Portfolio Management puzzles, solved step by step

Stake 10% of capital each time; staking 20% drives long-run growth to about zero. For an even-money bet the growth-maximising, or Kelly, stake is p minus q, 0.55 minus 0.45, which is 10%. That earns about 0.50% a bet in log terms. At 20%, 0.55 x log 1.2 plus 0.45 x log 0.8 is about zero, so twice the stake gives no growth at all despite double the expected profit per bet.

Why does betting more of a winning edge ever make you poorer?

A shopkeeper who wins more often than she loses still goes under if one bad month can wipe out half the shop, because rebuilding from half takes a doubling. Wealth compounds, so what matters over many bets is the average log return, and a loss hurts the log more than an equal-sized gain helps it. With a small stake, the edge dominates. With a large stake, the losses' extra damage grows faster than the edge, and past a point it wins.

How do you find the 10% and check the 20%?

Write growth per bet as g(f) = p log(1 + f) + q log(1 minus f), set its slope to zero, and the answer for even money is f = p minus q. The Kelly fraction for an even-money bet is simply the edge: 55% minus 45% is 10%. At 10%, g is 0.55 x 0.0953 minus 0.45 x 0.1054, about 0.50%. At 20% it is 0.55 x 0.1823 minus 0.45 x 0.2231, about -0.014%: essentially zero, and in fact a touch below it. Over 100 bets at the Kelly stake, capital grows by a factor of about 1.65 on the typical path; at 20% it goes nowhere.

Practitioners usually bet a fraction of Kelly. Half Kelly, 5%, keeps about 75% of the peak growth with half the volatility, and it protects against the real problem: in markets you never know p exactly. If the true edge were 52.5% rather than 55%, the 10% stake would already sit near zero growth.

About 8% a year, so the survivor average overstates the category by 3 points. Weight each group by its share of the 100 funds: 0.8 x 11% plus 0.2 x minus 4% is 8.8% minus 0.8%, or 8%. The survivors' 11% leaves out exactly the funds that did worst, so it describes the winners, not what the average investor in the category earned.

Why does a survivors-only average flatter the category?

Look at a school's alumni wall of fame and you would think every student became a success. The wall shows who is on it, not who was in the class. Funds that do badly are closed or merged into better ones, so a list of funds that exist today is a list of funds that did well enough to survive, and its average is biased upward. The bias is not random: the missing funds are exactly the ones with the worst numbers.

How do you compute the true figure and the bias?

Take a weighted average by number of funds. The true category average is 80% of 11 plus 20% of minus 4, which is 8%, and the bias is the gap: 3 points a year. A quick way to see it: the closed funds were 15 points behind the survivors, and they were a fifth of the category, so they pull the average down by a fifth of 15, which is 3.

State the simplifications. This is an equal-weighted average across funds; an asset-weighted figure could differ, since closed funds are often small. The closed funds also ran for less than five years, so a precise study would compound each fund's returns over the time it existed. Neither changes the direction of the bias, and in real data sets the bias is larger over longer windows because more funds disappear.

Value rises about 14.3%, from Rs 100 crore to Rs 114.3 crore, and the owner's equity rises about 35.7%. Rs 8 crore divided by 0.07 is Rs 114.3 crore. Debt stays at Rs 60 crore, so equity goes from Rs 40 crore to Rs 54.3 crore. At 60% debt the equity is 2.5 times as sensitive as the building, and the same leverage works in reverse if cap rates rise.

Why does a one-point fall in the cap rate lift value by 14%, not by one point?

A cap rate is a yield, and value is income divided by it, just as a bond's price rises when its yield falls. Going from 8% to 7% multiplies value by 8 over 7, because the same income is now capitalised at a lower yield: the building is worth 14.3 years of income instead of 12.5. A one-point move in a single-digit cap rate is a large move in value.

How does leverage turn 14% into 36%?

Buy a Rs 50 lakh flat with Rs 10 lakh of your own and a Rs 40 lakh loan. If the flat rises 10% to Rs 55 lakh, the loan is unchanged and your stake has risen from 10 to 15 lakh, or 50%. Debt is a fixed claim, so the whole change in asset value lands on the equity, and the equity's percentage move is the asset's move times value over equity. Here value over equity is 100 over 40, or 2.5, so 14.3% becomes 35.7%.

Always give the reverse case, because the interviewer is testing whether you see both directions. If cap rates rise to 9%, value falls to Rs 88.9 crore, 11.1% down, and equity falls to Rs 28.9 crore, 27.8% down. The loan-to-value ratio climbs from 60% to about 68%, which may breach a lending covenant. This ignores interest, fees and any loan amortisation, which change the numbers but not the shape.

About -0.4% a year, a small real loss. Tax at 30% takes 2.4 points of the 8% interest, leaving 5.6%. Inflation at 6% then shrinks what that buys: 1.056 over 1.06 minus 1 is about -0.38%. The quick version, 5.6 minus 6, gives minus 0.4. Before tax the real return was about 1.9%, so the tax took more than all of it. Confirm current tax rates for any real case.

Why can a positive-looking rate lose purchasing power?

Imagine your rent rises 6% a year and your savings account adds 8% a year, but a third of that 8% goes to tax. You are left with 5.6% more money to meet 6% higher prices. Tax is charged on the nominal interest, including the part that only compensates for inflation, so it can take more than the whole real gain. The deposit looks like it earns 2 points above inflation; after tax it earns slightly less than inflation.

How do you get the exact figure, and what does the tax really cost?

Divide growth factors rather than subtracting rates: the money grows by 1.056 while prices grow by 1.06, so real wealth changes by 1.056 over 1.06, a factor of 0.9962. Measured against the real return, the 30% tax rate works like a 120% tax, because it is levied on the inflation part of the interest too. Before tax, the real return was 1.08 over 1.06 minus 1, or 1.89%; after tax it is -0.38%. Rs 10 lakh on deposit buys about Rs 9.96 lakh of today's goods a year later.

Keep the numbers as illustrations. Tax slabs, deposit rates and inflation all change, and the treatment of interest depends on the depositor's total income, so confirm the current figures before advising anyone. The shape does not change: the higher inflation and the tax rate are, the more a taxed nominal return overstates what the saver keeps.

Roughly Rs 6,000 crore a day, on stated assumptions. Assume about 2.5 lakh ATMs, each handling about 80 withdrawals a day: 2 crore withdrawals. Check from demand: 30 crore regular users withdrawing twice a month is 60 crore a month, also about 2 crore a day. At an average of Rs 3,000 a withdrawal, the total is about Rs 6,000 crore a day. Confirm against published payment statistics for a real figure.

Why build the count from the machines and from the people?

If you wanted to know how many cups of tea a railway station sells, you could count the stalls and ask how many cups each pours, or count the passengers and ask how many buy one. A supply-side estimate and a demand-side estimate rest on different assumptions, so when they land close together, each one checks the other. Here both roads meet at about 2 crore withdrawals a day, which is the number that matters; the rupee value is one multiplication away.

Which assumptions need the most care?

Every number here is an assumption to be stated, not a fact to be quoted. The average ticket size is the loosest input: Rs 2,000 against Rs 4,000 moves the answer from Rs 4,000 crore to Rs 8,000 crore a day, a factor of two. Withdrawals per machine vary widely, too, between a busy city branch and a rural machine, so treat 80 as an average across very different sites. The user count is the easiest to reason about from the number of adults with bank cards who still rely on cash.

Say what would change the answer over time. Growth in digital payments lowers withdrawal counts; festivals, salary days and month ends raise them sharply, so an average day hides large swings. Converting to a year, about 365 times the daily figure, is a useful sanity check against any published annual total the interviewer might mention.

For an asset management desk, the point is less the number than the method: a stock-and-flow structure, two independent builds, and a clear statement of which input you trust least. That is the same discipline as sizing a company's addressable market before building a revenue forecast.

Reroll on 1, 2 or 3; keep 4, 5 or 6. The game is worth Rs 4,250. A reroll is worth 3.5 on average, so you keep any face above 3.5 and swap any face below it. The kept faces contribute 4, 5 and 6 each with a one-in-six chance, 2.5 in all; the other half of the time you get 3.5. The total is 2.5 plus 1.75, which is 4.25 thousand.

How do you decide whether to reroll?

Think of a job offer in hand against the option to keep interviewing. You take the offer if it beats what you expect to get by walking away. The rule is to keep a result only when it is worth more than the option you give up, and here the option is a fresh roll worth 3.5. A 3 is below 3.5, so reroll it. A 4 is above, so keep it. Nothing about how you feel about a 4 enters into it; the comparison is one number against another.

Why is this called working backwards?

Because you value the last decision first. The second roll has no choice left in it, so it is worth 3.5. Only once you know that can you decide the first roll. Every multi-stage game is solved from the final stage back to the first, because each decision depends on what the later stages are worth. Allow two rerolls and the same logic stacks: the last reroll is worth 3.5, the middle stage is worth 4.25, so on the first roll you keep only a 5 or 6, and the game is worth 4.67.

Rerolling only on 1 and 2, a common cautious answer, gives 4.17: you keep the 3 when a reroll would have averaged half a point more. The portfolio version is the same idea. A position is kept only while its expected return beats what the capital would earn redeployed, which is an opportunity cost, not a sunk cost.

Fund A, by about two points a year: roughly 8.5% against 6.5%. Long-run compound growth is approximately the average return less half the variance. For A that is 9% less half of 0.10 squared, 0.5 points. For B it is 11% less half of 0.30 squared, 4.5 points. Over twenty years a rupee in A grows to about Rs 5.11 and in B to about Rs 3.52.

Why is the average return not what you earn?

A shop that marks a price up 30% and then down 30% does not end where it started. Rs 100 becomes Rs 130 and then Rs 91, although the average move is zero. Returns compound by multiplying, so a swing up and an equal swing down always leave you behind, and the loss grows with the size of the swing. For moderate returns the loss is close to half the variance, half of volatility squared. That is why a fund's compound annual growth sits below its average annual return, and the gap widens sharply as volatility rises.

How precise is the half-variance rule?

It is an approximation, good when returns and volatility are modest, and it overstates the drag when volatility is large. A more careful calculation that treats returns as lognormal puts A at 8.5% and B at 7.2%. The size of B's handicap moves with the method, but the ranking does not: the higher-volatility fund compounds more slowly. Say that in the room. The rule is a tool for spotting the direction and rough size, not a precise forecast.

The portfolio consequence is the reason the question exists. Two funds with the same average return are not equal if one is far more volatile, and a client who holds a single volatile fund lives with the compound number, not the average in the brochure. Combining volatile assets that do not move together lowers portfolio variance, which is one of the few ways to raise compound growth without raising the average return.

35% for one stock, about 20% for ten, and a floor of 17.5% for infinitely many. Portfolio variance is the stock variance times one over n, plus the correlation times what is left. With ten stocks that is 0.1225 times 0.325, a volatility of 20.0%. As n grows the one-over-n part vanishes and only the correlation term remains: 35% times the square root of 0.25, or 17.5%.

Why does adding stocks lower risk at all?

Ten shops in ten different towns do not all have a bad week at once; ten shops in one mall often do. Some of what moves a stock is its own news and some is the market everyone shares. Stock-specific shocks cancel out as you add names, because one company's bad quarter is offset by another's good one, but the shared market shock hits every name together and does not cancel. Correlation measures how much of each stock's movement is shared.

How much of the benefit do the first ten stocks deliver?

Most of it. Going from one stock to ten cuts volatility from 35% to 20.0%, about 86% of the whole distance to the floor. Going from ten to thirty takes it only to 18.4%. Diversification pays off quickly and then almost stops, and the level where it stops is set by correlation, not by the number of holdings. That is why a manager worried about risk gains more from adding assets that are less correlated than from adding a fortieth stock of the same kind.

Say the limitation. Correlations are not fixed: in a sell-off they tend to rise together, which raises the floor exactly when diversification is needed. Real stocks also differ in volatility and correlation, so this uniform market is a teaching model; the shape of the curve survives, the exact numbers do not.

About Rs 600 crore for the operating business, before any value in the land. Next year's surplus is Rs 40 crore grown 5%, or Rs 42 crore. A surplus that grows forever at 5% and is valued at 12% is worth next year's amount divided by the 7-point gap: 42 over 0.07 is Rs 600 crore. The reason to buy is the durability of that surplus: steady demand for seats and fees that can rise with costs.

What is the question really asking?

It is a stock pitch in disguise. The interviewer wants two things in order: why this asset produces reliable cash, and what that cash is worth. Answer the why with the quality of the surplus, and the how much with a valuation you can do out loud. For a college the why is simple to say: students keep applying every year, fees are paid in advance, and a college with a good name can raise fees roughly in line with its costs. Those are the reasons the surplus can be treated as growing and durable.

Why divide by 12% less 5%?

Think of a rented flat whose rent rises every year. A buyer asking for a 12% return on a rent that grows 5% needs only 7% from the current rent; the other 5% arrives through growth. A cash flow growing at g forever, valued at a required return r, is worth next year's cash flow divided by r minus g. Here that is Rs 42 crore over 0.07, which is Rs 600 crore. Dividing this year's Rs 40 crore instead gives Rs 571 crore, a common small slip: the buyer receives next year's surplus, not this year's.

Now say what the number is sensitive to. One point on the required return moves the value by Rs 100 crore up or Rs 75 crore down, because a 7-point gap becoming 6 or 8 is a large change in proportion. The land and buildings may be worth more than the operating surplus, so a seller would also ask what the campus fetches as property. And check the structure before promising anyone the surplus: many colleges, in India among other places, are run by trusts or societies, and whether an owner can take surplus out at all is a legal question to confirm.

About 1,200 a year, within a range of roughly 1,000 to 1,500. Size the fleet first: about 100,000 flights a day, three quarters on narrow-bodies, at five flights per jet a day, needs about 15,000 jets. Yearly demand is replacement plus growth: 15,000 over a 25-year life is 600, and 4% growth adds 600. Every input is an assumption to state and then check against published fleet data.

Where do you start with a plane that lasts 25 years?

Think of refrigerators in a city. Almost every home has one, but the shops sell only the ones that break down plus the ones new homes need. For a long-lived asset, yearly sales are a small flow out of a large stock: the stock divided by its life, plus the stock times its growth rate. So the job splits into two parts. Size the fleet from how the planes are used, then turn the fleet into a yearly flow.

How do you size the fleet without knowing it?

From usage, which is easier to guess. Take about 100,000 commercial flights a day worldwide, a round number to state as an assumption. Most flights are short and medium routes, so put three quarters on narrow-bodies: 75,000. A narrow-body on short routes flies several sectors a day; call it five. That needs 15,000 jets. Each assumption is a round number said out loud, so the interviewer can challenge one without losing the structure.

Now sanity-check. 1,200 deliveries is about 8% of the fleet, which sounds right for an asset that lasts decades. If the question asks about one maker, split by an assumed market share; two main makers sharing the market would each deliver around 600. For an asset manager the useful point is the sensitivity: deliveries swing with the growth assumption far more than the fleet does, which is why aircraft makers' order books are so cyclical. Treat every number here as an assumption to check against published fleet and delivery data before using it.