can make new boxes earns the gap. Only authorised participantsLarge brokers or market makers allowed by the fund house to create and redeem ETF units in bulk, by exchanging baskets of the underlying shares. can make boxes, and only in creation-unit sizes, here 50,000 units.

Why does the premium shrink rather than persist?

Every round of the trade adds new units to the market, and that extra supply pushes the ETF price down towards the value of its basket. Buying the basket nudges the shares up a little too. The participant stops when the gap no longer pays for brokerage, taxes, impact and the risk of prices moving mid-trade. If those costs are 0.25% of the basket, about Rs 30,700, the net gain is about Rs 64,300, and the trade stops once the price is within about Rs 0.61 of NAV. The reverse trade works on a discount: buy units cheap, redeem them for the basket, sell the shares.

When does the mechanism fail to hold the price?

The arbitrage is only as good as the participants' ability to trade the basket. If the underlying shares are illiquid, or a market is shut while the ETF trades, or participants step back in a stressed market, premiums and discounts can stay wide for days. The iNAV is itself an estimate, refreshed every few seconds, so a small gap may be noise. The 0.25% cost is an assumption for the arithmetic; real costs depend on the basket.

About 7.2% a year: the portfolio's yield to maturity less the expense ratio. Holding to the fund's maturity removes most of the price risk, so the bonds earn roughly their 7.4% yield and the fund keeps 0.2% a year. It will not be exactly 7.2% because coupons are reinvested at future yields nobody knows, the fund tracks its index imperfectly, and the yield you lock in is the one on the day you invest. On these numbers the drift is about plus or minus 0.13% a year.

Why is yield to maturity a fair starting point?

A fixed deposit tells you its rate because the bank promises to hold it to maturity for you. A bond does the same if you hold it to maturity: price swings along the way wash out, and what remains is the yield to maturityThe single discount rate that makes the present value of a bond coupons and principal equal its price today; the return earned if the bond is held to maturity and coupons are reinvested at that same rate.. A target maturity fund holds bonds that mature near one date and then pays out, so an investor who stays to that date earns close to the portfolio's yield to maturity, less the fund's costs. 7.4% less 0.2% is about 7.2% a year.

Why will the realised return not be exactly 7.2%?

Yield to maturity quietly assumes every coupon is reinvested at the same yield, and the future reinvestment rate is unknown. Take a 5-year bond bought at par with a 7.4% coupon. If its coupons can only be reinvested at 6.4%, the realised return is about 7.27% a year; at 8.4% it is about 7.53%. After the 0.2% fee that is 7.07% to 7.33%. The fund also holds bonds that do not mature on exactly one date, keeps some cash, and replicates its index imperfectly, each worth a few basis points either way.

What else should the investor be told?

Three things. The estimate holds only to maturity: an investor who exits after two years takes the market price then, which can be well above or below the path to 7.2%. The yield locked in is the one on the day of investment, not the one quoted at launch. And the figure is before tax; how a debt fund's gains are taxed has changed in recent years, so confirm the current treatment before turning 7.2% into an after-tax number. The estimate is the honest best guess, not a promise.

His average cost is Rs 75 a unit and he needs a 25% rise from 60 to break even, but the fund is no better than before. Rs 1 lakh bought 1,000 units at 100 and 1,667 units at 60, so Rs 2 lakh buys 2,667 units, Rs 75 each. Before the top-up he needed 66.7%. Averaging down moves the break-even point, not the fund's prospects, and it doubles the money exposed to the next move.

Why is the average 75 and not 80?

Spend Rs 100 on mangoes at Rs 100 a kilo and another Rs 100 at Rs 60 a kilo. You have 1 kilo plus 1.67 kilos, 2.67 kilos for Rs 200, which is Rs 75 a kilo. Equal rupees buy more units when the price is low, so the average cost leans towards the lower price; it is the harmonic mean of the two NAVs, not the simple average. That is the same arithmetic that makes rupee-cost averaging in an SIP work, applied here to two purchases.

Does a lower break-even mean the decision was good?

No, and this is the behavioural point. The break-even number is about his purchase history; the fund's next move does not know or care what he paid. His position today is 2,667 units worth Rs 1.6 lakh, with an unrealised loss of Rs 40,000, the same loss he had before the top-up. A further 10% fall now costs Rs 16,000 instead of Rs 6,000. The question that decides whether to add is whether he would buy this fund today at 60 with fresh money if he had never owned it, given his goal and his allocation.

There are good reasons to add after a fall: a disciplined rebalance back to a target equity weight, or an SIP that keeps running through the dip. There are bad ones: wanting to feel closer to breaking even, or refusing to accept that the first purchase was a mistake. The arithmetic is the same in both cases; only the reason differs, and the adviser's job is to ask which one it is.

About 0.48% a year, roughly half a percent. Turnover of 120% means the fund sells and rebuys 1.2 times its portfolio in a year. Each rupee replaced costs one sale and one purchase, 0.4% together, so the drag is 1.2 x 0.4%. That cost shows up as a lower NAV, never as a line in the expense ratio, so the investor pays it without seeing it.

Why does a cost this size not appear on the factsheet?

Think of a shopkeeper who keeps rearranging the stock. The rent is printed on the lease, but every time he returns goods and reorders, the supplier keeps a small handling margin, and that never appears on any bill he shows you. The expense ratioThe annual charge for running a fund: management fee, administration, distribution and similar costs, shown as a percentage of assets. is the rent. Trading costs are the handling margin: brokerage, taxes on trades and the impact costThe amount a price moves against a large buyer or seller while the order is being filled. It is paid through a worse price, never through a bill. of moving prices, all paid through the prices the fund gets, so they land in the NAV and not in the expense ratio. Which explicit trading charges may be loaded inside the expense ratio is set by regulation and should be checked for the market you are in; impact cost is never there.

How do you turn turnover into a cost without double counting?

Portfolio turnoverThe share of a portfolio replaced in a year, usually measured as the smaller of purchases or sales divided by average assets. counts how much of the book is replaced. At 120%, every rupee of assets is sold and rebought 1.2 times in the year. Because the 0.4% already covers both legs of a replacement, the cost is simply turnover times the round-trip cost: 1.2 x 0.4% = 0.48%. The common slip is to say a replacement has a buy and a sell and double it to 0.96%, counting each leg twice. A fund turning over 20% of its book pays only 0.08%.

Put rupees on it, because half a percent sounds small. Rs 10 lakh compounding for 20 years at 11.3% grows to about Rs 85.1 lakh; at 10.82% it grows to about Rs 78.0 lakh. The gap is about Rs 7.0 lakh, paid quietly. The limit of the estimate: impact cost depends on how liquid the stocks are and how large the fund is, so the same turnover costs a small-cap fund of Rs 20,000 crore far more than a large-cap fund of Rs 2,000 crore.

You are indifferent at about 7.18% a year. Rs 50 lakh at 50 matches Rs 1 crore at 60 only if it doubles in ten years, so (1 + r) to the 10th = 2, which gives 7.18%. The rule of 72 says money doubles in 72 / rate years, so doubling in 10 years needs about 72 / 10 = 7.2%. At 8% the early money grows to Rs 107.9 lakh; at 7% only Rs 98.4 lakh.

What is the question really asking?

An uncle offers you his old car now or a new one in ten years. You cannot answer until you know what you would do with the car meanwhile. Money received earlier can be invested, so the fair comparison is what the Rs 50 lakh grows to by 60, set against the Rs 1 crore paid then. At 8%, Rs 50 lakh x 1.08 to the 10th is Rs 107.9 lakh, more than a crore, so the early money wins. At 7%, it is Rs 98.4 lakh, just short, so the crore wins. The answer flips somewhere between.

Why is the crossover the rule of 72?

The crore is exactly twice the 50 lakh, and the wait is ten years. So the choice reduces to one question: can you double your money in ten years? The rate that does that is 7.18%, and the rule of 72A shortcut for compounding: money doubles in roughly 72 divided by the yearly rate in years, so 7.2% doubles in about ten years. gives 72 / 10 = 7.2% in your head. The rule works because the doubling time is ln 2 / ln(1 + r), and ln 2 is 0.693; using 72 instead of 69.3 corrects for the rates people usually quote being near 8%.

The arithmetic is not the whole decision, and the interviewer will want you to say so. The rate that matters is the after-tax rate you can actually earn, not a headline rate. Rs 1 crore promised at 60 carries the risk that the promiser does not pay; Rs 50 lakh in hand does not. And a person who needs the money at 50, for a child's education say, values it more than the arithmetic does.

Expected value says take the coin toss on the gain and the sure loss, the opposite of what most people do. The toss is worth Rs 60,000 against a sure Rs 50,000; on losses it costs Rs 60,000 against Rs 50,000. Combined, the popular pair gives +Rs 50,000 or -Rs 70,000, which is worse in every outcome than +Rs 70,000 or -Rs 50,000. In a portfolio the same pattern sells winners early and holds losers.

What does expected value say for each choice?

Weight each outcome by its probability and add. In choice 1 the coin toss is worth 0.5 x Rs 1,20,000 = Rs 60,000, Rs 10,000 more than the sure thing. In choice 2 the toss costs 0.5 x Rs 1,20,000 = Rs 60,000, Rs 10,000 worse than the sure loss. So a person who maximises expected value gambles on the gain and accepts the loss, and most people do the reverse. Taking a sure Rs 50,000 is defensible on its own if the person cannot afford to walk away with nothing; the oddity is the switch to gambling the moment the frame becomes a loss.

Why is the popular pair worse in every outcome, not just on average?

Put the two choices together, because in real life they arrive together. The popular pair, sure +50,000 plus the loss gamble, ends at +50,000 if the toss goes well and -70,000 if it goes badly. The expected value pair, the gain gamble plus the sure loss, ends at +70,000 or -50,000. The second pair beats the first by Rs 20,000 whichever way the coin falls, so the popular choice is not a matter of taste; it is a mistake caused by judging each choice in its own frame. This is the pattern Kahneman and Tversky described in prospect theoryA description of how people actually choose under risk: they judge outcomes as gains or losses from a reference point and feel losses more sharply than equal gains..

Now the portfolio. A stock up 30% is a gain, so the investor wants the sure thing and sells. A stock down 30% is a loss, so the investor prefers to gamble on a recovery and holds. This is the disposition effectThe habit of selling investments that have risen too early and holding investments that have fallen too long., and it leaves a portfolio full of its weakest holdings while the strongest have been cashed out. The same pull shows up in fund redemptions: investors exit a fund that has done well to lock in the gain and stay in one that has done badly to avoid booking a loss. The limit of the lesson: expected value ignores how much a person can bear to lose, and for a large loss relative to wealth a sure thing can be the right choice.

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.