Mutual Fund Mastery puzzles, solved step by step

Allocation cost 0.70% and selection added 0.75%, a net active return of about plus 0.05%. The IT overweight is 10 points in a sector that trailed the benchmark by 7 points: 0.10 x (5% minus 12%) is minus 0.7%. The bank stocks beat their index by 3 points on a 25% weight: 0.25 x 3% is plus 0.75%. Good stock picking almost exactly paid for a poor sector bet.

What is the difference between allocation and selection?

Picture a selector who picks four spinners for a pitch that suits pace, but whose four spinners bowl better than any other spinners in the country would have. Two separate decisions: how many of each kind, and which ones. Allocation measures the first decision, sector weights against the benchmark's weights; selection measures the second, how the stocks chosen inside a sector did against that sector. Splitting them tells a fund research team whether a manager's skill lies in calling sectors or in picking stocks, which matters more than the total when deciding what to trust next.

Why is allocation measured against the whole benchmark rather than against zero?

Because the extra 10% in IT had to come from somewhere, and the alternative was the benchmark itself. An overweight in a sector that makes money still costs you if that sector made less than everything else you could have held. Say the assumptions behind the split out loud: the bank weight matches the benchmark's, so banks carry no allocation effect; the IT stocks held matched the IT index, so IT carries no selection effect; and the 10 points taken from other sectors came from sectors that earned the benchmark's 12%.

One detail an interviewer may probe. Using the fund's 25% weight in the selection term folds in what the BrinsonThe Brinson method, named after the authors who set it out in the 1980s, splits a fund active return into allocation, selection and an interaction term. framework calls the interaction effect; the textbook version uses the benchmark's weight and reports interaction separately. With the bank weights equal here, both give the same 0.75%.

About 1.35%: beta is the correlation times the ratio of the two volatilities. Correlation of 0.9 says the fund's moves line up closely with the index; the ratio 24 over 16 says the fund's moves are 1.5 times as large. Together, 0.9 x 1.5 is 1.35. Correlation measures how tightly the points hug the line, beta measures how steep the line is, and a fund can have a high correlation with a beta below one.

What is the difference between how tight and how steep?

Two friends walking a dog on a lead: the lead's length sets how far the dog can stray, and the dog's pace sets how far it travels for each step you take. A short lead says nothing about whether the dog walks faster or slower than you. Correlation is the lead, how closely the fund's moves track the index's; beta is the pace, how far the fund moves for each 1% the index moves. They are related, but they answer different questions, and confusing them is the commonest slip in fund statistics.

Can two funds have the same beta and very different correlations?

Yes, and the figure shows it. Fund B has a correlation of only 0.6 but a volatility of 36%, so its beta is 0.6 x 36 / 16, also 1.35. Both funds move 1.35% for a 1% index move on average, but fund B's actual months scatter far from that line, because most of its swings come from things the index does not explain. The square of correlation measures the share explained: 81% of fund A's variance comes from the index against 36% of fund B's. For a fund researcher that changes the reading of beta itself: fund A's 1.35 is a reliable guide to a typical month, fund B's is an average around which a single month can land almost anywhere.

One more asymmetry worth having ready. Correlation is the same whichever way round you put the two series, but the slope is not: regressing the index on the fund gives 0.9 x 16 / 24, or 0.6. Beta always needs a direction, and for a fund it is the fund's return explained by the index. Both figures are estimates from past returns and shift with the window chosen, so quote them with the period they came from.

About Rs 70.7 lakh, under half of the Rs 1.5 crore paid out. Discount each of the 300 monthly payments of Rs 50,000 at 7% a year, about 0.58% a month, and add them up. Payments in the first years are worth almost their face value; payments in year 25 are worth less than a fifth of it. Lump sum and pension are equivalent because Rs 70.7 lakh invested at 7% would pay exactly Rs 50,000 a month for 25 years.

Why is a rupee paid later worth less today?

A relative offers you Rs 1 lakh either today or in ten years. Taking it today is better even with no inflation at all, because you could put it in a deposit and have well over Rs 1 lakh in ten years. The value today of a future payment is the amount you would need to invest now to produce it, so the longer the wait, the smaller that amount. A pension is a long string of future payments, and each one has its own wait. The lump-sum value is the sum of all those discounted payments.

How do you add up 300 discounted payments without a spreadsheet?

Use the annuity formula, which is the sum of a series that shrinks by the same factor each month. With a monthly rate of 7% divided by 12 and 300 payments, each rupee of monthly pension is worth about Rs 141.5 today, so Rs 50,000 a month is worth about Rs 70.7 lakh. A quick check by years: Rs 6 lakh a year for 25 years at 7%, discounted once a year, gives about Rs 69.9 lakh, close because monthly payments arrive a little earlier on average.

Where does most of the gap come from?

From the back half of the pension. At 7%, money doubles in about ten years, so every payment after roughly year 10 is worth less than half its face value today, and the last ones are worth under a fifth. The first ten years pay Rs 60 lakh and are worth about Rs 43.1 lakh; the last fifteen pay Rs 90 lakh and are worth only about Rs 27.7 lakh. For an adviser comparing a pension offer with a lump sum, this is the number that matters, and the rate is the assumption that moves it most: at a lower rate the lump sum needed rises, at a higher one it falls.

Two limits to state. The 7% must be a rate the client could realistically earn with similar safety; using a risky return to discount a guaranteed pension flatters the lump sum. And Rs 50,000 a month is fixed in rupees, so its buying power in year 25 is far smaller than today; if the pension rises with inflation, its value is much higher.

About 0.56% a year when stocks return 13%, and the cash pays for itself only when stocks return less than 6%. The cash earns 6% instead of 13% on 8% of the fund: 0.08 x 7 points is 0.56%, so the fund returns 12.44%. If stocks fall 20%, the same cash means the fund loses 17.92%, a cushion of 2.08 points. Cash is a cost whenever stocks beat cash and a cushion when they do not.

Why is the cost the gap and not the whole return?

Keeping some money in a savings account rather than a share portfolio does not cost you the shares' return; it costs the difference between the shares' return and the interest the account pays. Cash drag is the cash weight multiplied by the gap between what the stocks earned and what the cash earned, not by the stocks' whole return. With 8% in cash, 13% on stocks and 6% on cash, the fund loses 0.08 x 7, or 0.56 points, against a fully invested version of itself.

When does the cash earn its keep?

Whenever stocks return less than cash. The breakeven is exactly the cash rate, 6% here: above it the cash costs, below it the cash helps. In a year when stocks fall 20%, the fund falls only 17.92%, because 8% of it earned plus 6% instead of minus 20%. Over ten years of 13% stock returns, though, the drag compounds: Rs 1 lakh grows to about Rs 3.39 lakh fully invested and about Rs 3.23 lakh with the cash, a difference of about Rs 16,500.

Say why funds hold cash at all before judging it. Some is needed to pay redeeming investors without forced selling; some is new money not yet invested; some is a deliberate call that prices are high. The first two are a cost of running an open-ended fund; only the third is a market view, and it should be judged like any other bet, by whether it paid over a full cycle. A fund that holds a lot of cash and calls it caution is making a market-timing bet, whatever it calls it.

Rs 16,000: the redemption uses all 1,000 January units and 200 June units. The January units gain Rs 15 each, Rs 15,000; the 200 June units gain Rs 5 each, Rs 1,000. An average cost of Rs 25 gives Rs 12,000, because it spreads the cheap January units across units still held. It also loses the purchase dates, which decide each lot's holding period.

Which units does a redemption actually sell?

Think of a shop's milk shelf: staff push the oldest cartons to the front so they sell first. Under first-in-first-out, a redemption is matched against the oldest units still held, lot by lot, so each unit sold carries the price and the date of the purchase it came from. That is the convention commonly used for mutual fund units in India; confirm how it applies to the investor's own folio. Here the January lot of 1,000 goes first, and only then 200 from June.

Why does the average-cost figure mislead?

Because it pretends every unit cost Rs 25. The units actually sold were mostly the cheap January ones, so the true gain is higher now, Rs 16,000 against Rs 12,000. The total gain over the whole holding does not change, Rs 20,000 once all 2,000 units are sold at Rs 35; what changes is how much of it is counted now and how much later. The 800 units left have a cost of Rs 30 under first-in-first-out, Rs 24,000, not the Rs 20,000 an average-cost view would record.

The dates matter as much as the amounts. Suppose she redeems the following March and the line between short and long term is twelve months, an assumption here; confirm the current rule for the fund type. Then the January units are long term and the 200 June units short term, so the Rs 15,000 and the Rs 1,000 may be taxed differently. An average-cost view cannot even ask the question, because it has thrown away the dates. Registrars track every purchase as a separate lot for exactly this reason.

About Rs 58,000 a year before tax, and more after it. On the matching Rs 2 lakh she pays 36%, Rs 72,000, while that slice of the deposit earns 7%, Rs 14,000. Clearing the card from the deposit saves the difference, and leaves Rs 3 lakh in the emergency fund. People keep both because they treat the deposit and the debt as separate pots with separate jobs, when money is interchangeable.

Why does holding both cost money?

Imagine filling a bucket at 7 litres a minute while a hole at the bottom drains 36. Keeping the tap on does not save the water; plugging the hole does. Savings and debt in the same household net against each other: Rs 2 lakh in the deposit and Rs 2 lakh owed on the card leave her net worth exactly where it would be with neither, except that one pays her 7% and the other charges her 36%. Her net worth is Rs 3 lakh whichever way she arranges it; the only thing the arrangement changes is the interest bill.

Why do sensible people do it anyway?

Because people keep money in mental pots with labels, an idea behavioural economists call mental accounting. The deposit is labelled 'emergency', and spending it feels like breaking a promise; the card is labelled 'monthly spending', and its balance feels like a bill to be dealt with later. There is a real worry underneath too: if she empties the deposit, where does the money come from in a crisis? The answer is that once the card is cleared, its unused limit is itself an emergency line, and she still has Rs 3 lakh in the deposit.

Say what the simple figure leaves out. Deposit interest is taxed, so at an assumed 30% slab she keeps only Rs 9,800 of the Rs 14,000, and the cost rises to about Rs 62,200. Card interest is usually charged monthly, which makes 36% a year closer to 43% when compounded. Every refinement makes holding both more expensive, not less, which is why this is among the cheapest wins an adviser can find. Confirm the client's actual tax slab and card terms before quoting her a figure.

Reroll on 1, 2 or 3; keep 4, 5 or 6. The game is worth 4.25 thousand, Rs 4,250. A reroll is a new die worth 3.5 on average, so keep any face above 3.5. Half the time you reroll and expect 3.5; the other half you keep 4, 5 or 6, worth 5 on average. So the game is 0.5 x 3.5 + 0.5 x 5, or 4.25. The right to reroll adds Rs 750 to a plain roll's Rs 3,500.

How do you decide whether to keep a roll?

A friend offers you a sealed envelope known to hold Rs 350 on average, in exchange for the Rs 300 note in your hand. You swap; if the note were Rs 500, you would keep it. Keep any outcome that beats what the alternative is expected to give, and the alternative here is a fresh roll worth 3.5 on average. So 4, 5 and 6 are kept and 1, 2 and 3 are thrown back. The decision depends only on comparing the face in front of you with 3.5.

Why is the option worth Rs 750, and what if there are more rerolls?

Without the reroll you get 3.5 on average. The option lets you throw away the low outcomes and replace them with an average one, so it lifts the value to 4.25: an option is worth something because you choose after seeing the outcome, and its value comes entirely from the bad outcomes it lets you escape. With two rerolls, work backwards: the last reroll is worth 3.5, so the middle roll is kept on 4 or more and is worth 4.25; the first roll is then kept only on 5 or 6, because 4 is below 4.25. The game rises to 4.67.

The fund-desk parallel is any decision with a later choice built in: a redemption right, a switch option, a stop-loss. Each one is worth what it lets you avoid, and each has a threshold set by what the alternative is expected to give. Saying the general rule, keep what beats the continuation value, is what lifts the answer above a piece of arithmetic.

About 114% of the market, so a 10% fall would be expected to cost the fund about 11.4%. Effective exposure is the invested weight times the beta of what is held: 0.95 x 1.2 = 1.14. The 5% in cash looks cautious, but the stocks are more sensitive than the market, and that more than cancels the cash. A typical 10% fall in the index maps to about -11.4% for the fund, before stock-specific moves.

Why can a fund with cash carry more market risk than the index?

Two drivers on the same road: one drives only 95% of the distance but at 1.2 times the speed of traffic, the other drives all the way at traffic speed. The first covers more ground per hour despite stopping short. Market exposure is how much money is in stocks multiplied by how strongly those stocks move with the market, so a small cash buffer can be more than undone by holding high-beta stocks. Here 95% invested at a beta of 1.2 gives an exposure of 114%, more than a fully invested index fund.

What should the investor expect in a 10% fall, and how firm is that?

About -11.4%, as a central estimate. The cash adds a sliver of interest, about 0.025% over a month at an assumed 6% a year, too small to change the answer. Beta is an average relationship, not a promise: in any one fall, the fund's own stocks can do better or worse than the beta implies, and betas measured from calm periods often rise in a sell-off. So -11.4% is the expected fund move, with a range around it that depends on how much of the fund's risk is stock-specific.

For a risk desk the lesson is about which number to monitor. A fund manager who says 'we are defensive, we hold 5% cash' may be running more market risk than a fully invested peer. Report exposure as invested weight times beta, not cash level, when judging how a fund will behave in a fall. The fully invested fund with a beta of 0.9 in the figure has 90% exposure and would be expected to fall about 9% in the same move.

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.