Mutual Fund Mastery puzzles, solved step by step

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.

Selling B is cheaper on tax by up to about Rs 14,103, and the right question is which fund he would buy today. Rs 2 lakh of A carries a gain of Rs 46,154, so Rs 5,769 of tax at the assumed 12.5%. Rs 2 lakh of B carries a loss of Rs 66,667: no tax, and a loss that can offset other gains. The urge to sell the winner is the disposition effect.

Why does selling the winner feel right?

Picture someone clearing out a wardrobe who gives away the shirts he likes and keeps the ones that never fit, because giving those away would admit the purchase was a mistake. The disposition effect is the habit of selling what has gone up to enjoy the gain and holding what has gone down to avoid the regret, and it decides on the past price rather than on the future. The purchase price of each fund is a fact about the client, not about the funds.

How much does the instinct cost in tax?

Only the gain inside the units sold is taxed. A is worth 1.3 times its cost, so 0.3 over 1.3, about 23%, of any rupee sold is gain: Rs 46,154 on Rs 2 lakh, and Rs 5,769 of tax at the assumed 12.5%. B is worth 0.75 times its cost, so every Rs 2 lakh sold realises a loss of Rs 66,667, pays nothing now and can be set against other gains, worth up to Rs 8,333 at the same rate. The full swing between the two is up to Rs 14,103.

Is tax the whole answer?

No, and saying so is what earns the marks. The test that cuts through the feeling is this: if the client held only cash today, would he buy A, B, or neither in these amounts? If B fell because its strategy has stopped working, selling it fixes the portfolio and saves tax. If B fell with its whole category and A has simply had a good run, the portfolio question is about weights and concentration, not about which one he feels good selling.

State the limits. The 12.5% rate is an assumption for the arithmetic; real rules distinguish short and long holding periods, can carry exemption thresholds, and restrict which gains a loss may be set against. Confirm the current rules before using any rate with a client. And this is a framework for a conversation, not an instruction about any real fund.

About Rs 16,691 a month now, and about Rs 27,048 after a three-year wait, 62% more. At 11% a year, taken as 0.917% a month and paid at the start of each month, 144 payments of Rs 16,691 grow to Rs 50 lakh. With 108 payments the SIP must be Rs 27,048. The early years carry the most growth, so losing them costs far more than their share of time.

How do you get the starting SIP?

A monthly SIP is a stream of equal payments, each growing until the goal date. The first rupee grows for 144 months, the last for one. Add up what one rupee a month becomes, and the required SIP is the goal divided by that sum. At 11% a year, taken as 11 over 12, about 0.917% a month, one rupee paid at the start of each month for 144 months grows to about 300. Rs 50 lakh divided by 299.6 is Rs 16,691.

Why is the cost of waiting not proportional to the wait?

Think of planting trees for shade in twelve years. A tree planted in year one is huge by the deadline; one planted in year ten is a sapling. Skip the first three years of planting and you cannot make it up by planting a few extra saplings later. The payments a delay removes are the ones that would have compounded the longest, so the later SIP must replace both the missing payments and the growth they would have earned. Starting now, Rs 24.0 lakh paid in becomes Rs 50 lakh; starting late, Rs 29.2 lakh has to be paid in, because growth contributes Rs 20.8 lakh instead of Rs 26.0 lakh.

The two guesses are worth naming. Twenty-five per cent comes from three years being a quarter of twelve. Thirty-three per cent comes from spreading the same total contributions over nine years instead of twelve, Rs 22,255 a month. Both ignore growth. The true figure, Rs 27,048, sits well above both.

Say the limits. Eleven per cent is an assumption, not a forecast, and equity returns arrive unevenly; a plan built on one smooth rate should be checked against lower rates and reviewed as the date nears. Monthly compounding at 11 over 12 slightly overstates an 11% annual rate; using the exact monthly equivalent raises the SIP to about Rs 17,300, and the ratio of about 1.6 survives either way.

About -10.5% with duration alone and about -9.6% with convexity. Duration gives -7 times 0.015, which is -10.5%. Convexity adds one half times 80 times 0.015 squared, or +0.90%. The price-yield curve bends upward, so a straight-line duration estimate overstates losses when yields rise and understates gains when they fall, and the gap grows with the square of the move.

Why is duration alone not enough for a big move?

Think of estimating how far a ball thrown upward will fall back by looking only at its speed at the start. For a moment the straight line is fine; over a longer stretch the curve of its path takes over. Duration is the slope of the price-yield curve at today's yield, so it is exact for tiny moves and drifts further from the true price the larger the move. The curve bends upward for an ordinary bond, so it always sits above the tangent.

How do you compute both estimates quickly?

Write the move as a decimal first, 0.015. The duration term is minus duration times the move, -7 times 0.015, which is -10.5%; the convexity term is half the convexity times the move squared, 40 times 0.000225, which is +0.90%. Add them: -9.60%. For a 150 basis point fall the same two terms give +10.5% plus 0.90%, a gain of 11.40%, so convexity helps in both directions.

The squared term is why size matters. At 50 basis points the convexity term is only +0.10%, easy to ignore; at 300 basis points it is +3.6%, too large to leave out. The limitation is that this is still a two-term approximation of a curve; for very large moves, or for bonds with call features whose convexity can turn negative, a desk reprices the bond in full rather than trusting the formula.

One third of each withdrawal, about Rs 16,667, is gain, and only that part is taxable. Rs 50,000 at NAV 60 redeems 833.33 units that cost 40 each, so Rs 33,333 is the investor's own capital returning. Deposit interest of Rs 50,000 is income in full. At any single assumed rate t, tax on the SWP is t times Rs 16,667 against t times Rs 50,000.

Is a withdrawal income?

Think of selling three of your own mangoes that you bought at Rs 40 each and can now sell at Rs 60. You receive Rs 180, but only Rs 60 of it is profit; Rs 120 is your money coming back. An SWP is a monthly sale of units, and each unit sold returns its purchase cost plus the gain on it, so only the gain portion is a taxable gain. At cost 40 and NAV 60, the gain is 20 out of 60, one third of every rupee withdrawn.

Why does it hold at any tax rate?

Apply one assumed rate t to both. The SWP's taxable amount is Rs 16,667 and the deposit's is Rs 50,000, so whatever t is, the SWP's tax is one third of the deposit's. At an assumed 30%, that is Rs 5,000 against Rs 15,000 a month. In practice the two are taxed under different heads and can face different rates and holding-period rules, so confirm the current rules; the structural point is that a withdrawal is mostly capital and interest is entirely income.

Now say the limitation honestly. The comparison is not like with like: the deposit's Rs 50,000 leaves the principal untouched, while the SWP is selling units, so the holding shrinks unless the fund grows faster than the withdrawals. And the gain share rises over time: as the NAV climbs, a larger part of every unit sold is gain, so the tax advantage narrows the longer the plan runs. A light tax bill early on is not a measure of whether the withdrawal rate is sustainable.

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.

B has the better Sharpe ratio, 0.6 against 0.5, but without borrowing a client who needs a 16% expected return can only get it from A. A earns 10 points over cash for 20 of volatility; B earns 6 for 10. With borrowing, B scaled to 20% volatility would offer 18%, beating A. Without it, B tops out at 12%.

What does the Sharpe ratio measure?

Think of two delivery riders. One earns Rs 1,000 a day riding 200 km; the other earns Rs 600 riding 100 km. The first earns more, the second earns more per kilometre. The Sharpe ratio is return above cash per unit of volatility, so it ranks how efficiently a fund turns risk into reward, not how much reward it delivers. A earns 10 points over cash for 20 points of volatility, 0.5; B earns 6 for 10, 0.6.

Why can the lower-Sharpe fund still be the one to hold?

The Sharpe ratio assumes you can slide along the line from cash. Mix A with cash half and half and you get 11% at 10% volatility, worse than B's 12% at the same risk; that is B's Sharpe advantage at work. Going the other way, beyond B's own risk, requires borrowing, so an investor who cannot borrow and needs more than 12% expected return has to take the less efficient fund. With borrowing at the cash rate, B levered to 20% volatility would return 6% plus 0.6 times 20, which is 18%, beating A's 16%.

Say the limits. Volatility treats upside and downside swings alike, and a fund with rare large losses can show a flattering Sharpe ratio until one arrives. Figures from a few years of history are noisy estimates, so a 0.5 against 0.6 difference may not be meaningful. And the answer to which to hold depends on the client's required return and tolerance for swings; the Sharpe ratio ranks the funds, it does not choose for the client.

Waiting saves Rs 5,000, and it puts the whole Rs 5 lakh at five more weeks of market risk, about Rs 27,900 either way at one standard deviation. The load is 1% of the redemption value. At an assumed 18% annual volatility, a five-week move has a standard deviation of 5.6%. With no drift assumed, there is about a 43% chance of falling more than the 1% saved. The load is a known cost to weigh, not an automatic reason to wait.

What exactly does waiting buy?

Think of a train ticket with a cancellation fee that disappears if you wait until tomorrow. If you were going to travel anyway, waiting costs nothing. If you need the money for something else tonight, waiting has a cost the fee schedule does not show. An exit load is a certain cost on one side; staying invested for the waiting period is an uncertain gain or loss on the other, and the two have to be sized against each other. The certain side here is 1% of Rs 5 lakh, Rs 5,000.

How big is the uncertain side?

Scale annual volatility to five weeks with the square root of time. Eighteen per cent times the square root of 5 over 52 is 5.58%, about Rs 27,900 on Rs 5 lakh: the possible swing is several times the load. If the market has no drift over those weeks, the chance of losing more than 1% is the chance of a move below -1%, which is 43%. Waiting is close to a coin toss with a small Rs 5,000 tilt in its favour.

The right answer depends on why the money is leaving. If the investor would stay invested anyway, waiting is close to free and the Rs 5,000 is worth having. If the money is needed for a fixed purpose, or he has decided he no longer wants this exposure, five weeks of risk on the full amount is a real cost. Check one more thing: holding periods can also change how a gain is taxed, so confirm whether the same date crosses a tax threshold under current rules. The 18% volatility is an assumption for the arithmetic.

About 23%. Picture 1,000 managers. The 100 skilled ones produce 100 times 0.7 cubed, 34.3 three-year streaks. The 900 unskilled produce 900 times 0.5 cubed, 112.5. Of 146.8 streak holders, 34.3 are skilled, 23.4%. The streak more than doubles the odds of skill, from 10%, yet most streak holders are still lucky.

Why does a three-year streak prove so little?

Think of a test for a rare condition that catches most real cases but also flags plenty of healthy people. If the condition is rare, most positive results are false alarms. When the thing you are looking for is rare, even good evidence leaves most positives coming from the larger group, so the starting proportion matters as much as the evidence. Here skill is the rare condition, at 10%, and a three-year streak is a test that unskilled managers pass one time in eight.

How do you set it up fast in the room?

Use counts, not formulas. Pick 1,000 managers, split them by skill, then split each group by whether it produced the streak, and the answer is one box over the sum of two boxes. Skilled: 100 times 0.343 is 34.3. Unskilled: 900 times 0.125 is 112.5. The answer is 34.3 over 146.8, 23.4%. The same result in odds form: prior odds of 1 to 9, times a likelihood ratiohow many times more likely the evidence is if the manager is skilled than if not of 0.343 over 0.125, about 2.74, gives 2.74 to 9.

Push it once to show judgement. A ten-year streak changes the picture: 100 times 0.7 to the tenth against 900 times 0.5 to the tenth gives about 76% skilled, because the unskilled pass a ten-year test only about once in a thousand. The limitation is that the 10% and 70% are assumptions, and real years are not independent coin tosses; managers with a style that suits a market phase can string wins together without skill.