Portfolio Management puzzles, solved step by step

About 25% for one year and about 1.75% for the ten-year average. One year: zero is 10 over 15, or 0.67 standard deviations below the mean, and the normal table gives 25.2%. The ten-year average has a standard deviation of 15 over the square root of 10, which is 4.74, so zero is 2.11 standard deviations away and the chance falls to 1.75%.

Why is a losing year so common when the portfolio expects 10%?

Think of a bus that is due every 10 minutes but can be 15 minutes early or late on a normal day. Being late is not rare; it happens about one day in four. A 10% expected return with 15% volatility means zero sits only two thirds of a standard deviation below the average, and about a quarter of any normal distribution lies further out than that. So a client holding this portfolio should expect a losing year roughly one year in four, even if nothing is wrong.

What does the square root of ten do, and what does it not do?

Averaging independent years cancels part of the noise: good and bad years offset. The standard deviation of an average of n years is the one-year figure divided by the square root of n, so ten years takes 15 down to 4.74. The chance of a losing average falls sharply with time, but the spread of the total amount you end up with keeps growing. The total ten-year return has a standard deviation of 15 times the square root of 10, about 47 points, against 15 for one year. A longer horizon makes a loss less likely, not smaller when it comes.

Name the assumptions when you give the number. Real yearly returns have fatter tails than a normal curve and are not fully independent; losing years tend to cluster. Both make the true ten-year figure higher than 1.75%, so treat it as a floor, not a promise.

The average trade loses 0.2%, and break-even needs a 75% hit rate. Expected value is 0.7 x 1% minus 0.3 x 3%, which is 0.7% minus 0.9%, or minus 0.2%. To break even the winners must pay for the losers: p x 1 = (1 minus p) x 3, so p is 3 over 4, or 75%. The trader is right more often than wrong and still loses money.

Why does a 70% hit rate not settle the question?

A shopkeeper who makes a small profit on seven sales out of ten but sells the other three at a big loss can still close the month in the red. Counting the happy sales tells you nothing until you know how big each one was. Expected return is each outcome's probability times its size, added up, so a hit rate only means something next to the payoff ratio. Here the losers are three times the size of the winners, and that ratio is doing all the damage.

How do you find the break-even hit rate in one line?

Set expected value to zero and solve. The break-even hit rate is the loss size divided by the sum of the win and loss sizes: 3 over 1 plus 3, which is 75%. Every point of hit rate is worth 0.04% a trade here, because moving one trade in a hundred from loser to winner swings 4 points in total. So the trader is 5 points of hit rate, or 0.2% a trade, short of break-even.

Then say what you would change. The trader can cut losers sooner, let winners run further, or be more selective; lifting the average win to 1.5% with the same losses moves break-even to 3 over 4.5, about 67%. Averages also hide transaction costs, which push break-even higher still.

About Rs 75.1. The formula C over r gives the value one year before the first payment. With the first payment in year 4, the stream is worth 10 over 0.10, or Rs 100, at year 3. Discount that back three years: 100 over 1.1 cubed is Rs 75.1. As a check, Rs 100 less the present value of the three missing payments, Rs 24.9, gives the same answer.

Where does the perpetuity formula put its answer in time?

Think of a pension that starts paying at retirement. Its value on the day before the first cheque is one number; its value to a 30-year-old is that number shrunk by three decades of waiting. C over r values a level perpetuity exactly one period before its first payment, so a delayed stream is the ordinary perpetuity discounted from that point. Here the first payment is in year 4, so C over r lands at year 3, where it reads Rs 100. One more discounting step gets it to today.

How do you check it without the formula?

Start from what you know: Rs 10 a year from year 1 is worth Rs 100. The delayed stream is that same stream with the first three payments cut out. Their present values are 9.09, 8.26 and 7.51, which add to 24.87, and Rs 100 less 24.87 is 75.13. Two methods agreeing to the paisa is what makes the answer safe to say out loud.

This is the same step as a terminal value in a discounted cash flow. The Gordon growth value at the end of year 5 is a year 5 number and must be discounted five years, not six. Counting the periods wrong by one is the most frequent error in valuation models, and this puzzle is the cleanest place to catch it.

The Sharpe ratio is about 0.44 and the information ratio is 0.5. Sharpe divides the excess over cash, 14 minus 6 or 8 points, by total volatility of 18: 0.44. The information ratio divides the excess over the benchmark, 14 minus 12 or 2 points, by the tracking error of 4: 0.5. Sharpe judges the whole portfolio's risk; the information ratio judges only the manager's active bet.

Why are there two ratios for one fund?

Picture judging a cook. One question is whether the whole meal was worth its price. Another is whether the chef's changes to the standard recipe made it better. The Sharpe ratio asks whether the fund's total return beat cash by enough to justify all its risk, and the information ratio asks whether the manager's departures from the benchmark earned enough to justify that active risk. An investor choosing an asset mix cares about the first; one who has already chosen the market and is picking a manager cares about the second.

How do you read 0.44 and 0.5 once you have them?

The tracking errorThe standard deviation of the difference between a fund's return and its benchmark's return, a measure of how far the fund strays. is small against total volatility because most of the fund's ups and downs are the market's, which the benchmark shares. A 0.5 information ratio from a single year is respectable on paper but statistically weak: one year of 2 points against a 4-point tracking error is half of one standard deviation. It would take many years at that rate before anyone could separate skill from luck with confidence. The Sharpe ratio is also best compared with the benchmark's own Sharpe over the same period, not read alone.

Say one limitation in the room: both ratios assume returns are roughly normal. A fund that sells insurance-like options can post a smooth, high Sharpe for years and then lose a large amount in one month, which neither ratio sees in advance.

Light rope 1 at both ends and rope 2 at one end, together. When rope 1 burns out, light the other end of rope 2; when rope 2 burns out, 45 minutes have passed. Rope 1 lasts 30 minutes because two flames share its 60 minutes of burning. At that moment rope 2 has 30 minutes left, and lighting its other end halves that to 15. 30 plus 15 is 45.

Why does uneven burning not spoil the halving?

Two people eating one plate of food from opposite sides finish it in half the time one person would take, whether the food is piled high on one side or spread evenly. They simply meet off-centre. A rope holds 60 minutes of burning in total, and two flames consume it twice as fast, so it is gone in 30 minutes wherever the flames happen to meet. The unevenness decides the place, never the time. That is the only fact the puzzle needs.

How do you know rope 2 has exactly 30 minutes left at the half-hour?

Rope 2 has been burning from one end for 30 minutes, so it has used 30 of its 60 minutes of material, whatever length that turned out to be. You never measure length; you only ever track time used and time left. Lighting the far end of what remains halves the 30 minutes left, and the rope finishes 15 minutes later. Say the timeline in that order and the answer is audible.

Interviewers often extend it: with the same two ropes you can also time 15, 30, 60 and 90 minutes, each by deciding which ends are burning at which moment. Answering one extension shows you own the principle rather than a memorised trick.

Why would a real asset desk ask this? It is a clean test of whether you separate the thing you can measure from the thing that is noisy. A rent roll is lumpy month to month; the annual total is what the valuation rests on. Saying that link in one sentence costs nothing.

About Rs 19.5 lakh less, and about 62% of that gap opens in the last five years. Rs 48 lakh paid in grows to about Rs 1.73 crore at 11.5% net in the direct plan and Rs 1.54 crore at 10.5% in the regular plan. The gap is Rs 7.4 lakh after 15 years, then more than doubles in five. The fee is a percentage of a balance that is largest at the end.

Why does a one-point fee gap cost so little early and so much late?

Think of a small leak in a water tank that is being filled from a tap. When the tank is nearly empty, the leak loses a trickle; when it is almost full, the same size of hole loses a lot more, because more water is pressing on it. A fund's costs are a percentage of the balance, and a monthly plan's balance is tiny in the early years and large in the last few, so most of the rupee cost falls at the end. On top of that, every rupee lost early would itself have compounded, so the gap grows faster than the balance.

How do you estimate the gap without a spreadsheet?

Use the future value of a monthly annuity at each net rate and subtract. The gap is about 13% of the regular plan's final value, which is far more than the one-point fee suggests, because the fee compounds for the whole life of every instalment. A quick mental check: the average rupee is invested for about ten years, and 1% a year for ten years is roughly 10% of the pot. The exact figure is a little higher because the early instalments sit in the fund for up to twenty years.

State the assumptions: a steady 12% gross return every year, costs that never change, and no tax. Real returns arrive unevenly, and expense ratios differ by scheme and change over time, so confirm the current figures for any real scheme. The limitation does not change the shape: whatever the numbers, the drag on a monthly plan is back-loaded.

About 8.2%. The bond pays 7 a year and also rises from 95 to 100 by maturity, which is about 1 point a year over five years. That is 8 a year of return on money that averages about 97.5 invested, and 8 over 97.5 is 8.21%. The exact yield with annual coupons is 8.26%, so the shortcut is within a few hundredths of a point.

Where does the return on a discount bond come from?

Buy a Rs 100 gift voucher for Rs 95 that also pays you Rs 7 each year until it can be cashed at full value in five years. You collect the Rs 7 every year, and you also pocket the Rs 5 discount at the end. A discount bond's yield is its coupon plus the discount it recovers as the price is pulled to par, spread across the years to maturity. Spread evenly, that is 1 point a year, so the bond earns roughly 8 a year.

Why divide by the average price rather than 95?

The amount you have invested is not fixed at 95: in this rough picture the bond's value drifts up towards 100 over the five years, so the capital at work averages about 97.5. Dividing by the average price corrects most of the error from spreading the discount in a straight line. Dividing 8 by 95 instead gives 8.42%, too high. As a check, a bond priced at 8.21% comes out at 95.21, within a fraction of 95.

Know when the shortcut drifts. It is close for short bonds near par and gets worse for long maturities or deep discounts, where the true discounting curve bends away from a straight line. For a quick answer in the room, 8.2% with the one-line derivation is what the interviewer is after; then say that the exact figure is a touch higher.

About 47%, close to a coin toss. One manager beats the benchmark five years running with probability one half to the fifth, or 1 in 32. The chance that none of the 20 does it is 31/32 to the 20th power, about 53%. So the chance that at least one does is about 47%. In a crowd of unskilled managers, a five-year streak is almost as likely as not.

Why is a five-year streak weak evidence of skill?

Put 20 people in a room and ask each to call five coin tosses. Somebody calling all five correctly would look gifted, but with that many people, somebody usually does. The chance of a streak for one named person is small, but the chance that someone in a crowd produces one is large, and fund selection looks at the crowd. An investor who screens a category for managers with five straight winning years is sampling exactly the lucky tail of this distribution.

How do you compute at least one without adding up cases?

Use the complement. The chance that at least one of many independent trials succeeds is one minus the chance that every trial fails. Each manager fails the streak with probability 31/32. All 20 fail with probability (31/32) to the 20th, which is about 0.53. One minus that is 0.47. The expected number of streaks is 20 over 32, or 0.625, which is a separate and also useful number: less than one streak on average, yet close to even odds of seeing at least one.

The limitation runs in two directions. Real managers are not independent, since many hold similar stocks, which makes streaks cluster and the calculation rougher. And some managers do have skill; the point is only that a streak on its own cannot tell the skilled from the lucky. That takes longer records, consistent process and a view on why the edge should persist.

About 112 rupees to the dollar. Relative PPP says the higher-inflation currency loses value at the inflation gap. Each year the rate rises by 1.05 over 1.02, about 2.94%. Over ten years that compounds to a factor of 1.336, and 84 times that is about 112.2. The simple 3% shortcut gives 112.9, close enough in the room.

Why should higher inflation weaken a currency?

If a cup of tea costs Rs 20 this year and Rs 21 next year while the same cup abroad goes from 1 dollar to 1.02, the rupee has lost more buying power than the dollar. If money is to buy roughly the same basket in both countries over time, the exchange rate has to move by the gap between the two inflation rates. That is relative purchasing power parity: it does not say what the rate should be today, only how it should drift.

What does this mean for an investor holding assets abroad?

For a rupee-based investor, a US asset earns its local return plus the rupee's depreciation. A US bond yielding 3 points less than an Indian bond is not worse on this reasoning, because the expected currency drift gives back roughly the same gap. Measured the other way, the rupee loses about 25% of its dollar value over the ten years, which is what a dollar-based investor in Indian assets must earn back through higher local returns.

Say the limitation plainly. PPP is a weak guide over one or two years: capital flows, interest rate moves and risk appetite can push the rate far from this path and hold it there. Over a decade the evidence for the drift is better, but it is a tendency, not a rule. The inflation rates themselves are assumptions; for a real view, confirm current figures and consider what each central bank is targeting.

About 0.6 points a year here. Cash drag is the cash weight times the gap between what equities and cash earn: 0.08 x (14 minus 6.5) is 0.6. The fund returns 13.4% instead of 14%. The drag is largest in strong markets and turns into a cushion when equities fall: in a year equities lose 10%, the same cash saves about 1.3 points.

Why is the cost the gap, not the whole equity return?

Suppose you keep a little of your salary in a savings account instead of a higher-paying deposit. You have not lost the whole deposit rate on that money, only the difference between the two rates. Cash in an equity fund still earns something, so the return given up is the cash weight times the gap between the equity return and the cash return. Here that is 8% of 7.5 points, which is 0.6 of a point for the whole fund.

Why does it matter most in strong markets?

The drag scales with the equity return. In a 30% year the same 8% of cash costs about 1.9 points, in a 14% year 0.6, and in a year equities lose 10% it adds about 1.32 points. That is why cash drag shows up most in the years a fund is judged hardest: a fund that trails its benchmark in a strong rally often simply held more cash than the index, which holds none.

Funds hold cash for real reasons: redemptions, new money not yet invested, and dry powder for opportunities. The implementation answer is not zero cash but equitising it, for example by holding index futures against the cash balance so that the fund keeps market exposure while holding liquidity. Say that as the practical close, and note that futures carry their own costs and margin needs.